Abstract. We describe space-efficient algorithms for solving problems related to finding maxima among points in two and three dimensions. Our algorithms run in optimal O(n log n) time and occupy only constant extra space in addition to the space needed for representing the input. 1

Abstract We revisit a classic problem in computational geometry: preprocessing a planar n-point set to answer nearest neighbor queries. In SoCG 2004, Br"onnimann, Chan, and Chen showed that it is possible to design an efficient data structure that takes no extra space at all other than the input array holding a permutation of the points. The best query time known for such "in-place data structures " is O(log 2 n). In this paper, we break the O(log 2 n) barrier by providing a method that answers nearest neighbor queries in time O((log n) log3=2 2 log log n) = O(log

Abstract. Slope selection is a well-known algorithmic tool used in the context of computing robust estimators for fitting a line to a collection P of n points in the plane. We demonstrate that it is possible to perform slope selection in expected O(n log n) time using only constant extra space in addition to the space needed for representing the input. Our solution is based upon a space-efficient variant of Matouˇsek’s randomized interpolation search, and we believe that the techniques developed in this paper will prove helpful in the design of space-efficient randomized algorithms using samples. To underline this, we also sketch how to compute the repeated median line estimator in an in-place setting. 1