A Randomized Linear-Time A lgorithm to Find Minimum S paning Trees

A Randomized Linear-Time A lgorithm to Find Minimum S paning Trees

黃則翰 R96922141 蘇承祖 R96922077 張紘睿 R96922136 許智程 D95922022 戴于晉 R96922171

David R. Karger Philip N. Klein Robert E. Tarjan

Outline

 Introduction

 Basic Property & Definition

 Algorithm

 Analysis

Outline

 Introduction

 Basic Property & Definition

 Algorithm

 Analysis

Introduction

 [Borůvka 1962] O(m log n)

 Gabow et al.[1984] O(m log β(m,n) )

◦β(m,n)= min { i |log⁽ⁱ⁾n <= m/n}

 Verification algorithm

◦King[1993] O(m)

 A randomize algorithm runs in O(m) tim e with high probability

Outline

 Introduction

 Basic Property & Definition

 Algorithm

 Analysis

Cycle property

 For any cycle C in a graph, the heavie st edge in C dose not appear in the mi nimum spanning forest.

2

3

5

6

2

3

5

6

Cut Property

2

3

5

6

 For any proper nonempty subset X of the vertices, the lightest edge with exactly one endpoint in X belongs to the minimum

spanning tree

X

Definition

 Let G be a graph with weighted edges.

◦w(x,y)

 The weight of edge {x,y}

 If F is a forest of a subgraph in G

◦F(x, y)

 the path (if any) connecting x and y in F

◦w_F(x, y)

 the maximum weight of an edge on F(x , y)

◦w_F(x, y)=∞

 If x and y are not connected in F

F-heavy & F-light

 An edge {x,y} is F-heavy if w(x,y) > w_F (x,y)

and F-light otherwise

 Edge of F are all F-light

A C

B D

2

3

5

6

E G

F H

2

3

5

W(B,D)=6 6

W_F(B,D)=max{2,3, 5}

F-heavy

W(F,H)=6 W_F(F,H)= ∞ F-light

W(C,D)=5 W_F(C,D)=5 F-light

 No F-heavy edge can be in the minimum spanning forest of G (cycle property)

 Discard edge that cannot be in the min imum spanning tree

 F-light edge can be the candidate edge for the minimum spanning tree of G

Observation

Outline

 Introduction

 Basic Property & Definition

 Algorithm

 Analysis

Boruvka Algorithm

 For each vertex, select the minimum-we ight edge incident to the vertex.

 Replace by a single vertex each connec ted component defined by the selected edges.

 Delete all resulting isolated vertice s, loops, and all but the lowest-weigh t edge among each set of multiple edge s.

Algorithm Step1

 Apply two successive Boruvka steps to the graph, thereby reducing the number of vertices by at least a factor of fo ur.

Algorithm Step2

 Choose a subgraph H by selecting each edge independently with probability ½.

 Apply the algorithm recursively to H, producing a minimum spanning forest F of H.

 Find all the F-heavy edges and delete them from the contracted graph.

Algorithm Step3

 Apply the algorithm recursively to the remaining graph to compute a spanning forest F’. Return those edges contrac ted in Step1 together with the edges o f F’.

G

H

Boruvka × 2

G*

Original Problem

G

’

Return minimum forest F of H

Delete F-heavy edges from G*

Left Sub- problem

Sample with F’

p=0.5

Correctness

 By the cut property, every edge contra cted during Step1 is in the MSF.

 By the cycle property, the edges delet ed in Step2 do NOT belong to the MSF.

 By the induction hypothesis, the MSF o f the remaining graph is correctly det ermined in the recursive call of Step 3.

Candidate Edge of MST

 The expected number of F-light edges i n G is at most n/p (negative binomial)

 For every sample graph H, the expected candidate edge for MST in G is at most n/p (F-light edge)

Random-sampling

 To help discard some edge that cannot be in the minimum spanning tree

 Construct the sample graph H

◦Process the edges in increasing order

◦To process an edge e

◦1. Test whether both endpoints of e i n same component

◦2. Include the edge in H with probabi lity p

◦3. If e is in H and is F-light, add e to the Forest F

Random-sampling

C E

D F

6

5

11

9

A G

4

3

10

14

13 B

7

C E

D F

6

5

11

9

A G

4

3

10

14

13 B

7

G H

F

W(E,G)=14

W_F(E,G)=max{5,6, 9,13}

F-heavy

W(E,F)=11

W_F(E,F)=max{5,6, 9}

F-heavy W(D,F)=9 W_F(D,F)=9 F-light

W(A,B)=7 W_F(A,B)= ∞ F-light

Random-sampling

C E

D F

6

5

11 9

A G

4

3

10

14

13 7 B

G

F

1. Increasing Order 2. If F-light

Throw If

Select 3. Else

Throw

Don’t select

1. Random select edges to H 2. Find F of H

C E

D F

6

5

11 9

A G

4

3

10

14

13 7 B

G

 No F-heavy edge can be in the minimum spanning forest of G (cycle property)

