scholarly journals The Lower Tail of the Random Minimum Spanning Tree

10.37236/1004 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Abraham D. Flaxman
Keyword(s):  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Large Deviation ◽  
Random Variable ◽  
Independent Random Variables ◽  
Large Deviation Probability ◽  
Deviation Probability ◽  
Lower Tail ◽  
The Subject ◽  
The Cost

Consider a complete graph $K_n$ where the edges have costs given by independent random variables, each distributed uniformly between 0 and 1. The cost of the minimum spanning tree in this graph is a random variable which has been the subject of much study. This note considers the large deviation probability of this random variable. Previous work has shown that the log-probability of deviation by $\varepsilon$ is $-\Omega(n)$, and that for the log-probability of $Z$ exceeding $\zeta(3)$ this bound is correct; $\log {\rm Pr}[Z \geq \zeta(3) + \varepsilon] = -\Theta(n)$. The purpose of this note is to provide a simple proof that the scaling of the lower tail is also linear, $\log {\rm Pr}[Z \leq \zeta(3) - \varepsilon] = -\Theta(n)$.

scholarly journals Probabilities of very large deviations

1978 ◽  
Vol 25 (3) ◽  
pp. 332-347 ◽  
Author(s):  
Stephen A. Book
Keyword(s):  
Generating Functions ◽  
Large Deviation ◽  
Random Variables ◽  
Large Deviation Probability ◽  
Deviation Probability ◽  
Absolutely Continuous ◽  
The Central Limit Theorem ◽  
The Subject ◽  
Elementary Form ◽  
Independent Identically Distributed

If {Xn: 1 ≦ n < ∞} are independent, identically distributed random variables having E(X1) = 0 and Var(X1) = 1, the most elementary form of the central limit theorem implies that P(n-½Sn≧ zn) → 0 as n → ∞, where Sn = Σnk=1 X,k, for all sequences {zn:1 ≧ n gt; ∞} for which zn → ∞. The probability P(n-½ Sn ≧ zn) is called a “large deviation probability”, and the rate at which it converges to 0 has been the subject of much study. The objective of the present article is to complement earlier results by describing its asymptotic behavior when n-½zn → ∞ as n → ∞, in the case of absolutely continuous random variables having moment-generating functions.

scholarly journals Algorithms for the power-p Steiner tree problem in the Euclidean plane

2018 ◽  
Vol 25 (4) ◽  
pp. 28
Author(s):  
Christina Burt ◽  
Alysson Costa ◽  
Charl Ras
Keyword(s):  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Valid Inequalities ◽  
Minimum Cost ◽  
Performance Ratio ◽  
Mixed Integer ◽  
Steiner Tree Problem ◽  
Steiner Points ◽  
Tree Construction ◽  
The Cost

We study the problem of constructing minimum power-$p$ Euclidean $k$-Steiner trees in the plane. The problem is to find a tree of minimum cost spanning a set of given terminals where, as opposed to the minimum spanning tree problem, at most $k$ additional nodes (Steiner points) may be introduced anywhere in the plane. The cost of an edge is its length to the power of $p$ (where $p\geq 1$), and the cost of a network is the sum of all edge costs. We propose two heuristics: a ``beaded" minimum spanning tree heuristic; and a heuristic which alternates between minimum spanning tree construction and a local fixed topology minimisation procedure for locating the Steiner points. We show that the performance ratio $\kappa$ of the beaded-MST heuristic satisfies $\sqrt{3}^{p-1}(1+2^{1-p})\leq \kappa\leq 3(2^{p-1})$. We then provide two mixed-integer nonlinear programming formulations for the problem, and extend several important geometric properties into valid inequalities. Finally, we combine the valid inequalities with warm-starting and preprocessing to obtain computational improvements for the $p=2$ case.

A Low Cost MST-FSM Obfuscation Method for Hardware IP Protection

2020 ◽  
Vol 29 (13) ◽  
pp. 2050208
Author(s):  
Yuejun Zhang ◽  
Zhao Pan ◽  
Pengjun Wang ◽  
Xiaowei Zhang
Keyword(s):  
Integrated Circuit ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Hamming Distance ◽  
Low Cost ◽  
The Arts ◽  
Finite State ◽  
Ip Cores ◽  
Building Program ◽  
The Cost

