Mathematical Thinking
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1 Mathematical Thinking Chapter 2 Hamiltonian Circuits and Spanning Trees It often happens in mathematics that what appears to be a minor detail in the statement of a problem can have a profound effect on its complexity.
2 Hamiltonian Circuits A Hamiltonian circuit is a circuit that includes every vertex exactly once. Hundreds of practical applications routing delivery trucks moving an industrial robot arm
3 Hamiltonian Circuits A Hamiltonian circuit is a circuit that includes every vertex exactly once. Hundreds of practical applications routing delivery trucks moving an industrial robot arm
4 Hamiltonian Circuits It s easy to construct graphs that have no Hamiltonian circuits.
5 Hamiltonian Circuits It s easy to construct graphs that have no Hamiltonian circuits.
6 Hamiltonian Circuits It s easy to construct graphs that have no Hamiltonian circuits. Grid graphs are often of interest when modeling a network of roads. Consider the problem of inspecting traffic lights after a storm. Can you explain why this graph has no Hamiltonian circuit? The vertex colors are a hint.
7 Hamiltonian Circuits It s easy to construct graphs that have no Hamiltonian circuits. This is our textbook s idea for an infinite family of graphs without Hamiltonian circuits. Bipartite graphs
8 Famous Conjecture There does not exist a computationally feasible procedure for finding a Hamiltonian circuit in a given graph, or even for deciding if a graph has a Hamiltonian circuit. If you find such a procedure, or prove that one does not exist, you will become very famous.
9 Min-Cost Hamiltonian Circuits In a weighted graph, find the Hamiltonian circuit of least cost. Cleveland St. Louis 300 Minneapolis 425 Chicago
10 Min-Cost Hamiltonian Circuits In a weighted graph, find the Hamiltonian circuit of least cost. Cleveland St. Louis 300 Minneapolis 425 Chicago
11 Traveling Salesman Problem In a complete graph, every pair of vertices is connected by an edge. The problem of finding a min-cost Hamiltonian circuit in a complete weighted graph is known as the TSP. No known feasible solution.
12 Traveling Salesman Problem Scheduling deliveries (UPS, FedEx, Dominos,...) Shuttling customers between JFK and NYC Soldering connections on a printed circuit Organizing political campaign stops Order-picking in a warehouse Inspecting local franchises Computer network wiring Automated drilling Bus scheduling
13 Solving TSP Cleveland CLE MIN STL CHI 562 STL CHI MIN CHI MIN STL 562 St. Louis CHI STL CHI MIN STL MIN CLE CLE CLE CLE CLE CLE Minneapolis Chicago So all we have to do is add up the edge weights for every tour, keeping track of the smallest sum as we go...?
14 Solving TSP How many different Hamiltonian circuits are there in a complete graph with five vertices? What about the general case of n vertices? How big would the graph have to be for the number of traveling salesman tours to exceed the number of atoms in the observable universe? 100 billion galaxies, each with about a trillion stars on average. Each star has about atoms on average. 80 About 10 atoms in total. Sources: 1 2 How fast do factorials grow?
15 Nearest-Neighbor Heuristic Choose a starting vertex. Go to nearest unvisited neighbor NO Has every vertex been visited? YES Return to starting vertex.
16 Nearest-Neighbor Heuristic Choose a starting vertex. A 70 B Go to nearest unvisited neighbor NO E 75 C Has every vertex been visited? YES D Return to starting vertex.
17 Nearest-Neighbor Heuristic Choose a starting vertex. A 70 B Go to nearest unvisited neighbor NO E 75 C Has every vertex been visited? YES D Return to starting vertex.
18 Nearest-Neighbor Heuristic Choose a starting vertex. A 70 B Go to nearest unvisited neighbor NO E 75 C Has every vertex been visited? YES D Return to starting vertex.
19 Nearest-Neighbor Heuristic Choose a starting vertex. A 70 B Go to nearest unvisited neighbor NO E 75 C Has every vertex been visited? YES D Return to starting vertex.
20 Nearest-Neighbor Heuristic Choose a starting vertex. A 70 B Go to nearest unvisited neighbor NO E 75 C Has every vertex been visited? YES D Return to starting vertex.
21 Nearest-Neighbor Heuristic Choose a starting vertex. A 70 B Go to nearest unvisited neighbor NO E 75 C Has every vertex been visited? YES D Return to starting vertex.