 F-light edge can be the candidate edge for the minimum spanning tree of G

 The forest F produced is the forest th at would be produced by Kruskal and in lcude all possible MSF of G

Observation

Observation

 The size of F is at most n-1

 The expected number of F-light edges i n G is at most n/p (negative binomial)

k

n p

k p n p k

n k

f 1 (1 )

)

;

;

(  



 



  



p p

k p n

k _{n k}

) 1

) ( 1

( ₁

 



 



  

 ^

p n1 p



  p n n1 p

p n

Mean k = Expected n =

Outline

 Introduction

 Basic Property & Definition

 Algorithm

 Analysis

Analysis of the Algorithm

 The worst case.

 The expectations running time.

 The probability of the expectations ru nning time.

Running time Analysis

 Total running time= running time in ea ch steps.

 Step(1): 2 steps Boruvka’s algorithm

 Step(2):Dixon-Rauch-Tarjan verificatio n algorithm.

 All takes linear time to the number of edges.

◦Estimate the total number of edges.

Observe the recursion tree

 G=(V,E) |V| = n, |E|=m .

◦m≧n/2 since there is no isolate vert ices.

 Each problem generates at most 2 subpr oblems.

◦At depth d, there is at most 2^d node s.

◦Each node in depth d has at most n/4^d vertices.

 The depth d is at most log₄n.

◦There are at most vertices in all sub problems







₀2 /4  _{d 0} /2^d  2

d

d

dn n n

The worst case

 Theorem 4.1 The worst-case running ti me of the minimum-spanning-forest alg orithm is O(min{n²,m log n}), the same as the bound for Boruvka’s algorith

m.

 Proof: There is two different estimat e ways.

1. A subproblem at depth contains at m ost (n/4^d)²/2 edges.

 Total edges in all subproblems i s:



0

2 2

) ( 2 2

) 4 / (

The worst case

2. Consider a subprolbem G=(V,E) after step(1), we have a G^’=(V^’ ,E^’),|E^’|≦

|E| - |V|/2, |V^’| ≦|V|/4 Edges in left-child = |H|

Edges in right-child ≦ |E^’| - |H| +

|F|

so edges in two subproblem is less th en:

(|H|) + (|E^’| - |H| + |F|)

=|E^’| +|F|≦|E|-|V|/2 + |V|/4≦|E|

The two sub problem at most contains | E| edges.

The worst case

m edges

edges

m

edges

m

edges

m n

log

The worst case

 The depth is at most log₄n and each lev el has at most m edges, so there are a t most (m log n) edges.

 The worst-case running time of the min imum-spanning-forest algorithm is O(mi n{n²,m log n}).

Analysis of the Algorithm

 The worst case.

 The expectations running time.

 The probability of the expectations ru nning time.

Analysis – Average Case

8)



Theorem: the expected running time o f the minimum spanning forest algori thm is O ( m )

◦Calculating the expected total number of edges for all left path problemsOriginal Problem

Left Sub-problem Right Sub-problem

Left Subsub-problem Right Subsub-problem

Analysis – Average Case

8)



Calculating the expected total edge number for one left path started at one problem with m’ edges



Evaluating the total edge number for all right sub-problems

= m’

# of edges

= m’

Expected total edge number

≤ 2m’

Expected total edge number

≤ 2m’

Analysis – Average Case

8)

G

H G

’

Boruvka × 2

G*

Sample with p=0.5

1. E[edge number of H] = 0.5 × edge number of G*

Original Problem

Left

Sub-problem

Right Sub-problem

2. ∵ Boruvka × 2

∴ edge number of G* ≤ edge number of G

E[edge number of H] ≤ 0.5 × edge number of G



Calculating the expected total edge numbe

r for one left path started at one proble

m with m’ edges

Analysis – Average Case

8)

G

H G

’

