 NSFPAR ID:
 10350674
 Date Published:
 Journal Name:
 The Electronic Journal of Combinatorics
 Volume:
 29
 Issue:
 1
 ISSN:
 10778926
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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null (Ed.)Motivated by the increasing need to understand the distributed algorithmic foundations of largescale graph computations, we study some fundamental graph problems in a messagepassing model for distributed computing where k ≥ 2 machines jointly perform computations on graphs with n nodes (typically, n >> k). The input graph is assumed to be initially randomly partitioned among the k machines, a common implementation in many realworld systems. Communication is pointtopoint, and the goal is to minimize the number of communication rounds of the computation. Our main contribution is the General Lower Bound Theorem , a theorem that can be used to show nontrivial lower bounds on the round complexity of distributed largescale data computations. This result is established via an informationtheoretic approach that relates the round complexity to the minimal amount of information required by machines to solve the problem. Our approach is generic, and this theorem can be used in a “cookbook” fashion to show distributed lower bounds for several problems, including nongraph problems. We present two applications by showing (almost) tight lower bounds on the round complexity of two fundamental graph problems, namely, PageRank computation and triangle enumeration . These applications show that our approach can yield lower bounds for problems where the application of communication complexity techniques seems not obvious or gives weak bounds, including and especially under a stochastic partition of the input. We then present distributed algorithms for PageRank and triangle enumeration with a round complexity that (almost) matches the respective lower bounds; these algorithms exhibit a round complexity that scales superlinearly in k , improving significantly over previous results [Klauck et al., SODA 2015]. Specifically, we show the following results: PageRank: We show a lower bound of Ὼ(n/k 2 ) rounds and present a distributed algorithm that computes an approximation of the PageRank of all the nodes of a graph in Õ(n/k 2 ) rounds. Triangle enumeration: We show that there exist graphs with m edges where any distributed algorithm requires Ὼ(m/k 5/3 ) rounds. This result also implies the first nontrivial lower bound of Ὼ(n 1/3 ) rounds for the congested clique model, which is tight up to logarithmic factors. We then present a distributed algorithm that enumerates all the triangles of a graph in Õ(m/k 5/3 + n/k 4/3 ) rounds.more » « less

We present a new technique for efficiently removing almost all short cycles in a graph without unintentionally removing its triangles. Consequently, triangle finding problems do not become easy even in almost kcycle free graphs, for any constant k≥ 4. Triangle finding is at the base of many conditional lower bounds in P, mainly for distance computation problems, and the existence of many 4 or 5cycles in a worstcase instance had been the obstacle towards resolving major open questions. Hardness of approximation: Are there distance oracles with m1+o(1) preprocessing time and mo(1) query time that achieve a constant approximation? Existing algorithms with such desirable time bounds only achieve superconstant approximation factors, while only 3− factors were conditionally ruled out (Pătraşcu, Roditty, and Thorup; FOCS 2012). We prove that no O(1) approximations are possible, assuming the 3SUM or APSP conjectures. In particular, we prove that kapproximations require Ω(m1+1/ck) time, which is tight up to the constant c. The lower bound holds even for the offline version where we are given the queries in advance, and extends to other problems such as dynamic shortest paths. The 4Cycle problem: An infamous open question in finegrained complexity is to establish any surprising consequences from a subquadratic or even lineartime algorithm for detecting a 4cycle in a graph. This is arguably one of the simplest problems without a nearlinear time algorithm nor a conditional lower bound. We prove that Ω(m1.1194) time is needed for kcycle detection for all k≥ 4, unless we can detect a triangle in √ndegree graphs in O(n2−δ) time; a breakthrough that is not known to follow even from optimal matrix multiplication algorithms.more » « less

null (Ed.)Triangle counting is a fundamental technique in network analysis, that has received much attention in various input models. The vast majority of triangle counting algorithms are targeted to static graphs. Yet, many realworld graphs are directed and temporal, where edges come with timestamps. Temporal triangles yield much more information, since they account for both the graph topology and the timestamps. Temporal triangle counting has seen a few recent results, but there are varying definitions of temporal triangles. In all cases, temporal triangle patterns enforce constraints on the time interval between edges (in the triangle). We define a general notion (δ1,3,δ1,2,δ2,3)temporal triangles that allows for separate time constraints for all pairs of edges. Our main result is a new algorithm, DOTTT (Degeneracy Oriented Temporal Triangle Totaler), that exactly counts all directed variants of (δ1,3,δ1,2,δ2,3)temporal triangles. Using the classic idea of degeneracy ordering with careful combinatorial arguments, we can prove that DOTTT runs in O(mκlogm) time, where m is the number of (temporal) edges and κ is the graph degeneracy (max core number). Up to log factors, this matches the running time of the best static triangle counters. Moreover, this running time is better than existing. DOTTT has excellent practical behavior and runs twice as fast as existing stateoftheart temporal triangle counters (and is also more general). For example, DOTTT computes all types of temporal queries in Bitcoin temporal network with half a billion edges in less than an hour on a commodity machine.more » « less

null (Ed.)Triangle counting is a fundamental technique in network analysis, that has received much attention in various input models. The vast majority of triangle counting algorithms are targeted to static graphs. Yet, many realworld graphs are directed and temporal, where edges come with timestamps. Temporal triangles yield much more information, since they account for both the graph topology and the timestamps. Temporal triangle counting has seen a few recent results, but there are varying definitions of temporal triangles. In all cases, temporal triangle patterns enforce constraints on the time interval between edges (in the triangle). We define a general notion (δ1,3,δ1,2,δ2,3)temporal triangles that allows for separate time constraints for all pairs of edges. Our main result is a new algorithm, DOTTT (Degeneracy Oriented Temporal Triangle Totaler), that exactly counts all directed variants of (δ1,3,δ1,2,δ2,3)temporal triangles. Using the classic idea of degeneracy ordering with careful combinatorial arguments, we can prove that DOTTT runs in O(mκlogm) time, where m is the number of (temporal) edges and κ is the graph degeneracy (max core number). Up to log factors, this matches the running time of the best static triangle counters. Moreover, this running time is better than existing. DOTTT has excellent practical behavior and runs twice as fast as existing stateoftheart temporal triangle counters (and is also more general). For example, DOTTT computes all types of temporal queries in Bitcoin temporal network with half a billion edges in less than an hour on a commodity machine.more » « less

Abstract A graph is ‐
free if it has no induced subgraph isomorphic to , and G  denotes the number of vertices of . A conjecture of Conlon, Sudakov and the second author asserts that:For every graph , there exists such that in every ‐free graph with 
This is equivalent to:G  there are two disjoint sets of vertices, of sizes at least and , complete or anticomplete to each other.The “sparse linear conjecture”: For every graph , there exists such that in every ‐free graph with , either some vertex has degree at least , or there are two disjoint sets of vertices, of sizes at least and , anticomplete to each other.
The sparse linear conjecture holds for all almost‐bipartite graphs .
For every almost‐bipartite graph , there exist such that for every graph with and maximum degree less than , and for every with , either contains induced copies of , or there are two disjoint sets with and , and with at most edges between them.
For every graph , there exists such that in every ‐free graph with vertices, either some vertex has degree at least , or there are two disjoint sets of vertices with , anticomplete to each other.