Approximating the Minimum Equivalent Digraph

The MEG (minimum equivalent graph) problem is, given a directed graph, to find a small subset of the edges that maintains all reachability relations between nodes. The problem is NP-hard. This paper gives an approximation algorithm with performance guarantee of pi^2/6 ~ 1.64. The algorithm and its analysis are based on the simple idea of contracting long cycles. (This result is strengthened slightly in ``On strongly connected digraphs with bounded cycle length'' (1996).) The analysis applies directly to 2-Exchange, a simple ``local improvement'' algorithm, showing that its performance guarantee is 1.75.Comment: conference version in ACM-SIAM Symposium on Discrete Algorithms (1994

Super cyclically edge connected graphs with two orbits of the same size

对于图$G$,如果$G -; F$是不连通的且至少有两个分支含有圈,则称$F$为图$G$的圈边割.如果图$G$有圈边割,则称其为圈可分的.最小圈边割的基数叫作圈边连通度.如果; 去除任何一个最小圈边割,总存在一分支为最小圈,则图$G$为超圈边连通的.设$G = \left( {{G_1},{G_2},\left(; {{V_1},{V_2}} \right)} \right)$为双轨道图,最小度$\delta \left( G \right) \ge; 4$,围长$g\left( G \right) \ge 6$且$\left| {{V_1}} \right| = \left| {{V_2}}; \right|$.假设${G_i}$是${k_i}$-正则的,${k_1} \le; {k_2}$且${{G_1}}$包含一个长度为$g$的圈,则$G$是超圈边连通的.For a graph $G$, an edge set $F$ is a cyclic edge-cut if ($G - F$) is; disconnected and at least two of its components contain cycles. If $G$; has a cyclic edge-cut, it is said to be cyclically separable. The cyclic; edge-connectivity is cardinality of a minimum cyclic edgecut of $G$. A; graph is super cyclically edge-connected if removal of any minimum; cyclic edge-cut makes a component a shortest cycle. Let $G = \left(; {{G_1},{G_2},\left( {{V_1},{V_2}} \right)} \right)$ be a doubleorbit; graph with minimum degree $\delta \left( G \right) \ge 4$, girth $g \ge; 6$ and $\left| {{V_1}} \right| = \left| {{V_2}} \right|$. Suppose; ${G_i}$ is ${k_i}$-regular, ${k_1} \le {k_2}$ and ${{G_1}}$ contains a; cycle of length $g$, then $G$ is super cyclically edge connected.国家自然科学基金资助项

Aspects of practical implementations of PRAM algorithms

The PRAM is a shared memory model of parallel computation which abstracts away from inessential engineering details. It provides a very simple architecture independent model and provides a good programming environment. Theoreticians of the computer science community have proved that it is possible to emulate the theoretical PRAM model using current technology. Solutions have been found for effectively interconnecting processing elements, for routing data on these networks and for distributing the data among memory modules without hotspots. This thesis reviews this emulation and the possibilities it provides for large scale general purpose parallel computation. The emulation employs a bridging model which acts as an interface between the actual hardware and the PRAM model. We review the evidence that such a scheme crn achieve scalable parallel performance and portable parallel software and that PRAM algorithms can be optimally implemented on such practical models. In the course of this review we presented the following new results: 1. Concerning parallel approximation algorithms, we describe an NC algorithm for finding an approximation to a minimum weight perfect matching in a complete weighted graph. The algorithm is conceptually very simple and it is also the first NC-approximation algorithm for the task with a sub-linear performance ratio. 2. Concerning graph embedding, we describe dense edge-disjoint embeddings of the complete binary tree with n leaves in the following n-node communication networks: the hypercube, the de Bruijn and shuffle-exchange networks and the 2-dimcnsional mesh. In the embeddings the maximum distance from a leaf to the root of the tree is asymptotically optimally short. The embeddings facilitate efficient implementation of many PRAM algorithms on networks employing these graphs as interconnection networks. 3. Concerning bulk synchronous algorithmics, we describe scalable transportable algorithms for the following three commonly required types of computation; balanced tree computations. Fast Fourier Transforms and matrix multiplications