In this work, we estimate the number of hyperedges in a hypergraph ${\cal H}(U({\cal H}), {\cal F}({\cal H}))$, where $U({\cal H})$ denotes the set of vertices and ${\cal F}({\cal H}))$ denotes the set of hyperedges. We assume a query oracle access to the hypergraph ${\cal H}$. Estimating the number of edges, triangles or small subgraphs in a graph is a well studied problem. Beame \etal~and Bhattacharya \etal~gave algorithms to estimate the number of edges and triangles in a graph using queries to the {\sc Bipartite Independent Set} ({\sc BIS}) and the {\sc Tripartite Independent Set} ({\sc TIS}) oracles, respectively. We generalize the earlier works by estimating the number of hyperedges using a query oracle, known as the {\bf Generalized $d$-partite independent set oracle ({\sc GPIS})}, that takes $d$ (non-empty) pairwise disjoint subsets of vertices $A_1,\ldots,A_d \subseteq U({\cal H})$ as input, and answers whether there exists a hyperedge in ${\cal H}$ having (exactly) one vertex in each $A_i, i \in \{1,2,\ldots,d\}$. We give a randomized algorithm for the hyperedge estimation problem using the {\sc GPIS} query oracle to output $\widehat{m}$ for $m({\cal H})$ satisfying $(1-ε) \cdot m({\cal H}) \leq \widehat{m} \leq (1+ε) \cdot m({\cal H})$. The number of queries made by our algorithm, assuming $d$ to be a constant, is polylogarithmic in the number of vertices of the hypergraph.
The suffix array $SA[1..n]$ of a text $T$ of length $n$ is a permutation of $\{1,\ldots,n\}$ describing the lexicographical ordering of suffixes of $T$, and it is considered to be among of the most important data structures in string algorithms, with dozens of applications in data compression, bioinformatics, and information retrieval. One of the biggest drawbacks of the suffix array is that it is very difficult to maintain under text updates: even a single character substitution can completely change the contents of the suffix array. Thus, the suffix array of a dynamic text is modelled using …
We present a near-linear time algorithm that approximates the edit distance between two strings within a polylogarithmic factor; specifically, for strings of length n and every fixed epsilon>0, it can compute a (log n)^O(1/epsilon) approximation in n^(1+epsilon) time. This is an exponential improvement over the previously known factor, 2^(O (sqrt(log n))), with a comparable running time (Ostrovsky and Rabani J.ACM 2007; Andoni and Onak STOC 2009). Previously, no efficient polylogarithmic approximation algorithm was known for any computational task involving edit distance (e.g., nearest neighbo…
Estimating the number of triangles in a graph is one of the most fundamental problems in sublinear algorithms. In this work, we provide an algorithm that approximately counts the number of triangles in a graph using only polylogarithmic queries when \emph{the number of triangles on any edge in the graph is polylogarithmically bounded}. Our query oracle {\em Tripartite Independent Set} (TIS) takes three disjoint sets of vertices $A$, $B$ and $C$ as inputs, and answers whether there exists a triangle having one endpoint in each of these three sets. Our query model generally belongs to the class …
In this work, we consider $d$-{\sc Hyperedge Estimation} and $d$-{\sc Hyperedge Sample} problem in a hypergraph $\mathcal{H}(U(\mathcal{H}),\mathcal{F}(\mathcal{H}))$ in the query complexity framework, where $U(\mathcal{H})$ denotes the set of vertices and $\mathcal{F}(\mathcal{H})$ denotes the set of hyperedges. The oracle access to the hypergraph is called {\sc Colorful Independence Oracle} ({\sc CID}), which takes $d$ (non-empty) pairwise disjoint subsets of vertices $A_1,\ldots,A_d \subseteq U(\mathcal{H})$ as input, and answers whether there exists a hyperedge in $\mathcal{H}$ having (exa…
Dynamic and Streaming Algorithms for Union Volume Estimation
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Indian Institute of Technology Bombay（印度理工学院孟买分校）；Saarland University（萨尔兰大学）；Max-Planck-Institute for Informatics（马克斯·普朗克 informatics 研究所）
