20240816/2401.11622v4.json

{
    "title": "The Markov-Chain Polytope with Applications to Binary AIFV-\ud835\udc5a Coding11footnote 1Work of both authors partially supported by RGC CERG Grant 16212021.",
    "abstract": "This paper is split into two parts.",
    "sections": [
        {
            "section_id": "1",
            "parent_section_id": null,
            "section_name": "Introduction",
            "text": "Let  be an -state Markov chain. For convenience,\n will denote the set of indices  See Figure 1  ###reference_###.\nFor  let  be the probability of transitioning from state  to state . Assume further that\n has a unique stationary distribution\nAdditionally, each state  has an associated reward or cost  the average steady-state cost or gain of  is then defined [7  ###reference_b7###] as\nA state  is fully defined by the  values  and\n###figure_1### Next suppose that, for each  instead of\nthere being only one  there exists a large set  containing all permissible \u201ctype-\u2019 states. Fix\nto be their Cartesian product,\nand further assume that   has a unique stationary distribition.\nThe problem is to find the Markov chain in  with smallest cost.\nThis problem first arose in the context of binary AIFV- coding [4  ###reference_b4###, 22  ###reference_b22###, 23  ###reference_b23###] (described later in detail in Section 6  ###reference_###),\nin which a code is an -tuple\n of binary coding trees for a size  source alphabet; for each  there are different restrictions on\nthe structure of \nThe cost of  is the cost of a corresponding -state Markov chain, so the problem of finding the minimum-cost binary AIFV- code reduced to finding a minimum-cost Markov chain\n[5  ###reference_b5###].\nThis same minimum-cost Markov chain approach was later used to find better parsing trees\n[17  ###reference_b17###], lossless codes for finite channel coding [18  ###reference_b18###] and AIFV codes for unequal bit-cost coding [15  ###reference_b15###].\nNote that in all of these problems, the input size  was relatively small, e.g., a set of  probabilities, but the associated  had size exponential in \nThe previous algorithms developed for solving the problem were iterative ones\nthat moved from Markov chain to Markov chain in , in some non-increasing cost order. For the specific applications mentioned, they ran in exponential (in ) time. Each iteration step also required solving a local optimization procedure which was often polynomial time (in ).\n[8  ###reference_b8###, 10  ###reference_b10###] developed a different approach for solving the binary AIFV- coding problem, corresponding to a -state Markov chain, in weakly polynomial time using a simple binary search. In those papers, they noted that they could alternatively solve the problem in weakly polynomial time via the Ellipsoid algorithm for linear programming [12  ###reference_b12###] on a two-dimensional polygon.\nThey hypothesized that this latter technique could be extended to  but only with a better understanding of the geometry of the problem.\nThat is the approach followed in this paper, in which we define a mapping of type- states to type- hyperplanes in .\nWe show that the unique intersection of any  hyperplanes, where each is of a different type, always exists. We call such an intersection point \u201cdistinctly-typed\u201d and prove that its \u201cheight\u201d is equal to the cost of its associated Markov chain. The solution to the minimum-cost Markov-chain problem is thus the lowest height of any \u201cdistinctly-typed\u2019 intersection point.\nWe then define the Markov-Chain polytope  to be the lower envelope of the hyperplanes associated with all possible states and note that some lowest-height distinctly-typed intersection point is a highest point on \nThis transforms the problem of finding the cheapest Markov chain to the linear programming one of finding a highest point of\nThe construction and observations described above will be valid for ALL Markov chain problems.\nIn the applications mentioned earlier, the polytope  is defined by an exponential number of constraints. But, observed from the proper perspective, the local optimization procedures used at each step of the iterative algorithms in [4  ###reference_b4###, 22  ###reference_b22###, 23  ###reference_b23###, 5  ###reference_b5###, 17  ###reference_b17###] can be repurposed as polynomial time separation oracles for . This permits\nusing the Ellipsoid algorithm approach of [12  ###reference_b12###]\nto solve the binary AIFV- problem in weakly polynomial time instead of exponential time.\nThe remainder of the paper is divided into two distinct parts. Part 1 consists of Sections 2  ###reference_###-5  ###reference_###. These develop a procedure for solving the generic minimum-cost Markov Chain problem.\nSection 2  ###reference_### discusses how to map the problem into a linear programming one and\nhow to interpret the old iterative algorithms\nfrom this perspective.\nSection 3  ###reference_### states our new results while Section 4  ###reference_### discusses their algorithmic implications. In particular, Lemma 4.6  ###reference_numbered6### states sufficient conditions on  that guarantees a polynomial time algorithm for finding the minimum cost Markov chain.\nSection 5  ###reference_### then completes Part 1 by proving the main results stated in Section 3  ###reference_###.\nPart 2, in Sections 6  ###reference_###-8  ###reference_###,\nthen discusses how to apply Part 1\u2019s techniques to construct best binary AIFV- codes in weakly polynomial time.\nSection 6  ###reference_### provides necessary background, defining binary AIFV- codes and deriving their important properties.\nSection 7  ###reference_### describes how to apply the techniques from Section 4  ###reference_### to binary AIFV- coding.\nSection 8  ###reference_### proves a very technical lemma specific to binary AIFV- coding required to show that its associated Markov Chain polytope has a polynomial time separation oracle, which is the last piece needed to apply the Ellipsoid method.\nFinally, Section 9  ###reference_### concludes with a quick discussion of other applications of the technique and possible directions for going further."
        },
        {
            "section_id": "2",
            "parent_section_id": null,
            "section_name": "Markov Chains",
            "text": ""
        },
        {
            "section_id": "2.1",
            "parent_section_id": "2",
            "section_name": "The Minimum-Cost Markov Chain problem",
            "text": "Fix\nA state\n is defined by a set of  transition probabilities  along with a cost   and\nlet  be some finite given set of states,\nsatisfying that \nThe states in  are known as type- states.\nMarkov Chain  is permissible if\nDefine  to be the set of permissible Markov chains.\nThe actual composition and structure of each \nis different from problem to problem and, within a fixed problem, different for different \nThe only universal constraint is that stated in (b), that\n  This implies that\n is an ergodic unichain, with one aperiodic recurrent class (containing ) and, possibly, some transient states.  therefore has a unique stationary distribution,\n(where  if and only if  is a transient state.)\nLet  be a permissible -state Markov chain.\nThe average steady-state cost of  is defined to be\nWhat has been described above is a Markov Chain with rewards, with  being its gain\n[7  ###reference_b7###].\nThe minimum-cost Markov chain problem is to find  satisfying\nComments:\n(i) In the applications motivating this problem, each  has size exponential in  so the search space  has size exponential in \n(ii) The requirement in (b) that  (which is satisfied by all the motivating applications,) guarantees that  has a unique stationary distribution.\nIt is also needed later in other places in the analysis. Whether this condition is necessary is unknown; this is discussed in Section 9  ###reference_###."
