A New Eecient Radix Sort

We present new improved algorithms for the sorting problem. The algorithms are not only eecient but also clear and simple. First, we introduce Forward Radix Sort which combines the advantages of traditional left-to-right and right-to-left radix sort in a simple manner. We argue that this algorithm will work very well in practice. Adding a preprocessing step, we obtain an algorithm with attractive theoretical properties. For example, n binary strings can be sorted in n log B n log n + 2 time, where B is the minimum number of bits that have to be inspected to distinguish the strings. This is an improvement over the previously best known result by Paige and Tarjan. The complexity may also be expressed in terms of H, the entropy of the input: n strings from a stationary er-godic process can be sorted in ? n log ? 1 H + 1 time, an improvement over the result recently presented by Chen and Reif. 1 Background A common idea in the design of eecient algorithms is to take the distribution of the input into account. Well known examples are interpolation search 9, 21, 22], trie structures 8, 10, 13], and bucketing algorithms 6, 7]. When describing the complexity of such algorithms, it is common to rely on the assumption that the input elements are independently and randomly drawn from a speciic distribution, for instance the uniform or normal distribution. A more general approach is to assume no particular distribution, but to express the complexity of the algorithm in terms of certain properties of the input. In this article, we study this more general approach in the context of sorting. Speciically, we study the problem of arranging binary strings in lexicographic order. A new approach to this problem was recently presented by Chen and Reif 4]. They introduced a randomized sorting algorithm for binary strings from a stationary ergodic process (see Section 3.2) with en-tropy H, yielding an expected cost n log log n H + 2 : This was claimed to be the rst algorithm to adapt to an unknown input distribution. We take a more general approach based on distinguishing preexes. Let B denote the total length of all distinguishing preexes, i.e. the minimum number of bits that have to be examined in order to sort the strings. Let B = B=n. For a stationary ergodic process satisfying a certain mixing condition (see Section …