Succinct data structures for nearest colored node in a tree

Highlights

• We give a data-structure that stores a tree with colors on the nodes. Given a node x and a color α, the structure finds the nearest node to x with color α.

• The structure is succinct, namely its space complexity is very near to the optimal space required to store the tree.

• This structure improves the $O(nlogn)$-bits structure.

Abstract

We give succinct data structures that store a tree with colors on the nodes. Given a node x and a color α, the structures find the nearest node to x with color α. Our results improve the $O(n\log n)$-bits structure of Gawrychowski et al. (2016) [12].

Introduction

In the nearest colored node problem the goal is to store a tree with colors on the nodes such that given a node x and a color α, the nearest node to x with color α can be found efficiently. Gawrychowski et al. [12] gave a data structure for this problem that uses $O(n\log n)$ bits and answers queries in $O(\log \log n)$ time, where n is the number of nodes in the tree.

In this paper we give succinct structures for the nearest colored node problem. Our results are given in the following theorem.

Theorem 1

Let T be a colored tree with n nodes and colors from $[1,\sigma ]$, and let $P_T$ be a string containing the colors of the nodes in preorder.

• For $\sigma =o(\log n/{(\log \log n)}^2)$, for any $k=o(\log n/{\log }^2\sigma )$, there is a representation of T that uses $n{H}_k(P_T)+2n+o(n)$ bits and answers nearest colored node queries in $O(1)$ time, where $H_k(P_T)$ is the k-th order entropy of $P_T$.

• For $\sigma =w^{O(1)}$ (where w is the word size), for any function $f(n)=\omega (1)$, there is a representation of T that uses $n{H}_0(P_T)+2n+o(n)$ bits and answers nearest colored node queries in $O(f(n))$ time.

• For $\sigma \le n$, there is a representation of T that uses $n{H}_0(P_T)+2n+o(n{H}_0(P_T))+o(n)$ bits and answers nearest colored node queries in $O(\log \frac{\log \sigma }{\log w})$ time.

Theorem 1 improves both the space complexity and the query time complexity of the structure of Gawrychowski et al. [12].

Gawrychowski et al. [12] also considered a dynamic version of the nearest colored node problem in which the colors of the nodes can be changed. For this problem they gave an $O(n\log n)$ bits structure that supports updates and queries in $O(\log n)$ time. They also gave a structure with $O(n{\log }^{2+ϵ}n)$ space, optimal $O(\log n/\log \log n)$ query time, and $O({\log }^{1+ϵ}n)$ update time.

Several papers studied data structures for storing colored trees with support for various queries [4], [6], [9], [13], [14], [19], [20]. In particular, the problem of finding the nearest ancestor with color α was considered in [6], [13], [14], [19], [20]. In order to solve the nearest colored node problem, we combine techniques from the papers above and from Gawrychowski et al. [12].

Another related problem is to find an approximate nearest node with color α. This problem has been studied in general graphs [7], [15], [16] and planar graphs [1], [17], [18].