Motivated by the increasing need for fast distributed processing of large-scale graphs such as the Web graph and various social networks, we study a number of fundamental graph problems in the message-passing model, where we have k machines that jointly perform computation on an arbitrary n-node (typically, n ≫ k) input graph. The graph is assumed to be randomly partitioned among the k ≥ 2 machines (a common implementation in many real world systems). The communication is point-to-point, and the goal is to minimize the time complexity, i.e., the number of communication rounds, of solving various fundamental graph problems. We present lower bounds that quantify the fundamental time limitations of distributively solving graph problems. We first show a lower bound of Ω(n/k) rounds for computing a spanning tree (ST) of the input graph. This result also implies the same bound for other fundamental problems such as computing a minimum spanning tree (MST), breadth-first tree (BFS), and shortest paths tree (SPT). We also show an Ω(n/k 2 ) lower bound for connectivity, ST verification and other * Division of Mathematical Sciences, Nanyang Technological University, Singapore 637371 & Centre for Quantum Technologies, Singapore 117543. E-mail: hklauck@gmail.com. This work is funded by the Singapore Ministry of Education (partly through the Academic Research Fund Tier 3 MOE2012-T3-1-009) and by the Singapore National Research Foundation. † KTH Royal Institute of Technology, Sweden, and University of Vienna, Austria E-mail: danupon@gmail.com. Work done while at ICERM, Brown University, USA, and Nanyang Technological University, Singapore. To complement our lower bounds, we also give algorithms for various fundamental graph problems, e.g., PageRank, MST, connectivity, ST verification, shortest paths, cuts, spanners, covering problems, densest subgraph, subgraph isomorphism, finding triangles, etc. We show that problems such as PageRank, MST, connectivity, and graph covering can be solved inÕ(n/k) time (the notationÕ hides polylog(n) factors and an additive polylog(n) term); this shows that one can achieve almost linear (in k) speedup, whereas for shortest paths, we present algorithms that run inÕ(n/ √ k) time (for (1 + ǫ)-factor approximation) and inÕ(n/k) time (for O(log n)-factor approximation) respectively. Our results are a step towards understanding the complexity of distributively solving large-scale graph problems.
In this model REF shows a low bound of Ω(n/k) for computing shortest paths, where k is the number of machines.
Distributed Computation of Large-scale Graph Problems
arXiv: Distributed, Parallel, and Cluster Computing
Computer Science