The nearest colored node in a tree

Paweł Gawrychowski,Gad M Landau,Shay Mozes,Oren Weimann

doi:10.1016/j.tcs.2017.08.021

Paweł Gawrychowski, Gad M Landau + Show 2 more

Open Access

https://doi.org/10.1016/j.tcs.2017.08.021

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Abstract

We start a systematic study of data structures for the nearest colored node problem on trees. Given a tree with colored nodes and weighted edges, we want to answer queries (v,c) asking for the nearest node to node v that has color c. This is a natural generalization of the well-known nearest marked ancestor problem. We give an O(n)-space O(log⁡log⁡n)-query solution and show that this is optimal. We also consider the dynamic case where updates can change a node's color and show that in O(n) space we can support both updates and queries in O(log⁡n) time. We complement this by showing that O(polylogn) update time implies Ω(log⁡nlog⁡log⁡n) query time. Finally, we consider the case where updates can change the edges of the tree (link-cut operations). There is a known (top-tree based) solution that requires update time that is roughly linear in the number of colors. We show that this solution is probably optimal by showing that a strictly sublinear update time implies a strictly subcubic time algorithm for the classical all pairs shortest paths problem on a general graph. We also consider versions where the tree is rooted, and the query asks for the nearest ancestor/descendant of node v that has color c, and present efficient data structures for both variants in the static and the dynamic setting.

Highlights

We consider a number of problems on trees with colored nodes
This holds even when the number of colors is only two. We show that this lower bound extends to both the nearest colored node and the nearest colored descendant
We show that the nearest colored node problem can be solved with O(n)-space, O(log n)-update, and O(log n)-query

Summary

Introduction

We consider a number of problems on trees with colored nodes. Each of these problems can be either static, meaning the color of every node of a tree T on n nodes is fixed, or dynamic, meaning that an update can change a node’s color (but the tree itself does not change). We show that the nearest colored node problem can be solved with O(n)-space, O(log n)-update, and O(log n)-query. The original top-trees of Alstrup-Holm-de Lichtenberg-Thorup [2] were designed for only two colors (i.e., for the nearest marked node problem) They achieve O(log n) query and update and support updates that insert and delete edges (i.e., maintain a forest under link-cut operations). A predecessor structure is a data structure that stores a set of n integers S ⊆ [0, U ], so that given x ∈ [0, U ] we can determine the largest y ∈ S such that y ≤ x It is known [15] that for U = n2 any predecessor structure of O(n polylog n)-space requires Ω(log log n)-query, and that linear-size structures with such query-time exist [16, 18]. There exists O(n)-space, O(1)-query perfect hash structure [10], which can be made dynamic with O(1)-update (expected amortized) [9]

Static Upper Bounds

Dynamic Upper Bounds

Lower Bounds