Combined Data Structure for Previous- and Next-Smaller-Values

Let $A$ be a static array storing $n$ elements from a totally ordered set. We present a data structure of optimal size at most $n\log_2(3+2\sqrt{2})+o(n)$ bits that allows us to answer the following queries on $A$ in constant time, without accessing $A$: (1) previous smaller value queries, where given an index $i$, we wish to find the first index to the left of $i$ where $A$ is strictly smaller than at $i$, and (2) next smaller value queries, which search to the right of $i$. As an additional bonus, our data structure also allows to answer a third kind of query: given indices $i<j$, find the position of the minimum in $A[i..j]$. Our data structure has direct consequences for the space-efficient storage of suffix trees.Comment: to appear in Theoretical Computer Scienc

LRM-Trees: Compressed Indices, Adaptive Sorting, and Compressed Permutations

LRM-Trees are an elegant way to partition a sequence of values into sorted consecutive blocks, and to express the relative position of the first element of each block within a previous block. They were used to encode ordinal trees and to index integer arrays in order to support range minimum queries on them. We describe how they yield many other convenient results in a variety of areas, from data structures to algorithms: some compressed succinct indices for range minimum queries; a new adaptive sorting algorithm; and a compressed succinct data structure for permutations supporting direct and indirect application in time all the shortest as the permutation is compressible.Comment: 13 pages, 1 figur

Linear-Space Data Structures for Range Mode Query in Arrays

A mode of a multiset $S$ is an element $a \in S$ of maximum multiplicity; that is, $a$ occurs at least as frequently as any other element in $S$. Given a list $A[1:n]$ of $n$ items, we consider the problem of constructing a data structure that efficiently answers range mode queries on $A$. Each query consists of an input pair of indices $(i, j)$ for which a mode of $A[i:j]$ must be returned. We present an $O(n^{2-2\epsilon})$-space static data structure that supports range mode queries in $O(n^\epsilon)$ time in the worst case, for any fixed $\epsilon \in [0,1/2]$. When $\epsilon = 1/2$, this corresponds to the first linear-space data structure to guarantee $O(\sqrt{n})$ query time. We then describe three additional linear-space data structures that provide $O(k)$, $O(m)$, and $O(|j-i|)$ query time, respectively, where $k$ denotes the number of distinct elements in $A$ and $m$ denotes the frequency of the mode of $A$. Finally, we examine generalizing our data structures to higher dimensions.Comment: 13 pages, 2 figure

Towards Tight Lower Bounds for Range Reporting on the RAM

In the orthogonal range reporting problem, we are to preprocess a set of $n$ points with integer coordinates on a $U \times U$ grid. The goal is to support reporting all $k$ points inside an axis-aligned query rectangle. This is one of the most fundamental data structure problems in databases and computational geometry. Despite the importance of the problem its complexity remains unresolved in the word-RAM. On the upper bound side, three best tradeoffs exists: (1.) Query time $O(\lg \lg n + k)$ with $O(nlg^{\varepsilon}n)$ words of space for any constant $\varepsilon>0$. (2.) Query time $O((1 + k) \lg \lg n)$ with $O(n \lg \lg n)$ words of space. (3.) Query time $O((1+k)\lg^{\varepsilon} n)$ with optimal $O(n)$ words of space. However, the only known query time lower bound is $\Omega(\log \log n +k)$, even for linear space data structures. All three current best upper bound tradeoffs are derived by reducing range reporting to a ball-inheritance problem. Ball-inheritance is a problem that essentially encapsulates all previous attempts at solving range reporting in the word-RAM. In this paper we make progress towards closing the gap between the upper and lower bounds for range reporting by proving cell probe lower bounds for ball-inheritance. Our lower bounds are tight for a large range of parameters, excluding any further progress for range reporting using the ball-inheritance reduction

Stable Noncrossing Matchings

Given a set of $n$ men represented by $n$ points lying on a line, and $n$ women represented by $n$ points lying on another parallel line, with each person having a list that ranks some people of opposite gender as his/her acceptable partners in strict order of preference. In this problem, we want to match people of opposite genders to satisfy people's preferences as well as making the edges not crossing one another geometrically. A noncrossing blocking pair w.r.t. a matching $M$ is a pair $(m,w)$ of a man and a woman such that they are not matched with each other but prefer each other to their own partners in $M$, and the segment $(m,w)$ does not cross any edge in $M$. A weakly stable noncrossing matching (WSNM) is a noncrossing matching that does not admit any noncrossing blocking pair. In this paper, we prove the existence of a WSNM in any instance by developing an $O(n^2)$ algorithm to find one in a given instance.Comment: This paper has appeared at IWOCA 201

Towards Tight Lower Bounds for Range Reporting on the RAM

In the orthogonal range reporting problem, we are to preprocess a set of n points with integer coordinates on a UxU grid. The goal is to support reporting all k points inside an axis-aligned query rectangle. This is one of the most fundamental data structure problems in databases and computational geometry. Despite the importance of the problem its complexity remains unresolved in the word-RAM. On the upper bound side, three best tradeoffs exist, all derived by reducing range reporting to a ball-inheritance problem. Ball-inheritance is a problem that essentially encapsulates all previous attempts at solving range reporting in the word-RAM. In this paper we make progress towards closing the gap between the upper and lower bounds for range reporting by proving cell probe lower bounds for ball-inheritance. Our lower bounds are tight for a large range of parameters, excluding any further progress for range reporting using the ball-inheritance reduction