We present space-efficient algorithms for computing the convex hull of a simple polygonal line in-place, in linear time. It turns out that the problem is as hard as stable partition, i.e., if there were a truly simple solution then stable partition would also have a truly simple solution, and vice versa. Nevertheless, we present a simple self-contained solution that uses O(log n) space, and indicate how to improve it to O(1) space with the same techniques used for stable partition. If the points inside the convex hull can be discarded, then there is a truly simple solution that uses a single call to stable partition, and even that call can be spared if only extreme points are desired (and not their order). If the polygonal line is closed, then the problem admits a very simple solution which does not call for stable partitioning at all.

We present a space-efficient algorithm for reporting all k intersections induced by a set of n line segments in the plane. Our algorithm is an in-place variant of Balaban’s algorithm and, in the worst case, runs in O(n log2 n+k) time using O(1) extra words of memory in addition to the space used for the input to the algorithm.

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. 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

It is shown that an array of n elements can be sorted using O(1) extra space, O(n log n= log log n) element moves, and n log 2 n+O(n log log n) comparisons. This is the first in-place sorting algorithm requiring o(n log n) moves in the worst case while guaranteeing O(n log n) comparisons but, due to the constant factors involved, the algorithm is predominantly of theoretical interest. Key words: In-place algorithms, sorting, merging, mergesort, multiway merge 1

Abstract. We present an algorithm for asymptotically efficient multiway blockwise in-place merging. Given an array A containing sorted subsequences A1,..., Ak of respective lengths n1,..., nk, where ∑k ni = n, we assume i=1 that extra k·s elements (so called buffer elements) are positioned at the very end of array A, and that the lengths n1,..., nk are positive integer multiples of some parameter s (i.e., multiples of a given block of length s). The number of input sequences k is a fixed constant parameter, not dependent on the lengths of input sequences. Then our algorithm merges the subsequences A1,..., Ak into a single sorted sequence, performing Θ(log k·n) + O((n/s) 2) + O(s· log s) element comparisons and 3·n + O(s·log s) element moves. 1 Then, for s = ⌈n 2/3 /(log n) 1/3 ⌉, this gives an algorithm performing Θ(log k·n) + O((n·log n) 2/3) comparisons and 3·n + O((n·log n) 2/3) moves. That is, our algorithm runs in linear time, with an asymptotically optimal number of comparisons and with the number of moves independent on the number of input sequences. Moreover, our algorithm is “almost in-place”, it requires only k extra blocks of size s = o(n). 1