TY - GEN
T1 - (1 + ϵ)-Approximate Shortest Paths in Dynamic Streams
AU - Elkin, Michael
AU - Trehan, Chhaya
N1 - Funding Information: This research was supported by ISF grant 2344/19. Publisher Copyright: © Michael Elkin and Chhaya Trehan.
PY - 2022/9/15
Y1 - 2022/9/15
N2 - Computing approximate shortest paths in the dynamic streaming setting is a fundamental challenge that has been intensively studied. Currently existing solutions for this problem either build a sparse multiplicative spanner of the input graph and compute shortest paths in the spanner offline, or compute an exact single source BFS tree. Solutions of the first type are doomed to incur a stretch-space tradeoff of 2κ - 1 versus n1+1/κ, for an integer parameter κ. (In fact, existing solutions also incur an extra factor of 1 + ϵ in the stretch for weighted graphs, and an additional factor of logO(1) n in the space.) The only existing solution of the second type uses n1/2-O(1/κ) passes over the stream (for space O(n1+1/κ)), and applies only to unweighted graphs. In this paper we show that (1 + ϵ)-approximate single-source shortest paths can be computed with Õ(n1+1/κ) space using just constantly many passes in unweighted graphs, and polylogarithmically many passes in weighted graphs. Moreover, the same result applies for multi-source shortest paths, as long as the number of sources is O(n1/κ). We achieve these results by devising efficient dynamic streaming constructions of (1 + ϵ, β)-spanners and hopsets. On our way to these results, we also devise a new dynamic streaming algorithm for the 1-sparse recovery problem. Even though our algorithm for this task is slightly inferior to the existing algorithms of [26, 11], we believe that it is of independent interest.
AB - Computing approximate shortest paths in the dynamic streaming setting is a fundamental challenge that has been intensively studied. Currently existing solutions for this problem either build a sparse multiplicative spanner of the input graph and compute shortest paths in the spanner offline, or compute an exact single source BFS tree. Solutions of the first type are doomed to incur a stretch-space tradeoff of 2κ - 1 versus n1+1/κ, for an integer parameter κ. (In fact, existing solutions also incur an extra factor of 1 + ϵ in the stretch for weighted graphs, and an additional factor of logO(1) n in the space.) The only existing solution of the second type uses n1/2-O(1/κ) passes over the stream (for space O(n1+1/κ)), and applies only to unweighted graphs. In this paper we show that (1 + ϵ)-approximate single-source shortest paths can be computed with Õ(n1+1/κ) space using just constantly many passes in unweighted graphs, and polylogarithmically many passes in weighted graphs. Moreover, the same result applies for multi-source shortest paths, as long as the number of sources is O(n1/κ). We achieve these results by devising efficient dynamic streaming constructions of (1 + ϵ, β)-spanners and hopsets. On our way to these results, we also devise a new dynamic streaming algorithm for the 1-sparse recovery problem. Even though our algorithm for this task is slightly inferior to the existing algorithms of [26, 11], we believe that it is of independent interest.
KW - Approximate Distances
KW - Dynamic Streams
KW - Hopsets
KW - Shortest Paths
KW - Spanners
UR - http://www.scopus.com/inward/record.url?scp=85139088795&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.APPROX/RANDOM.2022.51
DO - 10.4230/LIPIcs.APPROX/RANDOM.2022.51
M3 - Conference contribution
T3 - Leibniz International Proceedings in Informatics, LIPIcs
SP - 51:1--51:23
BT - Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2022
A2 - Chakrabarti, Amit
A2 - Swamy, Chaitanya
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 25th International Conference on Approximation Algorithms for Combinatorial Optimization Problems and the 26th International Conference on Randomization and Computation, APPROX/RANDOM 2022
Y2 - 19 September 2022 through 21 September 2022
ER -