Compact Routing on Planar Graphs

Abstract

This thesis delves into the exploration of shortest path queries in planar graphs, with an emphasis on the utilization of space-efficient data structures. Our investigation primarily targets connected, undirected, static pointer planar graphs, focusing on scenarios where queries predominantly start or end at a select subset of nodes.

The shortest path problem, central to our study, boasts a rich historical context and has profound real-world implications in diverse fields such as web mapping, robotics, and VLSI circuit design. Our research is pivoted on the space-efficient representation of planar graphs, a critical consideration in 2D visualizations and city map representations.

In this thesis, shortest path queries are delineated into three categories: shortest path, distance oracle, and port queries, each with distinct computational characteristics and storage requirements. A significant portion of our research is focused on center-based configurations in graphs, where a small subset of nodes, designated as ‘centers,’ plays a pivotal role. These centers are crucial, either due to their strategic importance within the graph, which necessitates more prompt responses to queries, or due to their high frequency in the query list. We explore various scenarios within this configuration. Our approach prioritizes handling queries involving these centers more efficiently, aiming to provide rapid responses for strategically important queries and to enhance overall query processing speed. This method is particularly effective, as addressing the queries linked to these relatively few but significant centers can substantially improve the efficiency of the entire system. Such prioritization reflects practical applications like urban navigation, where focusing on key locations can significantly expedite overall navigation and operational efficiency.

For shortest path queries in a center-based configuration, we have developed a data structure that efficiently answers queries from other nodes to centers in O(length of the path) time. In the first scenario, where all queries are from or to a center, the space requirement is 3n+2m+2km+o(nk), where n represents the number of nodes, m the number of edges, and k the number of centers. Additionally, our approach supports distributed storage and processing, facilitating parallel computing.

For distance oracle queries in unweighted graphs within a center-based configuration, our methods manage responses in O(log^(1+ϵ) n) time with an additional o(nk) space requirement. In general, for unweighted graphs without any specific configuration, the distance oracle requires 2n + 2m + 2nm + o(n) bits of space, offering responses in a similar time frame. The strength of our approach lies in its distributability across multiple servers, which enhances concurrent query processing, a feature particularly beneficial in center-based configurations.

Moreover, we introduce a specialized data structure for distributed routing tables, capable of responding to port queries in constant time. This structure efficiently utilizes space, limiting the aggregate bit requirement for all routing tables within graph G to 3.2n^2+o(n^2) bits.