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Register a new userA Simple Sublinear Algorithm for Gap Edit Distance
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Added by
Joshua Brakensiek
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We study the problem of estimating the edit distance between two $n$-character strings. While exact computation in the worst case is believed to require near-quadratic time, previous work showed that in certain regimes it is possible to solve the following {em gap edit distance} problem in sub-linear time: distinguish between inputs of distance $le k$ and $>k^2$. Our main result is a very simple algorithm for this benchmark that runs in time $tilde O(n/sqrt{k})$, and in particular settles the open problem of obtaining a truly sublinear time for the entire range of relevant $k$. Building on the same framework, we also obtain a $k$-vs-$k^2$ algorithm for the one-sided preprocessing model with $tilde O(n)$ preprocessing time and $tilde O(n/k)$ query time (improving over a recent $tilde O(n/k+k^2)$-query time algorithm for the same problem [GRS20].
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In the subgraph counting problem, we are given a input graph $G(V, E)$ and a target graph $H$; the goal is to estimate the number of occurrences of $H$ in $G$. Our focus here is on designing sublinear-time algorithms for approximately counting occurrences of $H$ in $G$ in the setting where the algorithm is given query access to $G$. This problem has been studied in several recent papers which primarily focused on specific families of graphs $H$ such as triangles, cliques, and stars. However, not much is known about approximate counting of arbitrary graphs $H$. This is in sharp contrast to the closely related subgraph enumeration problem that has received significant attention in the database community as the database join problem. The AGM bound shows that the maximum number of occurrences of any arbitrary subgraph $H$ in a graph $G$ with $m$ edges is $O(m^{rho(H)})$, where $rho(H)$ is the fractional edge-cover of $H$, and enumeration algorithms with matching runtime are known for any $H$. We bridge this gap between subgraph counting and subgraph enumeration by designing a sublinear-time algorithm that can estimate the number of any arbitrary subgraph $H$ in $G$, denoted by $#H$, to within a $(1pm epsilon)$-approximation w.h.p. in $O(frac{m^{rho(H)}}{#H}) cdot poly(log{n},1/epsilon)$ time. Our algorithm is allowed the standard set of queries for general graphs, namely degree queries, pair queries and neighbor queries, plus an additional edge-sample query that returns an edge chosen uniformly at random. The performance of our algorithm matches those of Eden et.al. [FOCS 2015, STOC 2018] for counting triangles and cliques and extend them to all choices of subgraph $H$ under the additional assumption of edge-sample queries. We further show that our algorithm works for the more general database join size estimation problem and prove a matching lower bound for this problem.
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