The input/output complexity of triangle enumeration

Abstract

We consider the well-known problem of enumerating all triangles of an undirected graph. Our focus is on determining the input/output (I/O) complexity of this problem. Let E be the number of edges, M < E the size of internal memory, and B the block size. The best results obtained previously are sort(E 3/2 ) I/Os (Dementiev, PhD thesis 2006) and O E 2 /(M B) I/Os (Hu et al., SIGMOD 2013), where sort(n) denotes the number of I/Os for sorting n items. We improve the I/O complexity to O E 3/2 /( √ M B) expected I/Os, which improves the previous bounds by a factor min( E/M , √ M ). Our algorithm is cache-oblivious and also I/O optimal: We show that any algorithm enumerating t distinct triangles must always use Ω t/( √ M B) I/Os, and there are graphs for which t = Ω E 3/2 . Finally, we give a deterministic cache-aware algorithm using O E 3/2 /( √ M B) I/Os assuming M ≥ E ε for a constant ε > 0. Our results are based on a new color coding technique, which may be of independent interest.

MapReduce Triangle Enumeration With Guarantees

“…CTTP, howerver, partitions vertexes according with a coloring function randomly selected from a 4-wise independent family of functions, as suggested in [28] for triangle enumeration in external memory. By just requiring 4-wiseness to the coloring function, our technique overcomes previous analyses that assume a random input graph, and guarantees that the claimed result applies to any input graph.…”

“…In particular, Chu et al and Hu et al [7,17] have proposed deterministic algorithms requiring O E 2 /(mB) I/Os, while Pagh and Silvestri [28] proposed an optimal randomized algorithm re- [39] for a reduction to matrix multiplication, and Jafargholi and Viola [18] for 3SUM/3XOR. Triangle listing in certain classes of random graphs has been addressed recently by Berry et al [3] to explain the empirically good behavior of simple triangle listing algorithms.…”

“…We observe that triangles with at least one high degree vertex can be enumerated using sorting. Specifically, for each very high degree vertex v, all triangles containing v are found by suitably sorting three times the edge sets E as shown in [28,Lemma 1]. The sorting operation can be carried out using the MapReduce algorithm proposed in [14], which requires O (1) rounds as soon as the reducer size m is larger than √ E. Since E aggregate space is required for finding all triangles with a given very high degree vertex, M/E very high degree vertexes can be processed in parallel.…”

“…Triangle enumeration has been widely studied in the last years, in particular in sequential platforms [17,28,7,5]. However, as the graph size continues to increase, it is effectively impossible to enumerate triangles in massive graphs.…”

Accurately and quickly calculating the weighted spectral distribution

“…7c, the time complexities are O(n 2 ) for the FWSD and O(n 4 ) for the restricted algorithm. Enumerating all triangles (3-cycles) in a graph is useful for understanding graph structures [40] and has been widely utilized in social networks [41] and community detection [42]. Triangle enumeration is quick and suitable for largescale complex networks.…”