Expander Decomposition with Fewer Inter-Cluster Edges Using a Spectral Cut Player
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2022
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866929693887823872 |
|---|---|
| author | Agassy, Daniel Dorfman, Dani Kaplan, Haim |
| author_facet | Agassy, Daniel Dorfman, Dani Kaplan, Haim |
| contents | A $(ϕ,ε)$-expander-decomposition of a graph $G$ (with $n$ vertices and $m$ edges) is a partition of $V$ into clusters $V_1,\ldots,V_k$ with conductance $Φ(G[V_i]) \ge ϕ$, such that there are at most $εm$ inter-cluster edges. Such a decomposition plays a crucial role in many graph algorithms. We give a randomized $\tilde{O}(m/ϕ)$ time algorithm for computing a $(ϕ, ϕ\log^2 {n})$-expander decomposition. This improves upon the $(ϕ, ϕ\log^3 {n})$-expander decomposition also obtained in $\tilde{O}(m/ϕ)$ time by [Saranurak and Wang, SODA 2019] (SW) and brings the number of inter-cluster edges within logarithmic factor of optimal.
One crucial component of SW's algorithm is non-stop version of the cut-matching game of [Khandekar, Rao, Vazirani, JACM 2009] (KRV): The cut player does not stop when it gets from the matching player an unbalanced sparse cut, but continues to play on a trimmed part of the large side. The crux of our improvement is the design of a non-stop version of the cleverer cut player of [Orecchia, Schulman, Vazirani, Vishnoi, STOC 2008] (OSVV). The cut player of OSSV uses a more sophisticated random walk, a subtle potential function, and spectral arguments. Designing and analysing a non-stop version of this game was an explicit open question asked by SW. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_10301 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Expander Decomposition with Fewer Inter-Cluster Edges Using a Spectral Cut Player Agassy, Daniel Dorfman, Dani Kaplan, Haim Data Structures and Algorithms A $(ϕ,ε)$-expander-decomposition of a graph $G$ (with $n$ vertices and $m$ edges) is a partition of $V$ into clusters $V_1,\ldots,V_k$ with conductance $Φ(G[V_i]) \ge ϕ$, such that there are at most $εm$ inter-cluster edges. Such a decomposition plays a crucial role in many graph algorithms. We give a randomized $\tilde{O}(m/ϕ)$ time algorithm for computing a $(ϕ, ϕ\log^2 {n})$-expander decomposition. This improves upon the $(ϕ, ϕ\log^3 {n})$-expander decomposition also obtained in $\tilde{O}(m/ϕ)$ time by [Saranurak and Wang, SODA 2019] (SW) and brings the number of inter-cluster edges within logarithmic factor of optimal. One crucial component of SW's algorithm is non-stop version of the cut-matching game of [Khandekar, Rao, Vazirani, JACM 2009] (KRV): The cut player does not stop when it gets from the matching player an unbalanced sparse cut, but continues to play on a trimmed part of the large side. The crux of our improvement is the design of a non-stop version of the cleverer cut player of [Orecchia, Schulman, Vazirani, Vishnoi, STOC 2008] (OSVV). The cut player of OSSV uses a more sophisticated random walk, a subtle potential function, and spectral arguments. Designing and analysing a non-stop version of this game was an explicit open question asked by SW. |
| title | Expander Decomposition with Fewer Inter-Cluster Edges Using a Spectral Cut Player |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2205.10301 |