On approximating the depth and related problems

In this paper, we study the problem of finding a disk covering the largest number of red points, while avoiding all the blue points. We reduce it to the question of finding a deepest point in an arrangement of pseudodisks and provide a near-linear expected-time randomized approximation algorithm for this problem. As an application of our techniques, we show how to solve linear programming with violations approximately. We also prove that approximate range counting has roughly the same time and space complexity as answering emptiness range queries.

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