Beyond Bidimensionality: Parameterized Subexponential Algorithms on Directed Graphs

We develop two different methods to achieve subexponential time parameterized algorithms for problems on sparse directed graphs. We exemplify our approaches with two well studied problems. For the first problem, {\sc $k$-Leaf Out-Branching}, which is to find an oriented spanning tree with at least $k$ leaves, we obtain an algorithm solving the problem in time $2^{O(\sqrt{k} \log k)} n+ n^{O(1)}$ on directed graphs whose underlying undirected graph excludes some fixed graph $H$ as a minor. For the special case when the input directed graph is planar, the running time can be improved to $2^{O(\sqrt{k})}n + n^{O(1)}$. The second example is a generalization of the {\sc Directed Hamiltonian Path} problem, namely {\sc $k$-Internal Out-Branching}, which is to find an oriented spanning tree with at least $k$ internal vertices. We obtain an algorithm solving the problem in time $2^{O(\sqrt{k} \log k)} + n^{O(1)}$ on directed graphs whose underlying undirected graph excludes some fixed apex graph $H$ as a minor. Finally, we observe that for any $\epsilon>0$, the {\sc $k$-Directed Path} problem is solvable in time $O((1+\epsilon)^k n^{f(\epsilon)})$, where $f$ is some function of \ve. Our methods are based on non-trivial combinations of obstruction theorems for undirected graphs, kernelization, problem specific combinatorial structures and a layering technique similar to the one employed by Baker to obtain PTAS for planar graphs

Memory and I/O optimized rectilinear steiner minimum tree routing for VLSI

As the size of devices are scaling down at rapid pace, the interconnect delay play a major part in performance of IC chips. Therefore minimizing delay and wire length is the most desired objective. FLUTE (Fast Look-Up table) presented a fast and accurate RSMT (Rectilinear Steiner Minimum Tree) construction for both smaller and higher degree net. FLUTE presented an optimization technique that reduces time complexity for RSMT construction for both smaller and larger degree nets. However for larger degree net this technique induces memory overhead, as it does not consider the memory requirement in constructing RSMT. Since availability of memory is very less and is expensive, it is desired to utilize memory more efficiently which in turn results in reducing I/O time (i.e. reduce the number of I/O disk access). The proposed work presents a Memory Optimized RSMT (MORSMT) construction in order to address the memory overhead for larger degree net. The depth-first search and divide and conquer approach is adopted to build a Memory optimized tree. Experiments are conducted to evaluate the performance of proposed approach over existing model for varied benchmarks in terms of computation time, memory overhead and wire length. The experimental results show that the proposed model is scalable and efficient

A Novel Analysis of Clustering for Minimum Spanning Tree using Divide

Because of their capability to distinguish groups with sporadic limits, least spanning treebased grouping calculations have been generally utilized within practice. Be that as it may, in such bunching calculations, the quest for closest neighbour in the development of least spanning trees is the primary wellspring of processing and the standard results take O(N 2) time. In this paper, we exhibit a quick least spanning tree-motivated grouping calculation, which, by utilizing a proficient execution of the cut and the cycle property of the least spanning trees, can have much preferable execution than O(N 2)

A Pre-Programming Approach to Algorithmic Thinking in High School Mathematics

Given the impact of computers and computing on almost every aspect of society, the ability to develop, analyze, and implement algorithms is gaining more focus. Algorithms are increasingly important in theoretical mathematics, in applications of mathematics, in computer science, as well as in many areas outside of mathematics. In high school, however, algorithms are usually restricted to computer science courses and as a result, the important relationship between mathematics and computer science is often overlooked (Henderson, 1997). The mathematical ideas behind the design, construction and analysis of algorithms, are important for students' mathematical education. In addition, exploring algorithms can help students see mathematics as a meaningful and creative subject. This study provides a review of the history of algorithms and algorithmic complexity, as well as a technical monograph that illustrates the mathematical aspects of algorithmic complexity in a form that is accessible to mathematics instructors at the high school level. The historical component of this study is broken down into two parts. The first part covers the history of algorithms with an emphasis on how the concept has evolved from 3000 BC through the Middle Ages to the present day. The second part focuses on the history of algorithmic complexity, dating back to the text of Ibn al-majdi, a fourteenth century Egyptian astronomer, through the 20th century. In particular, it highlights the contributions of a group of mathematicians including Alan Turing, Michael Rabin, Juris Hartmanis, Richard Stearns and Alan Cobham, whose work in computability theory and complexity measures was critical to the development of the field of algorithmic complexity. The technical monograph which follows describes how the complexity of an algorithm can be measured and analyzes different types of algorithms. It includes divide-and-conquer algorithms, search and sort algorithms, greedy algorithms, algorithms for matching, and geometric algorithms. The methods used to analyze the complexity of these algorithms is done without the use of a programming language in order to focus on the mathematical aspects of the algorithms, and to provide knowledge and skills of value that are independent of specific computers or programming languages. In addition, the study assesses the appropriateness of these topics for use by high school teachers by submitting it for independent review to a panel of experts. The panel, which consists of mathematics and computer science faculty in high school and colleges around the United States, found the material to be interesting and felt that using a pre-programming approach to teaching algorithmic complexity has a great deal of merit. There was some concern, however, that portions of the material may be too advanced for high school mathematics instructors. Additionally, they thought that the material would only appeal to the strongest students. As per the reviewers' suggestions, the monograph was revised to its current form

Constructive Heuristics for the Minimum Labelling Spanning Tree Problem: a preliminary comparison

This report studies constructive heuristics for the minimum labelling spanning tree (MLST) problem. The purpose is to find a spanning tree that uses edges that are as similar as possible. Given an undirected labeled connected graph (i.e., with a label or color for each edge), the minimum labeling spanning tree problem seeks a spanning tree whose edges have the smallest possible number of distinct labels. The model can represent many real-world problems in telecommunication networks, electric networks, and multimodal transportation networks, among others, and the problem has been shown to be NP-complete even for complete graphs. A primary heuristic, named the maximum vertex covering algorithm has been proposed. Several versions of this constructive heuristic have been proposed to improve its efficiency. Here we describe the problem, review the literature and compare some variants of this algorithm