Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Tuesday, August 07, 2012, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Micha Sharir (Tel Aviv University)

Waiting for the ceiling to fall: On the longest path in the randomized incremental point-location DAG

A simple and classical solution for point location amid n (nonintersecting) segments in the plane is via a randomized incremental construction: One insertes the segments one by one in a random order, and maintains a vertical trapezoidal decomposition of the plane for the segments inserted so far. In addition, one maintains a History DAG, where each nonfinal trapezoid points to the trapezoids that have ``stepped'' on it at the insertion step that destroyed it. A point location query is implemented simply by searching with the query point q through the path in the DAG of the trapezoids that contain q. The expected length of the path, for any query point q, is O(\log n). Suppose one runs the algorithm and wants to decide whether the produced DAG is ``good'', in the sense that every query has a search path of logarithmic depth? The simplest solution is to check that the depth of the DAG is O(\log n), but unfortunately, as recently observed by Hemmer, Kleinbort, and Halperin, the depth may be much larger, even though the DAG is good, due to the existence of paths that are unrealizable by any query point. The main open question here is whether the expected maximum depth of the DAG is O(\log n). In a work in progress (joint with Haim Kaplan and others) we examine this question, and make a few observations concerning the expected maximum depth of the DAG. One goal of this talk is to solicit ideas from the audience, on this innocently looking but seemingly difficult problem.

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