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Minimum-Spanning Trees

Minimum- Spanning TreesMinimum- Spanning Trees1. Concrete example: computer connection

2. Definition of a Minimum- Spanning Tree

3. The Crucial Fact about Minimum- Spanning Trees

4. Algorithms to find Minimum- Spanning Trees - Kruskal‘s Algorithm - Prim‘s Algorithm - Barůvka‘s Algorithm

Minimum-Spanning Trees

Imagine: You wish to connect all the computers in an office building using the least amount of cable

a weighted graph problem !!

- Each vertex in a graph G represents a computer- Each edge represents the amount of cable needed to connect all computers

Concrete exampleConcrete example

Minimum-Spanning Tree

We are interested in:

Finding a tree T that contains all the verticesof a graph G spanning treespanning treeand has the least total weight over allsuch trees minimum-spanning treeminimum-spanning tree (MST)(MST)

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Crucial Fact

min-weight“bridge“ edge

The Crucial Fact about MSTThe Crucial Fact about MST

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Crucial Fact

The Crucial Fact about MST-

The basis of the following algorithms

Proposition: Let G = (V,E) be a weighted graph, and let and

be two disjoint nonempty sets such that .

Furthermore, let e be an edge with minimum weight from

among those with one vertex in and the other in .

There is a minimum- spanning tree T that has e as one of

its edges.

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Crucial Fact

The Crucial Fact about MST-

The basis of the following algorithms

Justification: There is no minimum- spanning tree that has e as one of

of ist edges. The addition of e must create a cycle.

There exists an edge f (one endpoint in the other in ).

Choose: . By removing f from , a

spanning tree is created, whose total weight is no more than

before. A new MSTcontaining e

Contradiction!!!

There is a MST containing e after all!!!

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MST-Algorithms

Input: A weighted connected graph G = (V,E) with n vertices and m edges

Output: A minimum- spanning tree T

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MST-AlgorithmsMST-Algorithms

Kruskal's Algorithm

Kruskal‘s AlgorithmKruskal‘s Algorithm

1. Each vertex is in its own cluster

2. Take the edge e with the smallest weight - if e connects two vertices in different clusters, then e is added to the MST and the two clusters, which are connected by e, are merged into a single cluster - if e connects two vertices, which are already in the same cluster, ignore it

3. Continue until n-1 edges were selected

Kruskal's Algorithm

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Kruskal's Algorithm

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cycle!!

Kruskal's Algorithm

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Kruskal's Algorithm

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minimum- spanning tree

Kruskal's Algorithm

The correctness of Kruskal‘s AlgorithmThe correctness of Kruskal‘s Algorithm

Crucial Fact about MSTs

Running time: O ( m log n )

By implementing queue Q as a heap, Q could be initialized in O ( m ) time and a vertex could be extracted in each iteration in O ( log n ) time

Kruskal's Algorithm

Input: A weighted connected graph G with n vertices and m edges Output: A minimum-spanning tree T for G

for each vertex v in G do Define a cluster C(v) {v}.Initialize a priority queue Q to contain all edges in G, using weights as keys.T while Q do Extract (and remove) from Q an edge (v,u) with smallest weight. Let C(v) be the cluster containing v, and let C(u) be the cluster containing u. if C(v) C(u) then Add edge (v,u) to T. Merge C(v) and C(u) into one cluster, that is, union C(v) and C(u).return tree T

Code FragmentCode Fragment

Prim's Algorithm

Prim‘s AlgorithmPrim‘s Algorithm

1. All vertices are marked as not visited

2. Any vertex v you like is chosen as starting vertex and is marked as visited (define a cluster C)

3. The smallest- weighted edge e = (v,u), which connects one vertex v inside the cluster C with another vertex u outside of C, is chosen and is added to the MST.

4. The process is repeated until a spanning tree is formed

Prim's Algorithm

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Prim's Algorithm

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We could delete these edges because of Dijkstra‘s label D[u] for each vertex outside of the cluster

Prim's Algorithm

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Prim's Algorithm

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Prim's Algorithm

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Prim's Algorithm

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Prim's Algorithm

The correctness of Prim‘s AlgorithmThe correctness of Prim‘s Algorithm



By implementing queue Q as a heap, Q could be initialized in O ( m ) time and a vertex could be extracted in each iteration in O ( log n ) time

Baruvka's Algorithm

BarBarůůvka‘s Algorithmvka‘s Algorithm

1. For all vertices search the edge with the smallest weightof this vertex and mark these edges

2. Search connected vertices (clusters) and replace them by a “new“ vertex (cluster)

3. Remove the cycles and, if two vertices are connected by more than one edge, delete all edges except the “cheapest“

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Baruvka's Algorithm

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Baruvka's Algorithm

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Baruvka's Algorithm

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Baruvka's Algorithm

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Baruvka's Algorithm

The correctness of BarThe correctness of Barůvkaůvka‘s Algorithm‘s Algorithm



The number of edges is at least reduced by half in each step.

Number of steps: O ( log n )

Comparison

ComparisonComparison

Kruskal‘s, Prim‘s, and Borůvka‘s

algorithm

Comparison

ComparisonComparison

Although each of the above algorithms has the same worth-case running time, each one achieves this running time using different data structures and different approaches to build the MST.

there is no clear winner among these three algorithms

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