Non-local Probes Do Not Help with Many Graph Problems

Abstract

This work bridges the gap between distributed and centralised models of computing in the context of sublinear-time graph algorithms. A priori, typical centralised models of computing (e.g., parallel decision trees or centralised local algorithms) seem to be much more powerful than distributed message-passing algorithms: centralised algorithms can directly probe any part of the input, while in distributed algorithms nodes can only communicate with their immediate neighbours. We show that for a large class of graph problems, this extra freedom does not help centralised algorithms at all: for example, efficient stateless deterministic centralised local algorithms can be simulated with efficient distributed message-passing algorithms. In particular, this enables us to transfer existing lower bound results from distributed algorithms to centralised local algorithms. arXiv:1512.05411v1 [cs.DS] 16 Dec 2015A lot of recent work on efficient graph algorithms for massive graphs can be broadly classified in one of the following categories:1. Probe-query models [1, 3, 7-9, 16, 22, 31]: Here typical applications are related to large-scale network analysis: we have a huge storage system in which the input graph is stored, and a computer that can access the storage system. The user of the computer can make queries related to the properties of the graph.Conceptually, we have two separate entities: the input graph and a computer. Initially, the computer is unaware of the structure of the graph, but it can probe it to learn more about the structure of the graph. Typically, the goal is to answer queries after a sublinear number of probes. 2.Message-passing models [6,14,15,19,21,26,29,34]: In message-passing models, typical applications are related to controlling large computer networks: we have a computer network (say, the Internet) that consists of a large number of network devices, and the devices need to collaborate to solve a graph problem related to the structure of the network so that each node knows its own part of the solution when the algorithm stops.Conceptually, each node of the input graph is a computational entity. Initially, the nodes are only aware of their own identity and the connections to their immediate neighbours, but the nodes can exchange messages with their neighbours in order to learn more about the structure of the graph. Typically, the goal is to solve graph problems in a sublinear number of communication rounds.

The Randomized Local Computation Complexity of the Lovász Local Lemma

“…This finishes the lower bound proof for the VOLUME model. This directly leads to the same lower bound for the LCA model by a result of [17].…”

Online Algorithms with Lookaround

“…On the one hand, OLOCAL algorithms did not even need to look at the input graph, while the problem was not solvable at all in SLOCAL or LOCAL models. This problem was not a "nice" graph problem in the sense of Göös et al [24], and hence it also does not play well with local computation algorithms, either. But we can easily address all of these issues as follows: Definition 4.3 (componentwise leader election).…”

“…It is known that for a broad family of graph problems (that includes LCLs), we can w.l.o.g. assume that whenever the adversary queries a node v, the LCA will make probes to learn a connected subgraph around node v [24]. Hence for such problems, an OLOCAL algorithm with locality T is at least as strong as an LCA that makes T probes per query:…”

Fast Distributed Algorithms for Testing Graph Properties

“…The wider family of local computation algorithms (LCA) is known to have connections with distributed computing, as shown by Parnas and Ron [30] and later used by others. A recent study by Göös et al [23] proves that under some conditions, the fact that a centralized algorithm can query distant vertices does not help with speeding up computation. However, they consider the LOCAL model, and their results apply to certain properties that are not influenced by distances.…”