On the Range Maximum-Sum Segment Query Problem

Abstract

The range minimum query problem, RMQ for short, is to preprocess a sequence of real numbers A[1 . . . n] for subsequent queries of the form: "Given indices i, j, what is the index of the minimum value of A[i . . . j]?" This problem has been shown to be linearly equivalent to the LCA problem in which a tree is preprocessed for answering the lowest common ancestor of two nodes. It has also been shown that both the RMQ and LCA problems can be solved in linear preprocessing time and constant query time under the unit-cost RAM model. This paper studies a new query problem arising from the analysis of biological sequences. Specifically, we wish to answer queries of the form: "Given indices i and j, what is the maximum-sum segment of A[i . . . j]?" We establish the linear equivalence relation between RMQ and this new problem. As a consequence, we can solve the new query problem in linear preprocessing time and constant query time under the unit-cost RAM model. We then present alternative linear time solutions for two other biological sequence analysis problems to demonstrate the utilities of the techniques developed in this paper.

Space-Efficient Preprocessing Schemes for Range Minimum Queries on Static Arrays

“…For an array A [1, n] of n objects from a totally ordered universe and two indices i and j with 1 ≤ i ≤ j ≤ n, a Range Minimum Query 1 rmq A (i, j) returns the position of a minimum element in the sub-array A[i, j]; in symbols: rmq A (i, j) = argmin i≤k≤j A [k] . Given the ubiquity of arrays and the fundamental nature of this question, it is not surprising that RMQs have a wide range of applications in various fields of computing: text indexing [23,52], pattern matching [2,12], string mining [21,34], text compression [9,45], document retrieval [42,53,59], trees [4,6,38], graphs [28,49], bioinformatics [58], and in other types of range queries [10,56], to mention just a few.…”

Path Maximum Query and Path Maximum Sum Query in a Tree

“…Chen and Chao [2] proves the linear equivalence between RMQ and RMSQ, and, as a consequence, gives a solution with O(n) preprocessing time and O(1) query time.…”

Efficient Data Structures for Online QoS-Constrained Data Transfer Scheduling

“…Extended variants of these structures have been used in many fields, like computational geometry [6], machine vision, data management in OLAP applications [3,4,5] and advance resource reservations [13]. Usually, they only solve particular problems, for only one type of update and query, or even just for the static case (without any updates) [7]. Moreover, the most commonly studied case corresponds to range queries and point updates.…”