minimum spanning trees. a b d f g e c a b d f g e c
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Minimum Spanning Trees
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Minimum spanning tree weight = 0
Systematic Approach:
-Grow a spanning tree A
-Always add a safe edge to A that we know belongs to some minimal spanning tree
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There must go exactly one edge from b to the minimum spanning tree. Which edge should we choose?Are all three a possibility (maybe suboptimal)?Could the 7 edge be optimal to choose?What about the two 5 edges?
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There must go exactly one edge from b to the spanning tree. Which edge should we choose?Are all three a possibility? Yes, every vertex will connect to the tree Could the 7 edge be optimal to choose? No, we must connect with a light edgeWhat about the two 5 edges? The red is possible, but they both are !!
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R’’’
Kruskal’s Algorithm
Idea:
Rethink the previous algorithm from vertices to edges
-Initially each vertex forms a singleton tree
-In each iteration add the lightest edge connecting two trees in the forest.
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R’’’
MSTkruskal(G,w)
1 A Ø
2 for each vertex v V[G]
3 do MakeSet(v)
4 sort the edges of E into nondecreasing order by weight w
5 for each edge (u,v) E, taken in nondecreasing order by weight
6 do if FindSet(u) FindSet(v)
7 then A A {(u,v)}
8 Union(u,v)
9 return A
MSTkruskal(G,w)
1 A Ø
2 for each vertex v V[G]
3 do MakeSet(v)
4 sort the edges of E into nondecreasing order by weight w
5 for each edge (u,v) E, taken in nondecreasing order by weight
6 do if FindSet(u) FindSet(v)
7 then A A {(u,v)}
8 Union(u,v)
9 return A
Time?
MSTkruskal(G,w)
1 A Ø
2 for each vertex v V[G]
3 do MakeSet(v)
4 sort the edges of E into nondecreasing order by weight w
5 for each edge (u,v) E, taken in nondecreasing order by weight
6 do if FindSet(u) FindSet(v)
7 then A A {(u,v)}
8 Union(u,v)
9 return A
Time?
O(E*lgE)
MSTkruskal(G,w)
1 A Ø
2 for each vertex v V[G]
3 do MakeSet(v)
4 sort the edges of E into nondecreasing order by weight w
5 for each edge (u,v) E, taken in nondecreasing order by weight
6 do if FindSet(u) FindSet(v)
7 then A A {(u,v)}
8 Union(u,v)
9 return A
Time?
O(E*lgE)
Theorem 21.13: m FindSet, Union, and MakeSet operations of which n are MakeSet operations takes O(m*α(n))
MSTkruskal(G,w)
1 A Ø
2 for each vertex v V[G]
3 do MakeSet(v)
4 sort the edges of E into nondecreasing order by weight w
5 for each edge (u,v) E, taken in nondecreasing order by weight
6 do if FindSet(u) FindSet(v)
7 then A A {(u,v)}
8 Union(u,v)
9 return A
Time?
O(E*lg(E))
O(V+E*α(V)) =O(V+E*lg(V)) = O(V+E*lg(E)) =O(E*lg(E))
MSTkruskal(G,w)
1 A Ø
2 for each vertex v V[G]
3 do MakeSet(v)
4 sort the edges of E into nondecreasing order by weight w
5 for each edge (u,v) E, taken in nondecreasing order by weight
6 do if FindSet(u) FindSet(v)
7 then A A {(u,v)}
8 Union(u,v)
9 return A
O(E*lg(E))
O(E*lg(E))
O(V+E*α(V)) =O(V+E*lg(V)) = O(V+E*lg(E)) =O(E*lg(E))
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Obviously tree. Which edge should we choose?
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