We present an $O(n^{1.5})$-space distance oracle for directed planar graphs that answers distance queries in $O(\log n)$ time. Our oracle both significantly simplifies and significantly improves the recent oracle of Cohen-Addad, Dahlgaard and Wulff-Nilsen [FOCS 2017], which uses $O(n^{5/3})$-space and answers queries in $O(\log n)$ time. We achieve this by designing an elegant and efficient point location data structure for Voronoi diagrams on planar graphs. We further show a smooth tradeoff between space and query-time. For any $S\in [n,n^2]$, we show an oracle of size $S$ that answers querie…
We consider differentially private range queries on a graph where query ranges are defined as the set of edges on a shortest path of the graph. Edges in the graph carry sensitive attributes and the goal is to report the sum of these attributes on a shortest path for counting query or the minimum of the attributes in a bottleneck query. We use differential privacy to ensure that the release of these query answers provide protection of the privacy of the sensitive edge attributes. Our goal is to develop mechanisms that minimize the additive error of the reported answers with the given privacy bu…
The problem of sparsifying a graph or a hypergraph while approximately preserving its cut structure has been extensively studied and has many applications. In a seminal work, Benczúr and Karger (1996) showed that given any $n$-vertex undirected weighted graph $G$ and a parameter $\varepsilon \in (0,1)$, there is a near-linear time algorithm that outputs a weighted subgraph $G'$ of $G$ of size $\tilde{O}(n/\varepsilon^2)$ such that the weight of every cut in $G$ is preserved to within a $(1 \pm \varepsilon)$-factor in $G'$. The graph $G'$ is referred to as a {\em $(1 \pm \varepsilon)$-approxima…
We present a simple sublinear-time algorithm for sampling an arbitrary subgraph $H$ \emph{exactly uniformly} from a graph $G$ with $m$ edges, to which the algorithm has access by performing the following types of queries: (1) degree queries, (2) neighbor queries, (3) pair queries and (4) edge sampling queries. The query complexity and running time of our algorithm are $\tilde{O}(\min\{m, \frac{m^{ρ(H)}}{\# H}\})$ and $\tilde{O}(\frac{m^{ρ(H)}}{\# H})$, respectively, where $ρ(H)$ is the fractional edge-cover of $H$ and $\# H$ is the number of copies of $H$ in $G$. For any clique on $r$ vertices…
We consider the problem of estimating the number of triangles in a graph. This problem has been extensively studied in both theory and practice, but all existing algorithms read the entire graph. In this work we design a {\em sublinear-time\/} algorithm for approximating the number of triangles in a graph, where the algorithm is given query access to the graph. The allowed queries are degree queries, vertex-pair queries and neighbor queries. We show that for any given approximation parameter $0<ε<1$, the algorithm provides an estimate $\widehat{t}$ such that with high constant probability, $(1…
The Discrepancy of a hypergraph is the minimum attainable value, over two-colorings of its vertices, of the maximum absolute imbalance of any hyperedge. The Hereditary Discrepancy of a hypergraph, defined as the maximum discrepancy of a restriction of the hypergraph to a subset of its vertices, is a measure of its complexity. Lovasz, Spencer and Vesztergombi (1986) related the natural extension of this quantity to matrices to rounding algorithms for linear programs, and gave a determinant based lower bound on the hereditary discrepancy. Matousek (2011) showed that this bound is tight up to a p…
Given a set $P$ of coloured points on the real line, we study the problem of answering range $α$-majority (or "heavy hitter") queries on $P$. More specifically, for a query range $Q$, we want to return each colour that is assigned to more than an $α$-fraction of the points contained in $Q$. We present a new data structure for answering range $α$-majority queries on a dynamic set of points, where $α\in (0,1)$. Our data structure uses O(n) space, supports queries in $O((\lg n) / α)$ time, and updates in $O((\lg n) / α)$ amortized time. If the coordinates of the points are integers, then the quer…
