Optimal time-space trade-offs for sorting

Abstract

We study the fundamental problem of sorting in a sequential model of computation and in particular consider the time-space trade-off (product of time and space) for this problem.Beame has shown a lower bound of ª´Ò ¾ µ for this product leaving a gap of a logarithmic factor up to the previously best known upper bound of Ç´Ò ¾ ÐÓ Òµ due to Frederickson. Since then, no progress has been made towards tightening this gap.The main contribution of this paper is a comparison based sorting algorithm which closes the gap by meeting the lower bound of Beame. The time-space product Ç´Ò ¾ µ upper bound holds for the full range of space bounds between ÐÓ Ò and Ò ÐÓ Ò. Hence in this range our algorithm is optimal for comparison based models as well as for the very powerful general models considered by Beame.

Space-Efficient Biconnected Components and Recognition of Outerplanar Graphs

“…As usual, we assume that for a given instances of size n, the word size is Ω(log n). One of the first problems considered for space-efficient algorithms is sorting [17], which was finally solved to optimality [6,18]. Other researchers considered problems in geometry [1,3,5].…”

Multi-Pass Geometric Algorithms

“…Read-only algorithms have been previously considered for the sorting problem [5], [33] but are apparently less often studied. The main advantage is that if the user needs the input to be left in its original state, we do not need to duplicate the input first before sending it to the algorithm.…”

Selection from read-only memory with limited workspace

“…In contrast to sorting, the exact complexity of selection is still open. The time-space trade-off for sorting is known to be Θ(N 2 /S +N lg S) [2,19], where S is the size of the workspace in bits, lg N ≤ S ≤ N/ lg N . The optimal bound for sorting can even be realized using a natural priority-queue-based algorithm [1].…”