Minimum-weight spanning trees
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1 Spannin Trees Minimum-weit spannin trees Motivation: Create a connected network as ceaply as possible. Tink: Settin up electrical rid or road network.
2 Spannin Trees Minimum-weit spannin trees Motivation: Create a connected network as ceaply as possible. Tink: Settin up electrical rid or road network. Some connections are ceaper tan oters.
3 Spannin Trees Minimum-weit spannin trees Motivation: Create a connected network as ceaply as possible. Tink: Settin up electrical rid or road network. Some connections are ceaper tan oters. Only need to minimally connect te vertices.
4 Spannin Trees Minimum-weit spannin trees Motivation: Create a connected network as ceaply as possible. Tink: Settin up electrical rid or road network. Some connections are ceaper tan oters. Only need to minimally connect te vertices. Deinition: A weited rap consists o a rap G =(V, E) and weit unction w : E R deined on te edes o G. Te weit o a subrap H o G is te sum o te edes in H. Example. a 2 b 7 c e 6 d a 2 b 7 c e 6 d a 2 b 7 c e 6 d
5 Spannin Trees Minimum-weit spannin trees Motivation: Create a connected network as ceaply as possible. Tink: Settin up electrical rid or road network. Some connections are ceaper tan oters. Only need to minimally connect te vertices. Deinition: A weited rap consists o a rap G =(V, E) and weit unction w : E R deined on te edes o G. Te weit o a subrap H o G is te sum o te edes in H. Example. a 2 b 7 c e 6 d a 2 b 7 c e 6 d a 2 b 7 c e 6 d Deinition: For a rap G,aspannin tree T is a subrap o G wic is a tree and contains every vertex o G.
6 Spannin Trees Minimum-weit spannin trees Motivation: Create a connected network as ceaply as possible. Tink: Settin up electrical rid or road network. Some connections are ceaper tan oters. Only need to minimally connect te vertices. Deinition: A weited rap consists o a rap G =(V, E) and weit unction w : E R deined on te edes o G. Te weit o a subrap H o G is te sum o te edes in H. Example. a 2 b 7 c e 6 d a 2 b 7 c e 6 d a 2 b 7 c e 6 d Deinition: For a rap G,aspannin tree T is a subrap o G wic is a tree and contains every vertex o G. Goal: For a weited rap G, ind a minimum-weit spannin tree.
7 Spannin Trees Kruskal s aloritm Kruskal s Aloritm inds a minimum-weit spannin tree in a weited rap. 1 Initialization: Order te edes rom lowest to iest weit: w(e 1 ) w(e 2 ) w(e 3 ) w(e k ).
8 Spannin Trees Kruskal s aloritm Kruskal s Aloritm inds a minimum-weit spannin tree in a weited rap. 1 Initialization: Order te edes rom lowest to iest weit: w(e 1 ) w(e 2 ) w(e 3 ) w(e k ). 2 Step 1: DeineT = {e 1 } and row te tree as ollows:
9 Spannin Trees Kruskal s aloritm Kruskal s Aloritm inds a minimum-weit spannin tree in a weited rap. 1 Initialization: Order te edes rom lowest to iest weit: w(e 1 ) w(e 2 ) w(e 3 ) w(e k ). 2 Step 1: DeineT = {e 1 } and row te tree as ollows: 3 Step i: Determine i addin e i to T would create a cycle. I not, add e i to te set T. I so, do notin.
10 Spannin Trees Kruskal s aloritm Kruskal s Aloritm inds a minimum-weit spannin tree in a weited rap. 1 Initialization: Order te edes rom lowest to iest weit: w(e 1 ) w(e 2 ) w(e 3 ) w(e k ). 2 Step 1: DeineT = {e 1 } and row te tree as ollows: 3 Step i: Determine i addin e i to T would create a cycle. I not, add e i to te set T. I so, do notin. I you ave a spannin tree, STOP. You ave a m.w.s.t. Oterwise, continue onto step i + 1.