        },
        {
            "section_id": "2.2",
            "parent_section_id": "2",
            "section_name": "Associated Hyperplanes and Polytopes",
            "text": "The next set of definitions map type- states into type- hyperplanes in  and then defines lower envelopes of those hyperplanes. In what follows,  denotes a vector   is a shorthand denoting that  and \n denotes a state\nLet .\nDefine the type- hyperplanes  as follows:\nFor all  define  and\nand Markov chain\nFor later use, we note that by definition\nFinally, for all  define.\nmaps a type- state  to a type- hyperplane\n in  For fixed   is the\nlower envelope of all of the type- hyperplanes\nEach  maps point  to the lowest type- hyperplane evaluated at   maps point  to the Markov chain\nwill be the lower envelope of the .\nSince both  and  are lower envelopes of hyperplanes, they are the upper surface of convex polytopes in \nThis motivates defining the following polytope:\nThe Markov Chain Polytope in  corresponding to  is"
        },
        {
            "section_id": "2.3",
            "parent_section_id": "2",
            "section_name": "The Iterative Algorithm",
            "text": "[4  ###reference_b4###, 5  ###reference_b5###, 22  ###reference_b22###, 23  ###reference_b23###] present an iterative algorithm\nthat was first formulated for finding minimum-cost binary AIFV- codes\nand then generalized into a procedure for finding minimum-cost Markov Chains.\nThe algorithm starts with an arbitrary  and iterates, at each step constructing an  such that\n The algorithm terminates at step  if\nAlthough it does not run in polynomial time, it is noteworthy because the fact of its termination\n(Lemma 2.6  ###reference_numbered6###) will be needed later (in Corollary 3.4  ###reference_numbered4###) to prove that  contains some point corresponding to a minimum-cost Markov chain.\n[23  ###reference_b23###] proves that the algorithm always terminates when\n and, at termination,  is a minimum-cost Markov Chain.\n[4  ###reference_b4###, 5  ###reference_b5###, 22  ###reference_b22###] prove that222They also claim that the algorithm always terminates. The proofs of correctness there are only sketches and missing details but they all\nseem to implictly assume that  for all  and not just \n,\nfor  if the algorithm terminates, then  is a minimum-cost Markov Chain.\nA complete proof of termination (and therefore solution of the minimum-cost Markov Chain problem) is provided in [1  ###reference_b1###]. This states\n[Theorem 1, [1  ###reference_b1###]]\nNo matter what the starting value  there always exists  such that  Furthermore, for that\nNotes/Comments:\nThe algorithm in [4  ###reference_b4###, 5  ###reference_b5###, 22  ###reference_b22###, 23  ###reference_b23###] looks different than the one described in [1  ###reference_b1###] but they are actually identical, just expressed in a different coordinate system. The relationship between the two coordinate systems is shown in [11  ###reference_b11###]. The coordinate system in [1  ###reference_b1###] is the same as the one used in this paper.\nEach step of the iterative algorithm requires calculating  for the current  and all \nIn applications,\nfinding  is very problem specific and is usually\na combinatorial optimization problem. For example, the first papers on AIFV- coding [23  ###reference_b23###] and the most current papers on finite-state channel coding [18  ###reference_b18###]\ncalculate them using integer linear programming, as does a recent paper on constructing AIFV codes for unequal bit costs [15  ###reference_b15###].\nThe more recent papers on both AIFV- coding [16  ###reference_b16###, 22  ###reference_b22###, 11  ###reference_b11###] and AIVF coding\n[17  ###reference_b17###]\nuse dynamic programming.\nAs noted, the fact that if the iterative algorithm terminates at some  then  is a minimum-cost Markov Chain was proven directly in [4  ###reference_b4###, 5  ###reference_b5###, 22  ###reference_b22###, 23  ###reference_b23###]. An alternative proof of this fact is given in Lemma 3.3  ###reference_numbered3### in this paper.\nThe proof of termination given in [1  ###reference_b1###] Theorem 1, strongly depends upon condition (b) from Definition 2.1  ###reference_numbered1###, i.e.,"
        },
        {
            "section_id": "3",
            "parent_section_id": null,
            "section_name": "The Main Results",
            "text": "This section states our two main lemmas, Lemma 3.1  ###reference_numbered1### and Lemma 3.7  ###reference_numbered7###, and their consequences. Their proofs are deferred to Section 5  ###reference_###.\n\n###figure_2### Let  be a permissible Markov Chain.\nThe  -dimensional hyperplanes  \nintersect at a unique point .\nWe call such a  point a distinctly-typed intersection point.\n\nThe intersection point  is on or above the lower envelope of\nthe  hyperplanes   i.e.,\nThe intersection point  also satisfies\nThe\nLemma immediately implies that\nfinding a minimum-cost Markov Chain is equivalent to finding a minimum \u201cheight\u201d (\u2019th-coordinate) distinctly-typed intersection point. It also implies the following corollary.\nLet .\nThis in turn, permits, proving\nIf\nfor some  and  then\nand\nBy the definition of the  and  if  satisfies (2  ###reference_###),\nRecall that for all  \nThus, (2  ###reference_###), implies that the hyperplanes  intersect at the point\n\nApplying Lemma 3.1  ###reference_numbered1### (b) proves (3  ###reference_###).\nCombining (3  ###reference_###) with Corollary 3.2  ###reference_numbered2### immediately implies\nBecause the leftmost and rightmost values in (5  ###reference_###) are identical, the two inequalities in (5  ###reference_###) must be equalities, proving (4  ###reference_###).\n\u220e\nCondition (2  ###reference_###) in Lemma 3.3  ###reference_numbered3###\nmeans that the  different lower envelopes ,  must simultaneously intersect at a point .\nIt is not a-priori obvious that such an  should always exist but\nLemma 2.6  ###reference_numbered6### tells us that the iterative algorithm always terminates at such an , immediately proving333For completness, we note that\nLemma 7.4  ###reference_numbered4### later also directly proves that condition (2  ###reference_###) in Lemma 3.3  ###reference_numbered3### holds for the special case of AIFV- coding. Thus this paper provides a\nfully self-contained proof of Corollary 3.4  ###reference_numbered4### for AIFV- coding.\nThere exists  satisfying (2  ###reference_###) in Lemma 3.3  ###reference_numbered3###. This  satisfies\n and\n###figure_3### This suggests a new approach to solving the minimum-cost Markov chain problem.\nUse linear programming to find a highest point\nStarting from  find a distinctly-typed intersection point\nReturn\nFrom Corollary 3.4  ###reference_numbered4###, an  in step 2 must exist;\nfrom Lemma 3.3  ###reference_numbered3### any such  found would yield a  solving the\nminimum-cost Markov chain problem.\nWhile it would usually not be difficult to move to a highest vertex  when starting from a highest point  the added complication that  must be distinctly-typed complicates step 2. These complications can be sidestepped using the extension to -restricted search spaces and the pruning procedure described below.\nLet\n denote the set of all subsets of  that contain \u201c0\u201d, i.e.,\n\nFor all  and all \ndefine  the set of all states to which  can transition. Since\nNow fix  and \nDefine\nto be the subset of states in  that only transition to states in\nFurther define\nNote that  and . Also note that if  then, if   is a transient state in\nLet  and\nFor all  define  and\n as\nand\nNote that ,  and  are, respectively, equivalent to\n,  and  i.e, all types permitted.\nThese definitions permit moving from any \u201chighest\u201d point on  to a distinctly-typed intersection point at the same height.\nSuppose  satisfies\nLet  denote any minimum-cost Markov chain\nand\n the set of its recurrent indices .\nLet \nThen\nIf  , then\n\nEquivalently, this implies \nis a minimum-cost Markov chain.\nIf , then\n.\nSuppose  satisfies\nThen the procedure  in Algorithm 6  ###reference_6### terminates in at most  steps. At termination,  is a minimum-cost Markov chain.\nAt the start of the algorithm, \nAs long as , Lemma 3.7  ###reference_numbered7### (b) implies that\n If , the size of  decreases, so the process can not run more than  steps. At termination,  so from Lemma 3.7  ###reference_numbered7### (a),  is a minimum-cost Markov chain.\n\u220e"
        },
        {
            "section_id": "4",
            "parent_section_id": null,
            "section_name": "Algorithmic Implications",
            "text": "Stating our running times requires introducing further definitions.\nLet  be a hyperrectangle in  Define\nis the maximum time required to calculate  for any\nis the maximum time required to calculate \nfor any  and any\nNote that for any  \nAlso,  and  will denote  and\nIn addition,\ndenotes the number of iterative steps made by the iterative algorithm. The only known bound for this is  the number of permissible Markov Chains.\nThe iterative algorithm requires  time. Improvements to its running time have focused on improving  in specific applications.\nFor binary AIFV- coding, of  source-code words,  was first solved using integer linear programming [23  ###reference_b23###] so  was exponential in \nFor the specific case of  this was improved to polynomial time using different dynamic programs. More specifically, for  [16  ###reference_b16###] showed that\n, improved to  by [9  ###reference_b9###]; for  [22  ###reference_b22###]\nshowed that\n, improved to  by [11  ###reference_b11###]. These sped up the running time of the iterative algorithm under the (unproven) assumption that the iterative algorithm always stayed within .\nFor AIVF codes, [17  ###reference_b17###] proposed using a modification of a dynamic programming algorithm due to\n[3  ###reference_b3###], yielding that for any fixed ,  is polynomial time in  the number of words permitted in the parse dictionary.\nIn all of these cases, though, the running time of the algorithm is still exponential because \ncould be exponential.\nNote: Although  has been studied, nothing was previously known about  This is simply because there was no previous need to define and construct . In all the known cases in which algorithms for constructing  exist, it is easy to slightly modify them to construct\n in the same running time. So\nWe now see how the properties of the Markov chain polytope will, under some fairly loose conditions, permit finding the minimum-cost Markov chain in polynomial time. This will be done via\nthe Ellipsoid method of Gr\u00f6tschel, Lov\u00e1sz and Schrijver [12  ###reference_b12###, 13  ###reference_b13###, 20  ###reference_b20###], which, given a polynomial time separation oracle for a polytope, permits solving a linear programming problem on the polytope in polynomial time. The main observation is that  provides a separation oracle for  so the\nprevious application specific algorithms for constructing  can be reused to derive polynomial time algorithms."
