We prove NP-hardness results for five of Nintendo's largest video game franchises: Mario, Donkey Kong, Legend of Zelda, Metroid, and Pokémon. Our results apply to generalized versions of Super Mario Bros. 1-3, The Lost Levels, and Super Mario World; Donkey Kong Country 1-3; all Legend of Zelda games; all Metroid games; and all Pokémon role-playing games. In addition, we prove PSPACE-completeness of the Donkey Kong Country games and several Legend of Zelda games.

We prove that for every centrally symmetric convex polygon Q, there exists a constant α such that any αk-fold covering of the plane by translates of Q can be decomposed into k coverings. This improves on a quadratic upper bound proved by Pach and Tóth (SoCG'07). The question is motivated by a sensor network problem, in which a region has to be monitored by sensors with limited battery lifetime.

Several measures of data depth have been proposed, each attempting to maintain certain robustness properties. This paper lists the main approaches known to the computer science community. Properties and algorithms are mentioned, for computing the depth of a point and the location of the deepest point.

Abstract. Given a set of points in R 2 or R 3 , we aim to color them such that every region of a certain family (for instance disks) containing at least a certain number of points contains points of many different colors. Using k colors, it is not always possible to ensure that every region containing k points contains all k colors. Thus, we introduce two relaxations: either we allow the number of colors to increase to c(k), or we require that the number of points in each region increases to p(k). We give upper bounds on c(k) and p(k) for halfspaces, disks, and pseudodisks. We also consider the dual question, where we want to color regions instead of points. This is related to previous results of Pach, Tardos and Tóth on decompositions of coverings.

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For a set R of n red points and a set B of n blue points, a BR-matching is a non-crossing geometric perfect matching where each segment has one endpoint in B and one in R. Two BRmatchings are compatible if their union is also non-crossing. We prove that, for any two distinct BRmatchings M and M , there exists a sequence of BR-matchings M = M 1 , . . . , M k = M such that M i−1 is compatible with M i . This implies the connectivity of the compatible bichromatic matching graph containing one node for each BR-matching and an edge joining each pair of compatible BR-matchings, thereby answering the open problem posed by Aichholzer et al. in [5].

We consider the theoretical model of Crystalline robots, which have been introduced and prototyped by the robotics community. These robots consist of independently manipulable unit-square atoms that can extend/contract arms on each side and attach/detach from neighbors. These operations suffice to reconfigure between any two given (connected) shapes. The worstcase number of sequential moves required to transform one connected configuration to another is known to be Θ(n). However, in principle, atoms can all move simultaneously. We develop a parallel algorithm for reconfiguration that runs in only O(log n) parallel steps, although the total number of operations increases slightly to Θ(n log n). The result is the first (theoretically) almost-instantaneous universally reconfigurable robot built from simple units. * This result, constrained to 2D, was presented by the authors at the 19th International Symposium on Algorithms and Computation [3].

A highway H is a line in the plane on which one can travel at a greater speed than in the remaining plane. One can choose to enter and exit H at any point. The highway time distance between a pair of points is the minimum time required to move from one point to the other, with optional use of H.The highway hull H(S, H) of a point set S is the minimal set containing S as well as the shortest paths between all pairs of points in H(S, H), using the highway time distance.We provide a Θ(n log n) worst-case time algorithm to find the highway hull under the L1 metric, as well as an O(n log 2 n) time algorithm for the L2 metric which improves the best known result of O(n 2 ) [15,20]. We also define and construct the useful region of the plane: the region that a highway must intersect in order that the shortest path between at least one pair of points uses the highway.