Fun-Sort-or the chaos of unordered binary search

Usually, binary search only makes sense in sorted arrays. We show that insertion sort based on repeated "binary searches" in an initially unsorted array also sorts n elements in time θ(n²log n). If n is a power of two, then the expected termination point of a binary search in a random permutation of n elements is exactly the cell where the element should be if the array was sorted. We further show that we can sort in expected time θ(n²log n) by always picking two random cells and swapping their contents if they are not ordered correctly.