CS 157 : Assignment 3

In this write-up, we extend quicksort to the task of fuzzy sorting of intervals. In many situations the precise value of a quantity is uncertain (e.g., any physical measurement is subject to noise). For such situations we may represent a measurement i as a closed interval [ai, bi], where ai ≤ bi. A fuzzysort is defined as a permutation 〈i1, i2, . . . , in〉 of the intervals such that there exist cj ∈ [aij , bij ] satisfying c1 ≤ c2 ≤ . . . ≤ cn. This document summarizes the design of a fuzzy sorting algorithm which operates in-place on two arrays containing the interval endpoints. The proposed algorithm has the general structure of randomized quicksort. As a result, its expected running time is Θ(n lg n) for disjoint intervals. The presence of overlapping intervals is detected and reduces the expected running time. In fact, the proposed algorithm has an expected running time Θ(n) when all intervals overlap at some point. This document is organized in four sections. In Section 1 we introduce the basic strategy for fuzzy-sorting and a linear-time algorithm to detect the intersection of a set of intervals. In Section 2 we modify randomized quicksort to allow fuzzy sorting. In Section 3 we present a proof of correctness. Finally, in Section 4 we analyze the expected running time of the proposed algorithm.