[PDF] Randomized Minimum Spanning Tree Algorithms Using





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Randomized Algorithm for Minimum Spanning Trees

If the graph G is not connected a MST does not exist and we generalize the notion of minimum spaning tree to that of the minimum spanning forest. 3 Properties 



A randomized linear-time algorithm to find minimum spanning trees

We present a randomized linear-time algorithm to find a minimum spanning tree in a connected graph with edge weights. The algorithm uses random sampling in 





A Randomized Algorithm to Find Minimum Spanning Tree

5 déc. 2013 The problem of finding such tree T for a graph G is known as minimum spanning tree problem. 3 Non-Randomized Algorithms. We show three ...



A Randomized Linear-Time Algorithm for Finding Minimum

The algorithm presented by the paper delivers an algorithm for finding the minimum spanning tree (MST) as the title implies but it also solves the slightly more 



Randomized Minimum Spanning Tree Algorithms Using

focus of this article is a linear-time randomized minimum spanning tree Concepts from the Karger-Klein-Tarjan algorithm



a randomized time-work optimal parallel algorithm for finding a

We also give a simple general processor allocation scheme for tree-like computations. Key words. parallel algorithm



Analysis of haplotype networks: The randomized minimum spanning

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A Randomized Linear-Time Algorithm to Find Minimum Spanning

7 juil. 1994 Abstract. We present a randomized linear-time algorithm to nd a minimum spanning tree in a connected graph with edge weights.



Lecture 18 - Duke University

Figure 1: Kruskal’s algorithm and Prim’s algorithm for minimum spanning tree The red edges are added this iteration 2 1 Kruskal’s Algorithm Kruskal’s algorithm maintains a spanning forest (starting with only singletons) and on each step connects two components with the globally lightest edge between components This is typically



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Randomized Minimum Spanning Tree Algorithms Using Exponentially Fewer Random Bits Randomized Minimum Spanning Tree Algorithms Using Exponentially Fewer Random Bits SETH PETTIE University of Michigan and VIJAYA RAMACHANDRAN The University of Texas at Austin



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A tree is an undirected graph Gwhich satis es any two of the three following statements such a graph also then satis es the other property as well 2 1 Gconnected 2 Ghas no cycles 3 Ghas exactly n 1 edges The minimum spanning tree (MST) of Gis T = (V;E T;w) such that P e2E T w(e) P e2E w(e) for any other spanning tree T



Lecture 12: Greedy Algorithms and Minimum Spanning Tree

Minimum Spanning Tree problem: Given an undirected graph G = (VE) and edge weights W: E ? R ?nd a spanning tree T of minimum weight e?T w (e) A naive algorithm The obvious MST algorithm is to compute the weight of every tree and return the tree of minimum weight Unfortunately this can take exponential time in the worst case



1 Minimum Spanning Tree - University of California Berkeley

1 MinimumSpanningTree RecallfromCS170thede?nitionoftheminimumspanningtree: givenanun-directedgraphGwithweightsonitsedges aspanningtreeconsistsofasubsetofedgesfromGthat connectsallthevertices Amongthem aminimumspanningtree(MST)hastheminimumtotalweight overitsedges Byconvention let nbethenumberofverticesinGandmbethenumberofedges



Searches related to randomized minimum spanning tree algorithm filetype:pdf

In thisreport we present 3 algorithms to compute the minimum spanning tree (MST) or minimum spanning for-est (MSF) for large graphs which do not ?t in the memory of a single machine We analyze the theoreticalprocessing time communication cost and communication time for these algorithms under certain assump-tions

What is the minimum spanning tree algorithm?

    The algorithm returns a set containing the edge ids of all the edges in the MST. The code was tested on Python 3.3.2. Implementation of Prim's Minimum Spanning Tree algorithm, along with a library for the heap data structure.

How many spanning trees can be in a graph?

    There can be many spanning trees for any given graph. By assigning a weight to each edge, the different spanning trees are assigned a number for the total weight of their edges. The minimum spanning tree is then the spanning tree whose edges have the least total weight.

What is minimum bottleneck spanning tree?

    • Minimum Bottleneck Spanning Tree-Sometimes we seek a spanning tree which minimizes the maximum edge weight over all such trees. In fact, the minimum spanning tree has this property. The proof follows directly from the correctness of Kruskal’s algorithm.

What is the MST algorithm in machine learning?

    The obvious MST algorithm is to compute the weight of every tree, and return the tree of minimum weight. Unfortunately, this can take exponential time in the worst case. Consider the following example:

Last compiled January 5, 2023 at 12:08 Noon.

Randomized Algorithms, CS588, Fall 2022

Lectures:4:30 to 5:45 PM, on Tuesdays and Thursdays, in Lawson room 1106. Staff:Email(@purdue.edu)Office hoursKent QuanrudkrqAfter class, LWSN 1211 Jawad "JD" Raheel (TA)jraheelWed., 10-11AM, Haas G50

Please see the syllabus (page

384
) for more information.

Homework

•Homework0 due A ugust31.

•Homework1 due Septem ber16.

•Homework2 due Septem ber28.

•Homework3 due Octob er21.

•Homework4 due No vember4.

•Homework5 due No vember18.

•Homework6 due Decem ber9.

Class schedule

1

1.August 23.Randomized SAT and quicksort (chapter1 ).

2.August 25.Hashing and heavy hitters (chapter2 ).

3.August 30.Hash tables and linear probing (chapter3 ).

4.September 1.Randomized minimum cut (sections4.1 -4.3).

5.September 6.Random graphs (chapter5 ).

6.September 8.Randomized rounding (chapter6 ).1All future dates are tentative. Lecture materials and video recordings can be found at the end of

each chapter. 1

Kent Quanrud, December 20

7.September 13.Distinct elements (chapter7 ).

