Minimum Spanning Tree

Transcription

1 Minimum Spanning Tree Section 4.5 Minimum Spanning Tree p Minimum Spanning Tree Given a connected graph G = (V,E) with positive edge-costs c e, an MST is a sub-graph (V,T) such that T is a spanning tree with the minimum total edgecost 1

2 Minimum Spanning Tree p Minimum Spanning Tree Given a connected graph G = (V,E) with positive edge-costs c e, an MST is a sub-graph (V,T) such that T is a spanning tree with the minimum total edgecost Applications p Network Design Telephone, electrical, water, cable, computer, road Steiner network (connect only a subset of nodes) p Indirect applications Max bottleneck paths Error correction Data compression 2

3 Greedy algorithms for MST p Start with T = Φ. Consider edges in ascending order of their cost. Insert edge in T unless doing so would create a cycle. Greedy algorithms for MST p Start with T = Φ. Consider edges in ascending order of their cost. Insert edge in T unless doing so would create a cycle. p Start with T = E. Consider edges in descending order of cost. Delete e from T unless doing so would disconnect T. 3

4 Greedy algorithms for MST p Start with T = Φ. Consider edges in ascending order of their cost. Insert edge in T unless doing so would create a cycle. p Start with T = E. Consider edges in descending order of cost. Delete e from T unless doing so would disconnect T. p Start with some root node s and greedily grow a tree from s outward. Let the current set of nodes in the tree be S. At each step, add the cheapest edge to T such that e has one end-point in S and another outside S. All three are optimal! 4

5 All three are optimal! p [Kruskal s Algo] Start with T = Φ. Consider edges in ascending order of their cost. Insert edge in T unless doing so would create a cycle. p [Reverse-Delete Algo] Start with T = E. Consider edges in descending order of cost. Delete e from T unless doing so would disconnect T. p [Prim s Algo] Start with some root node s and greedily grow a tree from s outward. Let the current set of nodes in the tree be S. At each step, add the cheapest edge to T such that e has one end-point in S and another outside S. Demo p l 5

6 Snapshot of Prim sand Kruskal salgos Analysis p When is it safe to include an edge in the MST? p When can we guarantee an edge is not in the MST? 6

7 Analysis p When is it safe to include an edge in the MST? Whenever an edge satisfies the cut property p When can we guarantee an edge is not in the MST? Whenever an edge satisfies the cycle property Cycles p Cycle A cycle is a set of edges of the form (a,b), (b,c), (c,d),..., (z,a) Path = Cycle = {1, 2}, {2, 3}, {3, 4}, {4, 5}, {5, 6}, {6, 1} 7 8 7

8 Cuts p Cut The cut induced by a subset of nodes S is the set of all arcs with exactly one endpoint in S S = {4, 5, 8} Cut = {5, 6}, {5, 7}, {3, 4}, {3, 5}, {7, 8} 7 8 Cut-Cycle Intersection p [CCC: Cut-Cycle Claim] A cycle and a cut intersect in an even number of edges. 8

9 Cycle-Cut Intersection Intersection = {3, 4}, {5, 6} 7 8 S Cut and Cycle Property p [4.17: Cut Property] Let S be any (proper) subset of nodes, and let e be the min cost edge with exactly one point in S. Then every MST contains e. p [4.20: Cycle Property] Let C be any cycle, and let e be the max cost edge belonging to C. Then no MST contains e. 9

10 Cycle and Cut property 1 2 g 3 6 f e 8 Simplifying assumption p All edge costs c e are distinct and positive (> 0) 10

11 Incorrect proof of the Cut property p Let e = (v,w) be the minimum cost edge with one end-point in S and another in V - S plet T be a minimum spanning tree that does not contain e p T must have an edge e with one end-point in S and another in V S (why?) p Since e is the min cost edge with this property, c e < c e p Create a new spanning tree T = T {e } {e} p Cost of T < Cost of T (a contradiction!) Incorrect proof of the cut property 11

12 Proof of the cut property p Let e = (v,w) be the minimum cost edge with one end-point in S and another in V - S plet T be a minimum spanning tree that does not contain e p Create T = T {e}; this creates a cycle C in T which contains the edge e p C must have another edge e e with exactly one end-point in S (from [cut-cycle claim]) p Since e is the min cost edge with this property, c e < c e p Create a new spanning tree T = T {e } {e} p Cost of T < Cost of T (a contradiction!) Exchange argument for cut property 12

13 Proof of cycle property p Let C be a cycle and let e = (v,w) be the max cost edge in C. Let T be an MST containing e p Delete e from T. This creates two components S and V-S (each of which contains an end-point of e) p C must have another edge e e with exactly one end-point in S (from [cut-cycle claim]) p Since e is the max cost edge in cycle C, c e < c e p Create a new spanning tree T = T {e} {e } p Cost of T < Cost of T (a contradiction!) Exchange argument for cycle property 13

14 Optimality of Prim s, Kruskal s and Reverse-Delete Algos p All three algos produce a spanning tree p Prim s and Kruskal s algos always add only those edges which satisfy the cut property. Hence they produce MSTs p Reverse-Delete algo always deletes edges which satisfies cycle property. Hence it produces an MST. Implementing Prim s algorithm 14

15 Implementing Prim s algorithm Prim s algorithm d (v) = min e=(u,v):u S l e 15