Minimum spanning tree

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Minimum spanning tree

Jean Cousty MorphoGraph and Imagery

2011

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Outline of the lecture

1 Minimum spanning tree

2 Cut theorem for MST

3 Kruskal algorithm

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Minimum spanning tree

Edge weighted graph

G denotes a connected undirected graph (E,Γ)

W denotes a map from Γ into R

W(u) is called the weight (or value) of u (∀u ∈Γ) The pair (G,W) is called an(edge-) weighted graph

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Edge weighted graph

G denotes a connected undirected graph (E,Γ) W denotes a map from Γ into R

W(u) is called the weight (or value) of u (∀u ∈Γ)

The pair (G,W) is called an(edge-) weighted graph

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Edge weighted graph

G denotes a connected undirected graph (E,Γ) W denotes a map from Γ into R

W(u) is called the weight (or value) of u (∀u ∈Γ) The pair (G,W) is called an(edge-) weighted graph

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Spanning subgraph

Definition

A spanning subgraph ofG is a graph(E,Γ⁰) such thatΓ⁰ ⊆Γ

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Weight of a subgraph

Definition

Let G⁰ = (E,Γ⁰) be a spanning subgraph of G

The weight ofG⁰, denoted by W(G⁰), is the sum of the weights of the edges that belong to Γ⁰

W(G) =P

{W(u)|u∈Γ⁰}

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Minimum spanning tree (MST)

Definition

A minimum spanning tree (for (G,W))is a connected spanning subgraph G⁰= (E,Γ⁰) of G whose weight is minimum:

for any connected spanning subgraph G⁰⁰of G , if W(G⁰⁰)≤W(G⁰) then W(G⁰⁰) =W(G⁰)

Property

If G⁰ is an MST, then G⁰ is a tree

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Minimum spanning tree (MST)

Definition

A minimum spanning tree (for (G,W))is a connected spanning subgraph G⁰= (E,Γ⁰) of G whose weight is minimum:

for any connected spanning subgraph G⁰⁰of G , if W(G⁰⁰)≤W(G⁰) then W(G⁰⁰) =W(G⁰)

Property

If G⁰ is an MST, then G⁰ is a tree

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Illustration

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Maximum spanning tree (MaxST)

Definition

A maximum spanning tree (for (G,W))is a spanning tree(E,Γ⁰) of maximum weight:

for any spanning tree G⁰⁰, if W(G⁰⁰)≥W(G⁰) then W(G⁰⁰) =W(G⁰)

Property

G⁰ is a MaxST of(G,W)if and only if G⁰ is a MST of (G,−W)

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Maximum spanning tree (MaxST)

Definition

A maximum spanning tree (for (G,W))is a spanning tree(E,Γ⁰) of maximum weight:

for any spanning tree G⁰⁰, if W(G⁰⁰)≥W(G⁰) then W(G⁰⁰) =W(G⁰)

Property

G⁰ is a MaxST of(G,W)if and only if G⁰ is a MST of (G,−W)

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MST algorithm and safe edges

Generic MST Algorithm ( Data: (E,Γ,W) ;

Result: Γ⁰ such that (E,Γ⁰) is an MST)

n :=|E|; Γ⁰:=∅ ;k := 0 ; While k <n−1do

Find an edgeuthat issafefor Γ⁰ Γ⁰ := Γ⁰∪ {u}

Definition

Let(E,Γ⁰) be a subgraph of a MST Let u∈Γ\Γ⁰

u issafe forΓ⁰ if(E,Γ⁰∪ {u}) is also a subgraph of an MST

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MST algorithm and safe edges

Generic MST Algorithm ( Data: (E,Γ,W) ;

Result: Γ⁰ such that (E,Γ⁰) is an MST)

n :=|E|; Γ⁰:=∅ ;k := 0 ; While k <n−1do

Find an edgeuthat issafefor Γ⁰ Γ⁰ := Γ⁰∪ {u}

Definition

Let (E,Γ⁰) be a subgraph of a MST Let u ∈Γ\Γ⁰

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Cut theorem for MST

Cut and MST

Question

How to efficiency detect an edge that is safe?

