A Linear Time Algorithm for Computing Longest Paths in Cactus Graphs

Abstract

We propose an algorithm that computes the length of a longest path in a cactus graph. Our algorithm can easily be modified to output a longest path as well or to solve the problem on cacti with edge or vertex weights. The algorithm works on rooted cacti and assigns to each vertex a two-number label, the first number being the desired parameter of the subcactus rooted at that vertex. The algorithm applies the divide-and-conquer approach and computes the label of each vertex from the labels of its children. The time complexity of our algorithm is linear in the number of vertices, thus improving the previously best quadratic time algorithm.