Graph Algorithms: Minimum Spanning Tree

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Graph Algorithms Minimum Spanning Tree
We are given a weighted, undirected graph G (V,
E), with weight function w E ? R mapping edges
to real valued weights.
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Graph Algorithms Minimum Spanning Tree
We are given a weighted, undirected graph G (V,
E), with weight function w E ? R mapping edges
to real valued weights.
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A spanning tree T (V, E) is a subgraph of G
such that V V and T is a tree.
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Graph Algorithms Minimum Spanning Tree
We are given a weighted, undirected graph G (V,
E), with weight function w E ? R mapping edges
to real valued weights.
1
A spanning tree T (V, E) is a subgraph of G
such that V V and T is a tree.
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A minimum spanning tree is a spanning tree where
the sum of the weights in E' is minimal.
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Graph Algorithms Minimum Spanning Tree

• Kruskals algorithm for finding an MST
• Starting from an empty tree, T, pick the edge e
  with the
• minimum weight, and initialize E e.

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Add edge (1, 3) to E
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Graph Algorithms Minimum Spanning Tree

• Kruskals algorithm for finding an MST
• Starting from an empty tree, T, pick the edge e
  with the
• minimum weight, and initialize E e.

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2. Choose the edge e' not in E' such that, edge
weight of e' in E E' is minimal and no cycle
will be created if e' is added to E. Add e' to
E.
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Add edge (2, 6) to E
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Graph Algorithms Minimum Spanning Tree

• Kruskals algorithm for finding an MST
• Starting from an empty tree, T, pick the edge e
  with the
• minimum weight, and initialize E e.

1
2. Choose the edge e' not in E such that, edge
weight of e' in E E is minimal and no cycle
will be created if e' is added to E. Add e' to
E. Repeat step 2 until all edges are looked at.
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10
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3
8
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4
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Add edge (2, 5) to E
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Graph Algorithms Minimum Spanning Tree

• Kruskals algorithm for finding an MST
• Starting from an empty tree, T, pick the edge e
  with the
• minimum weight, and initialize E e.

1
1
2. Choose the edge e' not in E such that, edge
weight of e' in E E is minimal and no cycle
will be created if e' is added to E. Add e' to
E. Repeat step 2 until all edges are looked at.
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10
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8
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Edge (5, 6) is NOT added, since it creates a
cycle.
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5
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Graph Algorithms Minimum Spanning Tree

• Kruskals algorithm for finding an MST
• Starting from an empty tree, T, pick the edge e
  with the
• minimum weight, and initialize E e.

1
1
2. Choose the edge e' not in E such that, edge
weight of e' in E E is minimal and no cycle
will be created if e' is added to E. Add e' to
E. Repeat step 2 until all edges are looked at.
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5
10
10
1
1
8
3
8
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Add edge (3, 4) to E
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Graph Algorithms Minimum Spanning Tree

• Kruskals algorithm for finding an MST
• Starting from an empty tree, T, pick the edge e
  with the
• minimum weight, and initialize E e.

1
1
2. Choose the edge e' not in E such that, edge
weight of e' in E E is minimal and no cycle
will be created if e' is added to E. Add e' to
E. Repeat step 2 until all edges are looked at.
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5
10
10
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1
8
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8
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Add edge (4, 5) to E
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5
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Graph Algorithms Minimum Spanning Tree

• Kruskals algorithm for finding an MST
• Starting from an empty tree, T, pick the edge e
  with the
• minimum weight, and initialize E e.

1
1
2. Choose the edge e' not in E such that, edge
weight of e' in E E is minimal and no cycle
will be created if e' is added to E. Add e' to
E. Repeat step 2 until all edges are looked
at. Edge(1,4) is NOT added, since it creates a
cycle.
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1
8
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Graph Algorithms Minimum Spanning Tree

• Kruskals algorithm for finding an MST
• Starting from an empty tree, T, pick the edge e
  with the
• minimum weight, and initialize E e.

1
1
2. Choose the edge e' not in E such that, edge
weight of e' in E E is minimal and no cycle
will be created if e' is added to E. Add e' to
E. Repeat step 2 until all edges are looked
at. Edge(2,3) is NOT added, since it creates a
cycle.
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Graph Algorithms Minimum Spanning Tree

• Kruskals algorithm for finding an MST
• Starting from an empty tree, T, pick the edge e
  with the
• minimum weight, and initialize E e.

1
1
2. Choose the edge e' not in E such that, edge
weight of e' in E E is minimal and no cycle
will be created if e' is added to E. Add e' to
E. Repeat step 2 until all edges are looked
at. Edge(1, 2) is NOT added, since it creates a
cycle.
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5
10
10
1
1
8
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8
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4
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6
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5
2
2
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Graph Algorithms Minimum Spanning Tree

• Kruskals algorithm for finding an MST
• Starting from an empty tree, T, pick the edge e
  with the
• minimum weight, and initialize E e.

1
1
2. Choose the edge e' not in E such that, edge
weight of e' in E E is minimal and no cycle
will be created if e' is added to E. Add e' to
E. Repeat step 2 until all edges are looked
at. Done!!
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Sum of edge weights 1 1 1 3 4 10.