Characterization of Randomly k-Dimensional Graphs

For an ordered set $W=\{w_1,w_2,...,w_k\}$ of vertices and a vertex $v$ in a connected graph $G$, the ordered $k$-vector $r(v|W):=(d(v,w_1),d(v,w_2),.,d(v,w_k))$ is called the (metric) representation of $v$ with respect to $W$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. The set $W$ is called a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$. A minimum resolving set for $G$ is a basis of $G$ and its cardinality is the metric dimension of $G$. The resolving number of a connected graph $G$ is the minimum $k$, such that every $k$-set of vertices of $G$ is a resolving set. A connected graph $G$ is called randomly $k$-dimensional if each $k$-set of vertices of $G$ is a basis. In this paper, along with some properties of randomly $k$-dimensional graphs, we prove that a connected graph $G$ with at least two vertices is randomly $k$-dimensional if and only if $G$ is complete graph $K_{k+1}$ or an odd cycle.Comment: 12 pages, 3 figure

Advances in Engineering Software for Multicore Systems

The vast amounts of data to be processed by today’s applications demand higher computational power. To meet application requirements and achieve reasonable application performance, it becomes increasingly profitable, or even necessary, to exploit any available hardware parallelism. For both new and legacy applications, successful parallelization is often subject to high cost and price. This chapter proposes a set of methods that employ an optimistic semi-automatic approach, which enables programmers to exploit parallelism on modern hardware architectures. It provides a set of methods, including an LLVM-based tool, to help programmers identify the most promising parallelization targets and understand the key types of parallelism. The approach reduces the manual effort needed for parallelization. A contribution of this work is an efficient profiling method to determine the control and data dependences for performing parallelism discovery or other types of code analysis. Another contribution is a method for detecting code sections where parallel design patterns might be applicable and suggesting relevant code transformations. Our approach efficiently reports detailed runtime data dependences. It accurately identifies opportunities for parallelism and the appropriate type of parallelism to use as task-based or loop-based