Average-Case Complexity of Shellsort

We prove a general lower bound on the average-case complexity of Shellsort: the average number of data-movements (and comparisons) made by a $p$-pass Shellsort for any incremental sequence is \Omega (pn^{1 + 1/p) for all $p \leq \log n$. Using similar arguments, we analyze the average-case complexity of several other sorting algorithms.Comment: 11 pages. Submitted to ICALP'9

Zig-zag Sort: A Simple Deterministic Data-Oblivious Sorting Algorithm Running in O(n log n) Time

We describe and analyze Zig-zag Sort--a deterministic data-oblivious sorting algorithm running in O(n log n) time that is arguably simpler than previously known algorithms with similar properties, which are based on the AKS sorting network. Because it is data-oblivious and deterministic, Zig-zag Sort can be implemented as a simple O(n log n)-size sorting network, thereby providing a solution to an open problem posed by Incerpi and Sedgewick in 1985. In addition, Zig-zag Sort is a variant of Shellsort, and is, in fact, the first deterministic Shellsort variant running in O(n log n) time. The existence of such an algorithm was posed as an open problem by Plaxton et al. in 1992 and also by Sedgewick in 1996. More relevant for today, however, is the fact that the existence of a simple data-oblivious deterministic sorting algorithm running in O(n log n) time simplifies the inner-loop computation in several proposed oblivious-RAM simulation methods (which utilize AKS sorting networks), and this, in turn, implies simplified mechanisms for privacy-preserving data outsourcing in several cloud computing applications. We provide both constructive and non-constructive implementations of Zig-zag Sort, based on the existence of a circuit known as an epsilon-halver, such that the constant factors in our constructive implementations are orders of magnitude smaller than those for constructive variants of the AKS sorting network, which are also based on the use of epsilon-halvers.Comment: Appearing in ACM Symp. on Theory of Computing (STOC) 201

The Frobenius Problem in a Free Monoid

Given positive integers c1,c2,...,ck with gcd(c1,c2,...,ck) = 1, the Frobenius problem (FP) is to compute the largest integer g(c1,c2,...,ck) that cannot be written as a non-negative integer linear combination of c1,c2,...,ck. The Frobenius problem in a free monoid (FPFM) is a non-commutative generalization of the Frobenius problem. Given words x1,x2,...,xk such that there are only finitely many words that cannot be written as concatenations of words in {x1,x2,...,xk}, the FPFM is to find the longest such words. Unlike the FP, where the upper bound g(c1,c2,...,ck)≤max 1≤i≤k ci2 is quadratic, the upper bound on the length of the longest words in the FPFM can be exponential in certain measures and some of the exponential upper bounds are tight. For the 2FPFM, where the given words over Σ are of only two distinct lengths m and n with 1<m<n, the length of the longest omitted words is ≤g(m, m|Σ|n-m + n - m). In Chapter 1, I give the definition of the FP in integers and summarize some of the interesting properties of the FP. In Chapter 2, I give the definition of the FPFM and discuss some general properties of the FPFM. Then I mainly focus on the 2FPFM. I discuss the 2FPFM from different points of view and present two equivalent problems, one of which is about combinatorics on words and the other is about the word graph. In Chapter 3, I discuss some variations on the FPFM and related problems, including input in other forms, bases with constant size, the case of infinite words, the case of concatenation with overlap, and the generalization of the local postage-stamp problem in a free monoid. In Chapter 4, I present the construction of some essential examples to complement the theory of the 2FPFM discussed in Chapter 2. The theory and examples of the 2FPFM are the main contribution of the thesis. In Chapter 5, I discuss the algorithms for and computational complexity of the FPFM and related problems. In the last chapter, I summarize the main results and list some open problems. Part of my work in the thesis has appeared in the papers