Shortest path trees in disk intersection graphs can be computed in O(n log n) time, and in O(n log^2 n) time for intersection graphs of fat triangles.
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An $O(n\log n)$ Algorithm for Single-Source Shortest Paths in Disk Graphs
Shortest path trees in disk intersection graphs can be computed in O(n log n) time, and in O(n log^2 n) time for intersection graphs of fat triangles.