8.September 15.Dimensionality reduction (chapter8 ).

9.September 20.Locality sensitive hashing and approximate nearest neighbors

(chapter 9

10.September 22.L1-metric embeddings and sparsest cut (sections10.1 -10.3).

11.September 27.Randomized tree metrics (chapter11 ).

12.September 29.Sampling geometric range spaces (chapter12 ).

13.October 4.PAC learning (chapter13 ).

14.October 6.Entropy and error-correcting codes (chapter14 ).

Midterm on Thursday, October 13, at 8PM in Hampton Hall, room 1252. (See appendix C.4.1

15.October 18.Lovász local lemma (chapter15 ).

16.October 20.Pagerank (chapter16 ).

17.October 25.Discrepancy via Gaussian random walks (chapter17 ).

18.October 27.Connectivity via random walks (chapter18 ).

19.November 1.Spectral analysis of random walks (chapter19 ).

20.November 3.No class.

21.November 8.Mixing times, conductance, and sparsest cut (chapter20 ).

22.November 10.Deterministic log-space connectivity (chapter21 ).

23.November 15.Reducing randomness with random walks (chapter22 ).

24.November 17.PCP Theorem (chapter23 ).

(?)Thanksgiving break.

25.November 29.PCP theorem (cont"d).

26.December 1.Randomly testing boolean functions (chapter24 ).

2

Kent Quanrud, Fall 2022

27.December 6.Online algorithms (chapter25 ).

28.December 8.Sparsification (chapter26 ).

(?)Final on Thursday, December 15 at 8AM in Lawson, B151. 3

Contents

Basic information

1

Class schedule

1

1 Randomized searching and sorting

11

1.1 First randomized algorithms

11

1.2 Basic probability

14

1.3 Randomized approximations for SAT

21

1.4 Randomized sorting

25

1.5 Additional notes and materials

28

1.6 Exercises

29

2 Hashing and heavy hitters

34

2.1 Introduction

34

2.2 Hashing

37

2.3 Using hashing to approximate frequencies

38

2.4 Amplification

40

2.5 Extensions

41

2.6 Takeaways

42

2.7 Additional notes and references

43

2.8 Exercises

44

3 Hash tables and linear probing

48

3.1 Dictionaries

48

3.2 Hash tables with chaining

52

3.3 Linear probing

53

3.4 4-wise independence

58

3.5 Takeaways

61

3.6 Additional notes and materials

61

3.7 Exercises

61
4

ContentsK entQuanrud, F all2022

4 Sampling edges

64

4.1 Minimum cut

64

4.2 Amplification by branching

69

4.3 Randomized branching

72

4.4 Sparsification

73

4.5 Randomized Ford-Fulkerson

82

4.6 Additional notes and materials

87

4.7 Exercises

87

5 Random Sums and Graphs

90

5.1 Random sums

90

5.2 Random graphs

92

5.3 A gap in component size

96

5.4 Galton-Watson process with general branching factors

99

5.5 Additional notes and materials

100

5.6 Exercises

101

6 Randomized Rounding

102

6.1 SAT

104

6.2 Set cover

108

6.3 Covering integer programs

110

6.4 Additional notes and materials

113

6.5 Exercises

113

6.A Proof of the multiplicative Chernoff bound

115

7 Distinct elements

118

7.1 Unique visitors

118

7.2 Preliminaries on continuous random variables

120

7.3 Distinct elements, continued

121

7.4 Variance and Chebyshev"s inequality

124

7.5 Amplification: variance reduction and the median trick

124

7.6 Distinct elements with pairwise independent hash functions

128

7.7 Takeaways

128

7.8 Additional notes and materials

130

7.9 Exercises

130

8 Dimensionality Reduction

133

8.1 Large data sets and long vectors

133

8.2 Gaussian random variables: an interface

135
5

ContentsK entQuanrud, F all2022

8.3 Random Projections

138

8.4 Gaussians

139

8.5 Additional notes and materials

142

8.6 Exercises

142

9 Locality sensitive hashing and approximate nearest neighbors

145

9.1 Nearest neighbor search

145

9.2 Locality Sensitive Hashing for Angles

146

9.3 Locality Sensitive Hashing for Euclidean Distance

148

9.4 Additional notes and materials

153

9.5 Exercises

153

10L1-metric embeddings and sparsest cut155

10.1 LP duality, line embeddings, and max-flow min-cut

155

10.2 Sparsest cut

164

10.3 Rounding viaL1-metric embeddings. . . . . . . . . . . . . . . . . . . 167

10.4 Rounding via region growing (for uniform demands)

175

10.5 Application: Minimum bisection

178

10.6 Additional notes and materials

179

10.7 Exercises

180

11 Tree Metrics

183

11.1 Introduction

183

11.2 Low-Stretch Spanning Trees

184

11.3 Hierarchical Tree Metrics

187

11.4 Additional notes and materials

190

11.5 Exercises

190

12 Sampling geometric range spaces

192

12.1 Introduction

192

12.2 Proof of the?-sample theorem. . . . . . . . . . . . . . . . . . . . . . 195

12.3 VC dimension

197

12.4 Additional notes and materials

199

12.5 Exercises

199

12.A Proof of the additive Chernoff bound

200

13 PAC learning

203

13.1 The consistency model

203

13.2 The PAC learning model

205
6

ContentsK entQuanrud, F all2022

13.3 Generalization for finite hypothesis classes

206

13.4 Occam"s razor

207

13.5 Stronger generalization bounds via growth rate

209

13.6 Additional notes and materials

211

13.7 Exercises

212

14 Entropy and error-correcting codes

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