TY - GEN

T1 - Preprocess, Set, query!

AU - Porat, Ely

AU - Roditty, Liam

PY - 2011

Y1 - 2011

N2 - 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 (mm1/k) time and generate a compact data structure of size O(kn 1+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(n 1.5) size that produces in constant time estimated distances with stretch 3. Recently, Pǎtraşcu 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(n 5/3) 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(nm 1-ε′) and the query time is Õ(m1-ε′). Using it for sparse unweighted graphs we can get a data structure of size O(n 1.86) that can supply in O(n 0.86) time estimated distances with multiplicative stretch 1.75.

AB - 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 (mm1/k) time and generate a compact data structure of size O(kn 1+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(n 1.5) size that produces in constant time estimated distances with stretch 3. Recently, Pǎtraşcu 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(n 5/3) 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(nm 1-ε′) and the query time is Õ(m1-ε′). Using it for sparse unweighted graphs we can get a data structure of size O(n 1.86) that can supply in O(n 0.86) time estimated distances with multiplicative stretch 1.75.

UR - http://www.scopus.com/inward/record.url?scp=80052813485&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-23719-5_51

DO - 10.1007/978-3-642-23719-5_51

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AN - SCOPUS:80052813485

SN - 9783642237188

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 603

EP - 614

BT - Algorithms, ESA 2011 - 19th Annual European Symposium, Proceedings

T2 - 19th Annual European Symposium on Algorithms, ESA 2011

Y2 - 5 September 2011 through 9 September 2011

ER -