        },
        {
            "section_id": "4.1",
            "parent_section_id": "4",
            "section_name": "Separation oracles and",
            "text": "Recall the definition of a separation oracle.\nLet  be a closed convex set. A separation oracle444Some references label this a strong separation oracle. We follow the formulation of [20  ###reference_b20###] in not adding the word strong.\nfor  is a procedure that, for any\n either reports that  or, if , returns a hyperplane that separates  from  That is, it returns  such that\nIt is now clear that  provides a separation oracle for\nLet  be fixed and\n be the Markov Chain polytope. Let . Then\nknowing  provides a  time algorithm for either reporting that  or returning a hyperplane that separates  from\nif and only if\nThus knowing  immediately determines whether  or not. Furthermore if , i.e.,  let  be an index\nsatisfying\nThe hyperplane  then separates  from  because  is a supporting hyperplane of  at point \n\u220e"
        },
        {
            "section_id": "4.2",
            "parent_section_id": "4",
            "section_name": "The Ellipsoid Algorithm with Separation Oracles",
            "text": "The ellipsoid method of Gr\u00f6tschel, Lov\u00e1sz and Schrijver [12  ###reference_b12###, 13  ###reference_b13###] states that, given a polynomial-time separation oracle, (even if  is a polytope defined by an exponential number of hyperplanes) an approximate optimal solution to the \u201cconvex optimization problem\u201d can be found in polynomial time. If  is a rational polytope, then an exact optimal solution can be found in polynomial time.\nWe follow the formulation of [20  ###reference_b20###] in stating these results.\nOptimization Problem Let  be a rational polyhedron555 is a Rational Polyhedron if\n\nwhere the components of matrix  and vector  are all rational numbers.  is the set of rationals.\n\nin  Given the input  conclude with one of the following:\ngive a vector  with\ngive a vector  in  with\nassert that  is empty.\nNote that in this definition,  is the characteristic cone of  The characteristic cone of a bounded polytope is  [20  ###reference_b20###][Section 8.2] so, if  is a bounded nonempty polytope, the optimization problem is to find a vector  with\nThe relevant result of [12  ###reference_b12###, 13  ###reference_b13###] is\nThere exists an algorithm ELL such that if ELL is given the input\n where:\n\n         \n\n\n and  are natural numbers and SEP is a separation oracle for some rational polyhedron  in , defined by linear inequalities of size at most  and \n \n\nthen ELL solves the optimization problem for  for the input  in time polynomially bounded by , , the size of  and the running time of\nIn this statement, the size of a linear inequality is the number of bits needed to write the rational coefficients of the inequality, where the number of bits required to write rational  where  are relatively prime integers, is"
        },
        {
            "section_id": "4.3",
            "parent_section_id": "4",
            "section_name": "Solving the Minimum-Cost Markov Chain problem.",
            "text": "Combining all of the pieces, we can now prove our main result.\nGiven  let  be the maximum number of bits required to write any transition probability  or cost  of a permissible state .\nFurthermore, assume some known hyper-rectangle\n with the property that there exists  satisfying  and\n.\nThen the minimum-cost Markov chain problem can be solved in time polynomially bounded by ,  and\nWithout loss of generality, we assume that  To justify, recall that the minimum cost Markov chain  has cost\n where  is the \u2019th component of the stationary distribution of \nTrivially, if  for all  then\nThe original problem formulation does not require that . But, we can\nmodify a given input by adding the same constant  to  for every ,  This makes every  non-negative so the minimum cost Markov chain in this modified problem has non-negative cost.\nSince this modification adds  to the cost of every Markov chain, solving the modified problem solves the original problem. Note that this modification can at most double  so this does not break the statement of the lemma.\nWe may thus assume that .\nThus\n\nwhere\n\nSince  is bounded from above by the cost of any permissible Markov chain,  is a bounded non-empty polytope.\nSince\nNow consider the following separation oracle for  Let\nIn  time, first check whether  If no, then  and  is a separating hyperplane.\nOtherwise, in  time, check whether  If no,  and a separating hyperplane is just the corresponding side of  that  is outside of.\nOtherwise, calculate  in  time. From Lemma 4.3  ###reference_numbered3###, this provides a separation oracle.\nConsider solving the optimization problem on polytope  with  to find  satisfying\nSince  is a bounded non-empty polytope, we can apply\nTheorem 4.5  ###reference_numbered5### to find such an  in time polynomially bounded in  , and\nApplying Corollary 3.8  ###reference_numbered8### and its procedure \nthen produces a minimum cost Markov-Chain in another  time.\nThe final result follows from the fact that\n\n\u220e\nPart 2, starting in Section 6  ###reference_###, shows how to apply this Lemma to derive a polynomial time algorithm for constructing minimum-cost AIFV- codes.\nWe also note that [2  ###reference_b2###] recently applied Lemma 4.3  ###reference_### in a plug-and-play manner to derive the first polynomial time algorithms for constructing optimal AIVF codes. AIVF codes are a multi-tree generalization of Tunstall coding [17  ###reference_b17###, 18  ###reference_b18###], for which the previous constriction algorithms ran in exponential time."
        },
        {
            "section_id": "5",
            "parent_section_id": null,
            "section_name": "Proofs of Lemmas 3.1 and 3.7",
            "text": ""
        },
        {
            "section_id": "5.1",
            "parent_section_id": "5",
            "section_name": "Proof of Lemma 3.1",
            "text": "Before starting the proof, we note that (a) and the first equality of (b) are, after a change of variables, implicit in the analysis provided in [5  ###reference_b5###] of the convergence of their iterative algorithm. The derivation there is different than the one provided below, though, and is missing intermediate steps that we need for proving our later lemmas.\nIn what follows,  denotes , the transition matrix associated with  and  denotes , its unique stationary distribution.\nTo prove (a) observe that the intersection condition\ncan be equivalently rewritten as\nwhere the right-hand side of (7  ###reference_###) can be expanded into\nEquation (7  ###reference_###) can therefore be rewritten as\nwhere the matrix in (8  ###reference_###), denoted as\n is  after subtracting the identity matrix  and replacing the first column with s.\nTo prove (a) it therefore suffices to prove that\n\nis invertible.\nThe uniqueness of  implies that the kernel of  is -dimensional. Applying the rank-nullity theorem, the column span of  is -dimensional. Since , each column of  is redundant, i.e., removing any column of  does not change the column span.\nNext, observe that  implying that  is not orthogonal to . In contrast, each vector  in the column span of  is of the form\n for some . Thus , implying that  is orthogonal to . Therefore,  is not in the column span of .\nCombining these two observations, replacing the first column of  with  increases the rank of  by exactly one.\nHence,\n\nhas rank . This shows invertibility, and the proof of (a) follows.\nTo prove (b) and (c) observe that\nand that for all ,\nApplying these observations by setting  and left-multiplying by\nApplying these observations again, it follows that\nproving (b).\nSince the transition probabilities  are non-negative,\nproving (c).\n(d) follows by observing that\n\u220e"
        },
        {
            "section_id": "5.2",
            "parent_section_id": "5",
            "section_name": "Proof of Lemma 3.7",
            "text": "To prove (a)\nset \nAssume\nthat,\nBy definition,  cannot transition to any  where . This implies that\nLemma 3.1  ###reference_numbered1### (b) then implies\nThus,  is a minimum-cost Markov chain.\nTo prove (b)\nfirst note that, we are given that   Thus,  and , as required.\nThe Markov chain starts in state  where .\nWe claim that if  and  then  If not, , contradicting that\n Thus,  for all .\nNow, suppose that  Then, for all ,\nOn the other hand, using Lemma 3.1  ###reference_numbered1### (b) and the fact that \nis a minimum-cost Markov chain,\nThe left and right hand sides of (10  ###reference_###) are the same and  so .\nThis in turn forces all the inequalities in (9  ###reference_###) to be equalities, i.e., for all ,\nThus, , proving (b).\n\u220e"
        },
        {
            "section_id": "6",
            "parent_section_id": null,
            "section_name": "A Polynomial Time Algorithm for binary AIFV- Coding.",
            "text": "This second part of the paper introduces binary AIFV- codes and then applies Lemma 4.6  ###reference_numbered6### to find a minimum-cost code of this type in\ntime polynomial in , the number of source words to be encoded, and  the number of bits required to state the probability of any source word.\nThe remainder of this section\ndefines binary AIFV- codes\nand describes how they are a\nspecial case of the\nminimum-cost Markov chain problem.\nSection 7  ###reference_### describes how to apply Lemma 4.6  ###reference_numbered6###.\nThis first requires showing that  is polynomial in  and , which will be straightforward.\nIt also requires identifying a hyperrectangle  that contains a highest point  and\nfor which\n is polynomial in .\nThat is,\n  can be calculated in polynomial time.\nAs mentioned at the start of Section 4  ###reference_###, as part of improving the running time of the iterative algorithm, [16  ###reference_b16###, 9  ###reference_b9###, 22  ###reference_b22###, 11  ###reference_b11###] showed that, for\n\n is polynomial time. As will be discussed in Section 7.2  ###reference_###, the algorithms there can be easily modified to show that .\nCorollary 3.4  ###reference_numbered4### only tells us that there exists some  such that  is a highest point in . In order to use\nLemma 4.6  ###reference_numbered6###, we will need to show that there exists such an .\nProving this is the\nmost cumbersome part of the proof. It combines a case-analysis of the tree structures of AIFV- trees with the Poincare-Miranda theorem to show that the functions\n  must all mutually intersect at some point  From Lemma 3.3  ###reference_numbered3###, \nand is therefore the optimum point needed. Section\n8  ###reference_### develops the tools required for this analysis."
        },
        {
            "section_id": "6.1",
            "parent_section_id": "6",
            "section_name": "Background",
            "text": "Consider a stationary memoryless source with alphabet  in which symbol\n is generated with probability .\nLet  be a message generated by the source.\nBinary compression codes encode each  in  using a binary codeword.\nHuffman codes are known to be \u201coptimal\u201d such codes. More specifically, they are Minimum Average-Cost Binary Fixed-to-Variable Instantaneous codes. \u201cFixed-to-Variable\u201d denotes that the binary codewords corresponding to the different  can have different lengths. \u201cInstantaneous\u201d, that, in a bit-by-bit decoding process, the end of a codeword is recognized immediately after its last bit is scanned. The redundancy of a code is the difference between its average-cost and the Shannon entropy\n of the source. Huffman codes can have worst case redundancy of\nHuffman codes are often represented by a coding tree, with the codewords being the leaves of the tree.\nA series of recent work\n[4  ###reference_b4###, 5  ###reference_b5###, 6  ###reference_b6###, 14  ###reference_b14###, 16  ###reference_b16###, 22  ###reference_b22###, 23  ###reference_b23###] introduced Binary Almost-Instantaneous Fixed-to-Variable- (AIFV-) codes. Section 6.2  ###reference_### provides a complete definition as well as examples. These differ from Huffman codes in that they use  different coding trees.\nFurthermore, decoding might require reading ahead  bits before knowing that the end of a codeword has already been reached (hence \u201calmost\u201d-instantaneous). Since AIFV- codes include Huffman coding as a special case, they are never worse than Huffman codes.\nTheir advantage is that, at least for  they have worst-case redundancy  [14  ###reference_b14###, 6  ###reference_b6###], beating Huffman coding.666Huffman coding with blocks of size  will also provide worst case redundancy of\n But the block source coding alphabet, and thus the Huffman code dictionary, would then have size  In contrast, AIFV- codes have dictionary size\nHistorically, AIFV- codes were preceded by\n-ary Almost-Instantaneous FV (-AIFV) codes,\nintroduced in [21  ###reference_b21###].\n-ary AIFV codes used a  character encoding alphabet;\nfor , the procedure used  coding trees and had a coding delay of  bit. For  it used 2 trees and had a coding delay of  bits.\nBinary AIFV- codes\nwere introduced later in [14  ###reference_b14###]. These are binary codes that are comprised of an -tuple of binary coding trees and have decoding delay of at most  bits.\nThe binary AIFV- codes of [14  ###reference_b14###] are identical to the -ary AIFV codes of [21  ###reference_b21###].\nConstructing optimal777 An \u201coptimal\u201d -AIFV or AIFV- code is one with minimum average encoding cost over all such codes. This will be formally specified later in Definition 6.8  ###reference_numbered8###.\n-AIFV or binary AIFV- codes is much more difficult than constructing Huffman codes.\n[23  ###reference_b23###] described an iterative algorithm for constructing optimal binary AIFV- codes.\n[22  ###reference_b22###] generalized this\nand proved that, for , under some general assumptions, this algorithm would terminate and, at termination would produce an optimal binary AIFV- code. The same was later proven for  by\n[4  ###reference_b4###]\nand [22  ###reference_b22###]. This algorithm was later generalized to solve the Minimun Cost Markov chain problem in [5  ###reference_b5###] and is the iterative algoithm referenced in Section 2.3  ###reference_###."
