In this paper, we address the following problems: Given a sequence A of n real numbers, and four parameters I,J,X and Y with I≤ J and X≤ Y, find the longest (or shortest) subsequence of A such that its length is between I and J and its sum is between X and Y. We present an online and an offline algorithm for the problems, both run in O(nlog n) time, which are optimal.

Given a tree T with edge lengths and edge weights, and a value B, the length-constrained heaviest path problem is to find a path in T with maximum path weight whose path length is at most B. We present a linear time algorithm for the problem when the edge lengths are uniform, i.e., all one. This algorithm with slight modification can be used to find the heaviest path of length exactly B in T in linear time.

Let T be a node-weighted tree with n nodes, and let π(u,v) denote the path between two nodes u and v in T. We address two problems: (i) Path Maximum Query: Preprocess T so that, for a query pair of nodes u and v, the maximum weight on π(u,v) can be found quickly. (ii) Path Maximum Sum Query: Preprocess T so that, for a query pair of nodes u and v, the maximum weight sum subpath of π(u,v) can be found quickly. For the problems we present solutions with O(1) query time and O(n log n) preprocessing time.

Let T be a tree in which every edge is associated with a real number. The sum of a path in T is the sum of the numbers associated with the edges of the path and its length is the number of the edges in it. For two positive integers L1 ≤ L2 and two real numbers S1 ≤ S2, a path is feasible if its length is between L1 and L2 and its sum is between S1 and S2. We address the problem: Given a tree T, and four numbers, L1, L2, S1 and S2, find the longest feasible path of T. We provide an optimal O(n log n) time algorithm for the problem, where n =|T|.

Given an edge-weighted tree with n vertices and a positive integer L, the length-constrained maximum-density path problem is to find a path of length at least L with maximum density in the tree. The density of a path is the sum of the weights of the edges in the path divided by the number of edges in the path. We present an O(n) time algorithm for the problem. The previously known algorithms run in O(nL) or O(n log n) time.

Let T be a tree with n nodes, in which each edge is associated with a length and a weight. The density-constrained longest (heaviest) path problem is to find a path of T with maximum path length (weight) whose path density is bounded by an upper bound and a lower bound. The path density is the path weight divided by the path length. We show that both problems can be solved in optimal O(n log n) time.