Oracles for Distances Avoiding a Failed Node or Link

• Published in: SIAM journal on computing Vol. 37; no. 5; pp. 1299 - 1318
• Main Authors: Demetrescu, Camil, Thorup, Mikkel, Chowdhury, Rezaul Alam, Ramachandran, Vijaya
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2008
• Subjects: Algorithmics. Computability. Computer arithmetics; Algorithms; Applied sciences; Combinatorics; Combinatorics. Ordered structures; Computer science; control theory; systems; Exact sciences and technology; Failure; Graph theory; Graphs; Information retrieval. Graph; Mathematics; Miscellaneous; Queries; Sciences and techniques of general use; Theoretical computing; 68P05; Vertex; Query; 68W01; Node; 90B18; Network; data structures; Oracle; Distance; Failure; graph algorithms; Edge(graph); Construction; network failures; 05C85; Shortest path; Algorithm; Failures; Weighted graph; shortest paths; Network routing; Digraph; Application; Directed graph; Link; 05C38
• ISSN: 0097-5397; 1095-7111
• DOI: 10.1137/S0097539705429847

We consider the problem of preprocessing an edge-weighted directed graph $G$ to answer queries that ask for the length and first hop of a shortest path from any given vertex $x$ to any given vertex $y$ avoiding any given vertex or edge. As a natural application, this problem models routing in networks subject to node or link failures. We describe a deterministic oracle with constant query time for this problem that uses $O(n^2\log n)$ space, where $n$ is the number of vertices in $G$. The construction time for our oracle is $O(mn^{2} + n^{3}\log n)$. However, if one is willing to settle for $\Theta (n^{2.5})$ space, we can improve the preprocessing time to $O(mn^{1.5}+n^{2.5}\log n)$ while maintaining the constant query time. Our algorithms can find the shortest path avoiding a failed node or link in time proportional to the length of the path.