Boruvka × 2

G*

Sample with p=0.5

Original Problem

Left

Sub-problem

Right Sub-problem

E[edge number of H] ≤ 0.5 × edge number of G



Calculating the expected total edge numbe r for one left path started at one proble m with m’ edges

# of edges

= m’

# of edges

= m’

# of edges ≤ 0.5

× m’

# of edges ≤ 0.5

× m’

Expected total edge number ≤ = 2m’

Analysis – Average Case

 Calculating the 8) expected total edge numbe r for one left path L started at one prob lem with m’ edges

◦Expected total edge number on L ≤ 2m’

• Evaluating the total edge number of all right sub-problems

• E[total edges of all right sub-problem] ≤ n

K.O

.

Analysis – Average Case

8)

G

H G

’

Original Problem

Left

Sub-problem

Right Sub-problem

1. ∵ Boruvka × 2

∴ vertex number of G*

≤ 0.25 × vertex number of G

E[edge number of G’] ≤ 0.5×vertex number of G

 Evaluating the total edge number for all right su b-problems

◦ To prove : E[total edges of all right sub-problem] ≤ n

Boruvka × 2

G*

Sample with p=0.5

Return minimum forest F of H

Delete F-heavy edges from G*

2. Based on lemma 2.1:

E[edge number of G’] ≤ 2 × vertex number of G*

Analysis – Average Case

8)

E[edge number of G’] ≤ 0.5×vertex number of G

 Evaluating the total edge number for all right su b-problems

◦ To prove : E[total edges of all right sub-problem] ≤ n

G

H G

’

Original Problem

Left

Sub-problem

Right Sub-problem

Boruvka × 2

G*

Sample with

p=0.5 # of vertices of sub-

problems ≤ 2×n/4

# of vertices of sub- problems ≤ 4×n/4²

# of vertices of sub- problems ≤ 8×n/4³

# of vertices of sub- problems ≤ 16×n/4⁴

# of edges of right sub-problems ≤ n/2

# of edges of right sub-problems ≤ 2×n/8

# of vertices of original- problems=n

# of edges of right sub- problems ≤ 4×n/(4²×2)

# of edges of right sub- problems ≤ 8×n/(4³×2)

= n

Analysis – Average Case

 Calculating the 8) expected total edge number for on e left path started at one problem with m’ edges

◦Expected total edge number for one left path ≤ 2m’

 Evaluating the total edge number for all right su b-problems

◦E[total edges of all right sub-problem] ≤ n

# of edges

= m’

# of edges

= m’

Expected total edge number

≤ 2m’

Expected total edge number

≤ 2m’

E[processed edges in the original problem and all sub- problems]

=2×(m+n)

Analysis of the Algorithm

 The worst case.

 The expectations running time.

 The probability of the expectations ru nning time.

The Probability of Lineari ty



Theorem 4.3

◦The minimum spanning forest algorithm runs in Ο(m) time with probability 1 – exp(-Ω(m))

The Probability of Lineari ty

  _  



 

 ⁿ

1 i

At E etX_i

e A

X Pr Chernoff Bound:

Given x_i as i.d.d. random variables and 0< i  n, and X is the sum of all x_i, for t > 0, we have

Thus, the probability that less than s successes (each with chance p) within k trials is

   

12 12

) s ( Ω

k t st

k 1 i st tX

p and t

for ,

e

) pe ( e

e E e

s X

Pr ⁱ

























The Probability of Lineari ty



Right Subproblems

◦At most the number of vertices in all right subproblems: n/2 ( proved by th eorem 4.2 )

◦n/2 is the upper bound on the total n umber of heads in nickel-flips

Right Subproblems



The probability

◦It occurs fewer than n/2 heads in a s equence of 3m nickel-tosses



m + n ≦ 3m since n/2 ≦ m



The probability is exp (-Ω(m)) by

a Chernoff bound

The Probability of Lineari ty



Left Subproblem

◦Sequence: every sequence ends up with a tail, that is, HH…HHT

◦The number of occurrences of tails is at most the number of sequences

◦Assume that there are at most m’ edg es in the root problem and in all rig ht subproblems

Left Subproblems

The probability

◦It occurs m’ tails in a sequence of more than 3m’ coin-tosses



The probability is exp (-Ω(m)) by

a Chernoff bound

The Probability of Lineari ty



Combining Right & Left Subproblems

◦The total number of edges is Ο(m) w ith a high-probability bound 1 – exp (-Ω(m))