arXiv subjects

Huacheng Yu

Publications and source records attributed to Huacheng Yu.

46 records · Page 3Linked to original sources

arXiv2019

Pruning based Distance Sketches with Provable Guarantees on Random Graphs

Measuring the distances between vertices on graphs is one of the most fundamental components in network analysis. Since finding shortest paths requires traversing the graph, it is challenging to obtain distance information on large graphs very quickly. In this work, we present a preprocessing algorithm that is able to create landmark based distance sketches efficiently, with strong theoretical guarantees. When evaluated on a diverse set of social and information networks, our algorithm significantly improves over existing approaches by reducing the number of landmarks stored, preprocessing time, or stretch of the estimated distances. On Erdös-Rényi graphs and random power law graphs with degree distribution exponent $2 < β< 3$, our algorithm outputs an exact distance data structure with space between $Θ(n^{5/4})$ and $Θ(n^{3/2})$ depending on the value of $β$, where $n$ is the number of vertices. We complement the algorithm with tight lower bounds for Erdos-Renyi graphs and the case when $β$ is close to two.

cs.SI↗

arXiv2017

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 $O(\log n)$-time data structure. Using online communication complexity, we prove a tight cell-probe lower bound for each: spending $o(\log n)$ (even amortized) time per operation results in at best an $\exp(-δ^2 n)$ probability of correctly answering a $(1/2+δ)$-fraction of the $n$ queries.

cs.DS↗

arXiv2017

Fillable arrays with constant time operations and a single bit of redundancy

In the fillable array problem one must maintain an array A[1..n] of $w$-bit entries subject to random access reads and writes, and also a $\texttt{fill}(Δ)$ operation which sets every entry of to some $Δ\in\{0,\ldots,2^w-1\}$. We show that with just one bit of redundancy, i.e. a data structure using $nw+1$ bits of memory, $\texttt{read}/\texttt{fill}$ can be implemented in worst case constant time, and $\texttt{write}$ can be implemented in either amortized constant time (deterministically) or worst case expected constant (randomized). In the latter case, we need to store an additional $O(\log n)$ random bits to specify a permutation drawn from an $1/n^2$-almost pairwise independent family.

cs.DS↗

arXiv2017

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 method for proving dynamic cell probe lower bounds and use it to prove a $\tildeΩ(\log^{1.5} n)$ 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 $\mathbb{F}_2$ ([Pat07]). Proving an $ω(\lg n)$ lower bound for this problem was explicitly posed as one of five important open problems in the late Mihai Pǎtraşcu's obituary [Tho13]. This result also implies the first $ω(\lg n)$ 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 (Chebychev) polynomials which may be of independent interest, and offers an entirely new algorithmic angle on the "cell sampling" method of Panigrahy et al. [PTW10].

cs.DS↗

arXiv2016

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 $O((N/B)\lg_{M/B} N)$ 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 $O((N/B) \lg_2 N)$ 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 $Ω((N/B) \lg_{\lg N} B)$ 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.

cs.DS↗

arXiv2016

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 communication model that enables "accelerated", error-preserving simulations of dynamic data structures. We use this technique to prove an $Ω(n(\log n/\log\log n)^2)$ cell-probe lower bound for the dynamic 2D weighted orthogonal range counting problem (2D-ORC) with $n/\mathrm{poly}\log n$ updates and $n$ queries, that holds even for data structures with $\exp(-\tildeΩ(n))$ success probability. This result not only proves the highest amortized lower bound to date, but is also tight in the strongest possible sense, as a matching upper bound can be obtained by a deterministic data structure with worst-case operational time. This is the first demonstration of a "sharp threshold" phenomenon for dynamic data structures. Our broader motivation is that cell-probe lower bounds for exponentially small success facilitate reductions from dynamic to static data structures. As a proof-of-concept, we show that a slightly strengthened version of our lower bound would imply an $Ω((\log n /\log\log n)^2)$ lower bound for the static 3D-ORC problem with $O(n\log^{O(1)}n)$ space. Such result would give a near quadratic improvement over the highest known static cell-probe lower bound, and break the long standing $Ω(\log n)$ barrier for static data structures.

cs.DS↗

arXiv2015

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 $\mathcal{I}$ over $[0, n]$ with integer coordinates, supporting the following operations: - insert(a, b): add an interval $[a, b]$ to $\mathcal{I}$, provided that $a$ and $b$ are integers in $[0, n]$; - delete(a, b): delete a (previously inserted) interval $[a, b]$ from $\mathcal{I}$; - query(): return the total length of the union of all intervals in $\mathcal{I}$. 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 $O(n^{0.01})$ queries, for which any data structure with any constant error probability requires $Ω(n\log n)$ time in expectation. Interestingly, we use the sparse set disjointness protocol of Håstad and Wigderson [ToC'07] 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.

cs.DS↗

arXiv2015

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 that runs in $\hat{O}(n^3/\log^3 n)$ time. Our algorithm generalizes the divide-and-conquer strategy of Chan's 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 $n^3$.

cs.DS↗

arXiv2011

On a Conjecture of Butler and Graham

Motivated by a hat guessing problem proposed by Iwasawa \cite{Iwasawa10}, Butler and Graham \cite{Butler11} made the following conjecture on the existence of certain way of marking the {\em coordinate lines} in $[k]^n$: there exists a way to mark one point on each {\em coordinate line} in $[k]^n$, so that every point in $[k]^n$ is marked exactly $a$ or $b$ times as long as the parameters $(a,b,n,k)$ satisfies that there are non-negative integers $s$ and $t$ such that $s+t = k^n$ and $as+bt = nk^{n-1}$. In this paper we prove this conjecture for any prime number $k$. Moreover, we prove the conjecture for the case when $a=0$ for general $k$.

math.CO↗

arXiv2011

A New Variation of Hat Guessing Games

Several variations of hat guessing games have been popularly discussed in recreational mathematics. In a typical hat guessing game, after initially coordinating a strategy, each of $n$ players is assigned a hat from a given color set. Simultaneously, each player tries to guess the color of his/her own hat by looking at colors of hats worn by other players. In this paper, we consider a new variation of this game, in which we require at least $k$ correct guesses and no wrong guess for the players to win the game, but they can choose to "pass". A strategy is called {\em perfect} if it can achieve the simple upper bound $\frac{n}{n+k}$ of the winning probability. We present sufficient and necessary condition on the parameters $n$ and $k$ for the existence of perfect strategy in the hat guessing games. In fact for any fixed parameter $k$, the existence of perfect strategy can be determined for every sufficiently large $n$. In our construction we introduce a new notion: $(d_1,d_2)$-regular partition of the boolean hypercube, which is worth to study in its own right. For example, it is related to the $k$-dominating set of the hypercube. It also might be interesting in coding theory. The existence of $(d_1,d_2)$-regular partition is explored in the paper and the existence of perfect $k$-dominating set follows as a corollary.

math.CO↗