In this paper we study the problem of partitioning a tree with $n$ weighted vertices into $p$ connected components. For each component, we measure its gap, that is, the difference between the maximum and the minimum weight of its vertices, with the aim of minimizing the sum of such differences. We present an $O\left(n^3{p}^2\right)$ time and $O\left(n^{3}p\right)$ space algorithm for this problem. Then, we generalize it, requiring a minimum of $ϵ\ge 1$ nodes in each connected component, and provide an $O\left(n^3{p}^2{ϵ}^2\right)$ time and $O\left(n^{3}pϵ\right)$ space algorithm to solve this new problem version. We provide a refinement of our analysis involving the topology of the tree and an improvement of the algorithms for the special case in which the weights of the vertices have a heap structure. All presented algorithms can be straightforwardly extended to other similar objective functions. Actually, for the problem of minimizing the maximum gap with a minimum number of nodes in each component, we propose an algorithm which is independent of $ϵ$ and requires $O(n^{2}logn{p}^2)$ time and $O\left(n^{2}p\right)$ space.

在本文中，我们研究了用$n$加权顶点成$p$连接的组件。对于每个组件，我们测量其间隙，即其顶点的最大权重和最小权重之间的差异，目的是最小化这些差异的总和。我们提出一个$○\left(n^3{p}^2\right)$时间和$○\left(n^{3}p\right)$这个问题的空间算法。然后，我们对其进行推广，至少需要$\epsilon \ge 1$每个连接组件中的节点，并提供一个$○\left(n^3{p}^2{\epsilon }^2\right)$时间和$○\left(n^{3}p\epsilon \right)$空间算法来解决这个新的问题版本。我们对涉及树的拓扑结构的分析进行了改进，并对顶点权重具有堆结构的特殊情况的算法进行了改进。所有提出的算法都可以直接扩展到其他类似的目标函数。实际上，对于在每个组件中以最少的节点数最小化最大间隙的问题，我们提出了一种算法，它独立于$\epsilon$并要求$○(n^2日志n{p}^2)$时间和$○\left(n^{2}p\right)$空间。