        },
        {
            "section_id": "6.2",
            "parent_section_id": "6",
            "section_name": "Code Definitions, Encoding and Decoding",
            "text": "Note: We assume  and that each  can be represented using  bits, i.e., each probability is an integral multiple of . The running time of our algorithm will, for fixed  be polynomial in  and , i.e., weakly polynomial.\nA binary AIFV- code will be an -tuple  of\n binary code trees satisfying\nDefinitions 6.1  ###reference_numbered1### and 6.2  ###reference_numbered2### below.\nEach  contains  codewords. Unlike in Huffman codes, codewords can be internal nodes.\nFigure 4  ###reference_###.\nEdges in an AIFV- code tree are labelled as -edges or -edges.\nIf node  is connected to its child node  via a -edge (-edge) then  is \u2019s -child (-child).\nWe will often identify a node interchangeably with its associated (code)word. For example,  is the node reached by following the edges  down from the root.\nFollowing\n[14  ###reference_b14###], the nodes in AIFV- code trees can be classified as being exactly one of 3 different types:\nComplete Nodes. A complete node has two children: a -child and a -child.\nA complete node has no source symbol assigned to it.\nIntermediate Nodes. An intermediate node has no source symbol assigned to it and has exactly one child.\nAn intermediate node with a -child is called an intermediate- node; with a -child is called an intermediate- node.\nMaster Nodes. A master node has an assigned source symbol and at most one child node. Master nodes have associated degrees:\na master node of degree  is a leaf.\na master node  of degree  is connected to its unique child by a -edge. Furthermore,\nit has exactly  consecutive intermediate- nodes as its direct descendants, i.e.,  for  are intermediate- nodes while  is not an intermediate- node.\n###figure_4### Binary AIFV- codes are now defined as follows:\nSee Figure 5  ###reference_###. Let  be a positive integer. A binary AIFV- code is an ordered -tuple of  code trees  satisfying the following conditions:\nEvery node in each code tree is either a complete node, an intermediate node, or a master node of degree  where .\nFor , the code tree  has an intermediate- node connected to the root by exactly  -edges, i.e., the node  is an intermediate- node.\nConsequences of the Definitions:\nEvery leaf is a master node of degree . In particular, this implies that every code tree contains at least one master node of degree\nDefinition 6.2  ###reference_numbered2###, and in particular Condition (2), result in unique decodability (proven in [14  ###reference_b14###]).\nFor  the root of a  tree is permitted to be a master node. If a root is a master node, the associated codeword is the empty string (Figure\n5  ###reference_###)! The root of a  tree cannot be a master node.\nThe root of a  tree may be an intermediate- node.\nFor  every  tree must contain at least one intermediate- node, the node \nA  tree might not contain any intermediate- node.\nFor  the root of a  tree cannot be a intermediate- node. The root of a  tree is permitted to be an intermediate- node (but see Lemma 8.7  ###reference_numbered7###).\n###figure_5### We now describe the encoding and decoding procedures. These are illustrated in Figures 6  ###reference_### and 7  ###reference_###.\nA source sequence  is encoded as follows: Set  and\nLet  be a binary string that is the encoded message.\nSet  and\nNote that in order to identify , line 1 might require reading a few bits after the end of . The number of extra bits that must be read is known as the delay.\nBinary AIFV- codes are uniquely decodable with delay at most ."
        },
        {
            "section_id": "6.3",
            "parent_section_id": "6",
            "section_name": "The cost of AIFV- codes",
            "text": "Let  denote the set of all possible type- trees that can appear in a binary AIFV- code on  source symbols.  will be used to denote a tree .\nSet .\nwill be called a binary AIFV- code.\nFigure 8  ###reference_###.\nLet  and  be a source symbol.\ndenotes the length of the codeword in  for .\ndenotes the degree of the master node in  assigned to .\ndenotes the average length of a codeword in , i.e.,\nis the set of indices of source nodes that are assigned master nodes of degree  in \nSet\nso  is a probability distribution.\nIf a source symbol is encoded using a degree- master node in , then the next source symbol will be encoded using code tree . Since the source is memoryless, the transition probability of encoding using code tree  immediately after encoding using code tree  is .\n###figure_6### This permits viewing the process as a Markov chain whose states are the code trees. Figure 9  ###reference_### illustrates an example.\nFrom Consequence (a) following Definition 6.2  ###reference_numbered2###,  every  contains at least one leaf, so .\nThus, as described in Section 2.1  ###reference_###,\nthis implies that the associated Markov chain is a unichain whose unique recurrence class contains \nand whose associated transition matrix  has a unique stationary distribution .\nThus the Markov chain associated with any\n is permissible.\nLet  be some AIFV- code, \nbe the transition matrix of the associated Markov chain and\nbe \u2019s associated unique stationary distribution. Then the average cost of the code is the average length of an encoded symbol in the limit, i.e.,\nConstruct a binary AIFV- code\n\nwith minimum  i.e.,\nThis problem is exactly the minimum-cost Markov Chain problem introduced in Section 2.1  ###reference_### with .\nAs discussed in the introduction to Section 4  ###reference_###, this was originally solved in exponential time by using an iterative algorithm."
        },
        {
            "section_id": "7",
            "parent_section_id": null,
            "section_name": "Using Lemma 4.6 to derive a polynomial time algorithm for binary AIFV- coding",
            "text": "Because\nthe minimum-cost binary AIFV- coding problem is a special case of the minimum-cost Markov chain problem,\nLemma 4.6  ###reference_numbered6### can be applied to derive a polynomial time algorithm.\nIn the discussion below working through this application,  will denote, interchangeably, both a type- tree and a type- state with transition probabilities\n and cost  For example, when writing  (as in Definition 2.4  ###reference_numbered4###),  will denote the corresponding Markov chain state and not the tree.\nApplying Lemma 4.6  ###reference_numbered6###\nrequires showing that, for fixed  the  and  parameters in its statement are polynomial in  and ."
        },
        {
            "section_id": "7.1",
            "parent_section_id": "7",
            "section_name": "Showing that  is polynomial in  for AIFV- coding",
            "text": "Recall that  is the maximum number of bits needed to represent the coefficients of any linear inequality defining a constraint of\nShowing that  is polynomial in  and  is not difficult but will require the following fact proven later in Corollary 8.9  ###reference_numbered9###:\n           \nFor all  the height of  is at most\nis defined by inequalities of size  where  is the maximum number of bits needed to encode any of the\nNote that the definition of  can be equivalently written as\nwhere we set  to provide notational consistency between  and \nThus, the linear inequalities defining  are of the form\nSince each  can be represented with  bits,  for\nsome integral  This implies that  for\nsome integral  So the size of each  is\nCorollary 8.9  ###reference_numbered9###, to be proven later, states that the height of  is bounded by \nThus each  for all  and  can be written as\n for some integral  Thus,  has size\n Thus,\nthe size of every inequality (11  ###reference_###) is at most \n\u220e\nRecall that  is considered fixed. Thus"
        },
        {
            "section_id": "7.2",
            "parent_section_id": "7",
            "section_name": "Finding Appropriate  and Showing that  and  are Polynomial in  for AIFV- Coding",
            "text": "Recall that  Fix  and\nAs discussed at the starts of Section 4  ###reference_### and 6  ###reference_###, there are dynamic programming algorithms that, for  give\n [9  ###reference_b9###] and for    [11  ###reference_b11###]. For  the best known algorithms for calculating \nuse integer linear programming and run in exponential time.\nIn deriving the polynomial time binary search algorithm for , [10  ###reference_b10###] proved that  and could therefore use the  DP time algorithm for  as a subroutine. We need to prove something similar for\nThe main tool used will be the following highly technical lemma whose proof is deferred to the next section.\nLet  be fixed, \n and  Then\nIf\nIf\nThe proof also needs a generalization of the intermediate-value theorem:\nLet  be  continuous functions of  variables  such that for all indices ,  implies that  and  implies that . It follows that there exists a point  such that\n.\nCombining the two yields:\nLet  be fixed and  Then there exists  satisfying\nSet\n and  for all .\nFrom Lemma 7.2  ###reference_numbered2###, if  then  and if  then \nThe Poincar\u00e9-Miranda theorem then immediately implies the existence of  such that, , i.e., (12  ###reference_###).\n\u220e\nLemma 7.4  ###reference_numbered4### combined with\nLemma 3.3  ###reference_numbered3###, immediately show that if  there exists\n satisfying .\nAs noted earlier, for  there are dynamic programming algorithms for calculating  in  time when  [9  ###reference_b9###] and  when  [11  ###reference_b11###].\nThus  for  and  for\nThose dynamic programming algorithms work by building the  trees top-down. The status of nodes on the bottom level of the partially built tree, i.e., whether they are complete, intermediate nodes or master nodes of a particular degree, is left undetermined. One step of the dynamic programming algorithm then determines (guesses) the status of those bottom nodes and creates a new bottom level of undetermined nodes. It is easy to modify this procedure so that nodes are only assigned a status within some given\n The modified algorithms would then calculate  in the same running time as the original algorithms, i.e.,  time when  and  when . Thus, for ,"
        },
        {
            "section_id": "7.3",
            "parent_section_id": "7",
            "section_name": "The Final Polynomial Time Algorithm",
            "text": "Fix  and set . Since  is fixed, we may assume that  For smaller  the problem can be solved in  time by brute force.\nIn the notation of Lemma 4.6  ###reference_numbered6###, Section 7.1  ###reference_### shows that  where  is the maximum number of bits needed to encode any of the \nSection 7.2  ###reference_### shows that  when  and  when  and there always exists\n satisfying .\nThen, from Lemma 4.6  ###reference_numbered6###, the binary AIFV- coding problem\ncan be solved in time polynomially bounded by  and  i.e., weakly polynomial in the input."
