# Interval Query Problem on Cube-free Median Graphs

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(log^2 n) time. The required data structure is constructed in O(nlog^3 n) time and O(nlog^2 n) space. To obtain our algorithm, we introduce a new technique, named the stairs decomposition, to decompose an interval of cube-free median graphs into simpler substructures.

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