Simple Length-Constrained Expander Decompositions
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| Format: | Preprint |
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2025
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| _version_ | 1866915546703855616 |
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| author | Bodwin, Greg Haeupler, Bernhard Hershkowitz, D Ellis Tan, Zihan |
| author_facet | Bodwin, Greg Haeupler, Bernhard Hershkowitz, D Ellis Tan, Zihan |
| contents | Length-constrained expander decompositions are a new graph decomposition that has led to several recent breakthroughs in fast graph algorithms. Roughly, an $(h, s)$-length $ϕ$-expander decomposition is a small collection of length increases to a graph so that nodes within distance $h$ can route flow over paths of length $hs$ while using each edge to an extent at most $1/ϕ$. Prior work showed that every $n$-node and $m$-edge graph admits an $(h, s)$-length $ϕ$-expander decomposition of size $\log n \cdot s n^{O(1/s)} \cdot ϕm$.
In this work, we give a simple proof of the existence of $(h, s)$-length $ϕ$-expander decompositions with an improved size of $s n^{O(1/s)}\cdot ϕm$. Our proof is a straightforward application of the fact that the union of sparse length-constrained cuts is itself a sparse length-constrained cut. In deriving our result, we improve the loss in sparsity when taking the union of sparse length-constrained cuts from $\log ^3 n\cdot s^3 n^{O(1/s)}$ to $s\cdot n^{O(1/s)}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_10227 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Simple Length-Constrained Expander Decompositions Bodwin, Greg Haeupler, Bernhard Hershkowitz, D Ellis Tan, Zihan Data Structures and Algorithms Length-constrained expander decompositions are a new graph decomposition that has led to several recent breakthroughs in fast graph algorithms. Roughly, an $(h, s)$-length $ϕ$-expander decomposition is a small collection of length increases to a graph so that nodes within distance $h$ can route flow over paths of length $hs$ while using each edge to an extent at most $1/ϕ$. Prior work showed that every $n$-node and $m$-edge graph admits an $(h, s)$-length $ϕ$-expander decomposition of size $\log n \cdot s n^{O(1/s)} \cdot ϕm$. In this work, we give a simple proof of the existence of $(h, s)$-length $ϕ$-expander decompositions with an improved size of $s n^{O(1/s)}\cdot ϕm$. Our proof is a straightforward application of the fact that the union of sparse length-constrained cuts is itself a sparse length-constrained cut. In deriving our result, we improve the loss in sparsity when taking the union of sparse length-constrained cuts from $\log ^3 n\cdot s^3 n^{O(1/s)}$ to $s\cdot n^{O(1/s)}$. |
| title | Simple Length-Constrained Expander Decompositions |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2510.10227 |