22 Sorted-Edges Heuristic NO Sort the edges by weight. Select the next acceptable edge of lowest weight. Is the tour complete? An edge is acceptable if: It has not already been chosen. Adding it does not cause a vertex to have valence 3. Adding it does not form a circuit before the tour is complete.
23 Sorted-Edges Heuristic Sort the edges by weight. A 70 B Select the next acceptable edge of lowest weight NO E 75 C Is the tour complete? D
24 Sorted-Edges Heuristic Sort the edges by weight. A 70 B Select the next acceptable edge of lowest weight NO E 75 C Is the tour complete? D
25 Sorted-Edges Heuristic Sort the edges by weight. A 70 B Select the next acceptable edge of lowest weight NO E 75 C Is the tour complete? D
26 Sorted-Edges Heuristic Sort the edges by weight. A 70 B Select the next acceptable edge of lowest weight NO E 75 C Is the tour complete? D
27 Sorted-Edges Heuristic Sort the edges by weight. A 70 B Select the next acceptable edge of lowest weight NO E 75 C Is the tour complete? D
28 Sorted-Edges Heuristic Sort the edges by weight. A 70 B Select the next acceptable edge of lowest weight NO E 75 C Is the tour complete? D
29 Which Heuristic is Better? Results on 1000 complete graphs with random edge weights. Size of Graph Nearest Neighbor Sorted Edges
30 Trees A tree is a connected graph with no circuits.
31 Spanning Trees A spanning tree is a subgraph that is a tree and includes every vertex.
32 Spanning Trees A spanning tree is a subgraph that is a tree and includes every vertex.
33 Spanning Trees A minimum spanning tree in a weighted graph is a spanning tree of lowest possible cost Application: construction of video conferencing networks.
34 Kruskal s Algorithm Add edges in sorted order until a spanning tree is obtained
35 Kruskal s Algorithm Add edges in sorted order until a spanning tree is obtained Cost: 1
36 Kruskal s Algorithm Add edges in sorted order until a spanning tree is obtained Cost: = 2
37 Kruskal s Algorithm Add edges in sorted order until a spanning tree is obtained Cost: = 4
38 Kruskal s Algorithm Add edges in sorted order until a spanning tree is obtained Cost: = 6
39 Kruskal s Algorithm Add edges in sorted order until a spanning tree is obtained Cost: = 9
40 Kruskal s Algorithm Add edges in sorted order until a spanning tree is obtained Cost: = 12
41 Kruskal s Algorithm Add edges in sorted order until a spanning tree is obtained Cost: = 15
42 Kruskal s Algorithm Add edges in sorted order until a spanning tree is obtained Cost: = 19
43 Kruskal s Algorithm Add edges in sorted order until a spanning tree is obtained Cost: = 23
44 Kruskal s Algorithm Add edges in sorted order until a spanning tree is obtained Cost: = 27
45 Kruskal s Algorithm Add edges in sorted order until a spanning tree is obtained Cost: = 27
46 Kruskal s Algorithm Add edges in sorted order until a spanning tree is obtained Cost: = 27
47 Kruskal s Algorithm Add edges in sorted order until a spanning tree is obtained Cost: = 27
48 Kruskal s Algorithm Add edges in sorted order until a spanning tree is obtained Cost: = 33
49 Kruskal s Algorithm Add edges in sorted order until a spanning tree is obtained Cost: = 33
50 Kruskal s Algorithm Alternative interpretation: merging trees
51 Kruskal s Algorithm Alternative interpretation: merging trees
52 Kruskal s Algorithm Alternative interpretation: merging trees
53 Kruskal s Algorithm Alternative interpretation: merging trees
54 Kruskal s Algorithm Alternative interpretation: merging trees
55 Kruskal s Algorithm Alternative interpretation: merging trees
56 Kruskal s Algorithm Alternative interpretation: merging trees
57 Kruskal s Algorithm Alternative interpretation: merging trees
58 Kruskal s Algorithm Alternative interpretation: merging trees
59 Kruskal s Algorithm Alternative interpretation: merging trees
60 Kruskal s Algorithm Alternative interpretation: merging trees
61 Kruskal s Algorithm Alternative interpretation: merging trees
62 Kruskal s Algorithm Alternative interpretation: merging trees
63 Kruskal s Algorithm Alternative interpretation: merging trees
64 Kruskal s Algorithm Alternative interpretation: merging trees
65 Kruskal s Algorithm Alternative interpretation: merging trees
66 Proof Idea When merging two trees, it is always best to use the edge of smallest weight
67 Proof Idea When merging two trees, it is always best to use the edge of smallest weight. 3
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