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A Deterministic Algorithm for the FriezeKannan Regularity Lemma
, 2011
"... The FriezeKannan regularity lemma is a powerful tool in combinatorics. It has also found applications in the design of approximation algorithms and recently in the design of fast combinatorial algorithms for boolean matrix multiplication. The algorithmic applications of this lemma require one to ef ..."
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The FriezeKannan regularity lemma is a powerful tool in combinatorics. It has also found applications in the design of approximation algorithms and recently in the design of fast combinatorial algorithms for boolean matrix multiplication. The algorithmic applications of this lemma require one to efficiently construct a partition satisfying the conditions of the lemma. Williams [25] recently asked if one can construct a partition satisfying the conditions of the FriezeKannan regularity lemma in deterministic subcubic time. We resolve this problem by designing an Õ(nω) time algorithm for constructing such a partition, where ω < 2.376 is the exponent of fast matrix multiplication. The algorithm relies on a spectral characterization of vertex partitions satisfying the properties of the FriezeKannan regularity lemma.
Improved Quantum Algorithm for Triangle Finding via Combinatorial Arguments
"... Background. Triangle finding is a graphtheoretic problem whose complexity is deeply connected to the complexity of several other computational tasks in theoretical computer science, such as solving path or matrix problems [3, 8, 9, 13, 18, 17, 19]. In its standard version (sometimes called unweight ..."
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Background. Triangle finding is a graphtheoretic problem whose complexity is deeply connected to the complexity of several other computational tasks in theoretical computer science, such as solving path or matrix problems [3, 8, 9, 13, 18, 17, 19]. In its standard version (sometimes called unweighted triangle finding), it asks to find, given an undirected and unweighted graph G = (V,E), three vertices v1, v2, v3 ∈ V such that {v1, v2}, {v1, v3} and {v2, v3} are edges of the graph. Problems like triangle finding can be studied in the query complexity setting. In the usual model used to describe the query complexity of such problems, the set of edges E of the graph is unknown but can be accessed through an oracle: given two vertices u and v in V, one query to the oracle outputs one if {u, v} ∈ E and zero if {u, v} / ∈ E. In the quantum query complexity setting, one further assume that the oracle can be queried in superposition. Besides its intrinsic interest, the triangle finding problem has been one of the main problems that stimulated the development of new techniques in quantum query complexity, and the history of improvement of upper bounds on the query complexity of triangle finding parallels the development of general techniques in the quantum complexity setting, as we explain below. Grover search immediately gives, when applied to triangle finding as a search over the space of triples of vertices of the graph, a quantum algorithm with query complexity O(n3/2). Using amplitude
Triangle Detection Versus Matrix Multiplication: A Study of Truly Subcubic Reducibility
"... It is well established that the problem of detecting a triangle in a graph can be reduced to Boolean matrix multiplication (BMM). Many have asked if there is a reduction in the other direction: can a fast triangle detection algorithm be used to solve BMM faster? The general intuition has been that s ..."
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It is well established that the problem of detecting a triangle in a graph can be reduced to Boolean matrix multiplication (BMM). Many have asked if there is a reduction in the other direction: can a fast triangle detection algorithm be used to solve BMM faster? The general intuition has been that such a reduction is impossible: for example, triangle detection returns one bit, while a BMM algorithm returns n 2 bits. Similar reasoning goes for other matrix products and their corresponding triangle problems. We show this intuition is false, and present a new generic strategy for efficiently computing matrix products over algebraic structures used in optimization. We say an algorithm on n × n matrices (or nnode graphs) is truly subcubic if it runs in O(n 3−δ · poly(log M)) time for some δ> 0, where M is the absolute value of the largest entry (or the largest edge weight). We prove an equivalence between the existence of truly subcubic algorithms for any (min, ⊙) matrix product, the corresponding matrix product verification problem, and a corresponding triangle detection problem. Our work simplifies and unifies prior work, and has some new consequences: • The following problems either all have truly subcubic algorithms, or none of them do:
Adaptive Range Counting and Other FrequencyBased Range Query Problems
, 2012
"... I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii We consider variations of range searching in which, gi ..."
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I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii We consider variations of range searching in which, given a query range, our goal is to compute some function based on frequencies of points that lie in the range. The most basic such computation involves counting the number of points in a query range. Data structures that compute this function solve the wellstudied range counting problem. We consider adaptive and approximate data structures for the 2D orthogonal range counting problem under the wbit word RAM model. The query time of an adaptive range counting data structure is sensitive to k, the number of points being counted. We give an adaptive data structure that requires O(n log log n) space and O(log log n+logw k) query time. Nonadaptive data structures on the other hand require Ω(logw n) query time (Pătraşcu, 2007). Our specific bounds are interesting for two reasons. First, when k = O(1), our bounds
Finding Structures in Largescale Graphs
"... One of the most vexing challenges of working with graphical structures is that most algorithms scale poorly as the graph becomes very large. The computation is extremely expensive even for polynomial algorithms, thus making it desirable to devise fast approximation algorithms. We herein propose a fr ..."
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One of the most vexing challenges of working with graphical structures is that most algorithms scale poorly as the graph becomes very large. The computation is extremely expensive even for polynomial algorithms, thus making it desirable to devise fast approximation algorithms. We herein propose a framework using advanced tools 1–6 from random graph theory and spectral graph theory to address the quantitative analysis of the structure and dynamics of largescale networks. This framework enables one to carry out analytic computations of observable network structures and capture the most relevant and refined quantities of realworld networks. 1.
An Improved Combinatorial Algorithm for Boolean Matrix Multiplication
, 2015
"... We present a new combinatorial algorithm for triangle finding and Boolean matrix multiplication that runs in Ô(n3 / log4 n) time, where the O ̂ notation suppresses poly(loglog) factors. This improves the previous best combinatorial algorithm by Chan [4] that runs in Ô(n3 / log3 n) time. Our algor ..."
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We present a new combinatorial algorithm for triangle finding and Boolean matrix multiplication that runs in Ô(n3 / log4 n) time, where the O ̂ notation suppresses poly(loglog) factors. This improves the previous best combinatorial algorithm by Chan [4] that runs in Ô(n3 / log3 n) time. Our algorithm generalizes the divideandconquer 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 n3. 1