Huacheng Yu

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) factors. This improves the previous best combinatorial algorithm by Chan [4] that runs in \(\hat{O}(n^3/\log ^3 n)\) time. Our algorithm generalizes the divide-and-conquer strategy of Chan’s algorithm.

Crossing the logarithmic barrier for dynamic Boolean data structure lower bounds

This paper proves the first super-logarithmic lower bounds on the cell probe complexity of dynamic boolean (a.k.a. decision) data structure problems, a long-standing milestone in data structure lower bounds. We introduce a new approach and use it to prove a Ω(log1.5 n) lower bound on the operational time of a wide range of boolean data structure problems, most notably, on the query time of dynamic range counting over F2. Proving an ω(lgn) lower bound for this problem was explicitly posed as one of five important open problems in the late Mihai Pǎtraşcu’s obituary . This result also implies the first ω(lgn) lower bound for the classical 2D range counting problem, one of the most fundamental data structure problems in computational geometry and spatial databases. We derive similar lower bounds for boolean versions of dynamic polynomial evaluation and 2D rectangle stabbing, and for the (non-boolean) problems of range selection and range median. Our technical centerpiece is a new way of “weakly” simulating dynamic data structures using efficient one-way communication protocols with small advantage over random guessing. This simulation involves a surprising excursion to low-degree (Chebyshev) polynomials which may be of independent interest, and offers an entirely new algorithmic angle on the “cell sampling” method of Panigrahy et al. .

Beating brute force for systems of polynomial equations over finite fields

We consider the problem of solving systems of multivariate polynomial equations of degree k over a finite field. For every integer k ≥ 2 and finite field 𝔽q where q = pd for a prime p, we give, to the best of our knowledge, the first algorithms that achieve an exponential speedup over the brute force O(qn) time algorithm in the worst case. We present two algorithms, a randomized algorithm with running time qn+o(n) · q−n/O(k) time if q ≤ 24ekd, and [EQUATION] otherwise, where e = 2.718... is Napiers constant, and a deterministic algorithm for counting solutions with running time qn+o(n) ·q−n/O(kq6/7d). For the important special case of quadratic equations in 𝔽2, our randomized algorithm has running time O(20.8765n). For systems over 𝔽2 we also consider the case where the input polynomials do not have bounded degree, but instead can be efficiently represented as a ΣΠΣ circuit, i.e., a sum of products of sums of variables. For this case we present a deterministic algorithm running in time 2n−Δn for Δ = 1/O(log(s/n)) for instances with s product gates in total and n variables. Our algorithms adapt several techniques recently developed via the polynomial method from circuit complexity. 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)).

Cell-probe lower bounds for dynamic problems via a new communication model

In this paper, we develop a new communication model to prove a data structure lower bound for the dynamic interval union problem. The problem is to maintain a multiset of intervals I over [0, n] with integer coordinates, supporting the following operations: 1) insert(a, b), add an interval [a, b] to I, provided that a and b are integers in [0, n]; 2) delete(a, b), delete an (existing) interval [a, b] from I; 3) query(), return the total length of the union of all intervals in I. It is related to the two-dimensional case of Klee’s measure problem. We prove that there is a distribution over sequences of operations with O(n) insertions and deletions, and O(n0.01) queries, for which any data structure with any constant error probability requires Ω(nlogn) time in expectation. Interestingly, we use the sparse set disjointness protocol of Håstad and Wigderson to speed up a reduction from a new kind of nondeterministic communication games, for which we prove lower bounds. For applications, we prove lower bounds for several dynamic graph problems by reducing them from dynamic interval union.

DecreaseKeys are expensive for external memory priority queues

One of the biggest open problems in external memory data structures is the priority queue problem with DecreaseKey operations. If only Insert and ExtractMin operations need to be supported, one can design a comparison-based priority queue performing O((N/B)lgM/B N) I/Os over a sequence of N operations, where B is the disk block size in number of words and M is the main memory size in number of words. This matches the lower bound for comparison-based sorting and is hence optimal for comparison-based priority queues. However, if we also need to support DecreaseKeys, the performance of the best known priority queue is only O((N/B) lg2 N) I/Os. The big open question is whether a degradation in performance really is necessary. We answer this question affirmatively by proving a lower bound of Ω((N/B) lglgN B) I/Os for processing a sequence of N intermixed Insert, ExtraxtMin and DecreaseKey operations. Our lower bound is proved in the cell probe model and thus holds also for non-comparison-based priority queues.

Cell-probe lower bounds from online communication complexity

In this work, we introduce an online model for communication complexity. Analogous to how online algorithms receive their input piece-by-piece, our model presents one of the players, Bob, his input piece-by-piece, and has the players Alice and Bob cooperate to compute a result each time before the next piece is revealed to Bob. This model has a closer and more natural correspondence to dynamic data structures than classic communication models do, and hence presents a new perspective on data structures. We first present a tight lower bound for the online set intersection problem in the online communication model, demonstrating a general approach for proving online communication lower bounds. The online communication model prevents a batching trick that classic communication complexity allows, and yields a stronger lower bound. We then apply the online communication model to prove data structure lower bounds for two dynamic data structure problems: the Group Range problem and the Dynamic Connectivity problem for forests. Both of the problems admit a worst case O(logn)-time data structure. Using online communication complexity, we prove a tight cell-probe lower bound for each: spending o(logn) (even amortized) time per operation results in at best an exp(−δ2 n) probability of correctly answering a (1/2+δ)-fraction of the n queries.

An improved combinatorial algorithm for Boolean matrix multiplication

Abstract We present a new combinatorial algorithm for triangle finding and Boolean matrix multiplication that runs in O ˆ ( n 3 / log 4 ⁡ n ) time, where the O ˆ notation suppresses poly(loglog) factors. This improves the previous best combinatorial algorithm by Chan that runs in O ˆ ( n 3 / log 3 ⁡ n ) time. Our algorithm generalizes the divide-and-conquer strategy of Chans algorithm. Moreover, we propose a general framework for detecting triangles in graphs and computing Boolean matrix multiplication. Roughly speaking, if we can find the “easy parts” of a given instance efficiently, we can solve the whole problem faster than n 3 .