Bounds and Estimates on the Average Edit Distance

The edit distance is a metric of dissimilarity between strings, widely applied in computational biology, speech recognition, and machine learning. Let () denote the average edit distance between random, independent strings of n characters from an alphabet of a given size k. An open problem is the exact value of ()=()/. While it is known that, for increasing n, () approaches a limit , the exact value of this limit is unknown, for any 652. This paper presents an upper bound to based on the exact computation of some () and a lower bound to based on combinatorial arguments on edit scripts. Statistical estimates of () are also obtained, with analysis of error and of confidence intervals. The techniques are applied to several alphabet sizes k. In particular, for a binary alphabet, the rigorous bounds are 0.1742 642 640.3693 while the obtained estimate is 2 480.2888; for a quaternary alphabet, 0.3598 644 640.6318 and 4 480.5180. These values are more accurate than those previously published

On H-Topological Intersection Graphs

Biro, Hujter, and Tuza introduced the concept of H-graphs (1992), intersection graphs of connected subgraphs of a subdivision of a graph H. They naturally generalize many important classes of graphs, e.g., interval graphs and circular-arc graphs. Our paper is the first study of the recognition and dominating set problems of this large collection of intersection classes of graphs.We negatively answer the question of Biro, Hujter, and Tuza who asked whether H-graphs can be recognized in polynomial time, for a fixed graph H. Namely, we show that recognizing H-graphs is NP-complete if H contains the diamond graph as a minor. On the other hand, for each tree T, we give a polynomial-time algorithm for recognizing T-graphs and an O(n(4))-time algorithm for recognizing K-1,K-d-graphs. For the dominating set problem (parameterized by the size of H), we give FPT- and XP-time algorithms on K-1, d-graphs and H-graphs, respectively. Our dominating set algorithm for H-graphs also provides XP-time algorithms for the independent set and independent dominating set problems on H-graphs