$(1 + \epsilon,\beta)$-Spanner Constructions for General Graphs
SIAM Journal on Computing
Journal of the ACM (JACM)
Computing almost shortest paths
ACM Transactions on Algorithms (TALG)
Spanners and emulators with sublinear distance errors
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Ramsey partitions and proximity data structures
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
ACM Transactions on Algorithms (TALG)
Distance Oracles for Sparse Graphs
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Compact routing in power-law graphs
DISC'09 Proceedings of the 23rd international conference on Distributed computing
Distance Oracles beyond the Thorup-Zwick Bound
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Faster Algorithms for All-pairs Approximate Shortest Paths in Undirected Graphs
SIAM Journal on Computing
Fast, precise and dynamic distance queries
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
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Thorup and Zwick [J. ACM and STOC'01] in their seminal work introduced the notion of distance oracles. Given an n-vertex weighted undirected graph with m edges, they show that for any integer k ≥ 1 it is possible to preprocess the graph in Õ(mn1/k) time and generate a compact data structure of size O(kn1+1/k). For each pair of vertices, it is then possible to retrieve an estimated distance with multiplicative stretch 2k - 1 in O(k) time. For k = 2 this gives an oracle of O(n1.5) size that produces in constant time estimated distances with stretch 3. Recently, Patrascu and Roditty [FOCS'10] broke the long-standing theoretical status-quo in the field of distance oracles and obtained a distance oracle for sparse unweighted graphs of O(n5/3) size that produces in constant time estimated distances with stretch 2. In this paper we show that it is possible to break the stretch 2 barrier at the price of non-constant query time. We present a data structure that produces estimated distances with 1 + ε stretch. The size of the data structure is O(nm1-ε′) and the query time is Õ (m1-ε′). Using it for sparse unweighted graphs we can get a data structure of size O(n1.86) that can supply in O(n0.86) time estimated distances with multiplicative stretch 1.75.