Sparsifying Disk Intersection Graphs for Reliable Connectivity
DOI:
https://doi.org/10.57717/cgt.v3i1.66Abstract
The intersection graph induced by a set of n disks can be dense. It is thus natural to try and sparsify it, while preserving connectivity. Unfortunately, sparse graphs can always be made disconnected by removing a small number of vertices. In this work, we present a randomized sparsification algorithm that maintains connectivity between two regions in the computed graph, if the original graph remains "well-connected" even after removing an arbitrary `"attack'" set from both graphs. Thus, the new sparse graph has similar reliability to the original disk graph, and can withstand catastrophic failure of nodes while still providing a connectivity guarantee for the remaining graph. The new graph has near linear complexity, and can be constructed in near-linear time.
The algorithm extends to any collection of shapes in the plane with near linear union complexity.
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Copyright (c) 2024 Eliot Robson, Sariel Har-Peled
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