
$$\Pi\ \circ\Pi^{0}$$

nordering N N ( ( )) is MC-contradicting , if it is not min consistent

) is $\,M C$-$\epsilon$  . 
N,S ( N ) r N,S ( N )
. Two orderings 1 1 2 i the other one is

$N,\mathcal{S}_{0}\left(\begin{array}{cc}1&2\end{array}\right)$$N,\mathcal{S}_{0}\left(\begin{array}{cc}1\end{array}\right)$$N,\mathcal{S}_{0}\left(\begin{array}{cc}2\end{array}\right)$
set of orderings CN 
(when at least one min consistent ordering of   (  ) e ists)

$N,\mathcal{S}_{1}\left(\mathcal{C}_{N}\right)$$\exists$$N$$\left(\begin{array}{c}\left(\begin{array}{c}\right)\end{array}\right)$$N,\mathcal{S}_{1}\left(\begin{array}{c}\right)\end{array}\right)$$\exists$$N$$\mathcal{C}_{N}$$N,\mathcal{S}_{1}\left(\begin{array}{c}\right)$
n important propert  of valid sets is that a valid set of orderings of the root of
( o  ) must contain a minimal ordering of  o.  Indeed, in the root   (  )
and consistenc  becomes identit     lso,   (  )     , so that binderconsistenc  becomes consistenc , and min consistenc becomes minimalit . ince there alwa s e ists a minimal ordering of o, a valid set of orderings of the root must contain a minimal ordering of

## 43 The Outline Of The Divide-And-Conquer Algorithm

e propose an algorithm that is based on producing valid sets of orderings. ach node in a divisibilit tree produces a valid set for its associated set of subgoals, and passes it to its parent node. fter the valid set of the root node is found, we compare costs of all its members, and return the cheapest one.

The set of orderings produced b  the algorithm for a node    is called a candidate set Its members are called candidate orderings of   , or simpl  candidates.  To find C.

of a candidate set of , we first consider the set of all possible orderings of ( ) that are consistent with candidates of s children. This set is called the consistency set of iven the candidate sets of s children, the consistenc set of s is defined uni uel .

is usuall not uni ue.

candidate set of Definition: et be a node in a divisibilit tree of o. The consistency set of , denoted
( ), and the candidate set of , denoted
( ), are defined recursivel If is a leaf, its consistenc  set contains all permutations of  (  )
( )
( ))
is an N node, and its child nodes are 1.2 and k, we define the consistenc lf set of as the set of all possible orderings of ( ) consistent with candidates of l 2

( ) $\left\{\begin{array}{l}\mbox{\small$N$}\end{array}\right.$ ( ( )) $\left|\begin{array}{l}\mbox{\small$(1$}\end{array}\right.$ ) $\exists\mbox{\small$i$}\end{array}$ ( $\mbox{\small$i$}$ ) $\left(\begin{array}{l}\mbox{\small$i$}\end{array}\right.$ )
$$59$$