        },
        {
            "section_id": "8",
            "parent_section_id": null,
            "section_name": "Proof of Lemma 7.2 and Corollary 8.9",
            "text": "The polynomial running time of the algorithm rested upon the correctness of the technical Lemma 7.2  ###reference_numbered2### and Corollary 8.9  ###reference_numbered9###.\nThe proof of Lemma 7.2  ###reference_numbered2### requires deriving further technical properties of AIFV- trees. Corollary 8.9  ###reference_numbered9### will be a consequence of some of these derivations.\nThe main steps of the proof of Lemma 7.2  ###reference_numbered2### are:\ngeneralize binary AIFV- codes trees to extended binary code trees;\nprove Lemma 7.2  ###reference_numbered2### for these extended trees;\nconvert this back to a proof of the original lemma.\nWe first introduce the concept of extended binary code trees.\nFix \nAn extended binary AIFV- code tree   is defined exactly the same as a  except that it is permitted to have an arbitrary number of leaves,\ni.e., master nodes of degree \nassigned the empty symbol  The -labelled leaves are given source probabilities 888Since  satisfies Definitions 6.1  ###reference_numbered1### and 6.2  ###reference_numbered2###, just for a larger number of master nodes, it also satisfies Consequences (a)-(e) of those definitions. This fact will be used in the proof of Lemmas 8.7  ###reference_numbered7###.\nLet  denote the number of -labelled leaves in  Note that if\n, then .\nFor notational convenience, let\n,\n,\nand \nrespectively denote the extended versions of , , and\nFor ,\n(a) and (b) below always hold:\nThere exists a function  satisfying\nThere exists a function  satisfying\n(a) follows directly from the fact that, from Definition 6.2  ###reference_numbered2###,\n so\n. Thus, simply setting\n satisfes the required conditions.\n###figure_7### To see (b), given  consider the tree  whose root is complete, with the left subtree of the root being a chain of  intermediate- nodes, followed by one intermediate- node, and a leaf node assigned to , and whose right subtree is . See Figure 10  ###reference_###. Setting  satisfies the required conditions.\n\u220e\nThis permits proving:\nLet  Then\nLemma 8.2  ###reference_numbered2###\n(a) implies that for all ,\nBecause this is true for all , it immediately implies (i).\nSimilarly, Lemma 8.2  ###reference_numbered2###(b) implies that for all ,\nBecause this is true for all , it immediately implies (ii).\n\u220e\nPlugging  into (i) and  into (ii) proves\nLet  and Then\nIf\nIf\nNote that this is exactly Lemma 7.2  ###reference_numbered2### but written for extended binary AIFV- coding trees rather than normal ones.\nWhile it is not necessarily true that  , we can prove that if  is large enough, they coincide in the unit hypercube.\nLet  Then,\nfor all  and ,\nPlugging this Lemma into Corollary 8.4  ###reference_numbered4### immediately proves Lemma 7.2  ###reference_numbered2###. It therefore only remains to prove the correctness of Lemma 8.5  ###reference_numbered5### ."
        },
        {
            "section_id": "8.1",
            "parent_section_id": "8",
            "section_name": "Proving Lemma 8.5",
            "text": "The proof of Lemma 8.5  ###reference_numbered5### is split into two parts, The first justifies simplifying the structure of AIFV- trees. The second uses these properties to actually prove Lemma 8.5  ###reference_numbered5###.\nThis is not possible for  because .\nIt is also not possible for  because in that case,  has a -child so it can not be an intermediate- node.\nIf  then, trivially, Condition 2 from Definition 6.2  ###reference_numbered2### cannot be violated. If  then the transformation described leaves  as an intermediate- node so Condition 2 from Definition 6.2  ###reference_numbered2### is still not violated.\n###figure_8### This operation of removing an intermediate- node can be repeated until condition (e) is satisfied.\n\u220e\nThis lemma has two simple corollaries.\nThere exists a minimum-cost AIFV- code  such that  tree  satisfies conditions (a)-(e) of Lemma 8.7  ###reference_numbered7###.\nLet  be a minimum cost AIFV- code. For each  let  be the tree satisfying conditions (a)-(e) and equation (13  ###reference_###) and set\n Since   Since  ,\nBut  was a minimum-cost AIFV- code so  must be one as well.\n\u220e\nThe corollary implies that, our algorithmic procedures, for all  may assume that all  satisfy conditions (a)-(e) of Lemma 8.7  ###reference_numbered7###.\nThis assumption permits bounding the height of all  trees. The following corollary was needed in the proof of Lemma 7.1  ###reference_numbered1###.\nFor all  the height of  is at most\nLet  satisfy conditions (a)-(e) of Lemma 8.7  ###reference_numbered7###.\nLet  be the number of leaves in  Since all leaves are master nodes,  contains  non-leaf master nodes.\nEvery complete node in  must contain at least one leaf in each of its left and right subtrees, so the number of complete nodes in  is at most\ncontains no intermediate- node if  and one intermediate- node if\nIf  each intermediate- node in  can be written as  for some non-leaf master node  and \nIf  all intermediate- nodes in  with the possible exceptions of the left nodes   can be written as  for some non-leaf master node  and\nSo, the total number of intermediate- nodes in the tree is at most\nThe total number of non-leaf nodes in the tree is then at most\nThus any path from a leaf of  to its root has length at most \n\u220e\nWe note that this bound is almost tight. Consider a  tree which has only one leaf and with all of the other master nodes (including the root) being master nodes of degree  This tree is just a chain from the root to the unique leaf, and has of length"
        },
        {
            "section_id": "8.1.1",
            "parent_section_id": "8.1",
            "section_name": "8.1.1 Further Properties of minimum-cost  trees",
            "text": "Definitions 6.1  ###reference_numbered1### and 6.2  ###reference_numbered2### are very loose and technically permit many scenarios, e.g., the existence of more than one intermediate- node in a  tree or a chain of intermediate- nodes descending from the root of a  tree. These scenarios will not actually occur in trees appearing in minimum-cost codes. The next lemma lists some of these scenarios and justifies ignoring them. This will be needed in the actual proof of Lemma 8.5  ###reference_numbered5### in Section 8.1.2  ###reference_.SSS2###\nA node  is a left node of  if  corresponds to codeword  for some .\nNote that for   contains exactly\n left nodes. With the exception of  which must be an intermediate- node, the other left nodes can technically be any of complete, intermediate- or master nodes. By definition, they cannot\nbe intermediate- nodes.\nLet  and \nThen there exists\n\nsatisfying\nand the\nfollowing five conditions:\nThe root of  is not an intermediate- node;\nThe root of  is not an intermediate- node;\nIf  is an intermediate- node in  then the parent of \nis not an intermediate- node;\nIf  is a non-root intermediate- node in  and, furthermore, if  then  is not a left node, then the parent of \nis either a master node or an intermediate- node;\nIf  is an intermediate- node in  then  and \nThis implies that  is the unique intermediate- node in\nIf a  tree does not satisfy one of conditions (a)-(d), we first show that it can be replaced by a  tree with one fewer nodes satisfying (13  ###reference_###). Since this process cannot be repeated forever, a tree satisfying all of the conditions (a)-(d) and satisfying (13  ###reference_###) must exist.\nBefore starting, we emphasize that none of the transformations described below adds or removes -leaves, so\nIf condition (a) is not satisfied in  then, from Consequence (e) following Definition 6.2  ###reference_numbered2###, . Let  be the intermediate- root of  and  its child. Create  by removing  and making  the root. Then\n(13  ###reference_###) is valid (with ).\nIf condition (b) is not satisfied, let  be the intermediate- root of  and  its child. Create  by removing  and making  the root. Then again\n(13  ###reference_###) is valid (with ).\nNote that this argument fails for  because removing the edge from  to  would remove the node corresponding to word  and  would then no longer be in\nIf condition (c) is not satisfied in \nlet\n be a intermediate- node in  whose child  is also an intermediate- node.\nLet  be the unique child of  Now create  from  by pointing the -edge leaving  to  instead of  i.e., removing  from the tree. Then  and\n(13  ###reference_###) is valid.\n###figure_9### If condition (d) is not satisfied in  let  be an intermediate- node in  and suppose that its parent  is either an intermediate- node or a complete node.\nLet  be the unique child of \nNow create  from  by taking the pointer from  that was pointing to  and pointing it to  instead, i.e., again removing  from .\nAgain,  and\n(13  ###reference_###) is valid.\nNote that the condition that \u201cif  then  is not a left node\u201d, ensures that Condition 2 from Definition 6.2  ###reference_numbered2### is not violated.\nWe have shown that for any  there exists a  satisfying (a)-(d) and (13  ###reference_###).\nNow assume that conditions (a)-(d) are satisfied in  but condition\n(e) is not. From (a),  is not the root of  so  the parent of  exists.\nFrom (c),  is not an intermediate- node and from the definition of master nodes  is not a master node. Thus  is either a complete node or an intermediate- node.\nLet  be the unique (-child) of \nFrom (c),  cannot be an intermediate- node; from (d),  cannot be an intermediate- node. So  is either a complete or master node.\nNow create  from  by taking the pointer from  that was pointing to  and pointing it to  instead, i.e., again removing  from\nNote that after the transformation, it is easy to see that\n and\n(13  ###reference_###) is valid.\nNote that since  is either a complete or master node, pointing  to \ndoes not affect the master nodes above \nIt only remains to show that this pointer redirection is a permissible operation on  trees, i.e., that  does not violate Condition 2 from Definition 6.2  ###reference_numbered2###.\nSince (e) is not satisfied,  There are two cases:\nThis is not possible for  because .\nIt is also not possible for  because in that case,  has a -child so it can not be an intermediate- node.\nIf  then, trivially, Condition 2 from Definition 6.2  ###reference_numbered2###  ###reference_numbered2### cannot be violated. If  then the transformation described leaves  as an intermediate- node so Condition 2 from Definition 6.2  ###reference_numbered2###  ###reference_numbered2### is still not violated.\n###figure_10### This operation of removing an intermediate- node can be repeated until condition (e) is satisfied.\n\u220e\nThis lemma has two simple corollaries.\nThere exists a minimum-cost AIFV- code  such that  tree  satisfies conditions (a)-(e) of Lemma 8.7  ###reference_numbered7###  ###reference_numbered7###.\nLet  be a minimum cost AIFV- code. For each  let  be the tree satisfying conditions (a)-(e) and equation (13  ###reference_###  ###reference_###) and set\n Since   Since  ,\nBut  was a minimum-cost AIFV- code so  must be one as well.\n\u220e\nThe corollary implies that, our algorithmic procedures, for all  may assume that all  satisfy conditions (a)-(e) of Lemma 8.7  ###reference_numbered7###  ###reference_numbered7###.\nThis assumption permits bounding the height of all  trees. The following corollary was needed in the proof of Lemma 7.1  ###reference_numbered1###  ###reference_numbered1###.\nFor all  the height of  is at most\nLet  satisfy conditions (a)-(e) of Lemma 8.7  ###reference_numbered7###  ###reference_numbered7###.\nLet  be the number of leaves in  Since all leaves are master nodes,  contains  non-leaf master nodes.\nEvery complete node in  must contain at least one leaf in each of its left and right subtrees, so the number of complete nodes in  is at most\ncontains no intermediate- node if  and one intermediate- node if\nIf  each intermediate- node in  can be written as  for some non-leaf master node  and \nIf  all intermediate- nodes in  with the possible exceptions of the left nodes   can be written as  for some non-leaf master node  and\nSo, the total number of intermediate- nodes in the tree is at most\nThe total number of non-leaf nodes in the tree is then at most\nThus any path from a leaf of  to its root has length at most \n\u220e\nWe note that this bound is almost tight. Consider a  tree which has only one leaf and with all of the other master nodes (including the root) being master nodes of degree  This tree is just a chain from the root to the unique leaf, and has of length"
