Distance Queries over Dynamic Interval Graphs

Abstract

We design the first dynamic distance oracles for interval graphs, which are intersection graphs of a set of intervals on the real line, and for proper interval graphs, which are intersection graphs of a set of intervals in which no interval is properly contained in another. For proper interval graphs, we design a linear space data structure which supports distance queries (computing the distance between two query vertices) and vertex insertion or deletion in O (lg n ) worst-case time, where n is the number of vertices currently in G . Under incremental (insertion only) or decremental (deletion only) settings, we design linear space data structures that support distance queries in O (lg n ) worst-case time and vertex insertion or deletion in O (lg n ) amortized time, where n is the maximum number of vertices in the graph. Under fully dynamic settings, we design a data structure that represents an interval graph G in O ( n ) words of space to support distance queries in O ( n lg n/S ( n )) worst-case time and vertex insertion or deletion in O ( S ( n ) + lg n ) worst-case time, where n is the number of vertices currently in G and S ( n ) is an arbitrary function that satisfies S ( n ) = Ω(1) and S ( n ) = O ( n ). This implies an O ( n )-word solution with O ( √ n lg n )-time support for both distance queries and updates. All four data structures can answer shortest path queries by reporting the vertices in the shortest path between two query vertices in O (lg n ) worst-case time per vertex. We also study the hardness of supporting distance queries under updates over an intersection graph of 3D axis-aligned line segments, which generalizes our problem to 3D. Finally, we solve the problem of computing the diameter of a dynamic connected interval graph.