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 paper introduces PRISM, a chromatic-scheduling algorithm for executing dynamic data-graph computations. PRISM 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 PRISM to maintain a dynamic set of active vertices as an unordered set partitioned by color. We analyze PRISM 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] + Σ[subscript 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 PRISM performs updates in Q using O(Χ(lg (Q/Χ)+lgΔ)+ lgP) span and Θ(size(Q)+Χ+P) work. These theoretical guarantees are matched by good empirical performance. We modified GraphLab to incorporate PRISM and studied seven application benchmarks on a 12-core multicore machine. PRISM executes the benchmarks 1.2–2.1 times faster than GraphLab's nondeterministic lock-based scheduler while providing deterministic behavior. This paper also presents PRISM-R, a variation of PRISM that executes dynamic data-graph computations deterministically even when updates modify global variables with associative operations. PRISM-R satisfies the same theoretical bounds as PRISM, but its implementation is more involved, incorporating a multivector data structure to maintain an ordered set of vertices partitioned by color.