11 Spannin Trees Kruskal s aloritm Example. Run Kruskal s aloritm on te ollowin rap: a 2 b 7 c e 6 d
12 Spannin Trees Kruskal s aloritm Example. Run Kruskal s aloritm on te ollowin rap: a e 6 4 e 1 2 e 4 3 e 10 7 b e e 2 2 e 11 7 c e 12 9 e 5 3 e 3 3 e 7 5 e 8 5 e 9 6 d
13 Spannin Trees Kruskal s aloritm Example. Run Kruskal s aloritm on te ollowin rap: a e 6 4 e 1 2 e 4 3 e 10 7 b e e 2 2 e 11 7 c e 12 9 e 5 3 e 3 3 e 7 5 e 8 5 e 9 6 d
14 Spannin Trees Kruskal s aloritm Example. Run Kruskal s aloritm on te ollowin rap: a e 6 4 e 1 2 e 4 3 e 10 7 b e e 2 2 e 11 7 c e 12 9 e 5 3 e 3 3 e 7 5 e 8 5 e 9 6 d
15 Spannin Trees Kruskal s aloritm Example. Run Kruskal s aloritm on te ollowin rap: a e 6 4 e 1 2 e 4 3 e 10 7 b e e 11 7 c e 2 2 e 12 9 e 5 3 e 3 3 e 7 5 e 8 5 e 9 6 d
16 Spannin Trees Kruskal s aloritm Example. Run Kruskal s aloritm on te ollowin rap: a e 6 4 e 1 2 e 4 3 e 10 7 b e e 11 7 c e 2 2 e 12 9 e 5 3 e 3 3 e 7 5 e 8 5 e 9 6 d
17 Spannin Trees Kruskal s aloritm Example. Run Kruskal s aloritm on te ollowin rap: a e 6 4 e 1 2 e 4 3 e 10 7 b e e 11 7 c e 2 2 e 12 9 e 5 3 e 3 3 e 7 5 e 8 5 e 9 6 d
18 Spannin Trees Kruskal s aloritm Example. Run Kruskal s aloritm on te ollowin rap: a e 6 4 e 1 2 e 4 3 e 10 7 b e e 11 7 c e 2 2 e 12 9 e 5 3 e 3 3 e 7 5 e 8 5 e 9 6 d
19 Spannin Trees Kruskal s aloritm Example. Run Kruskal s aloritm on te ollowin rap: a e 6 4 e 1 2 e 4 3 e 10 7 b e e 11 7 c e 2 2 e 12 9 e 5 3 e 3 3 e 7 5 e 8 5 e 9 6 d
20 Spannin Trees Kruskal s aloritm Example. Run Kruskal s aloritm on te ollowin rap: a e 6 4 e 1 2 e 4 3 e 10 7 b e e 11 7 c e 2 2 e 12 9 e 5 3 e 3 3 e 7 5 e 8 5 e 9 6 d
21 Spannin Trees Kruskal s aloritm Example. Run Kruskal s aloritm on te ollowin rap: a e 6 4 e 1 2 e 4 3 e 10 7 b e e 11 7 c e 2 2 e 12 9 e 5 3 e 3 3 e 7 5 e 8 5 e 9 6 d
22 Spannin Trees Kruskal s aloritm Example. Run Kruskal s aloritm on te ollowin rap: a e 6 4 e 1 2 e 4 3 e 10 7 b e e 11 7 c e 2 2 e 12 9 e 5 3 e 3 3 e 7 5 e 8 5 e 9 6 d
23 Spannin Trees Kruskal s aloritm Example. Run Kruskal s aloritm on te ollowin rap: a e 6 4 e 1 2 e 4 3 e 10 7 b e e 11 7 c e 2 2 e 12 9 e 5 3 e 3 3 e 7 5 e 8 5 e 9 6 d
24 Spannin Trees Kruskal s aloritm Example. Run Kruskal s aloritm on te ollowin rap: a e 6 4 e 1 2 e 4 3 e 10 7 b e e 11 7 c e 2 2 e 12 9 e 5 3 e 3 3 e 7 5 e 8 5 e 9 6 d
25 Spannin Trees Kruskal s aloritm Example. Run Kruskal s aloritm on te ollowin rap: a e 6 4 e 1 2 e 4 3 e 10 7 b e e 11 7 c e 2 2 e 12 9 e 5 3 e 3 3 e e e 9 6 d
26 Spannin Trees Kruskal s aloritm Example. Run Kruskal s aloritm on te ollowin rap: a e 6 4 e 1 2 e 4 3 e 10 7 b e e 11 7 c e 2 2 e 12 9 e 5 3 e 3 3 e e e 9 6 d
27 Spannin Trees Kruskal s aloritm Example. Run Kruskal s aloritm on te ollowin rap: a e 6 4 e 1 2 e 4 3 e 10 7 b e e 11 7 c e 2 2 e 12 9 e 5 3 e 3 3 e 7 5 e 8 5 e 9 6 d
28 Spannin Trees Kruskal s aloritm Example. Run Kruskal s aloritm on te ollowin rap: a e 6 4 e 1 2 e 4 3 e 10 7 b e e 11 7 c e 2 2 e 12 9 e 5 3 e 3 3 e e e 9 6 d
29 Spannin Trees Kruskal s aloritm Example. Run Kruskal s aloritm on te ollowin rap: a e 6 4 e 1 2 e 4 3 e 10 7 b e e 11 7 c e 2 2 e 12 9 e 5 3 e 3 3 e e e 9 6 d
30 Spannin Trees Kruskal s aloritm Example. Run Kruskal s aloritm on te ollowin rap: a e 6 4 e 1 2 e 4 3 e 10 7 b e e 11 7 c e 2 2 e 12 9 e 5 3 e 3 3 e e e 9 6 d
31 Spannin Trees Notes on Kruskal s aloritm Proo o correctness similar to omework. Must additionally veriy tat te spannin tree is indeed minimum-weit.
