Finding a weight-constrained maximum-density subtree in a tree

Authors:
Sun-Yuan Hsieh;Ting-Yu Chou
Affiliations:
Department of Computer Science and Information Engineering, National Cheng Kung University, Tainan, Taiwan;Department of Computer Science and Information Engineering, National Cheng Kung University, Tainan, Taiwan
Venue:
ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
Year:
2005

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Abstract

Given a tree T=(V,E) of n vertices such that each node v is associated with a value-weight pair (valv,wv), where valuevalv is a real number and weightwv is a non-negative integer, the density of T is defined as ${\sum_{v \in V^{val_{v}}}} \over {\sum_{v \in V^{w_{v}}}}$. A subtree of T is a connected subgraph (V′,E′) of T, where V′⊆V and E′⊆E. Given two integers wmin and wmax, the weight-constrained maximum-density subtree problem on T is to find a maximum-density subtree T′=(V′,E′) satisfying $w_{min}{\leq} \Sigma_{v\in V^{\prime}}{w_{v}}{\leq} {w_{max}}$. In this paper, we first present an O(wmaxn)-time algorithm to find a weight-constrained maximum-density path in a tree, and then present an O(wmax2n)-time algorithm to find a weight-constrained maximum-density subtree in a tree. Finally, given a node subset S ⊂ V, we also present an O(wmax2n)-time algorithm to find a weight-constrained maximum-density subtree of T which covers all the nodes in S.