Virginia Vassilevska Williams

Multiplying matrices faster than coppersmith-winograd

We develop an automated approach for designing matrix multiplication algorithms based on constructions similar to the Coppersmith-Winograd construction. Using this approach we obtain a new improved bound on the matrix multiplication exponent ω<2.3727.

Popular Conjectures Imply Strong Lower Bounds for Dynamic Problems

We consider several well-studied problems in dynamic algorithms and prove that sufficient progress on any of them would imply a breakthrough on one of five major open problems in the theory of algorithms: 1) Is the 3SUM problem on n numbers in O(n2-ε) time for some ε > 0? 2) Can one determine the satisfiability of a CNF formula on n variables and poly n clauses in O((2 - ε)npoly n) time for some ε > 0? 3) Is the All Pairs Shortest Paths problem for graphs on n vertices in O(n3-ε) time for some ε > 0? 4) Is there a linear time algorithm that detects whether a given graph contains a triangle? 5) Is there an O(n3-ε) time combinatorial algorithm for n×n Boolean matrix multiplication? The problems we consider include dynamic versions of bipartite perfect matching, bipartite maximum weight matching, single source reachability, single source shortest paths, strong connectivity, subgraph connectivity, diameter approximation and some nongraph problems such as Paghs problem defined in a recent paper by Patrascu[STOC 2010].

Consequences of Faster Alignment of Sequences

The Local Alignment problem is a classical problem with applications in biology. Given two input strings and a scoring function on pairs of letters, one is asked to find the substrings of the two input strings that are most similar under the scoring function. The best algorithms for Local Alignment run in time that is roughly quadratic in the string length. It is a big open problem whether substantially subquadratic algorithms exist. In this paper we show that for all e > 0, an O(n 2 − e ) time algorithm for Local Alignment on strings of length n would imply breakthroughs on three longstanding open problems: it would imply that for some δ > 0, 3SUM on n numbers is in O(n 2 − δ ) time, CNF-SAT on n variables is in O((2 − δ) n ) time, and Max Weight 4-Clique is in O(n 4 − δ ) time. Our result for CNF-SAT also applies to the easier problem of finding the longest common substring of binary strings with don’t cares. We also give strong conditional lower bounds for the more general Multiple Local Alignment problem on k strings, under both k-wise and SP scoring, and for other string similarity problems such as Global Alignment with gap penalties and normalized Longest Common Subsequence.

Finding, Minimizing, and Counting Weighted Subgraphs

For a pattern graph

Tight Hardness Results for LCS and Other Sequence Similarity Measures

H

Simulating branching programs with edit distance and friends: or: a polylog shaved is a lower bound made

on

Manipulating stochastically generated single-elimination tournaments for nearly all players

k

Rigging tournament brackets for weaker players

nodes, we consider the problems of finding and counting the number of (not necessarily induced) copies of

Subtree isomorphism revisited

H

Who can win a single-elimination tournament?

in a given large graph