32 Spannin Trees Notes on Kruskal s aloritm Proo o correctness similar to omework. Must additionally veriy tat te spannin tree is indeed minimum-weit. Kruskal s aloritm is an example o a reedy aloritm. (It cooses te ceapest ede at eac point.)
33 Spannin Trees Notes on Kruskal s aloritm Proo o correctness similar to omework. Must additionally veriy tat te spannin tree is indeed minimum-weit. Kruskal s aloritm is an example o a reedy aloritm. (It cooses te ceapest ede at eac point.) Greedy aloritms don t always work.
34 Te Travelin Salesman Problem 134 Te travelin salesman problem Motivation: Visit all nodes and return ome as ceaply as possible.
35 Te Travelin Salesman Problem 134 Te travelin salesman problem Motivation: Visit all nodes and return ome as ceaply as possible. Least cost trip lyin between ive major cities. Optimal routes or deliverin mail, collectin arbae. Findin a trip to all buildins on campus, return.
36 Te Travelin Salesman Problem 134 Te travelin salesman problem Motivation: Visit all nodes and return ome as ceaply as possible. Least cost trip lyin between ive major cities. Optimal routes or deliverin mail, collectin arbae. Findin a trip to all buildins on campus, return. Goal: Find a minimum-weit Hamiltonian cycle in a weited rap.
37 Te Travelin Salesman Problem 134 Te travelin salesman problem Motivation: Visit all nodes and return ome as ceaply as possible. Least cost trip lyin between ive major cities. Optimal routes or deliverin mail, collectin arbae. Findin a trip to all buildins on campus, return. Goal: Find a minimum-weit Hamiltonian cycle in a weited rap. a b e c 1 d
38 Te Travelin Salesman Problem 134 Te travelin salesman problem Motivation: Visit all nodes and return ome as ceaply as possible. Least cost trip lyin between ive major cities. Optimal routes or deliverin mail, collectin arbae. Findin a trip to all buildins on campus, return. Goal: Find a minimum-weit Hamiltonian cycle in a weited rap. a b e c 1 d
39 Te Travelin Salesman Problem 134 Te travelin salesman problem Motivation: Visit all nodes and return ome as ceaply as possible. Least cost trip lyin between ive major cities. Optimal routes or deliverin mail, collectin arbae. Findin a trip to all buildins on campus, return. Goal: Find a minimum-weit Hamiltonian cycle in a weited rap. a b e c 1 d We can not use a reedy aloritm to ind tis TSP tour!
40 Te Travelin Salesman Problem 135 Te travelin salesman problem It is ard toindanoptimumsolution.
41 Te Travelin Salesman Problem 135 Te travelin salesman problem It is ard toindanoptimumsolution. Goal: Create an easy-to-ind pretty ood solution.
42 Te Travelin Salesman Problem 135 Te travelin salesman problem It is ard toindanoptimumsolution. Goal: Create an easy-to-ind pretty ood solution. Teorem. Wen te ede weits satisy te trianle inequality, te tree sortcut aloritm inds a tour tat costs at most twice te optimum tour. Recall. Te trianle inequality says tat i x, y, andz are vertices, ten wt(xy)+wt(yz) wt(xz).
