In this paper we initiate the study of expander decompositions of a graph G = (V, E) in the streaming model of computation. The goal is to find a partitioning C of vertices V such that the subgraphs of G induced by the clusters C ∈ C are good expanders, while the number of intercluster edges is small. Expander decompositions are classically constructed by a recursively applying balanced sparse cuts to the input graph. In this paper we give the first implementation of such a recursive sparsest cut process using small space in the dynamic streaming model. Our main algorithmic tool is a new type of cut sparsifier that we refer to as a power cut sparsifier - it preserves cuts in any given vertex induced subgraph (or, any cluster in a fixed partition of V ) to within a (δ, ϵ)-multiplicative/additive error with high probability. The power cut sparsifier uses Õ(n/ϵδ) space and edges, which we show is asymptotically tight up to polylogarithmic factors in n for constant δ.
|Title of host publication||14th Innovations in Theoretical Computer Science Conference, ITCS 2023|
|Editors||Yael Tauman Kalai|
|Publisher||Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing|
|State||Published - 1 Jan 2023|
|Event||14th Innovations in Theoretical Computer Science Conference, ITCS 2023 - Cambridge, United States|
Duration: 10 Jan 2023 → 13 Jan 2023
|Name||Leibniz International Proceedings in Informatics, LIPIcs|
|Conference||14th Innovations in Theoretical Computer Science Conference, ITCS 2023|
|Period||10/01/23 → 13/01/23|
Bibliographical noteFunding Information:
Funding Arnold Filtser: Supported by the ISRAEL SCIENCE FOUNDATION (grant No. 1042/22). Michael Kapralov: Supported in part by ERC Starting Grant 759471. Mikhail Makarov: Supported by ERC Starting Grant 759471.
© Arnold Filtser, Michael Kapralov, and Mikhail Makarov; licensed under Creative Commons License CC-BY 4.0.
- expander decomposition
- graph sparsifiers