Optimal Time-Space Trade-Offs for Non-Comparison-Based Sorting Rasmus Pagh Jakob Pagter January 2001

### Abstract:

We study the fundamental problem of sorting integers of bits on a unit-cost RAM with word size , and in particular consider the time-space trade-off (product of time and space in bits) for this problem. For comparison-based algorithms, the time-space complexity is known to be . A result of Beame shows that the lower bound also holds for non-comparison-based algorithms, but no algorithm has met this for time below the comparison-based lower bound.

We show that if sorting within some time bound is possible, then time can be achieved with high probability using space , which is optimal. Given a deterministic priority queue using amortized time per operation and space , we provide a deterministic algorithm sorting in time with . Both results require that . Using existing priority queues and sorting algorithms, this implies that we can deterministically sort time-space optimally in time for , and with high probability for .

Our results imply that recent lower bounds for deciding element distinctness in time are nearly tight

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