Sorting and searching revisted

Abstract

We discuss the complexity of sorting and searching from the following point of view: The n keys are binary strings (or integers) of length w and the employed computer has word size w. In this natural setting, and assuming that a traditional instruction set is available, a comparison-based algorithm is not the obvious choice. Indeed, the comparison-based algorithms are suboptimal in this model.

Rather than going into technical details, this invited talk aims at presenting some basic ideas. For more details, we give references to the literature. The following will be discussed:

• 1. Deterministic data structures and algorithms that use superlinear space

• tries are used to decrease key length;

• the resulting short keys are handled efficiently by packing them tightly. With these methods, the worst-case cost of sorting is O(n log log n) and the worst-case cost of searching is O(V√logn). (Tries use super-linear space, but with randomization, i.e. hash coding, the space can be reduced to linear).

• 2. Deterministic data structures and algorithms that use linear space

• the fusion tree; with this data structure the worst-case cost of sorting is O(n log n/log log n) and the amortized cost of searching is O (log n/log log n).

• a new data structure, the exponential search tree; the bounds on deterministic sorting and searching in linear space are improved to O(n√logn) and O(√log n), respectively.