Extending Search Phases in the Micali-Vazirani Algorithm

The Micali-Vazirani algorithm is an augmenting path algorithm that offers the best theoretical runtime of O(n^{0.5} m) for solving the maximum cardinality matching problem for non-bipartite graphs. This paper builds upon the algorithm by focusing on the bottleneck caused by its search phase structure and proposes a new implementation that improves efficiency by extending the search phases in order to find more augmenting paths. Experiments on different types of randomly generated and real world graphs demonstrate this new implementation\u27s effectiveness and limitations

Parallel Graph Connectivity in Log Diameter Rounds

We study graph connectivity problem in MPC model. On an undirected graph with $n$ nodes and $m$ edges, $O(\log n)$ round connectivity algorithms have been known for over 35 years. However, no algorithms with better complexity bounds were known. In this work, we give fully scalable, faster algorithms for the connectivity problem, by parameterizing the time complexity as a function of the diameter of the graph. Our main result is a $O(\log D \log\log_{m/n} n)$ time connectivity algorithm for diameter-$D$ graphs, using $\Theta(m)$ total memory. If our algorithm can use more memory, it can terminate in fewer rounds, and there is no lower bound on the memory per processor. We extend our results to related graph problems such as spanning forest, finding a DFS sequence, exact/approximate minimum spanning forest, and bottleneck spanning forest. We also show that achieving similar bounds for reachability in directed graphs would imply faster boolean matrix multiplication algorithms. We introduce several new algorithmic ideas. We describe a general technique called double exponential speed problem size reduction which roughly means that if we can use total memory $N$ to reduce a problem from size $n$ to $n/k$, for $k=(N/n)^{\Theta(1)}$ in one phase, then we can solve the problem in $O(\log\log_{N/n} n)$ phases. In order to achieve this fast reduction for graph connectivity, we use a multistep algorithm. One key step is a carefully constructed truncated broadcasting scheme where each node broadcasts neighbor sets to its neighbors in a way that limits the size of the resulting neighbor sets. Another key step is random leader contraction, where we choose a smaller set of leaders than many previous works do

A 2-3/4-Approximation Algorithm for the Shortest Superstring Problem

Given a collection of strings S={s_1,...,s_n} over an alphabet Sigma, a superstring alpha of S is a string containing each s_i as a substring, that is, for each i, 1\u3c=i\u3c=n, alpha contains a block of |s_i| consecutive characters that match s_i exactly. The shortest superstring problem is the problem of finding a superstring alpha of minimum length. The shortest superstring problem has applications in both computational biology and data compression. The problem is NP-hard [GallantMS80]; in fact, it was recently shown to be MAX SNP-hard [BlumJLTY91]. Given the importance of the applications, several heuristics and approximation algorithms have been proposed. Constant factor approximation algorithms have been given in [BlumJLTY91] (factor of 3), [TengY93] (factor of 2-8/9), [CzumajGPR94] (factor of 2-5/6) and [KosarajuPS94] (factor of 2-50/63). Informally, the key to any algorithm for the shortest superstring problem is to identify sets of strings with large amounts of similarity, or overlap. While the previous algorithms and their analyses have grown increasingly sophisticated, they reveal remarkably little about the structure of strings with large amounts of overlap. In this sense, they are solving a more general problem than the one at hand. In this paper, we study the structure of strings with large amounts of overlap and use our understanding to give an algorithm that finds a superstring whose length is no more than 2-3/4 times that of the optimal superstring. We prove several interesting properties about short periodic strings, allowing us to answer questions of the following form: given a string with some periodic structure, characterize all the possible periodic strings that can have a large amount of overlap with the first string

Distributed Scheduling in Finite Capacity Networks

We consider the problem of scheduling unit-sized jobs in a distributed network of processors. Each processor only knows the number of jobs it and its neighbors have. We give an analysis of intuitive algorithm and prove that the algorithm produces schedules that are within a logarithmic factor of the length of the optimal schedule given that the optimal schedule is sufficiently long

A 2-2/3 Approximation for the Shortest Superstring Problem

Given a collection of strings S={s_1, ..., s_n} over an alphabet \Sigma, a superstring \alpha of S is a string containing each s_i as a substring; that is, for each i, 1\u3c=i\u3c=n, \alpha contains a block of |s_i| consecutive characters that match s_i exactly. The shortest superstring problem is the problem of finding a superstring \alpha of minimum length. The shortest superstring problem has applications in both data compression and computational biology. In data compression, the problem is a part of a general model of string compression proposed by Gallant, Maier and Storer (JCSS \u2780). Much of the recent interest in the problem is due to its application to DNA sequence assembly. The problem has been shown to be NP-hard; in fact, it was shown by Blum et al.(JACM \u2794) to be MAX SNP-hard. The first O(1)-approximation was also due to Blum et al., who gave an algorithm that always returns a superstring no more than 3 times the length of an optimal solution. Several researchers have published results that improve on the approximation ratio; of these, the best previous result is our algorithm ShortString, which achieves a 2 3/4-approximation (WADS \u2795). We present our new algorithm, G-ShortString, which achieves a ratio of 2 2/3. It generalizes the ShortString algorithm, but the analysis differs substantially from that of ShortString. Our previous work identified classes of strings that have a nested periodic structure, and which must be present in the worst case for our algorithms. We introduced machinery to descibe these strings and proved strong structural properties about them. In this paper we extend this study to strings that exhibit a more relaxed form of the same structure, and we use this understanding to obtain our improved result

Scheduling in a Ring with Unit Capacity Links

We consider the problem of scheduling unit-sized jobs on a ring of processors with the objective of minimizing the completion time of the last job. Unlike much previous work we place restrictions on the capacity of the network links connecting processors. We give a polynomial time centralized algorithm that produces optimal length schedules. We also give a simple distributed 2-approximation algorithm