In this paper, we introduce the \emph{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(log2n) time. The required data structure is constructed in
O(nlog3n) time and O(nlog2n) 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.Comment: ISAAC'21, 21 page