        },
        {
            "section_id": "8.1.2",
            "parent_section_id": "8.1",
            "section_name": "8.1.2 The Actual Proof of Lemma 8.5",
            "text": "It now remains to prove Lemma 8.5  ###reference_numbered5###, i.e.,\nthat if \nthen, for all  and ,\n(of Lemma 8.5  ###reference_numbered5###.)\nRecall that  so,\nFix .\nNow let  be a code tree satisfying\nand\namong all trees satisfying (i),  is a tree minimizing the number of leaves assigned an .\nBecause\n(13  ###reference_###) in Lemma 8.7  ###reference_numbered7### keeps  the same and cannot increase  we may also assume that\n satisfies conditions (a)-(e) of Lemma 8.7  ###reference_numbered7###.\nSince ,\nto prove the lemma, it thus suffices to show that\n\nThis implies that  so\nSuppose to the contrary that , i.e.,  contains a leaf  assigned an . Let  denote the closest (lowest) non-intermediate- ancestor of  and\n be the child of  that is the root of the subtree containing\nBy the definition of  either  or\n where   are all intermediate- nodes for some \nNote that if  no left node is a leaf so, in particular neither  or  can be a left node.\nWe can therefore apply Lemma 8.7  ###reference_numbered7### (d) to deduce that, if  was an intermediate- node,  must be a master node. Thus, if  is a complete node or intermediate- node,  If  is a master node, then  and  so  is a master node of degree\nNow work through the three cases:\nis a complete node and  (See Figure 13  ###reference_### (a))\nLet  be the other child of  (that is not ).\nNow remove . If  was not already the -child of  make  the -child of\n###figure_11### The above transformation makes  an intermediate- node. The resulting tree remains a valid tree in  preserving the same cost  but reducing the number of leaves assigned to  by . This contradicts the minimality of , so this case is not possible.\nis a master node of degree  (See Figure 13  ###reference_### (b)) \nLet  be the source symbol assigned to  Next remove the path from  to \nconverting  to a leaf, i.e., a master node of degree . This reduces999This is the only location in the proof that uses   by  and the number of leaves assigned to  by . The resulting code tree is still in  and\ncontradicts the minimality of , so this case is also not possible.\n###figure_12### is an intermediate- node and\n (See Figure 14  ###reference_###) \nFrom Lemma 8.7  ###reference_numbered7### (a) and (e),  and , .\nSo,  is at depth .\nSince cases (a) and (b) cannot occur, \nis the unique leaf assigned an \nAll other master nodes must be assigned some source symbol.\nLet  be the deepest source symbol in the tree that is assigned to a non-left node\nand  be the master node to which  is assigned. Since  such a  must exist.\nSince every master node in\n except for  is assigned a source symbol, from\nconsequence (a) following Definition 6.2  ###reference_numbered2###,  is also a leaf.\nNow swap  and  i.e., assign  to  and  to the node that used to be\nThe resulting tree is still in  Furthermore, since the degree of all master nodes associated with source symbols remains unchanged, \nremains unchanged.\nNow consider\nIf  before the swap, then\n, and therefore\n are decreased by at least  by the swap.\nThis contradicts the minimality of , so this is not possible.\nIf  before the swap, then  and therefore\n remain unchanged. Thus the new tree satisfies conditions (i) and (ii) at the beginning of the proof. Since the original  satisfied conditions (a)-(e) of Lemma 8.7  ###reference_numbered7###, the new tree does as well. Because of the swap, the new tree does not satisfy case (c), so it must be in case (a) or (b). But, as already seen, neither of those cases can occur.\nSo,\nWe have therefore proven that, if , then it must satisfy case (c) with\n We now claim that if , then\n This, combined with  would prove the lemma.\nTo prove the claim, let  be some tree that maximizes the number of master nodes at depths  Call such a tree a -maximizing tree.\nSuppose  contained some non-leaf master node  of depth . Transform  into a complete node by giving it a -child that is a leaf. The resulting tree has the same number of master nodes, so it is also a -maximizing tree, but contains fewer non-leaf master nodes. Repeating this operation yields a -maximizing tree in which all of the master nodes at depth  are leaves.\nNext note that a -maximizing tree cannot contain any leaves of depth  because, if it did, those leaves could be changed into internal nodes with two leaf children, contradicting the definition of a -maximizing tree.\nWe have thus seen that there is a -maximizing tree  in which all of the master nodes are on level . But, any binary tree can have at most  total nodes on level  and  is not a master node in a type- tree, so  contains at most  master nodes.\nThis implies that any type- tree with  master nodes must have some (non-left) master node at depth , proving the claim and thus the lemma.\n\u220e"
        },
        {
            "section_id": "9",
            "parent_section_id": null,
            "section_name": "Conclusion and Directions for Further Work",
            "text": "The first part of this paper introduced the minimum-cost Markov chain problem. We then showed how to translate it into the problem of finding the highest point in the Markov Chain Polytope \nIn particular, Lemma 4.6  ###reference_numbered6### in Section 4.3  ###reference_### identified the problem specific information that is needed to use the Ellipsoid algorithm to solve the\nproblem in polynomial time.\nThis was written in a very general form so that it could be applied to solve problems other than binary Almost Instantaneous Fixed-to-Variable- (AIFV-) coding. For example, recent work\n[2  ###reference_b2###] uses this Lemma to derive polynomial time algorithms for AIVF-coding, a generalization of Tunstall coding\nwhich previously [17  ###reference_b17###, 18  ###reference_b18###] could only be solved in exponential time using an iterative algorithm.\nAnother possible problem use of Lemma 4.6  ###reference_numbered6### would be the construction of optimal codes for finite-state noiseless channels. [18  ###reference_b18###] recently showed how to frame this problem as a minimum cost Markov chain one and solve it using the iterative algorithm. This problem would definitely fit into the framework of Lemma 4.6  ###reference_numbered6###. Unfortunately, the calculation of the corresponding  in [18  ###reference_b18###], needed by Lemma 4.6  ###reference_numbered6### as a problem-specific separation oracle,\nis done using Integer Linear Programming and therefore requires exponential time. The development of a polynomial time algorithm for calculating the  would, by Lemma 4.6  ###reference_numbered6###, immediately yield polynomial time algorithms for solving the full problem.\nThe second part of the paper restricts itself to binary AIFV- codes, which were the original motivation for the\nminimum-cost Markov Chain problem.\nThese are -tuples of coding trees for lossless-coding that can be modelled as a Markov chain.\nWe derived properties of AIFV- coding trees that then permitted applying Lemma 4.6  ###reference_numbered6###. This yielded the first (weakly) polynomial time algorithm for constructing minimum cost binary AIFV- codes.\nThere are still many related open problems to resolve. The first is to note that\nour algorithm is only weakly polynomial, since its running time is\ndependent upon the actual sizes needed to encode the transition probabilities of the Markov Chain states in binary. For example, in AIFV- coding, this is polynomial in the number of bits needed to encode the probabilities of the words in the source alphabet.\nAn obvious goal would be, at least for AIFV- coding, to find a strongly polynomial time algorithm, one whose running time only depends upon\nA second concerns the definition of permissible Markov chains. Definition 2.1  ###reference_### requires that   This guarantees that every permissible Markov Chain  has a unique stationary distribution. It is also needed as a requirement for Theorem 1 in [1  ###reference_b1###], which is used to guarantee that there exists a distinctly-typed intersection point in  (Corollary 3.2  ###reference_numbered2### only guarantees that every distinctly-typed intersection point is on or above ). The open question is whether  is actually needed or whether the looser requirement that every  has a unique stationary distribution would suffice to guarantee that  contains some distinctly-typed intersection point.\nA final question\nwould return to the iterative algorithm approach of\n[23  ###reference_b23###, 22  ###reference_b22###, 4  ###reference_b4###, 5  ###reference_b5###].\nOur new polynomial time algorithm is primarily of theoretical interest and, like most Ellipsoid based algorithms, would be difficult to implement in practice.\nPerhaps the new geometric understanding of the problem developed here could improve the performance and analysis of the iterative algorithms.\nAs an example, the iterative algorithm of [4  ###reference_b4###, 5  ###reference_b5###, 22  ###reference_b22###, 23  ###reference_b23###]\ncan now be interpreted as moving from point to point in the set of distinctly-typed intersection points (of associated hyperplanes), never increasing the cost of\nthe associated Markov chain, finally terminating at the lowest point in this set.\nThis immediately leads to a better understanding of one of the issues with the iterative algorithms for the AIFV- problem.\nAs noted in Section 2.3  ###reference_###, the algorithm must solve for  at every step of the algorithm. As noted in Section 4  ###reference_###, this can be done in polynomial time if  but requires exponential time integer linear programming if  A difficulty with the iterative algorithm was that it was not able to guarantee that at every step, or even at the final solution,  With our new better understanding of the geometry of the Markov Chain polytope for the AIFV- problem, it might now be possible to prove that the condition\n always holds during the algorithm or develop a modified iterative algorithm in which the condition always holds.\nAcknowledgement: The authors would like to thank Reza Hosseini Dolatabadi and Arian Zamani for the generation of Figures 2  ###reference_### and 3  ###reference_###."
