Using spine decompositions to efficiently solve the length-constrained heaviest path problem for trees

a r t i c l e i n f o a b s t r a c t The length-constrained heaviest path (LCHP) in a weighted tree T , where each edge is assigned a weight and a length, is the path P in T with maximum total path weight and total path length bounded by a given value B. This paper presents an O (n log n) time LCHP algorithm which utilizes a data structure constructed from the spine decomposition of the input tree. This is an improvement over the existing algorithm by Wu et al. (1999), which runs in O (n log 2 n) time. Our method also improves on a previous O (n log n) time algorithm by Kim (2005) for the special case of finding a longest nonnegative path in a constant degree tree in that we can handle trees of arbitrary degree within the same time bounds.