Abstract
We present new distance oracles for computing distances of stretch less than 2 on general weighted undirected graphs. For the realistic case of sparse graphs and for any integer k, the new oracles return paths of stretch 1 + 1/k and exhibit a smooth three-way tradeoff of S ×T 1/k = O(n 2) between space S, stretch and query time T. This significantly improves the state-of-the-art for each point in the space-stretch-time tradeoff space, and matches the known space-time curve for stretch 2 and larger. We also present new oracles for stretch 1 + 1/(k + 0.5). A particularly interesting case is of stretch 5/3, where improving the query time of our oracles from T to T 1 − ε for any ε > 0 would lead to the first purely o(mn)-time combinatorial algorithm for Boolean Matrix Multiplication, a longstanding open problem.
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References
Abraham, I., Gavoille, C.: On approximate distance labels and routing schemes with affine stretch. In: Peleg, D. (ed.) Distributed Computing. LNCS, vol. 6950, pp. 404–415. Springer, Heidelberg (2011)
Agarwal, R., Caesar, M., Godfrey, P.B., Zhao, B.Y.: Shortest paths in less than a millisecond. In: SIGCOMM WOSN (2012)
Agarwal, R., Godfrey, P.B.: Brief announcement: A simple stretch 2 distance oracle. In: PODC (2013)
Agarwal, R., Godfrey, P.B.: Distance oracles for stretch less than 2. In: SODA (2013)
Agarwal, R., Godfrey, P.B., Har-Peled, S.: Approximate distance queries and compact routing in sparse graphs. In: INFOCOM (2011)
Agarwal, R., Godfrey, P.B., Har-Peled, S.: Faster approximate distance queries and compact routing in sparse graphs (2012)
Alon, N., Spencer, J.H.: The probabilistic method, vol. 57. Wiley Interscience (1992)
Awerbuch, B., Berger, B., Cowen, L., Peleg, D.: Near-linear time construction of sparse neighborhood covers. SIAM Journal on Computing 28(1), 263–277 (1998)
Bansal, N., Williams, R.: Regularity lemmas and combinatorial algorithms. In: FOCS (2009)
Baswana, S., Kavitha, T.: Faster algorithms for approximate distance oracles and all-pair small stretch paths. In: FOCS (2006)
Baswana, S., Sen, S.: Approximate distance oracles for unweighted graphs in expected O(n 2) time. ACM Transactions on Algorithms 2(4), 557–577 (2006)
Blelloch, G.E., Vassilevska, V., Williams, R.: A new combinatorial approach for sparse graph problems. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 108–120. Springer, Heidelberg (2008)
Chechik, S.: Approximate distance oracles with constant query time. In: STOC (2014)
Chen, W., Sommer, C., Teng, S.-H., Wang, Y.: A compact routing scheme and approximate distance oracle for power-law graphs. ACM Transactions on Algorithms 9(1), 4:1–4:26 (2012)
Cohen, E.: Fast algorithms for constructing t-spanners and paths with stretch t. SIAM Journal on Computing 28(1), 210–236 (1998)
Dor, D., Halperin, S., Zwick, U.: All pairs almost shortest paths. In: FOCS (1996)
Enachescu, M., Wang, M., Goel, A.: Reducing maximum stretch in compact routing. In: INFOCOM (2008)
Gubichev, A., Bedathur, S., Seufert, S., Weikum, G.: Fast and accurate estimation of shortest paths in large graphs. In: CIKM (2010)
Kawarabayashi, K.-I., Klein, P.N., Sommer, C.: Linear-space approximate distance oracles for planar, bounded-genus and minor-free graphs. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part I. LNCS, vol. 6755, pp. 135–146. Springer, Heidelberg (2011)
Kawarabayashi, K.-I., Sommer, C., Thorup, M.: More compact oracles for approximate distances in undirected planar graphs. In: SODA (2013)
Matoušek, J.: On the distortion required for embedding finite metric spaces into normed spaces. Israel Journal of Mathematics 93(1), 333–344 (1996)
Mendel, M., Naor, A.: Ramsey partitions and proximity data structures. Journal of European Mathematical Society 2(9), 253–275 (2007)
Pǎtraşcu, M., Roditty, L.: Distance oracles beyond the Thorup-Zwick bound. In: FOCS (2010)
Pǎtraşcu, M., Roditty, L., Thorup, M.: A new infinity of distance oracles for sparse graphs. In: FOCS (2012)
Porat, E., Roditty, L.: Preprocess, set, query!. In: Demetrescu, C., Halldórsson, M.M. (eds.) ESA 2011. LNCS, vol. 6942, pp. 603–614. Springer, Heidelberg (2011)
Potamias, M., Bonchi, F., Castillo, C., Gionis, A.: Fast shortest path distance estimation in large networks. In: CIKM (2009)
Roditty, L., Zwick, U.: Replacement paths and k simple shortest paths in unweighted directed graphs. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 249–260. Springer, Heidelberg (2005)
Sommer, C., Verbin, E., Yu, W.: Distance oracles for sparse graphs. In: FOCS (2009)
Thorup, M., Zwick, U.: Compact routing schemes. In: SPAA (2001)
Thorup, M.: Compact oracles for reachability and approximate distances in planar digraphs. Journal of the ACM 51(6), 993–1024 (2004)
Thorup, M., Zwick, U.: Approximate distance oracles. Journal of the ACM 52(1), 1–24 (2005)
Wulff-Nilsen, C.: Approximate distance oracles with improved query time. In: SODA (2013)
Wulff-Nilsen, C.: Approximate distance oracles with improved preprocessing time. In: SODA (2012)
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Agarwal, R. (2014). The Space-Stretch-Time Tradeoff in Distance Oracles. In: Schulz, A.S., Wagner, D. (eds) Algorithms - ESA 2014. ESA 2014. Lecture Notes in Computer Science, vol 8737. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44777-2_5
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DOI: https://doi.org/10.1007/978-3-662-44777-2_5
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