Selection and Sorting in the "Restore" Model

We consider the classical selection and sorting problems in a model where the initial permutation of the input has to be restored after completing the computation. While the requirement of the restoration is stringent compared to the classical versions of the problems, this model is more relaxed than a read-only memory where the input elements are not allowed to be moved within the input array. We first show that for a sequence of n integers, selection (finding the median or more generally the k -th smallest element for a given k ) can be done in O ( n ) time using O (lg n ) words 1 of extra space in this model. In contrast, no linear-time selection algorithm is known which uses polylogarithmic space in the read-only memory model. For sorting n integers in this model, we first present an O ( n lg n )-time algorithm using O (lg n ) words of extra space. When the universe size U is polynomial in n, we give a faster O ( n )-time algorithm (analogous to radix sort) which uses O ( n ∊ ) words of extra space for an arbitrarily small constant ∊ > 0. More generally, we show how to match the time bound of any word-RAM integer-sorting algorithms using O ( n ∊ ) words of extra space. In sharp contrast, there is an Ω( n 2 / S )-time lower bound for integer sorting using O ( S ) bits of space in the read-only memory model. Extension of our results to arbitrary input types beyond integers is not possible: for “indivisible” input elements, we can prove the same Ω( n 2 / S ) lower bound for sorting in our model. En route, we develop linear-time in-place algorithms to extract leading bits of the input array and to compress and decompress strings with low entropy; these techniques may be of independent interest.

Authors

• Timothy M. Chan
• J. Ian Munro
• Venkatesh Raman 0001

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