Showing papers on "Prim's algorithm published in 1993"

A direct combination of the Prim and Dijkstra constructions for improved performance-driven global routing

Abstract: Motivated by analysis of distributed RC delay in routing trees, a new tree construction is proposed for performance-driven global routing which directly trades off between Prim's minimum spanning tree algorithm and Dijkstra's shortest path tree algorithm. This direct combination of two objective functions and their corresponding optimal algorithms contrasts with the more indirect 'shallow-light' methods. The authors' method achieves routing trees which satisfy a given routing tree radius bound while using less wire than previous methods. Detailed simulations show that these wirelength savings translate into significantly improved delay over standard MST routing in both IC and multichip module (MCM) interconnect technologies. >

Balancing minimum spanning and shortest path trees

Abstract: Efficient algorithms are known for computing a minimum spann.ing tree, or a shortest path. tree (with a fixed vertex as the root). The weight of a shortest path tree can be much more than the weight of a minimum spa,nning tree. Conversely, the distance bet,ween the root, and any vertex in a minimum spanning tree may be much more than the distance bet#ween the two vertices in the graph. Consider the problem of balancing between the two kinds of trees: Does every graph contain a tree that is “light” (at most a constant times heavier than the minimum spanning t,ree), such that the distance from the root to any vertex in t,he tree is no more than a constant times the true distance? This paper answers the question in the affirmative. It is shown that there is a continuous tradeoff between the two parameters. For every y > 0, there is a tree in the graph whose total weight is at most 1 + $? times the weight of a minimum spanning tree, such that the di&nce in the tree between the root, and any vertex is at, most 1 + &y times the true distance. Efficient sequential and parallel algorithms achieving these factors are provided. The algorithms are shown to be optimal in two ways. First, it is shown that no algorithm can achieve better factors in all graphs, because there a.re graphs that do not have better trees. Second, it is shown that even on a per-graph basis, finding trees that achieve better factors is NP-hard.

Balancing minimum spanning trees and shortest-path trees

Abstract: We give a simple algorithm to find a spanning tree that simultaneously approximates a shortest-path tree and a minimum spanning tree. The algorithm provides a continuous tradeoff: given the two trees and aź>0, the algorithm returns a spanning tree in which the distance between any vertex and the root of the shortest-path tree is at most 1+ź2ź times the shortest-path distance, and yet the total weight of the tree is at most 1+ź2/ź times the weight of a minimum spanning tree. Our algorithm runs in linear time and obtains the best-possible tradeoff. It can be implemented on a CREW PRAM to run a logarithmic time using one processor per vertex.

A Linear Time Pattern Matching Algorithm Between a String and a Tree

Abstract: In this paper, we describe a linear time algorithm for testing whether or not there is a path of a tree T (¦V(T)¦= n) that coincides with a string s (¦s¦ = m). In the algorithm, O(n/m) vertices are selected from V(T) such that any path of length more than m −2 must contain at least one of the selected vertices. A search is performed using the selected vertices as ‘bases’. A suffix tree is used effectively in the algorithm. Although the size of the alphabet is assumed to be bounded by a constant in this paper, the algorithm can be applied to the case of unbounded alphabets by increasing the time complexity to O(n log n).

A parallel algorithm for multiple edge updates of minimum spanning trees

Abstract: The authors present a parallel algorithm for the multiple edge update problem on a minimum spanning tree. This problem is defined as follows: given a minimum spanning tree T(V,E/sub T/) of an undirected graph G(V,E), where mod V mod =n and E/sub T/ is the set of tree edges, recompute a new minimum spanning tree when (1) adding K new edges, (2) changing the weights of existent K edges, or (3) deleting a vertex of degree K in the tree, where 1 >

Incremental distributed synchronous algorithm for minimum spanning trees

Abstract: A distributed algorithm for updating a minimum spanning tree when a new vertex is added to the underlying network (a connected, undirected, weighted graph) is presented. The algorithm runs asynchronously and the processor at each vertex of the network is required to know only information concerning its adjacent edges. The number of messages transmitted is bounded by 6N and the time complexity is O(H) where N and H(⩽N) are the number of vertices and the height of the original minimum spanning tree, respectively. Each message contains at most an edge weight and a few bits. This result is superior to the previously known result as well as that of using the best known distributed minimum spanning tree algorithm to construct a minimum spanning tree for the new network from scratch. The algorithm is extended to handle k(⩾1) vertex insertions. The resulting time and message complexities are O (k ∗ max (H, H 1 ,…,H k−1 )) and O(kN + k2) respectively, where Hi is the height of the MST after i new vertices have been handled. The algorithm can be modified to handle edge insertions within same time and message bounds.

Minimum spanning tree generation with content-addressable memory

Abstract: An efficient parallel algorithm is proposed for finding a minimum spanning tree, with the aid of content-addressable memory. Operations such as minimum-value search and updating the active edges are implemented in an efficient manner. The algorithm has O(n) complexity, where n is the number of nodes. An example is given and compared with a conventional algorithm. Actual implementation shows approximately O(n) speedup. >