        }
    ],
    "appendix": [],
    "tables": {},
    "image_paths": {
        "1": {
            "figure_path": "2401.11622v4_figure_1.png",
            "caption": "Figure 1: The bottom half of the figure is a five-state Markov chain. Arrows represent non-zero transition probabilities. qj\u2062(Sk)subscript\ud835\udc5e\ud835\udc57subscript\ud835\udc46\ud835\udc58q_{j}(S_{k})italic_q start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) is the probability of transitioning from state Sksubscript\ud835\udc46\ud835\udc58S_{k}italic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT to state Sj.subscript\ud835\udc46\ud835\udc57S_{j}.italic_S start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT .\nNote that q0\u2062(Sk)>0subscript\ud835\udc5e0subscript\ud835\udc46\ud835\udc580q_{0}(S_{k})>0italic_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) > 0 for all k\u2208[5].\ud835\udc58delimited-[]5k\\in[5].italic_k \u2208 [ 5 ] .\nThe circles in the shaded rectangle \ud835\udd4aksubscript\ud835\udd4a\ud835\udc58{\\mathbb{S}}_{k}blackboard_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT represent the set of all permissible type-k\ud835\udc58kitalic_k states.\nEach Sksubscript\ud835\udc46\ud835\udc58S_{k}italic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT in the Markov Chain shown also belongs to the associated set \ud835\udd4ak.subscript\ud835\udd4a\ud835\udc58{\\mathbb{S}}_{k}.blackboard_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT .",
            "url": "http://arxiv.org/html/2401.11622v4/x1.png"
        },
        "2": {
            "figure_path": "2401.11622v4_figure_2.png",
            "caption": "Figure 2: An illustration of Lemma 3.1 (a) for a 3-state Markov chain \ud835\udc12=(S0,S1,S2).\ud835\udc12subscript\ud835\udc460subscript\ud835\udc461subscript\ud835\udc462\\mathbf{S}=(S_{0},S_{1},S_{2}).bold_S = ( italic_S start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) . The table lists the associated qj\u2062(Sk)subscript\ud835\udc5e\ud835\udc57subscript\ud835\udc46\ud835\udc58q_{j}(S_{k})italic_q start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) and \u2113\u2062(Sk)\u2113subscript\ud835\udc46\ud835\udc58\\ell(S_{k})roman_\u2113 ( italic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) values.\nThe green plane is f0\u2062(\ud835\udc31,S0)=9+x1/4+x2/4,subscript\ud835\udc530\ud835\udc31subscript\ud835\udc4609subscript\ud835\udc6514subscript\ud835\udc6524f_{0}(\\mathbf{x},S_{0})=9+x_{1}/4+x_{2}/4,italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( bold_x , italic_S start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) = 9 + italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT / 4 + italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT / 4 ,\nthe red plane f1\u2062(\ud835\udc31,S1)=11\u2212x1+x2/4,subscript\ud835\udc531\ud835\udc31subscript\ud835\udc46111subscript\ud835\udc651subscript\ud835\udc6524f_{1}(\\mathbf{x},S_{1})=11-x_{1}+x_{2}/4,italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( bold_x , italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = 11 - italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT / 4 ,\nand the blue plane f2\u2062(\ud835\udc31,S2)=14+x1/4\u2212x2.subscript\ud835\udc532\ud835\udc31subscript\ud835\udc46214subscript\ud835\udc6514subscript\ud835\udc652f_{2}(\\mathbf{x},S_{2})=14+x_{1}/4-x_{2}.italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( bold_x , italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = 14 + italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT / 4 - italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT .\nBy calculation, \ud835\udf45\u2062(\ud835\udc12)=(0.6,0.15,0.25)\ud835\udf45\ud835\udc120.60.150.25\\bm{\\pi}(\\mathbf{S})=(0.6,0.15,0.25)bold_italic_\u03c0 ( bold_S ) = ( 0.6 , 0.15 , 0.25 ) so\ncost\u2062(\ud835\udc12)=0.6\u22c59+0.15\u22c511+0.25\u22c514=10.55.cost\ud835\udc12\u22c50.69\u22c50.1511\u22c50.251410.55\\mbox{\\rm cost}(\\mathbf{S})=0.6\\cdot 9+0.15\\cdot 11+0.25\\cdot 14=10.55.cost ( bold_S ) = 0.6 \u22c5 9 + 0.15 \u22c5 11 + 0.25 \u22c5 14 = 10.55 . The planes intersect at unique point (x0,x1,y)=(1.6,4.6,10.55)subscript\ud835\udc650subscript\ud835\udc651\ud835\udc661.64.610.55(x_{0},x_{1},y)=(1.6,4.6,10.55)( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_y ) = ( 1.6 , 4.6 , 10.55 ).",
            "url": "http://arxiv.org/html/2401.11622v4/x2.png"
        },
        "3": {
            "figure_path": "2401.11622v4_figure_3.png",
            "caption": "Figure 3: An illustration of Lemma 3.3 and Corollary 3.4 for the case m=3.\ud835\udc5a3m=3.italic_m = 3 . 3000 states each of type 00, 1111 and 2222 were generated with associated \u2113\u2062(Sk)\u2113subscript\ud835\udc46\ud835\udc58\\ell(S_{k})roman_\u2113 ( italic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) values. The green, red and blue are, respectively the lower envelopes g0\u2062(\ud835\udc31)subscript\ud835\udc540\ud835\udc31g_{0}(\\mathbf{x})italic_g start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( bold_x ), g1\u2062(\ud835\udc31)subscript\ud835\udc541\ud835\udc31g_{1}(\\mathbf{x})italic_g start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( bold_x ), g2\u2062(\ud835\udc31),subscript\ud835\udc542\ud835\udc31g_{2}(\\mathbf{x}),italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( bold_x ) , i.e., the lower envelopes of the 3000 associated hyperplanes of each type. \u210d\u210d\\mathbb{H}blackboard_H is the lower envelope of those three lower envelopes. The three gi\u2062(\ud835\udc31)subscript\ud835\udc54\ud835\udc56\ud835\udc31g_{i}(\\mathbf{x})italic_g start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( bold_x ) intersect at a unique point (x1\u2217,x2\u2217,y\u2217)superscriptsubscript\ud835\udc651superscriptsubscript\ud835\udc652superscript\ud835\udc66(x_{1}^{*},x_{2}^{*},y^{*})( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT \u2217 end_POSTSUPERSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT \u2217 end_POSTSUPERSCRIPT , italic_y start_POSTSUPERSCRIPT \u2217 end_POSTSUPERSCRIPT ) which is a highest point in \u210d.\u210d\\mathbb{H}.blackboard_H . \ud835\udc12\u2217=\ud835\udc12\u2062((x1\u2217,x2\u2217))superscript\ud835\udc12\ud835\udc12superscriptsubscript\ud835\udc651superscriptsubscript\ud835\udc652\\mathbf{S}^{*}=\\mathbf{S}((x_{1}^{*},x_{2}^{*}))bold_S start_POSTSUPERSCRIPT \u2217 end_POSTSUPERSCRIPT = bold_S ( ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT \u2217 end_POSTSUPERSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT \u2217 end_POSTSUPERSCRIPT ) ) is a minimal cost Markov chain among the 30003superscript300033000^{3}3000 start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT permissible Markov chains in \ud835\udd4a\ud835\udd4a{\\mathbb{S}}blackboard_S and cost\u2062(\ud835\udc12\u2217)=y\u2217costsuperscript\ud835\udc12superscript\ud835\udc66{\\rm cost}(\\mathbf{S}^{*})=y^{*}roman_cost ( bold_S start_POSTSUPERSCRIPT \u2217 end_POSTSUPERSCRIPT ) = italic_y start_POSTSUPERSCRIPT \u2217 end_POSTSUPERSCRIPT.",
            "url": "http://arxiv.org/html/2401.11622v4/x3.png"
        },
        "4": {
            "figure_path": "2401.11622v4_figure_4.png",
            "caption": "Figure 4: Node types in a binary AIFV-3333 code tree: complete node (C\ud835\udc36Citalic_C), intermediate-00 and intermediate-1111 nodes (I0subscript\ud835\udc3c0I_{0}italic_I start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, I1subscript\ud835\udc3c1I_{1}italic_I start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT), master nodes of degrees 0,1,20120,1,20 , 1 , 2 (M0subscript\ud835\udc400M_{0}italic_M start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, M1subscript\ud835\udc401M_{1}italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, M2subscript\ud835\udc402M_{2}italic_M start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT), (M0subscript\ud835\udc400M_{0}italic_M start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is a leaf)and non-intermediate-00 nodes (N0subscript\ud835\udc410{N_{0}}italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT). The N0subscript\ud835\udc410{N_{0}}italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT nodes can be complete, master, or intermediate-1111 nodes, depending upon their location.",
