I/O-Efficient Path Traversal in Succinct Planar Graphs

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 lgN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lg {N}$$\end{document} denotes log2N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\log _2{N}$$\end{document}) each with an associated key of q=OlgN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q= \mathrm {O}\left( \lg N\right) $$\end{document} bits, can be stored in Nq+ON+oNq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Nq+ \mathrm {O}\left( N\right) + \mathrm {o}\left( Nq\right) $$\end{document} bits and traversing a path of length K takes OK/lgB\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {O}\left( K / \lg B\right) $$\end{document} 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 OK/lgB\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {O}\left( K / \lg B\right) $$\end{document} 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.