Paper by Erik D. Demaine

Reference:
Stelian Ciurea, Erik D. Demaine, Corina E. Pǎtraşcu, and Mihai Pǎtraşcu, “Finding a Divisible Pair and a Good Wooden Fence”, in Proceedings of the 3rd International Conference on Fun with Algorithms (FUN 2004), Isola d'Elba, Italy, May 26–28, 2004, pages 206–219.

Abstract:
We consider two algorithmic problems arising in the lives of Yogi Bear and Ranger Smith. The first problem is the natural algorithmic version of a classic mathematical result: any (n + 1)-subset of {1, …, 2n} contains a pair of divisible numbers. How do we actually find such a pair? If the subset is given in the form of a bit vector, we give a RAM algorithm with an optimal running time of O(n / lg n). If the subset is accessible only through a membership oracle, we show a lower bound of (4/3) n − O(1) and an almost matching upper bound of (4/3 + 1/24) n + O(1) on the number of queries necessary in the worst case.

The second problem we study is a geometric optimization problem where the objective amusingly influences the constraints. Suppose you want to surround n trees at given coordinates by a wooden fence. However, you have no external wood supply, and must obtain wood by chopping down some of the trees. The goal is to cut down a minimum number of trees that can be built into a fence that surrounds the remaining trees. We obtain efficient polynomial-time algorithms for this problem.

We also describe an unusual data-structural view of the Nim game, leading to an intriguing open problem.

Length:
The paper is 12 pages.

Availability:
The paper is available in PostScript (178k), gzipped PostScript (72k), and PDF (105k).
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Related papers:
DivisiblePair_SIGSAM (Finding a Divisible Pair)


See also other papers by Erik Demaine.
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Last updated November 16, 2017 by Erik Demaine.