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Cut theorem for MST

Cut

Definition

Let Γ⁰ ⊆Γ

The cut induced by Γ⁰ is the subset of Γthat contains the edges that link two distinct connected components of the graph (E,Γ⁰)

A subset C ofΓ is a cutif there existsΓ⁰⊆Γsuch that C is the cut induced byΓ⁰

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Cut theorem for MST

Cut

Definition

Let Γ⁰ ⊆Γ

The cut induced by Γ⁰ is the subset of Γthat contains the edges that link two distinct connected components of the graph (E,Γ⁰) A subset C ofΓ is a cutif there existsΓ⁰⊆Γsuch that C is the cut induced by Γ⁰

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Cut theorem for MST

Theorem

Let (E,Γ⁰) be a subgraph of an MST

Let C be a cut that is a subset of the cut induced byΓ⁰ Let u∈C be an edge of minimum weight in C (i.e.,∀v ∈C , W(u)≤W(v))

Then, u is safe forΓ⁰

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Cut theorem for MST

Theorem

Let C be a cut that is a subset of the cut induced by Γ⁰

Let u∈C be an edge of minimum weight in C (i.e.,∀v ∈C , W(u)≤W(v))

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Cut theorem for MST

Theorem

Let C be a cut that is a subset of the cut induced by Γ⁰ Let u ∈C be an edge of minimum weight in C (i.e.,∀v ∈C , W(u)≤W(v))

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Cut theorem for MST

Theorem

Let C be a cut that is a subset of the cut induced by Γ⁰ Let u ∈C be an edge of minimum weight in C (i.e.,∀v ∈C , W(u)≤W(v))

Then, u is safe for Γ⁰

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Kruskal algorithm

MST computation

KRUSKAL Algorithm ( Data: a connected graph (E,Γ,W) ; Result: Γ⁰ such that (E,Γ⁰) is an MST)

n :=|E|; Γ⁰:=∅ ;k := 0 ;i := 1;

Sort the edges in Γ in increasing order of weight:

Find the sequence (u₁, . . . ,u_m) such that Γ ={u₁, . . . ,u_m} and W(ui−1)≤W(ui),∀i ∈ {2, . . . ,m}

While k <n−1Do {xi,yi}:=ui ;

Ifx_i ∈/ the connected component containingy_i in (E,Γ⁰)Then Γ⁰:= Γ⁰∪ {ui};

k:=k+ 1;

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Kruskal algorithm

Execution example

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Kruskal algorithm

Proof of the algorithm

When the testIf is positive, the edgeu is safe for Γ⁰

This can be proved by the cut theorem applied to the cut induced by Γ⁰

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Kruskal algorithm

Complexity of Kruskal Algorithm

Complexity

Complexity of a sort (O(mlog(m))for quick sort)

Complexity of theWhile loop: O(m)

Complexity of theIf test: O(m×n)(connected component algorithm seen in the course)

Overall complexity: O(m×n+mlog(m))

The complexity of theIftest can be reduced to quasi linear time using Tarjan’s union of disjoint set algorithm

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Kruskal algorithm

Complexity of Kruskal Algorithm

Complexity

Complexity of a sort (O(mlog(m))for quick sort) Complexity of the While loop: O(m)

Complexity of theIf test: O(m×n)(connected component algorithm seen in the course)

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Kruskal algorithm

Complexity of Kruskal Algorithm

Complexity

Complexity of the If test: O(m×n)(connected component algorithm seen in the course)

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Kruskal algorithm

Complexity of Kruskal Algorithm

Complexity

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Kruskal algorithm

Complexity of Kruskal Algorithm

Complexity

The complexity of the Iftest can be reduced to quasi linear time using Tarjan’s union of disjoint set algorithm

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Kruskal algorithm

Prim and Boruvka Algorithm

These are other algorithms for computing an MST Also a version of the generic algorithm

Their invariants also rely on the cut theorem

Read http://www.ics.uci.edu/˜eppstein/161/960206.html