The Frequency Median Sequence and its Construction

A graph model based on single character operations is developed for constructing the median of two strings X and Y. Vertices represent stages of median construction and edges correspond to processing one or both characters of the two strings. In this paper, the median is defined as a string having the median character frequency of each character over the two strings and which minimizes the maximum distance to either string. While the graph has O(mn) vertices (product of the lengths of the two strings), an O(d∙p∙lg(p)) algorithm is presented for finding a median of length p where d = DE(X, Y) is the Levenstein or edit distance between X and Y. The algorithm uses a character frequency based bound and a priority queue to find a median.