Effective resistance to intellectual property (IP) piracy, overproduction and reverse engineering are becoming more and more necessary in the integrated circuit (IC) supply chain. To protect the hardware, the obfuscation methodology hides the original function by adding a large number of redundant states. However, existing hardware obfuscation approaches have hardware overhead and efficiency of obfuscation limitations. This paper proposed a novel methodology for IP security using the minimum spanning tree finite state machine (MST-FSM) obfuscation. In the minimum spanning tree (MST) algorithm, the Hamming distance defines the cost of obfuscated states. The Kruskal algorithm optimizes the connection relationship of obfuscated states by computing the Hamming distance of the MST-FSM. The proposed MST-FSM is automatically generated and embedded in the hardware IP with the self-building program. Finally, the MST-FSM is applied on the itc99 benchmark circuits and encryption standard IP cores. Compared with other state-of-the-arts, the obfuscation potency is improved by 3.57%, and the average hardware cost is decreased by about 6.01%.

scholarly journals Dynamic Shapley Value in the Game with Perishable Goods

2021 ◽  
Vol 14 ◽  
pp. 273-289
Author(s):  
Li Yin ◽  
◽  
Ovanes Petrosian ◽  
Zou Jinying ◽  
◽  
...  
Keyword(s):  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Time Consistency ◽  
Characteristic Functions ◽  
Cooperative Behaviour ◽  
Second Stage ◽  
Perishable Goods ◽  
Shapley Values ◽  
The Cost ◽  
Tree Games

The paper investigates two-stage stochastic minimum spanning tree games with perishable goods. The cooperative behaviour of the players is defined. At each stage, all players jointly take action to construct a network with a cost matrix. At the second stage, a particular player may leave the game, and the probability of this leaving depends on the cooperative behaviour of all players at the first stage. At each stage game, the total cost of the spanning tree is calculated to include the sum of the costs of the contained edges and the cost of the loss of perishable goods expended on that edge of the spanning tree. The characteristic functions in the game are considered, and the dynamic Shapley values are modified. The time consistency of the dynamic Shapley values is studied.

scholarly journals Pengujian Optimalisasi Jaringan Kabel Fiber Optic Di Universitas Islam Indonesia Menggunakan Minimum Spanning Tree

2014 ◽  
Vol 3 (1) ◽  
pp. 49
Author(s):  
Muchammad Abrori ◽  
Najib Ubaidillah
Keyword(s):  
Spanning Tree ◽  
Fiber Optic ◽  
Minimum Spanning Tree ◽  
Computer Network ◽  
Star Topology ◽  
Cable Length ◽  
Cable Network ◽  
Cable Lines ◽  
The Cost

Universitas Islam Indonesia (UII) intergrated campus computer network built since 1995. Development of UII integrated campus computer network is using a star topology and fiber optic (FO) cable. Considering that the star topology is the topology that requires a lot of wires, this study was conducted to determine and examine how the application of graph on the FO cable network UII integrated campus in order to minimize the cost, because FO cable network can be modeled by a graph where the buildings as points, while FO cable that connects to each building as a line. This type of research that is used here is a case study, in which data collection by observation, interviews, and documentation. This study used 4 algorithms, that is Kruskal algorithm, Prim, Boruvka and Solin algorithm to find the Minimum Spanning Tree. Based on the research that has been done, the conclution about the troubleshooting steps of optimization UII integrated campus FO cable network based graph theory has been got. From the four algorithms obtained the most optimal results FO cable length 4.700 meters long and is 1.590 meters cable lines. While the results of observations made, it is known that the existing computer network in UII integrated campus has a cable length of 6.120 meters and 2.050 meters long track. The results of the analysis showed that the resulrs of the study 23.2% more optimal than the existing computer networks in UII integrated campus.

scholarly journals A Randomly Weighted Minimum Spanning Tree with a Random Cost Constraint

10.37236/9445 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Alan Frieze ◽  
Tomasz Tkocz
Keyword(s):  
Complete Graph ◽  
Spanning Tree ◽  
Dual Problem ◽  
Minimum Spanning Tree ◽  
Random Variable ◽  
Independent Copy ◽  
Asymptotic Value ◽  
Optimum Weight ◽  
Minimum Spanning Tree Problem ◽  
Range Of Values

We study the minimum spanning tree problem on the complete graph $K_n$ where an edge $e$ has a weight $W_e$ and a cost $C_e$, each of which is an independent copy of the random variable $U^\gamma$ where $\gamma\leq 1$ and $U$ is  the uniform $[0,1]$ random variable. There is also a constraint that the spanning tree $T$ must satisfy $C(T)\leq c_0$. We establish, for a range of values for $c_0,\gamma$, the asymptotic value of the optimum weight via the consideration of a dual problem. 

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