External Memory Algorithms For Path Traversal in Graphs

TL;DR

This thesis develops efficient external memory data structures for path traversal in trees and planar graphs, with applications to mesh representations in R^2 and R^3, improving space efficiency and traversal performance.

Contribution

It introduces succinct external memory data structures for path traversal in trees and planar graphs, enhancing efficiency and space bounds, and applies these to mesh representations in R^2 and R^3.

Findings

Match or improve current best results for external memory path traversal.

Present succinct data structures reducing space usage.

Apply path traversal techniques to mesh representations in R^2 and R^3.

Abstract

This thesis presents a number of results related to path traversal in trees and graphs. In particular, we focus on data structures which allow such traversals to be performed efficiently in the external memory setting. In addition, for trees and planar graphs the data structures we present are succinct. Our tree structures permit efficient bottom-up path traversal in rooted trees of arbitrary degree and efficient top-down path traversal in binary trees. In the graph setting, we permit efficient traversal of an arbitrary path in bounded degree planar graphs. Our data structures for both trees and graphs match or slightly improve current best results for external memory path traversal in these settings while at the same time improving space bounds due to the succinct nature of our data structures. Employing our path traversal structure for bounded degree planar graphs, we describe a number of useful applications of this technique for triangular meshes in R^2. As an extension of the R^2 representation for triangular meshes we also present an efficient external memory representation for well-shaped tetrahedral meshes in R^3. The external memory representation we present is based on a partitioning scheme that matches the current best-known results for well-shaped tetrahedral meshes. We describe applications of path traversal in tetrahedral meshes which are made efficient in the external memory setting using our structure. Finally, we present a result on using jump-and-walk point location in well-shaped meshes in both R^2 and R^3. We demonstrate that, given an approximate nearest neighbour from among the vertices of a mesh, locating the simplex containing the query point involves a constant length walk (path traversal) in the mesh.