Executing Dynamic Data-Graph Computations Deterministically Using Chromatic Scheduling

Abstract

A data-graph computation—popularized by such programming systems as Galois, Pregel, GraphLab, PowerGraph, and GraphChi—is an algorithm that performs local updates on the vertices of a graph. During each round of a data-graph computation, an update function atomically modifies the data associated with a vertex as a function of the vertex’s prior data and that of adjacent vertices. A dynamic data-graph computation updates only an active subset of the vertices during a round, and those updates determine the set of active vertices for the next round. This article introduces P rism , a chromatic-scheduling algorithm for executing dynamic data-graph computations. P rism uses a vertex coloring of the graph to coordinate updates performed in a round, precluding the need for mutual-exclusion locks or other nondeterministic data synchronization. A multibag data structure is used by P rism to maintain a dynamic set of active vertices as an unordered set partitioned by color. We analyze P rism using work-span analysis. Let G = ( V , E ) be a degree-Δ graph colored with χ colors, and suppose that Q ⊆ V is the set of active vertices in a round. Define size ( Q )= | Q | + ∑ v ∈ Q deg( v ), which is proportional to the space required to store the vertices of Q using a sparse-graph layout. We show that a P -processor execution of P rism performs updates in Q using O (χ (lg ( Q/χ ) + lg Δ ) + lg P span and Θ( size ( Q ) + P ) work. These theoretical guarantees are matched by good empirical performance. To isolate the effect of the scheduling algorithm on performance, we modified GraphLab to incorporate P rism and studied seven application benchmarks on a 12-core multicore machine. P rism executes the benchmarks 1.2 to 2.1 times faster than GraphLab’s nondeterministic lock-based scheduler while providing deterministic behavior. This article also presents P rism -R, a variation of P rism that executes dynamic data-graph computations deterministically even when updates modify global variables with associative operations. P rism -R satisfies the same theoretical bounds as P rism , but its implementation is more involved, incorporating a multivector data structure to maintain a deterministically ordered set of vertices partitioned by color. Despite its additional complexity, P rism -R is only marginally slower than P rism . On the seven application benchmarks studied, P rism -R incurs a 7% geometric mean overhead relative to P rism .