Dynamic Edge Coloring of Forests
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909030893486080 |
|---|---|
| author | Kaplan, Haim Naori, David Sadeh, Yaniv |
| author_facet | Kaplan, Haim Naori, David Sadeh, Yaniv |
| contents | In the \emph{dynamic edge coloring} problem, one has to maintain a graph of maximum degree $Δ$ with at most $Δ+c$ colors, given updates to the edges of the graph. An important objective is to minimize the \emph{recourse}, which is the number of edges being recolored.
We study this problem on forests, which is a natural yet nontrivial restriction of the problem. We consider the problem in both \emph{incremental} (edges are only inserted) and \emph{fully dynamic} (edges may be deleted) models. In the deterministic setting, we show that the natural greedy algorithm achieves $O(\frac{1}{c + \sqrtΔ})$ amortized recourse in the incremental model, and this is tight up to tie-breaking. In contrast, in a fully dynamic forest, greedy can be forced to have $Ω(\log_Δn)$ amortized recourse. To partially alleviate this limitation of greedy, we show an optimal non-greedy algorithm with $O(1)$ amortized recourse for \emph{rooted} fully dynamic forests and $c = Δ- 2$. In the randomized setting, we give a natural distribution-maintaining algorithm that achieves $Θ(\frac{1}Δ)$ expected amortized recourse in the incremental model and $Θ(\min \{ \fracΔ{c}, \log_Δ n \})$ expected recourse in the dynamic model. These randomized results are optimal for $c=0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_09711 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Dynamic Edge Coloring of Forests Kaplan, Haim Naori, David Sadeh, Yaniv Data Structures and Algorithms In the \emph{dynamic edge coloring} problem, one has to maintain a graph of maximum degree $Δ$ with at most $Δ+c$ colors, given updates to the edges of the graph. An important objective is to minimize the \emph{recourse}, which is the number of edges being recolored. We study this problem on forests, which is a natural yet nontrivial restriction of the problem. We consider the problem in both \emph{incremental} (edges are only inserted) and \emph{fully dynamic} (edges may be deleted) models. In the deterministic setting, we show that the natural greedy algorithm achieves $O(\frac{1}{c + \sqrtΔ})$ amortized recourse in the incremental model, and this is tight up to tie-breaking. In contrast, in a fully dynamic forest, greedy can be forced to have $Ω(\log_Δn)$ amortized recourse. To partially alleviate this limitation of greedy, we show an optimal non-greedy algorithm with $O(1)$ amortized recourse for \emph{rooted} fully dynamic forests and $c = Δ- 2$. In the randomized setting, we give a natural distribution-maintaining algorithm that achieves $Θ(\frac{1}Δ)$ expected amortized recourse in the incremental model and $Θ(\min \{ \fracΔ{c}, \log_Δ n \})$ expected recourse in the dynamic model. These randomized results are optimal for $c=0$. |
| title | Dynamic Edge Coloring of Forests |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2605.09711 |