On-Line and Dynamic Shortest Paths Through Graph Decompositions

We describe algorithms for finding shortest paths and distances in a planar digraph which exploit the particular topology of the input graph. An important feature of our algorithms is that they can work in a dynamic environment, where the cost of any edge can be changed or the edge can be deleted. For outerplanar digraphs, for instance, the data structures can be updated after any such change in only $O(\log n)$ time, where $n$ is the number of vertices of the digraph. We also describe the first parallel algorithms for solving the dynamic version of the shortest path problem. Our results can be extended to hold for digraphs of genus $o(n)$

Parallel Max Cut Approximations

Given a graph with positive integer edge weights one may ask whether there exists an edge cut whose weight is bigger than a given number. This problem is NP-complete. We present here an approximation algorithm in NC which provides tight upper bounds to the proportion of edge cuts whose size is bigger than a given number. Our technique is based on the methods to convert randomized parallel algorithms into deterministic ones introduced by Karp and Wigderson. The basic idea of those methods is to replace an exponentially large sample space by one of polynomial size. In this work, we prove the interesting result that the statistical distance of random variables of the small sample space is bigger than the statistical distance of corresponding variables of the exponentially large space, which is the space of all edge cuts taken equiprobably

Quickest Paths: Faster Algorithms and Dynamization

Given a network $N=(V,E,{c},{l})$, where $G=(V,E)$, $|V|=n$ and $|E|=m$, is a directed graph, ${c}(e) \u3e 0$ is the capacity and ${l}(e) \ge 0$ is the lead time (or delay) for each edge $e\in E$, the quickest path problem is to find a path for a given source--destination pair such that the total lead time plus the inverse of the minimum edge capacity of the path is minimal. The problem has applications to fast data transmissions in communication networks. The best previous algorithm for the single--pair quickest path problem runs in time $O(r m+r n \log n)$, where $r$ is the number of distinct capacities of $N$ \cite{ROS}. In this paper, we present algorithms for general, sparse and planar networks that have significantly lower running times. For general networks, we show that the time complexity can be reduced to $O(r^{\ast} m+r^{\ast} n \log n)$, where $r^{\ast}$ is at most the number of capacities greater than the capacity of the shortest (with respect to lead time) path in $N$. For sparse networks, we present an algorithm with time complexity $O(n \log n + r^{\ast} n + r^{\ast} \tilde{\gamma} \log \tilde{\gamma})$, where $\tilde{\gamma}$ is a topological measure of $N$. Since for sparse networks $\tilde{\gamma}$ ranges from $1$ up to $\Theta(n)$, this constitutes an improvement over the previously known bound of $O(r n \log n)$ in all cases that $\tilde{\gamma}=o(n)$. For planar networks, the complexity becomes $O(n \log n + n\log^3 \tilde{\gamma}+ r^{\ast} \tilde{\gamma})$. Similar improvements are obtained for the all--pairs quickest path problem. We also give the first algorithm for solving the dynamic quickest path problem

A Note on Parallel Algorithms for Optional h-v Drawings of Binary Trees

In this paper we present a method to obtain optimal h-v drawings in parallel. Based on parallel tree contraction, our method computes optimal (with respect to a class of cost functions of the enclosing rectangle) drawings in O(log2 n) parallel time by using a polynomial number of EREW processors. The number of processors reduces substantially when we study minimum area drawings. Our work places the problem of obtaining optimal size h-v drawings in NC, presenting the first algorithm with polylogarithmic time complexity

Efficient Parallel Algorithms for some Tree Layout Problems

The minimum cut and minimum length linear arrangement problems usually occur in solving wiring problems and have a lot in common with job sequencing questions. Both problems are NP-complete for general graphs and in P for trees. We present here two algorithms in NC. The first solves the minimum length linear arrangement problem for unrooted trees in $O(\log^2 n)$ time and $O(n^2 3^{\log n})$ CREW PRAM processors. The second algorithm solves the minimum cut arrangement for unrooted trees of maximum degree $d$ in $O(d \log^2 n)$ time and $O(n^2 /\log n)$ CREW PRAM processors

Many-to-Many Routing on Trees via Matchings

In this paper we present an extensive study of many-to-many routing on trees under the matching routing model. Our study includes on-line and off-line algorithms. We present an asymptotically optimal on-line algorithm which routes k packets to their destination within d(k \Gamma 1) + d \Delta dist routing steps, where d is the degree of tree T on which the routing takes place and dist is the maximum distance any packet has to travel. We also present an off-line algorithm that solves the same problem within 2(k \Gamma 1)+dist steps. The analysis of our algorithms is based on the establishment of a close relationship between the matching and the hot-potato routing models that allows us to apply tools which were previously used exclusively in the analysis of hot-potato routing

On The Computation Of Fast Data Transmissions In Networks With Capacities And Delays

We examine the problem of transmitting in minimum time a given amount of data between a source and a destination in a network with finite channel capacities and non-zero propagation delays. In the absence of delays, the problem has been shown to be solvable in polynomial time. In this paper, we show that the general problem is NP-hard. In addition, we examine transmissions along a single path, called the quickest path, and present algorithms for general and sparse networks that outperform previous approaches. The first dynamic algorithm for the quickest path problem is also given

Dynamic Tree Routing under the "Matching with Consumption" Model

. In this paper we consider dynamic routing on trees under the "matching with consumption" routing model, a natural extension of the matching routing model introduced by Alon, Chung and Graham [1, 2], which allows for the consumption of packets when they reach their destination. We present an asymptotically optimal on-line algorithm that routes k packets to their destination within d(k \Gamma 1) + d \Delta dist routing steps where d is the degree of tree T on which the routing takes place and dist is the maximum distance some packet has to travel. We present an off-line algorithm that solves the same problem within 2(k \Gamma 1) + dist steps. Versions of both the on-line and the off-line algorithms which avoid live-lock situations are also provided. We establish a close relation between the "matching with consumption" and the hot-potato routing models, and we exploit it in the analysis of our routing algorithms. 1 Introduction In a packet routing problem on a connected undirected gra..