min spanning tree

8/9/2019 Min Spanning Tree

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Minimum Spanning Tree,

Disjoint-set Union,

Amortized analysis


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p2.

Minimum Spanning Tree

We model the wiring problem with a connected,

undirected graphs G = ( V, E) where V is the set

of pins, E is the set possible interconnections

between pairs of pins, and for each edge (u, v )

R, we have a weight w ( u, v ) specifying the cost

( amount of wire needed) to connect u and v.


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p3.


We then wish to find an acyclic subset T E that connects

all of the vertices and whose total weightw ( T ) = w ( u, v )

( u, v ) T

is minimized.

Since T is acyclic and connects all of the vertices, it must

form a tree, which we a spanning tree since it spans the

graph.


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p4.


G=(V,E): connected and undirected

w: ER, weight function

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p5.


The algorithm manages a set Abe that is

always a subset of some minimum spanning

tree. At each step, an edge (u,v) is

determined that can be added to A without

violating this invariant, in the sense that

A U {u, v}isalso a subset of a MST. We call

such an edge a safe for A, since it can be

safely added to Awithout destroying the

invariant.

A


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p6.


,

,

,

Generic-MST( ){

while A does not form a spanning tree

do find an edge ( ) that is safefor ;{( )};

G w

A

u vA A A u v

U

return}

A


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p7.


A cut (S, V-S) of an undirected graph G=(V, E) is apartition of V

An edge crosses the cut (S, V-S) if one of itsendpoints is in S and the other is in V-S

A cut respects the set A of edges if no edge in A crosses the

cut An edge is a light edge crossing a cut if its weight is the

minimum of any edge crossing the cut

What is the light edge in the above graph ?

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V-S

(u,v) E


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p8.


Thm1:

G=(V,E): connected, undirected

w: real-valued weight function on E

A: a subset of E and is included in some MST

(S, V-S): any cut of G and respects A

(u,v): a light edge crossing (S, V-S)

Then (u,v) is safe for A.

pf:

yu

x

v

S : {O}

V-S: {O}

A: { - }

T = T {(x,y)} U {(u,v)}

original MST


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p9.


w(T) = w(T) w(x,y) + w(u,v)

= w(T)

Thus, T is a MST

A T, and (x,y) AA {(u,v)} T'T' is a MST and (u, v) is safe for A

U


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p10.


Cor2:

G=(V,E): connected, undirected

w = real-valued weight function

and A is in some MST

C: a connected component in GA=(V,A)

if (u,v) is a light edge connecting C to some other

component in GA, then (u,v) is safe for A

Pf: The cut(C, V-C) respects A, and (u,v) is therefore a lightedge for this cut

A E

v

u

V-C

C


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p11.

Disjoint sets

{ , , ,....., }, ,

{{ }}, { , } { }

n i j

i j i j i j

i i

S S S S S S S i j

x S S xS S S S S S S S

x S S x S

= =

if

Operations:

Make-Set( )Union( )

Find-Set( ) return s.t.

I

U

U U

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p12.

Eg. Minimum spanning tree

G=(V,E): connected, undirected, edge-weighted graph

w: E R

,

Kruskals' algorithm:

T=for each

do Make-Set( )

sort E by increasing edge weight

for each ( ) (in sorted order)

do if Find-Set( )

v V

v

w

u v E

u

{ , }Find-Set( )

then T T ( )Union(Find-Set( ), Find-Set( ))

vu v

u v

U


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p13.

Kruskals algorithm8 7(a)

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414

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(b)


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p14.

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(c)

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(d)


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p15.

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(e)

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(f)


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p16.

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(g)

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(h)


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p17.

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(i)

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(j)


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p18.

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(k)

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(l)


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p19.

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(m)

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(n)


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p20.

Prims algorithm:

MTS-Prim(G, w, r)

{ Q V[G] /* Q: priority queue */for each u Q

do key[u][u] NIL

key[r] 0

while Qdo u Extrac

t-Min(Q)for each v adj[u]

do if v Q and w(u,v) < key[v]then [v] u

key[v] w(u,v)}

O(V)

O(V lg V)

O(E)

lg V

O(lg V), Decrease-key involves


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p21.

Analysis Binary heap: O(V lg V + E lg V)

= O(E lg V)

Fibonacci heap:

Decrease-key: O(1) amortized time

O(V lg V + E)


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p22.

Prims algorithm

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(a)

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(b)

r


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p23.

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(c)

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(d)


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p24.

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(e)

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(f)


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p25.

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(h)


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p26.

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(i)

min spanning tree

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