Encodings of Range Maximum-Sum Segment Queries and Applications

Abstract

Abstract. Given an array A containing arbitrary (positive and negative) numbers, we consider the problem of supporting range maximumsum segment queries on A: i.e., given an arbitrary range [i, j]Chen and Chao [Disc. App. Math. 2007] presented a data structure for this problem that occupies Θ(n) words, can be constructed in Θ(n) time, and supports queries in Θ(1) time. Our first result is that if only the indices [i , j ] are desired (rather than the maximum sum achieved in that subrange), then it is possible to reduce the space to Θ(n) bits, regardless the numbers stored in A, while retaining the same construction and query time. Our second result is to improve the trivial space lower bound for any encoding data structure that supports range maximum-sum segment queries from n bits to 1.89113n − Θ(lg n), for sufficiently large values of n. Finally, we also provide a new application of this data structure which simplifies a previously known linear time algorithm for finding k-covers: given an array A of n numbers and a number k, find k disjoint subranges [i1, j1], ..., [i k , j k ], such that the total sum of all the numbers in the subranges is maximized. As observed by Csürös [IEEE/ACM TCBB 2004], k-covers can be used to identify regions in genomes.

Encoding Two-Dimensional Range Top-k Queries

“…Now suppose that a p < b q (if a p > b q , we use G 21 and Q 21 instead of G 12 and Q 12 respectively, in the following procedure), and let a ′ p be a vertex with the largest index in G 12 which is incident to a p , if it exists. Note that we can find such a ′ p in O(k) time [8,18]. If there is no vertex incident to a p or a ′ p < a p , we show that A [1]…”

“…Remark Let S sorted and S unsorted be the space needed to encode the sorted and unsorted 1-sided -k queries respectively. For 1D array, Gawrychowski and Nicholson showed that S sorted ∕S unsorted ≤ 2 (thus, the space requirements are asymptotically same for both case) [8]. In contrast, in 2D array case when k = m , S sorted ∕S unsorted ≤ lg k by Theorems 1 and 2, which implies the gap between the space needed to encode the 1-sided -k queries for sorted and unsorted case for a 2D array is significantly more than the case for a 1D array.…”

“…Then by the definition fo G 12 , there are at least From the above lemma and the succinct representation of k-page graphs of Munro and Raman [18] (with minor modification as described in [8]), we can encode G 12 and G 21 using (4k + 4)n + k ⋅ o(n) bits in total, and for any vertex v in V(G 12 ) ∪ V(G 21 ) , we can find a vertex with the largest index which incident to v in O(k) time. Also to compare the elements in the same column, we maintain a bit string…”

“…Proof For 1 ≤ i ≤ j ≤ n , we first use 2S(n, k) bits to support sorted 2-sided -k(1, 1, i, j, A) and -k(2, 2, i, j, A) queries in T(n, k) time. To answer -k(1, 2, i, j, A) query, we maintain succinct representations of G 12 and G 21 [8,18] using (4k + 4)n + ko(n) bits, and P A , Q 12 , and Q 21 using 3n bits. Now for 1 ≤ p ≤ k , let a p (resp., b p ) be the position of the p-th largest value in A 1 [i … j] (resp., A 2 [i … j] ), which can be answered in O(T(n, k)) time using the encoding of -k queries on each row.…”

Encoding two-dimensional range top-k queries

“…For example we traverse the nodes of D 3 A in Figure 1 as [6,6]. During the traversal (in the above order), the position(s) picked at each node are: A in the modified level order.…”