Range Longest Increasing Subsequence and Its Relatives.

Longest increasing subsequence (LIS) is a classical textbook problem which is still actively studied in various computational models. In this work, we present a few results for the range longest increasing subsequence problem (Range-LIS) and its variants. The input to Range-LIS is a sequence S of n real numbers and a collection Q of m query ranges and for each query in Q, the goal is to report the LIS of the sequence S restricted to that query. Our two main results are for the following generalizations of the Range-LIS problem: 2D Range Queries: In this variant of the Range-LIS problem, each query is a pair of ranges, one of indices and the other of values, and we provide a randomized algorithm with running time(1) (O) over tilde (mn(1/2) + n(3/2)) + O(k), where k is the cumulative length of the m output subsequences. This improves on the elementary (O) over tilde (mn) runtime algorithm when m = Omega(root n). Previously, the only known result breaking the quadratic barrier was of Tiskin [SODA'10] which could only handle 1D range queries (i.e., each query was a range of indices) and also just outputted the length of the LIS (instead of reporting the subsequence achieving that length). Subsequent to our paper, Gawrychowski, Gorbachev, and Kociumaka in a preprint have extended Tiskin's approach to handle reporting 1D range queries in O(n(log n)(3) + m + k) time. Colored Sequences: In this variant of the Range-LIS problem, each element in S is colored and for each query in Q, the goal is to report a monochromatic LIS contained in the sequence S restricted to that query. For 2D queries, we provide a randomized algorithm for this colored version with running time (O) over tilde (mn(2/3) + n(5/3)) + O(k). Moreover, for 1D queries, we provide an improved algorithm with running time (O) over tilde (mn(1/2) + n(3/2)) + O(k). Thus, we again improve on the elementary (O) over tilde (mn) runtime algorithm. Additionally, we prove that assuming the well-known Combinatorial Boolean Matrix Multiplication Hypothesis, that the runtime for 1D queries is essentially tight for combinatorial algorithms. Our algorithms combine several tools such as dynamic programming (to precompute increasing subsequences with some desirable properties), geometric data structures (to efficiently compute the dynamic programming entries), random sampling (to capture elements which are part of the LIS), classification of query ranges into large LIS and small LIS, and classification of colors into light and heavy. We believe that our techniques will be of interest to tackle other variants of LIS problem and other range-searching problems. 2012 ACM Subject Classification Theory of computation -> Data structures design and analysis; Theory of computation -> Computational geometry