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# Interval Query Problem on Cube-Free Median Graphs

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LIPIcs.ISAAC.2021.18.pdf
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## Acknowledgements

We are grateful to our supervisor Prof. Hiroshi Hirai for supporting our work. He gave us a lot of ideas to improve our paper. In particular, he simplified the proofs and helped us improve the introduction and the overall structure of this paper.

## Cite As

Soh Kumabe. Interval Query Problem on Cube-Free Median Graphs. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 18:1-18:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ISAAC.2021.18

## Abstract

In this paper, we introduce the interval query problem on cube-free median graphs. Let G be a cube-free median graph and 𝒮 be a commutative semigroup. For each vertex v in G, we are given an element p(v) in 𝒮. For each query, we are given two vertices u,v in G and asked to calculate the sum of p(z) over all vertices z belonging to a u-v shortest path. This is a common generalization of range query problems on trees and grids. In this paper, we provide an algorithm to answer each interval query in O(log² n) time. The required data structure is constructed in O(n log³ n) time and O(n log² n) space. To obtain our algorithm, we introduce a new technique, named the staircases decomposition, to decompose an interval of cube-free median graphs into simpler substructures.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Combinatorial algorithms
##### Keywords
• Data Structures
• Range Query Problems
• Median Graphs

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