## Selection, Revisited

### November 30, 2012

We studied a selection algorithm in a previous exercise. The algorithm is the same as quicksort, except that recursion follows only the partition in which the target of the selection is located. We used a randomized algorithm to select the pivot, which gives an expected O(n) time complexity but a worst-case O(n^2) time complexity. In today’s exercise we examine an algorithm due to Blum, Floyd, Pratt, Rivest and Tarjan from their 1973 paper “Time Bounds for Selection” in Journal of Computer Systems Science that provides guaranteed O(n) time complexity (actually, the time complexity is sub-linear, but the only claim is that it is linear).

The algorithm is the same as the previous exercise except in the selection of the pivot. Instead of a random pivot, the algorithm partitions the input into blocks of five elements, finds the median of each block by comparisons, then chooses the median of the medians (which is computed recursively) as the pivot. Thus, the algorithm is known as “selection by the median of the medians of five.”

Your task is to write a function that selects the kth item from a list. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.