I/O-optimal algorithms on grid graphs

TL;DR

This paper introduces I/O-efficient algorithms for fundamental graph problems on grid graphs, achieving optimal I/O complexity of O(scan(n)) under certain memory conditions, improving upon previous Theta(sort(n)) algorithms.

Contribution

The paper presents the first I/O-optimal algorithms for several key graph problems on grid graphs, including shortest paths, MST, and topological sorting, with some being cache-oblivious.

Findings

Algorithms run in O(scan(n)) I/Os, matching the lower bound.

The minimum-spanning tree algorithm is cache-oblivious.

Estimated I/O volume suggests high practical efficiency.

Abstract

Given a graph of which the n vertices form a regular two-dimensional grid, and in which each (possibly weighted and/or directed) edge connects a vertex to one of its eight neighbours, the following can be done in O(scan(n)) I/Os, provided M = Omega(B^2): computation of shortest paths with non-negative edge weights from a single source, breadth-first traversal, computation of a minimum spanning tree, topological sorting, time-forward processing (if the input is a plane graph), and an Euler tour (if the input graph is a tree). The minimum-spanning tree algorithm is cache-oblivious. The best previously published algorithms for these problems need Theta(sort(n)) I/Os. Estimates of the actual I/O volume show that the new algorithms may often be very efficient in practice.