(Prefix) reversal distance for (signed) strings with few blocks or small alphabets

We study the String Reversal Distance problem, an extension of the well-known Sorting by Reversals problem. String Reversal Distance takes two strings S and T built on an alphabet Σ as input, and asks for a minimum number of reversals to obtain T from S. We consider four variants: String Reversal Distance, String Prefix Reversal Distance (a constrained version of the previous problem, in which any reversal must include the first letter of the string), and the signed variants of these problems, namely Signed String Reversal Distance and Signed String Prefix Reversal Distance. We study algorithmic properties of these four problems, in connection with two parameters of the input strings: the number of blocks they contain (a block being a maximal substring such that all letters in the substring are equal), and the alphabet size $|\mathrm{\Sigma }|$. Concerning the number of blocks, we show that the four problems are fixed-parameter tractable (FPT) when the considered parameter is the maximum number of blocks among the two input strings. Concerning the alphabet size, we first show that String Reversal Distance and String Prefix Reversal Distance are NP-hard even if the input strings are built on a binary alphabet $\mathrm{\Sigma }=\{0,1\}$, each 0-block has length at most two and each 1-block has length one. We also show that Signed String Reversal Distance and Signed String Prefix Reversal Distance are NP-hard even if the input strings have only one letter. Finally, when $|\mathrm{\Sigma }|=O(1)$, we provide a singly-exponential algorithm that computes the exact distance between any pair of strings, for a large family of distances that we call well-formed, which includes the four distances we study here.     

Analysis of particle methods for structured population models with nonlocal boundary term in the framework of bounded Lipschitz distance

Recently developed theoretical framework for analysis of structured population dynamics in the spaces of nonnegative Radon measures with a suitable metric provides a rigorous tool to study numerical schemes based on particle methods. The approach is based on the idea of tracing growth and transport of measures which approximate the solution of original partial differential equation. In this article, we present analytical and numerical study of two versions of Escalator Boxcar Train algorithm which has been widely applied in theoretical biology, and compare it to the recently developed split‐up algorithm. The novelty of this article is in showing well‐posedness and convergence rates of the schemes using the concept of semiflows on metric spaces. Theoretical results are validated by numerical simulations of test cases, in which distances between simulated and exact solutions are computed using flat metric. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1797–1820, 2014