The weight-constrained maximum-density subtree problem and related problems in trees

  • Authors:
  • Sun-Yuan Hsieh;Ting-Yu Chou

  • Affiliations:
  • Department of Computer Science and Information Engineering, National Cheng Kung University, Tainan, China 701;Department of Computer Science and Information Engineering, National Cheng Kung University, Tainan, China 701

  • Venue:
  • The Journal of Supercomputing
  • Year:
  • 2010

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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驴,E驴) of T, where V驴驴V and E驴驴E. Given two integers w min驴 and w max驴, the weight-constrained maximum-density subtree problem on T is to find a maximum-density subtree T驴=(V驴,E驴) satisfying w min驴驴驴 v驴V驴 w v 驴w max驴. In this paper, we first present an O(w max驴 n)-time algorithm to find a weight-constrained maximum-density path in a tree T, and then present an O(w max驴 2 n)-time algorithm to find a weight-constrained maximum-density subtree in T. Finally, given a node subset S驴V, we also present an O(w max驴 2 n)-time algorithm to find a weight-constrained maximum-density subtree in T which covers all the nodes in S.