The L∞ Hausdorff Voronoi Diagram Revisited

We revisit the L∞ Hausdorff Voronoi diagram of clusters of points, equivalently, the L∞ Hausdorff Voronoi diagram of rectangles, and present a plane sweep algorithm for its construction that generalizes and improves upon previous results. We show that the structural complexity of the L∞ Hausdorff Voronoi diagram is Θ(n+m), where n is the number of given clusters and m is the number of essential pairs of crossing clusters. The algorithm runs in O((n+M) logn) time and O(n + M) space where M is the number of potentially essential crossings; m,M are O(n), m ≤ M , but m = M , in the worst case. In practice m,M << n, as the total number of crossings in the motivating application is typically small. For non-crossing clusters, the algorithm is optimal running in O(n logn)-time and O(n)-space. The L∞ Hausdorff Voronoi diagram finds applications, among others, in the geometric min-cut problem, VLSI critical area analysis for via-blocks and open faults. Keywords-Voronoi diagram, Hausdorff distance, L∞ metric, plane sweep, point dominance, VLSI layout.