Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Partial-Match Queries with Random Wildcards: In Tries and Distributed
  Hash Tables   Consider an $m$-bit query $q$ to a bitwise trie $T$. A wildcard $*$ is an
unspecified bit in $q$ for which the query asks the membership for both cases
$*=0$ and $*=1$. It is common that such partial-match queries with wildcards
are issued in tries. With uniformly random occurrences of $w$ wildcards in $q$
assumed, the obvious upper bound on the average number of traversal steps in
$T$ is $2^w m$. We show that the average does not exceed \[ \frac{m+1}{w+1}
\left( 2^{w+2} - 2 w - 4 \right) + m = O \left( \frac{2^w m}{w} \right), \] and
equals the value exactly when $T$ includes all the $m$-bit keys as the worst
case. Here the query $q$ performs with the naive backtracking algorithm in $T$.
It is similarly shown that the average is $O \left( \frac{k^w m}{w} \right)$ in
a general trie of maximum out-degree $k$. Our analysis for tries is extended to
a distributed hash table (DHT), which is among the most frequently used
decentralized data structures in networking. We show, under a natural
probabilistic assumption for the largest class of DHTs, that the average number
of hops required by an $m$-bit query $q$ to a DHT $D$ with random $w$ wildcards
meets the same asymptotic bound. As a result, $q$ is answered with average $O
\left( \frac{2^w m}{w} \right)$ hops rather than $\Theta \left( 2^w m \right)$
in the four major DHTs Chord, Pastry, Tapestry and Kademlia. In addition, with
a uniform key distribution for sufficiently many entries, we prove that a
lookup request to the DHT Chord is answered correctly with $O(m)$ hops and
probability $1 - 2^{-\Omega (m)}$. To the author's knowledge, the probability
$1 - 2^{-\Omega (m)}$ of correct lookup in Chord has not been identified so
far.

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The categories of this paper are: physics
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The categories of this paper are: funct-an
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The categories of this paper are: nlin
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The categories of this paper are: atom-ph
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The categories of this paper are: econ
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The categories of this paper are: comp-gas
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The categories of this paper are: stat
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The categories of this paper are: hep-lat
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The categories of this paper are: ao-sci
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The categories of this paper are: q-alg
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The categories of this paper are: math-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-th
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The categories of this paper are: hep-ph
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The categories of this paper are: adap-org