I/O-Efficient Path Traversal in Succinct Planar Graphs

Abstract

We present a technique for representing bounded-degree planar graphs in a succinct fashion while permitting I/O-efficient traversal of paths. Using our representation, a graph with N vertices, (In this paper $$\lg {N}$$lgN denotes $$\log _2{N}$$log2N) each with an associated key of $$q= \mathrm {O}\left( \lg N\right) $$q=OlgN bits, can be stored in $$Nq+ \mathrm {O}\left( N\right) + \mathrm {o}\left( Nq\right) $$Nq+ON+oNq bits and traversing a path of length K takes $$\mathrm {O}\left( K / \lg B\right) $$OK/lgB I/Os, where B denotes the disk block size. By applying our construction to the dual of a terrain represented as a triangular irregular network, we can represent the terrain in the above space bounds and support path traversals on the terrain using $$\mathrm {O}\left( K / \lg B\right) $$OK/lgB I/Os, where K is the number of triangles visited by the path. This is useful for answering a number of queries on the terrain, such as reporting terrain profiles, trickle paths, and connected components.