In the classical binary search in a path the aim is to detect an unknown target by asking as few queries as possible, where each query reveals the direction to the target. This binary search algorithm has been recently extended by [Emamjomeh-Zadeh et al., STOC, 2016] to the problem of detecting a target in an arbitrary graph. Similarly to the classical case in the path, the algorithm of Emamjomeh-Zadeh et al. maintains a candidates' set for the target, while each query asks an appropriately chosen vertex-- the "median"--which minimises a potential $Φ$ among the vertices of the candidates' set.…
We characterize the complexity of minimizing $\max_{i\in[N]} f_i(x)$ for convex, Lipschitz functions $f_1,\ldots, f_N$. For non-smooth functions, existing methods require $O(Nε^{-2})$ queries to a first-order oracle to compute an $ε$-suboptimal point and $\tilde{O}(Nε^{-1})$ queries if the $f_i$ are $O(1/ε)$-smooth. We develop methods with improved complexity bounds of $\tilde{O}(Nε^{-2/3} + ε^{-8/3})$ in the non-smooth case and $\tilde{O}(Nε^{-2/3} + \sqrt{N}ε^{-1})$ in the $O(1/ε)$-smooth case. Our methods consist of a recently proposed ball optimization oracle acceleration algorithm (which …
In this paper, we first consider the subpath convex hull query problem: Given a simple path $π$ of $n$ vertices, preprocess it so that the convex hull of any query subpath of $π$ can be quickly obtained. Previously, Guibas, Hershberger, and Snoeyink [SODA 90'] proposed a data structure of $O(n)$ space and $O(\log n\log\log n)$ query time; reducing the query time to $O(\log n)$ increases the space to $O(n\log\log n)$. We present an improved result that uses $O(n)$ space while achieving $O(\log n)$ query time. Like the previous work, our query algorithm returns a compact interval tree representi…
Near-optimal quantum query lower bounds on bipartiteness and expansion testing in the bounded-degree graph model
Université Paris Cité, CNRS, IRIF（巴黎西岱大学、法国国家科学研究中心、IRIF）；Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）；Ludwig-Maximilians-Universität München & MCQST（慕尼黑路德维希-马克西米利安大学与MCQST）
For fixed $h \geq 2$, we consider the task of adding to a graph $G$ a set of weighted shortcut edges on the same vertex set, such that the length of a shortest $h$-hop path between any pair of vertices in the augmented graph is exactly the same as the original distance between these vertices in $G$. A set of shortcut edges with this property is called an exact $h$-hopset and may be applied in processing distance queries on graph $G$. In particular, a $2$-hopset directly corresponds to a distributed distance oracle known as a hub labeling. In this work, we explore centralized distance oracles b…
We design $f$-edge fault-tolerant diameter oracles ($f$-FDOs). We preprocess a given graph $G$ on $n$ vertices and $m$ edges, and a positive integer $f$, to construct a data structure that, when queried with a set $F$ of $|F| \leq f$ edges, returns the diameter of $G-F$. For a single failure ($f=1$) in an unweighted directed graph of diameter $D$, there exists an approximate FDO by Henzinger et al. [ITCS 2017] with stretch $(1+\varepsilon)$, constant query time, space $O(m)$, and a combinatorial preprocessing time of $\widetilde{O}(mn + n^{1.5} \sqrt{Dm/\varepsilon})$.We present an FDO for dir…
We study the problem of estimating the value of sums of the form $S_p \triangleq \sum \binom{x_i}{p}$ when one has the ability to sample $x_i \geq 0$ with probability proportional to its magnitude. When $p=2$, this problem is equivalent to estimating the selectivity of a self-join query in database systems when one can sample rows randomly. We also study the special case when $\{x_i\}$ is the degree sequence of a graph, which corresponds to counting the number of $p$-stars in a graph when one has the ability to sample edges randomly. Our algorithm for a $(1 \pm \varepsilon)$-multiplicative app…
