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More Applications of the Polynomial Method to Algorithm Design
TLDR
This paper extends the polynomial method to solve a number of problems in combinatorial pattern matching and Boolean algebra, considerably faster than previously known methods.
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Beating Brute Force for Systems of Polynomial Equations over Finite Fields
TLDR
The algorithm for systems of ΣΠΣ polynomials also introduces a new degree reduction method that takes an instance of the problem and outputs a subexponential-sized set of instances, in such a way that feasibility is preserved and every polynomial among the output instances has degree O(log(s/n).
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Matching Triangles and Basing Hardness on an Extremely Popular Conjecture
TLDR
Novel reductions from 3-SUM, APSP, and CNF-SAT are designed, and interesting consequences of this very plausible conjecture are derived, including tight n3-o(1) lower bounds for purely-combinatorial problems about the triangles in unweighted graphs and new conditional lower bound for the Single-Source-Max-Flow problem.
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Finding orthogonal vectors in discrete structures
TLDR
A very simple and efficient output-sensitive algorithm for matrix multiplication that works over any field and shows the randomized communication complexity of the problem is closely related to the sizes of matching vector families, which have been studied in the design of locally decodable codes.
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Finding Four-Node Subgraphs in Triangle Time
TLDR
A general randomized technique for finding any induced four-node subgraph, except for the clique or independent set on 4 nodes, in O(nω) time with high probability, which substantially improves on prior work.
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An Improved Combinatorial Algorithm for Boolean Matrix Multiplication
We present a new combinatorial algorithm for triangle finding and Boolean matrix multiplication that runs in \(\hat{O}(n^3/\log ^4 n)\) time, where the \(\hat{O}\) notation suppresses poly(loglog)…
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Optimal Lower Bounds for Distributed and Streaming Spanning Forest Computation
TLDR
It is proved that any data structure for fully dynamic spanning forest in which updates can insert or delete edges amongst a base set of $n$ vertices must use $\Omega(n\log^3 n)$ bits of memory.
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Amortized Dynamic Cell-Probe Lower Bounds from Four-Party Communication
This paper develops a new technique for proving amortized, randomized cell-probe lower bounds on dynamic data structure problems. We introduce a new randomized nondeterministic four-party…
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Succinct Filters for Sets of Unknown Sizes
TLDR
This work presents a filter data structure with improvements in two aspects: - it has constant worst-case time for all insertions and lookups with high probability; and - it uses space $(1+o(1) n(\log (1/\epsilon)+O(\log\log n)$ bits when $n>u^{0.001}$, achieving optimal leading constant for all $\ep silon=o( 1)$.
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Optimal succinct rank data structure via approximate nonnegative tensor decomposition
  • Huacheng Yu
  • Computer Science, Mathematics
    STOC
  • 5 November 2018
TLDR
This paper designs a new succinct rank data structure with r=n/(logn)Ω(t)+n1−c and query time O(t) and establishes an interesting connection between succinct data structures and approximate nonnegative tensor decomposition.
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