where conv(S) denotes the convex hull of S. In this paper, we consider the algorithmic problem of testing whether a given set S of n lattice points is digital convex. Although convex hull computation requires Ω(n log n) time even for dimension d = 2, we provide an algorithm for testing the digital convexity of S ⊂ Z 2 in O(n + h log r) time, where h is the number of edges of the convex hull and r is the diameter of S. This main result is obtained by proving that if S is digital convex, then the well-known quickhull algorithm computes the convex hull of S in linear time. In fixed dimension d, we present the first polynomial algorithm to test digital convexity, as well as a simpler and more practical algorithm whose running time may not be polynomial in n for certain inputs.Problem TestConvexity(S) Input: Set S ⊂ Z d of n lattice points given by their coordinates. Output: Determine whether S is digital convex or not.The input of TestConvexity(S) is an unstructured finite lattice set (without repeating elements).
In this article, we present our heuristic solutions to the problems of finding the maximum and minimum area polygons with a given set of vertices. Our solutions are based mostly on two simple algorithmic paradigms: greedy and local search. The greedy heuristic starts with a simple polygon and adds vertices one by one, according to a weight function. A crucial ingredient to obtain good solutions is the choice of an appropriate weight function that avoids long edges, since long edges impede the insertion of too many different edges later on. The local search part consists of moving consecutive vertices to another location in the polygonal chain. We also discuss the different implementation techniques that are necessary to reduce the running time.
This paper describes the heuristics used by the
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team for the CG:SHOP 2021 challenge. This year’s problem is to coordinate the motion of multiple robots in order to reach their targets without collisions and minimizing the makespan. It is a classical multi agent path finding problem with the specificity that the instances are highly dense in an unbounded grid. Using the heuristics outlined in this paper, our team won first place with the best solution to 202 out of 203 instances and optimal solutions to at least 105 of them. The main ingredients include several different strategies to compute initial solutions coupled with a heuristic called Conflict Optimizer to reduce the makespan of existing solutions.
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