Algorithms for two bottleneck optimization problems
Journal of Algorithms
Preprocessing an undirected planar network to enable fast approximate distance queries
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
A faster and simpler fully dynamic transitive closure
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Improved Dynamic Reachability Algorithms for Directed Graphs
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
A Fully Dynamic Approximation Scheme for All-Pairs Shortest Paths in Planar Graphs
WADS '93 Proceedings of the Third Workshop on Algorithms and Data Structures
A Fully Dynamic Data Structure for Reachability in Planar Digraphs
ESA '93 Proceedings of the First Annual European Symposium on Algorithms
A new approach to dynamic all pairs shortest paths
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
A new approach to all-pairs shortest paths on real-weighted graphs
Theoretical Computer Science - Special issue on automata, languages and programming
Approximating geometric bottleneck shortest paths
Computational Geometry: Theory and Applications
Compact oracles for reachability and approximate distances in planar digraphs
Journal of the ACM (JACM)
Journal of the ACM (JACM)
Planar graphs, negative weight edges, shortest paths, and near linear time
Journal of Computer and System Sciences - Special issue on FOCS 2001
Oracles for bounded-length shortest paths in planar graphs
ACM Transactions on Algorithms (TALG)
Faster Algorithms for Approximate Distance Oracles and All-Pairs Small Stretch Paths
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Approximate distance oracles for unweighted graphs in expected O(n2) time
ACM Transactions on Algorithms (TALG)
All-pairs bottleneck paths for general graphs in truly sub-cubic time
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
More algorithms for all-pairs shortest paths in weighted graphs
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
On bounded leg shortest paths problems
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Deterministic constructions of approximate distance oracles and spanners
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
Connectivity oracles for failure prone graphs
Proceedings of the forty-second ACM symposium on Theory of computing
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In a weighted, directed graph an L-bounded leg path is one whose constituent edges have length at most L. For any fixed L, computing L-bounded leg shortest paths is just as easy as the standard shortest path algorithm. In this paper we study approximate distance oracles (and reachability oracles) for bounded leg path problems, where the leg bound L is not known in advance, but forms part of the query. Bounded-leg path problems are more complicated than standard shortest path problems because the number of distinct shortest paths between two vertices (over all leg bounds) could be as large as the number of edges in the graph. The bounded leg constraint models situations where there is some limited resource that must be spent when traversing an edge. For example, the size of a fuel tank or the life of a battery places a hard limit on how far a vehicle can travel in one leg before refueling or recharging. Someone making a long road trip may place a hard limit on how many hours they are willing to drive in any one day. Our main result is a nearly optimal algorithm for preprocessing a directed graph in order to answer approximate bounded leg distance and bounded leg shortest path queries. In particular, we can preprocess any graph in Õ(n3) time, producing a data structure with size Õ(n2) that answers (1 + ∈)-approximate bounded leg distance queries in O(log log n) time. If the corresponding (1 + ∈)-approximate shortest path has l edges it can be returned in O(l log log n) time. These bounds are all within polylog(n) factors of the best standard all-pairs shortest path algorithm and improve substantially the previous best bounded leg shortest path algorithm, whose preprocessing time and space are O(n4) and Õ(n2.5). We also consider bounded leg oracles in other situations. In the context of planar directed graphs we give a time-space tradeoff for answering bounded leg reachability queries. For any k ≥ 2 we can build a data structure with size O(kn1+1/k) that answers reachability queries in time Õ(nk−1/2k).