Compact I/O-Efficient Representation of Separable Graphs and Optimal Tree Layouts

Abstract: Compact and I/O-efficient data representations play an important role in efficient algorithm design, as memory bandwidth and latency can present a significant performance bottleneck, slowing the computation by orders of magnitude. While this problem is very well explored in e.g. uniform numerical data processing, structural data applications (e.g. on huge graphs) require different algorithm-dependent approaches. Separable graph classes (i.e. graph classes with balanced separators of size $\mathcal{O}(n^c)$ with $c < 1$) include planar graphs, bounded genus graphs, and minor-free graphs. In this article we present two generalizations of the separator theorem, to partitions with small regions only on average and to weighted graphs. Then we propose I/O-efficient succinct representation and memory layout for random walks in(weighted) separable graphs in the pointer machine model, including an efficient algorithm to compute them. Finally, we present a worst-case I/O-optimal tree layout algorithm for root-leaf path traversal, show an additive (+1)-approximation of optimal compact layout and contrast this with NP-completeness proof of finding an optimal compact layout.