Compact Routing on Planar Graphs

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

Dynamic Complexity of Planar 3-connected Graph Isomorphism

Dynamic Complexity (as introduced by Patnaik and Immerman) tries to express how hard it is to update the solution to a problem when the input is changed slightly. It considers the changes required to some stored data structure (possibly a massive database) as small quantities of data (or a tuple) are inserted or deleted from the database (or a structure over some vocabulary). The main difference from previous notions of dynamic complexity is that instead of treating the update quantitatively by finding the the time/space trade-offs, it tries to consider the update qualitatively, by finding the complexity class in which the update can be expressed (or made). In this setting, DynFO, or Dynamic First-Order, is one of the smallest and the most natural complexity class (since SQL queries can be expressed in First-Order Logic), and contains those problems whose solutions (or the stored data structure from which the solution can be found) can be updated in First-Order Logic when the data structure undergoes small changes. Etessami considered the problem of isomorphism in the dynamic setting, and showed that Tree Isomorphism can be decided in DynFO. In this work, we show that isomorphism of Planar 3-connected graphs can be decided in DynFO+ (which is DynFO with some polynomial precomputation). We maintain a canonical description of 3-connected Planar graphs by maintaining a database which is accessed and modified by First-Order queries when edges are added to or deleted from the graph. We specifically exploit the ideas of Breadth-First Search and Canonical Breadth-First Search to prove the results. We also introduce a novel method for canonizing a 3-connected planar graph in First-Order Logic from Canonical Breadth-First Search Trees

Route Planning in Transportation Networks

We survey recent advances in algorithms for route planning in transportation networks. For road networks, we show that one can compute driving directions in milliseconds or less even at continental scale. A variety of techniques provide different trade-offs between preprocessing effort, space requirements, and query time. Some algorithms can answer queries in a fraction of a microsecond, while others can deal efficiently with real-time traffic. Journey planning on public transportation systems, although conceptually similar, is a significantly harder problem due to its inherent time-dependent and multicriteria nature. Although exact algorithms are fast enough for interactive queries on metropolitan transit systems, dealing with continent-sized instances requires simplifications or heavy preprocessing. The multimodal route planning problem, which seeks journeys combining schedule-based transportation (buses, trains) with unrestricted modes (walking, driving), is even harder, relying on approximate solutions even for metropolitan inputs.Comment: This is an updated version of the technical report MSR-TR-2014-4, previously published by Microsoft Research. This work was mostly done while the authors Daniel Delling, Andrew Goldberg, and Renato F. Werneck were at Microsoft Research Silicon Valle

Fast and Compact Exact Distance Oracle for Planar Graphs

For a given a graph, a distance oracle is a data structure that answers distance queries between pairs of vertices. We introduce an $O(n^{5/3})$-space distance oracle which answers exact distance queries in $O(\log n)$ time for $n$-vertex planar edge-weighted digraphs. All previous distance oracles for planar graphs with truly subquadratic space i.e., space $O(n^{2 - \epsilon})$ for some constant $\epsilon > 0$) either required query time polynomial in $n$ or could only answer approximate distance queries. Furthermore, we show how to trade-off time and space: for any $S \ge n^{3/2}$, we show how to obtain an $S$-space distance oracle that answers queries in time $O((n^{5/2}/ S^{3/2}) \log n)$. This is a polynomial improvement over the previous planar distance oracles with $o(n^{1/4})$ query time

Decremental Single-Source Reachability in Planar Digraphs

In this paper we show a new algorithm for the decremental single-source reachability problem in directed planar graphs. It processes any sequence of edge deletions in $O(n\log^2{n}\log\log{n})$ total time and explicitly maintains the set of vertices reachable from a fixed source vertex. Hence, if all edges are eventually deleted, the amortized time of processing each edge deletion is only $O(\log^2 n \log \log n)$, which improves upon a previously known $O(\sqrt{n})$ solution. We also show an algorithm for decremental maintenance of strongly connected components in directed planar graphs with the same total update time. These results constitute the first almost optimal (up to polylogarithmic factors) algorithms for both problems. To the best of our knowledge, these are the first dynamic algorithms with polylogarithmic update times on general directed planar graphs for non-trivial reachability-type problems, for which only polynomial bounds are known in general graphs