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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2503.11526 |
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| _version_ | 1866909537614692352 |
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| author | Luo, Ruixi Zhu, Taikun Jin, Kai |
| author_facet | Luo, Ruixi Zhu, Taikun Jin, Kai |
| contents | Path partition problems on trees have found various applications. In this paper, we present an $O(n \log n)$ time algorithm for solving the following variant of path partition problem: given a rooted tree of $n$ nodes $1, \ldots, n$, where vertex $i$ is associated with a weight $w_i$ and a cost $s_i$, partition the tree into several disjoint chains $C_1,\ldots,C_k$, so that the weight of each chain is no more than a threshold $w_0$ and the sum of the largest $s_i$ in each chain is minimized. We also generalize the algorithm to the case where the cost of a chain is determined by the $s_i$ of the vertex with the highest rank in the chain, which can be determined by an arbitrary total order defined on all nodes instead of the value of $s_i$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_11526 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sum-of-Max Chain Partition of a Tree Luo, Ruixi Zhu, Taikun Jin, Kai Data Structures and Algorithms Path partition problems on trees have found various applications. In this paper, we present an $O(n \log n)$ time algorithm for solving the following variant of path partition problem: given a rooted tree of $n$ nodes $1, \ldots, n$, where vertex $i$ is associated with a weight $w_i$ and a cost $s_i$, partition the tree into several disjoint chains $C_1,\ldots,C_k$, so that the weight of each chain is no more than a threshold $w_0$ and the sum of the largest $s_i$ in each chain is minimized. We also generalize the algorithm to the case where the cost of a chain is determined by the $s_i$ of the vertex with the highest rank in the chain, which can be determined by an arbitrary total order defined on all nodes instead of the value of $s_i$. |
| title | Sum-of-Max Chain Partition of a Tree |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2503.11526 |