Selection from read-only memory and sorting with minimum data movement

Abstract

Selecting an element of given rank, for example the median, is a fundamental problem in data organization and the computational complexity of comparison based problems. Here, we consider the scenario in which the data resides in an array of read-only memory and hence the elements cannot be moved within the array. Under this model, we develop efficient selection algorithms using very little extra space ( o( log n) extra storage cells). These include an O( n 1 + ε ) worst case algorithm and an O( n log log n) average case algorithm, both using a constant number of extra storage cells or indices. Our algorithms complement the upper bounds for the time-space tradeoffs obtained by Munro and Paterson [9] and Frederickson [4] who developed algorithms for selection in the same model when Ω((log n) 2) extra storage cells are available. We apply our selection algorithms to obtain sorting algorithms that perform the minimum number of data moves on any given array. We also derive upper bounds for time-space tradeoffs for sorting with minimum data movement.