            "url": "http://arxiv.org/html/2401.11622v4/x4.png"
        },
        "5": {
            "figure_path": "2401.11622v4_figure_5.png",
            "caption": "Figure 5: Example binary AIFV-3333 code for source alphabet {a,b,c,d}.\ud835\udc4e\ud835\udc4f\ud835\udc50\ud835\udc51\\left\\{a,b,c,d\\right\\}.{ italic_a , italic_b , italic_c , italic_d } . The small filled nodes are complete nodes; the small striped nodes are intermediate-1111 nodes and the small empty nodes are intermediate-00 ones. The large nodes are master nodes with their assigned source symbols. They are labelled to their sides as Misubscript\ud835\udc40\ud835\udc56M_{i}italic_M start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT nodes, indicating that they are master-i\ud835\udc56iitalic_i nodes. Note that T2subscript\ud835\udc472T_{2}italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT encodes a\ud835\udc4eaitalic_a, which is at its root, with an empty string!",
            "url": "http://arxiv.org/html/2401.11622v4/x5.png"
        },
        "9": {
            "figure_path": "2401.11622v4_figure_9.png",
            "caption": "Figure 9: Markov chain corresponding to AIFV-3333 code in Figure 5. Note that T1subscript\ud835\udc471T_{1}italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT contains no degree 1 master node, so there is no edge from T1subscript\ud835\udc471T_{1}italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT to T1.subscript\ud835\udc471T_{1}.italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT . Similarly, T2subscript\ud835\udc472T_{2}italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT contains no degree 2 master node, so there is no edge from T2subscript\ud835\udc472T_{2}italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT to T2.subscript\ud835\udc472T_{2}.italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT .",
            "url": "http://arxiv.org/html/2401.11622v4/x6.png"
        },
        "10": {
            "figure_path": "2401.11622v4_figure_10.png",
            "caption": "Figure 10: Illustration of Lemma 8.2 case (b), describing Tk\u2032\u2062[T0e\u2062x].subscriptsuperscript\ud835\udc47\u2032\ud835\udc58delimited-[]subscriptsuperscript\ud835\udc47\ud835\udc52\ud835\udc650T^{\\prime}_{k}\\left[T^{ex}_{0}\\right].italic_T start_POSTSUPERSCRIPT \u2032 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT [ italic_T start_POSTSUPERSCRIPT italic_e italic_x end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ] . Note that in Tk\u2032\u2062[T0e\u2062x],subscriptsuperscript\ud835\udc47\u2032\ud835\udc58delimited-[]subscriptsuperscript\ud835\udc47\ud835\udc52\ud835\udc650T^{\\prime}_{k}\\left[T^{ex}_{0}\\right],italic_T start_POSTSUPERSCRIPT \u2032 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT [ italic_T start_POSTSUPERSCRIPT italic_e italic_x end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ] , leaf 0k\u20621superscript0\ud835\udc5810^{k}10 start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT 1 is labelled with an \u03f5italic-\u03f5\\epsilonitalic_\u03f5 so the tree is in \ud835\udcafke\u2062x\u2062(m,n)subscriptsuperscript\ud835\udcaf\ud835\udc52\ud835\udc65\ud835\udc58\ud835\udc5a\ud835\udc5b\\mathcal{T}^{ex}_{k}(m,n)caligraphic_T start_POSTSUPERSCRIPT italic_e italic_x end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_m , italic_n ) but not in \ud835\udcafk\u2062(m,n)subscript\ud835\udcaf\ud835\udc58\ud835\udc5a\ud835\udc5b\\mathcal{T}_{k}(m,n)caligraphic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_m , italic_n ).",
            "url": "http://arxiv.org/html/2401.11622v4/x7.png"
        },
        "11": {
            "figure_path": "2401.11622v4_figure_11.png",
            "caption": "Figure 11: Illustration of transformations in cases (c) and (d) of Lemma 8.7. Note that the illustration of the second subcase of (d) assumes that v\ud835\udc63vitalic_v is the 1111-child of u\ud835\udc62uitalic_u. The case in which v\ud835\udc63vitalic_v is the 00-child is symmetric.",
            "url": "http://arxiv.org/html/2401.11622v4/x8.png"
        },
        "12": {
            "figure_path": "2401.11622v4_figure_12.png",
            "caption": "Figure 12: Illustration of transformation in case (e) of Lemma 8.7. v\ud835\udc63vitalic_v is always an intermediate-1111 node and w\ud835\udc64witalic_w can be either a master node or a complete node. z\ud835\udc67zitalic_z is a master node of degree t\ud835\udc61titalic_t. In the case in which u\ud835\udc62uitalic_u is a complete node, v\ud835\udc63vitalic_v may be either the left or right child of u.\ud835\udc62u.italic_u . Both cases are illustrated.",
            "url": "http://arxiv.org/html/2401.11622v4/x9.png"
        },
        "13": {
            "figure_path": "2401.11622v4_figure_13.png",
            "caption": "Figure 13: Illustration of the first two cases of the proof of Lemma 8.5. In case (a) a\u03f5subscript\ud835\udc4eitalic-\u03f5a_{\\epsilon}italic_a start_POSTSUBSCRIPT italic_\u03f5 end_POSTSUBSCRIPT is complete and c\u03f5subscript\ud835\udc50italic-\u03f5c_{\\epsilon}italic_c start_POSTSUBSCRIPT italic_\u03f5 end_POSTSUBSCRIPT can be its 1111-child or its 00-child. T\u2032superscript\ud835\udc47\u2032T^{\\prime}italic_T start_POSTSUPERSCRIPT \u2032 end_POSTSUPERSCRIPT is the subtree rooted at c\u03f5.subscript\ud835\udc50italic-\u03f5c_{\\epsilon}.italic_c start_POSTSUBSCRIPT italic_\u03f5 end_POSTSUBSCRIPT . In case (b) a\u03f5subscript\ud835\udc4eitalic-\u03f5a_{\\epsilon}italic_a start_POSTSUBSCRIPT italic_\u03f5 end_POSTSUBSCRIPT is a master node and \u2113\u03f5subscript\u2113italic-\u03f5\\ell_{\\epsilon}roman_\u2113 start_POSTSUBSCRIPT italic_\u03f5 end_POSTSUBSCRIPT is connected to it via a chain of intermediate-00 nodes.",
            "url": "http://arxiv.org/html/2401.11622v4/x10.png"
        },
        "14": {
            "figure_path": "2401.11622v4_figure_14.png",
            "caption": "Figure 14: Illustration of case (c) of the proof of Lemma 8.5. a\u03f5=0ksubscript\ud835\udc4eitalic-\u03f5superscript0\ud835\udc58a_{\\epsilon}=0^{k}italic_a start_POSTSUBSCRIPT italic_\u03f5 end_POSTSUBSCRIPT = 0 start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT is an intermediate-1111 node\nand \u03b2\u03f5=0k\u20621.subscript\ud835\udefditalic-\u03f5superscript0\ud835\udc581\\beta_{\\epsilon}=0^{k}1.italic_\u03b2 start_POSTSUBSCRIPT italic_\u03f5 end_POSTSUBSCRIPT = 0 start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT 1 . From definition 6.2 (2), this b\u03f5subscript\ud835\udc4fitalic-\u03f5b_{\\epsilon}italic_b start_POSTSUBSCRIPT italic_\u03f5 end_POSTSUBSCRIPT must exist in every Tk,subscript\ud835\udc47\ud835\udc58T_{k},italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , so it may not be removed.\nvisubscript\ud835\udc63\ud835\udc56v_{i}italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is a deepest leaf in Tk,subscript\ud835\udc47\ud835\udc58T_{k},italic_T start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , which is \u201cswapped\u201d with b\u03f5subscript\ud835\udc4fitalic-\u03f5b_{\\epsilon}italic_b start_POSTSUBSCRIPT italic_\u03f5 end_POSTSUBSCRIPT.",
            "url": "http://arxiv.org/html/2401.11622v4/x11.png"
        }
    },
    "validation": true,
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}