A Hybrid Sampling Scheme for Triangle Counting

Abstract

We study the problem of estimating the number of triangles in a graph stream. No streaming algorithm can get sublinear space on all graphs, so methods in this area bound the space in terms of parameters of the input graph such as the maximum number of triangles sharing a single edge. We give a sampling algorithm that is additionally parameterized by the maximum number of triangles sharing a single vertex. Our bound matches the best known turnstile results in all graphs, and gets better performance on simple graphs like G(n, p) or a set of independent triangles.We complement the upper bound with a lower bound showing that no sampling algorithm can do better on those graphs by more than a log factor. In particular, any insertion stream algorithm must use √ T space when all the triangles share a common vertex, and any sampling algorithm must take T 1/3 samples when all the triangles are independent. We add another lower bound, also matching our algorithm's performance, which applies to all graph classes. This lower bound covers "triangledependent" sampling algorithms, a subclass that includes our algorithm and all previous sampling algorithms for the problem.

The Sketching Complexity of Graph and Hypergraph Counting

“…The conclusion is that the expected fraction of triangles completely sampled is at most p 3/2 . This combinatorial lemma is different from the simple sampling lemma of [KP17] in several ways. First, it allows players to look at their entire inputs before deciding which edges to keep.…”

“…This was followed by alternative algorithms in [BFL + 06, JG05]. The lower bounds in [BOV13] and [KP17] were achieved by reductions to one-way communication complexity problems, the indexing problem and the Boolean Hidden Matching problem, respectively. Triangle detection has also been studied as a pure communication problem, for instance [FGO17], as well as in the adjacency-list [KMPT10, BFL + 06, MVV16], multi-pass [BOV13,CJ14], and query models [ELRS15].…”

“…First, it allows players to look at their entire inputs before deciding which edges to keep. Second, while the lemma of [KP17] was based on defining a fixed subset of edges to keep (so the number kept depended only on which edges were seen), in our lemma the players only need to keep a p fraction of their inputs on average, but the players do not have shared randomness. If they had shared randomness, there would be a trivial counterexample: with probability p every player samples every edge, giving the referee pT triangles in expectation.…”

“…If ∆ V bounds the maximum number of triangles to share a vertex, this bound becomes m √ ∆ V /T . The algorithm in [KP17] requires…”

Kirchhoff Index as a Measure of Edge Centrality in Weighted Networks: Nearly Linear Time Algorithms

“…triangles or shortest paths. As many previous theoretical studies [6,25], our work is also motivated by graph mining applications. In [6], a streaming algorithm was developed for analyzing large-scale rapidly-changing graphs, which maintains densest subgraphs in one pass and achieves time and space efficiency, whereas in our case, effective resistances are persevered under updates.…”