scholarly journals A distributed algorithm for directed minimum-weight spanning tree

2021 ◽  
Author(s):  
Orr Fischer ◽  
Rotem Oshman
Keyword(s):  
Distributed Algorithm ◽  
Spanning Tree ◽  
Minimum Weight ◽  
Minimum Weight Spanning Tree

The Problem of Scheduling Parallel Computations of Multibody Dynamic Analysis Is NP-Hard

1996 ◽  
Author(s):  
Jin-Fan Liu ◽  
Karim A. Abdel-Malek
Keyword(s):  
Dynamic Analysis ◽  
Spanning Tree ◽  
Minimum Weight ◽  
Computational Cost ◽  
Parallel Computations ◽  
Np Hard ◽  
Multibody Dynamic ◽  
Np Hard Problem ◽  
Np Complete ◽  
Minimum Weight Spanning Tree

Abstract A formulation of a graph problem for scheduling parallel computations of multibody dynamic analysis is presented. The complexity of scheduling parallel computations for a multibody dynamic analysis is studied. The problem of finding a shortest critical branch spanning tree is described and transformed to a minimum radius spanning tree, which is solved by an algorithm of polynomial complexity. The problems of shortest critical branch minimum weight spanning tree (SCBMWST) and the minimum weight shortest critical branch spanning tree (MWSCBST) are also presented. Both problems are shown to be NP-hard by proving that the bounded critical branch bounded weight spanning tree (BCBBWST) problem is NP-complete. It is also shown that the minimum computational cost spanning tree (MCCST) is at least as hard as SCBMWST or MWSCBST problems, hence itself an NP-hard problem. A heuristic approach to solving these problems is developed and implemented, and simulation results are discussed.

Minimum-weight spanning tree algorithms A survey and empirical study

2001 ◽  
Vol 28 (8) ◽  
pp. 767-785 ◽  
Author(s):  
Cüneyt F. Bazlamaçcı ◽  
Khalil S. Hindi
Keyword(s):  
Empirical Study ◽  
Spanning Tree ◽  
Minimum Weight ◽  
Tree Algorithms ◽  
Minimum Weight Spanning Tree

PARALLEL REAL-TIME OPTIMIZATION: BEYOND SPEEDUP

1999 ◽  
Vol 09 (04) ◽  
pp. 499-509 ◽  
Author(s):  
SELIM G. AKL ◽  
Stefan D. Bruda
Keyword(s):  
Parallel Algorithm ◽  
Real Time ◽  
Spanning Tree ◽  
Optimization Problems ◽  
Minimum Weight ◽  
Parallel Computer ◽  
Worst Case ◽  
Time Optimization ◽  
Minimum Weight Spanning Tree ◽  
Real Time Optimization

Traditionally, interest in parallel computation centered around the speedup provided by parallel algorithms over their sequential counterparts. In this paper, we ask a different type of question: Can parallel computers, due to their speed, do more than simply speed up the solution to a problem? We show that for real-time optimization problems, a parallel computer can obtain a solution that is better than that obtained by a sequential one. Specifically, a sequential and a parallel algorithm are exhibited for the problem of computing the best-possible approximation to the minimum-weight spanning tree of a connected, undirected and weighted graph whose vertices and edges are not all available at the outset, but instead arrive in real time. While the parallel algorithm succeeds in computing the exact minimum-weight spanning tree, the sequential algorithm can only manage to obtain an approximate solution. In the worst case, the ratio of the weight of the solution obtained sequentially to that of the solution computed in parallel can be arbitrarily large.

scholarly journals On the Rank of a Random Binary Matrix

10.37236/8092 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Colin Cooper ◽  
Alan Frieze ◽  
Wesley Pegden
Keyword(s):  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Applied Mathematics ◽  
Minimum Weight ◽  
Binary Matrix ◽  
The Matrix ◽  
Expected Length ◽  
Minimum Weight Spanning Tree ◽  
Random Hypergraph ◽  
Correct Estimate

We study the rank of a random $n \times m$ matrix $\mathbf{A}_{n,m;k}$ with entries from $GF(2)$, and exactly $k$ unit entries in each column, the other entries being zero. The columns are chosen independently and uniformly at random from the set of all ${n \choose k}$ such columns. We obtain an asymptotically correct estimate for the rank as a function of the number of columns $m$ in terms of $c,n,k$, and where $m=cn/k$. The matrix $\mathbf{A}_{n,m;k}$ forms the vertex-edge incidence matrix of a $k$-uniform random hypergraph $H$. The rank of $\mathbf{A}_{n,m;k}$ can be expressed as follows. Let $|C_2|$ be the number of vertices of the 2-core of $H$, and $|E(C_2)|$ the number of edges. Let $m^*$ be the value of $m$ for which $|C_2|= |E(C_2)|$. Then w.h.p. for $m<m^*$ the rank of $\mathbf{A}_{n,m;k}$ is asymptotic to $m$, and for $m \ge m^*$ the rank is asymptotic to $m-|E(C_2)|+|C_2|$. In addition, assign i.i.d. $U[0,1]$ weights $X_i, i \in {1,2,...m}$ to the columns, and define the weight of a set of columns $S$ as $X(S)=\sum_{j \in S} X_j$. Define a basis as a set of $n-𝟙 (k\text{ even})$ linearly independent columns. We obtain an asymptotically correct estimate for the minimum weight basis. This generalises the well-known result of Frieze [On the value of a random minimum spanning tree problem, Discrete Applied Mathematics, (1985)] that, for $k=2$,   the expected length of a minimum weight spanning tree tends to $\zeta(3)\sim 1.202$.

scholarly journals The Diameter of the Minimum Spanning Tree of a Complete Graph

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Louigi Addario-Berry ◽  
Nicolas Broutin ◽  
Bruce Reed
Keyword(s):  
Complete Graph ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Minimum Weight ◽  
Unique Minimum ◽  
International Audience ◽  
Minimum Weight Spanning Tree ◽  
Independent Identically Distributed

International audience Let $X_1,\ldots,X_{n\choose 2}$ be independent identically distributed weights for the edges of $K_n$. If $X_i \neq X_j$ for$ i \neq j$, then there exists a unique minimum weight spanning tree $T$ of $K_n$ with these edge weights. We show that the expected diameter of $T$ is $Θ (n^{1/3})$. This settles a question of [Frieze97].

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