On demand string sorting over unbounded alphabets

On-demand string sorting is the problem of preprocessing a set of n strings to allow subsequent queries of finding the k < n lexicographically smallest strings (and afterwards the next k etc.) This on-demand variant strongly resembles the search engine queries which give you the best k-ranked pages recurringly. We present a data structure that supports this in O(n) preprocessing time, and answers queries in O(log n) time. There is also a cost of O(N) time amortized over all operations, where N is the total length of the strings. Our data structure is a heap of strings, which supports heapify and delete-mins. As it turns out, implementing a full heap with all operations is not that simple. For the sake of completeness we propose a heap with full operations based on balanced indexing trees that supports the heap operations in optimal times.