Abstract
Given a tree T=(V,E) of n nodes such that each node v is associated with a value-weight pair (val v ,w v ), where value val v is a real number and weight w v is a non-negative integer, the density of T is defined as \(\frac{\sum_{v\in V}{\mathit{val}}_{v}}{\sum_{v\in V}w_{v}}\). A subtree of T is a connected subgraph (V′,E′) of T, where V′⊆V and E′⊆E. Given two integers w min and w max , the weight-constrained maximum-density subtree problem on T is to find a maximum-density subtree T′=(V′,E′) satisfying w min ≤∑v∈V′ w v ≤w max . In this paper, we first present an O(w max n)-time algorithm to find a weight-constrained maximum-density path in a tree T, and then present an O(w 2max n)-time algorithm to find a weight-constrained maximum-density subtree in T. Finally, given a node subset S⊂V, we also present an O(w 2max n)-time algorithm to find a weight-constrained maximum-density subtree in T which covers all the nodes in S.
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An extended abstract of this paper appeared in Proceedings of the 16th Annual International Symposium on Algorithms and Computation (ISAAC 2005), Lecture Notes in Computer Science, vol. 3827, pp. 944–953, 2005.
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Hsieh, SY., Chou, TY. The weight-constrained maximum-density subtree problem and related problems in trees. J Supercomput 54, 366–380 (2010). https://doi.org/10.1007/s11227-009-0328-z
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DOI: https://doi.org/10.1007/s11227-009-0328-z