Average Time Algorithm Research Articles

Computing Nonoverlapping Inversion Distance Between Two Strings in Linear Average Time.

Biological events like inversions are not automatically detected by the usual alignment algorithms. Alignment with inversions does not have a known polynomial time algorithm and Schöniger and Waterman introduced a simplification of the alignment problem with nonoverlapping inversions, where all regions will not be allowed to overlap. They presented an \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \cal O} ( {n^6} )$$ \end{document} algorithm to compute nonoverlapping inversion distance between two strings of length n. The time and space complexities were improved to \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \cal O} ( {n^3} )$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \cal O} ( {n^2} )$$ \end{document} later by Cho, Vellozo, and Ta. In this article, a linear space and linear average time algorithm to compute the inversion distance between two strings of the same length is presented. The recursive formula for this purpose is new to the best of our knowledge. The space costs of the algorithms to solve the same problem are quadratic in the literature, and thus our original algorithm is the first linear space and linear average time algorithm to solve the inversion distance problem.

A Simple Linear Space Algorithm for Computing Nonoverlapping Inversion and Transposition Distance in Quadratic Average Time.

In the sequence alignment problem, it is important to compare DNA sequences to retrieve relevant information and align these sequences. An inversion and a translocation are important operations in comparing DNA sequences in biosequence analysis. The alignment problem with nonoverlapping inversions and translocations is to find an alignment with nonoverlapping inversions and translocations for the given two strings X and Y. This problem has interesting application for finding a common sequence from two mutated sequences. A linear space and quadratic average time algorithm to compute the mutation distance between two strings of the same length under nonoverlapping inversions and transpositions is presented in this article. The recursive formula for this purpose is novel to the best of our knowledge. The space costs of the algorithms to solve the same problem are typically quadratic, and thus, our original algorithm is the first linear space algorithm to solve the mutation distance problem.

Generating permutations with given ups and downs

A permutation with a signature Q=( q 1, q 2,…, q n−1 ) where q i is either 1 or −1, is a permutation, P= π 1, π 2,…, π n , of the integers 1 to n such that q i ( π i+1 − π i ) is positive for all 1⩽ i⩽ n−1. Alternating permutations are an example of permutations with the signature (1,−1,1,−1,…). A constant average time algorithm is developed for generating all permutations with a given signature. Permutations are represented by a variant of their inversion tables. Ranking and unranking algorithms are also discussed.

Generating the Linear Extensions of Certain Posets by Transpositions

For poset $\mathcal{P}$ define the graph $G( \mathcal{P} )$ whose vertices are the linear extensions of $\mathcal{P}$ and where two vertices are connected by an edge if the corresponding linear extensions differ by a transposition. Let $\mathcal{P}$ be a poset and M a subset of its minimal elements for which $G( \mathcal{P} - M )$ has a Hamilton path (cycle). If no element of $\mathcal{P} - M$ has exactly one descendant in M then $G( \mathcal{P} )$ also has a Hamilton path (cycle). Given only $\mathcal{P}$ and a constant average time algorithm for building a path (cycle) in $G( \mathcal{P} - M )$, a Hamilton path (cycle) can also be constructed in $G( \mathcal{P} )$ in constant average time.As an application of the results stated above it is proved that the linear extensions of any ranked poser in which every nonmaximal element has at least two (upper) covers can be generated by transpositions in constant average time.

Generating binary trees by transpositions

Let T( n) denote the set of all bitstrings with n 1's and n 0's that satisfy the property that in every prefix the number of 0's does not exceed the number of 1's. This is a well known representation of binary trees. We consider algorithms for generating the elements of T( n) that satisfy one of the following constraints: (a) successive bitstrings differ by the transposition of two bits or (b) successive bitstrings differ by the transposition of two adjacent bits. In case (a) a constant average time generation algorithm is presented. In case (b) we show that such generation is possible if and only if n is even or n < 5. A constant average time algorithm is presented in this case as well.

Generating alternating permutations lexicographically

A permutation π1 π2 ... π n is alternating if π1 π3<π4 .... We present a constant average-time algorithm for generating all alternating permutations in lexicographic order. Ranking and unranking algorithms are also derived.

Finding a Minimum Circuit in a Graph

Finding minimum circuits in graphs and digraphs is discussed. An almost minimum circuit is a circuit which may have only one edge more than the minimum. To find an almost minimum circuit an $O(n^2 )$ algorithm is presented. A direct algorithm for finding a minimum circuit has an $O(ne)$ behavior. It is refined to yield an $O(n^2 )$ average time algorithm. An alternative method is to reduce the problem of finding a minimum circuit to that of finding a triangle in an auxiliary graph. Three methods for finding a triangle in a graph are given. The first has an $O(e^{3/2})$ worst case bound ($O(n)$ for planar graphs); the second takes $O(n^{5/3})$ time on the average; the third has an $O(n^{\log 7} )$ worst case behavior. For digraphs, results of Bloniarz, Fisher and Meyer are used to obtain an algorithm with $O(n^2 \log n)$ average behavior.