Interval Query Problem on Cube-free Median Graphs

In this paper, we introduce the \emph{interval query problem} on cube-free median graphs. Let \(G\) be a cube-free median graph and \(\mathcal{S}\) be a commutative semigroup. For each vertex \(v\) in \(G\), we are given an element \(p(v)\) in \(\mathcal{S}\). 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 \(O(n\log^3 n)\) time and \(O(n\log^2 n)\) space. To obtain our algorithm, we introduce a new technique, named the \emph{stairs decomposition}, to decompose an interval of cube-free median graphs into simpler substructures.