43 Te Travelin Salesman Problem 135 Te travelin salesman problem It is ard toindanoptimumsolution. Goal: Create an easy-to-ind pretty ood solution. Teorem. Wen te ede weits satisy te trianle inequality, te tree sortcut aloritm inds a tour tat costs at most twice te optimum tour. Recall. Te trianle inequality says tat i x, y, andz are vertices, ten wt(xy)+wt(yz) wt(xz). Example: Euclidean distances Non-example: Airares 0.47 y 0.75 x 1. z
44 Te Travelin Salesman Problem 136 Findin a ood TSP-tour Te Tree Sortcut Aloritm to ind a ood TSP-tour 1 Find a minimum-weit spannin tree (Use Kruskal s Aloritm) 2 Walk in a circuit around te edes o te tree. 3 Take sortcuts to ind a tour
45 Te Travelin Salesman Problem 136 Findin a ood TSP-tour Te Tree Sortcut Aloritm to ind a ood TSP-tour 1 Find a minimum-weit spannin tree (Use Kruskal s Aloritm) 2 Walk in a circuit around te edes o te tree. 3 Take sortcuts to ind a tour
46 Te Travelin Salesman Problem 136 Findin a ood TSP-tour Te Tree Sortcut Aloritm to ind a ood TSP-tour 1 Find a minimum-weit spannin tree (Use Kruskal s Aloritm) 2 Walk in a circuit around te edes o te tree. 3 Take sortcuts to ind a tour
47 Te Travelin Salesman Problem 136 Findin a ood TSP-tour Te Tree Sortcut Aloritm to ind a ood TSP-tour 1 Find a minimum-weit spannin tree (Use Kruskal s Aloritm) 2 Walk in a circuit around te edes o te tree. 3 Take sortcuts to ind a tour
48 Te Travelin Salesman Problem 136 Findin a ood TSP-tour Te Tree Sortcut Aloritm to ind a ood TSP-tour 1 Find a minimum-weit spannin tree (Use Kruskal s Aloritm) 2 Walk in a circuit around te edes o te tree. 3 Take sortcuts to ind a tour
49 Te Travelin Salesman Problem 136 Findin a ood TSP-tour Te Tree Sortcut Aloritm to ind a ood TSP-tour 1 Find a minimum-weit spannin tree (Use Kruskal s Aloritm) 2 Walk in a circuit around te edes o te tree. 3 Take sortcuts to ind a tour
50 Te Travelin Salesman Problem 136 Findin a ood TSP-tour Te Tree Sortcut Aloritm to ind a ood TSP-tour 1 Find a minimum-weit spannin tree (Use Kruskal s Aloritm) 2 Walk in a circuit around te edes o te tree. 3 Take sortcuts to ind a tour
51 Te Travelin Salesman Problem 136 Findin a ood TSP-tour Te Tree Sortcut Aloritm to ind a ood TSP-tour 1 Find a minimum-weit spannin tree (Use Kruskal s Aloritm) 2 Walk in a circuit around te edes o te tree. 3 Take sortcuts to ind a tour
52 Te Travelin Salesman Problem 137 Proo o teorem Teorem. Wen te ede weits satisy te trianle inequality, te tree sortcut aloritm inds a tour tat costs at most twice te optimum tour.
53 Te Travelin Salesman Problem 137 Proo o teorem Teorem. Wen te ede weits satisy te trianle inequality, te tree sortcut aloritm inds a tour tat costs at most twice te optimum tour. Proo. Deine: TSP A : TSP tour rom sortcuttin spannin tree CIRC A : Circuit constructed by doublin spannin tree MST : Minimum-weit spannin tree TSP : Minimum-weit TSP tour
54 Te Travelin Salesman Problem 137 Proo o teorem Teorem. Wen te ede weits satisy te trianle inequality, te tree sortcut aloritm inds a tour tat costs at most twice te optimum tour. Proo. Deine: TSP A : TSP tour rom sortcuttin spannin tree CIRC A : Circuit constructed by doublin spannin tree MST : Minimum-weit spannin tree TSP : Minimum-weit TSP tour Ten, wt(tsp A ) wt(circ A )=2wt(MST ) 2wt(TSP ).
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