Minimum Spanning Tree

1
Minimum Spanning Tree

• Assume a connected, undirected graph G (V, E).
• Weight w(u,v) on each edge (u,v) ? E

2
Minimum Spanning Tree

• A spanning tree whose weight is minimum over all
  spanning trees is called a minimum spanning tree,
  or MST.

3
Growing a Minimum Spanning Tree

• Assume a connected, undirected graph G (V, E).
• Assume a weight function w E ? R.
• Consider greedy generic algorithm that grows
  minimum spanning tree one edge at a time.

4
Generic Algorithm

• GENERIC-MST(G, w)
• A ? ?
• while A does not form a spanning tree
• do find an edge (u, v) that is safe for A
• A ? A ? (u, v)
• return A

5
Terminology

• A cut (S, V-S) of an undirected graph G (V, E)
  is a partition of V.
• Let C Si be a collection of nonempty sets. C
  forms a partition of a set S if
• the sets are pairwise disjoint, and
• their union is S.
• I.e., each element of S appears in exactly one Si
  (where Si ? C).

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Terminology (continued)

• Edge (u, v) ? E crosses the cut (S, V-S) if one
  of its endpoints is in S and the other is in V-S.
• A cut respects a 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.
• More generally, an edge is a light edge
  satisfying a given property if its weight is the
  minimum of any edge satisfying the property.

7
Safe Edges Theorem

• Let G (V, E) be a connected, undirected graph
  with a real-valued weight function w defined on
  E. Let A be a subset of E that is included in
  some minimum spanning tree for G, let (S, V-S) be
  any cut of G that respects A, and let (u, v) be a
  light edge crossing (S, V-S). Then edge (u, v) is
  safe for A.

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Proof of Safe Edges Theorem
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Corollary to Safe Edges Theorem

• Let G (V, E) be a connected, undirected graph
  with a real-valued weight function w defined on
  E. Let A be a subset of E that is included in
  some minimum spanning tree for G, and let C
  (VC, EC) be a connected component (tree) in the
  forest 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.

10
Kruskals Algorithm

• Let C1 and C2 be two trees.
• Let edge (u, v) be the edge of least weight that
  connects the two trees.
• Since (u, v) is a light edge connecting C1 to
  some other tree, it is a safe edge for C.

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Kruskals Algorithm

• A ? Ø
• for each vertex v ?VG
• do MAKE-SET(v)
• sort E into nondecreasing order by weight
• for each (u, v) ?E, taken in nondecreasing order
• do if FIND-SET(u) ? FIND-SET(v)
• then A ? A ? (u, v)
• UNION(u, v)
• return A

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Example of Kruskals Algorithm
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Example (continued)
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Example (continued)
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Example (continued)
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Example (continued)
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Example (continued)
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Example (continued)
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Prims Algorithm

• Let G be a connected graph.
• Let r be vertex that will serve as root of
  minimum cost spanning tree.
• At each step, add to the tree A a light edge that
  connects A to an isolated vertex of GA (V, A).
• By corollary, this adds only safe edges to A.
• Choose edge that adds minimum amount possible to
  trees weight.

20
Prims Algorithm

• MST-PRIM (G, w, r)
• for each u ?VG
• do keyu ? 8
• pu ? NIL
• keyr ? 0
• Q ? VG
• while Q ? ?
• do u ? EXTRACT-MIN(Q)
• for each v ?Adju
• do if v ?Q and w(u, v) lt keyv
• then pv ? u
• keyv ? w(u, v)

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Example of Prims Algorithm
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Example (continued)
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Example (continued)
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Example (continued)
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Example (continued)