Announcements. Quiz on Monday. Review Session at 5:00-6:30ish today in Padelford C-36. Project Proposals due one week from Monday Oct 22.

Transcription

1 Announcements Quiz on Monday. Review Session at 5:00-6:30ish today in Padelford C-36. Project Proposals due one week from Monday Oct 22.

2 Band Tour Problem You and the band want to tour the country this summer visiting several cities including Portland Reno Yellowstone Denver Salt Lake City San Francisco Flagstaff Decision Problem: What is the best route?

3 Two Subproblems What order should you visit the cities in? What is the best route between each pair of cities? You were asked to consider the best route from San Francisco to Salt Lake City. What do you recommend?

4 Network Map

5 Translating into a Math Model What is the best route between each pair of cities? The relevant data in the map is the cost of traveling between each pair of cities, not the geography. Here cost might mean distance, time, or expected number of delays due to construction.

6 Maps to Graph Theory Abstraction from Geography: From the map, focus only the points representing towns and the line segments representing what?

7 Graph Theory Def: A graph G=(V,E) has a vertex set V and an edge set E such that the edges are a subset of the set of all pairs of vertices. Example: V={1,2,3,4} E ={{2,3},{2,4},{3,4}}

8 Graph Theory Def: The neighbors of a vertex x in V are the set of other vertices y in V such that {x,y} is in E. Example: V={1,2,3,4} E ={{2,3},{2,4},{3,4}} and 3 are neighbors 1 and 2 are not neighbors 3 4

9 Graph Theory Graphs come in many flavors Simple graphs: edge sets are sets Multigraphs: edge sets are multisets, multiple edges and loops allowed between two vertices. Directed graphs: edges are ordered pairs so an edge from x to y represented as (x,y) is different than (y,x) Networks or labeled graphs: graphs where the edges have weights W=(w xy ).

10 Graph Theory A representation of the graph of the internet

11 Graph Theory Def: A path in a graph G=(V,E) is a sequence of edges in E of the form [(a,b),(b,c),(c,d),(d,e), ]. The weight of a path is the sum of the edge weights in the path. Example: V={1,2,3,4} E ={{2,3},{2,4},{3,4}} [(2,4),(4,3)] is a path from 2 to 3 of weight 14=6+8. [(1,4),(4,3)] is not a path so it has infinite weight.

12 Shortest Path Problem Decision Problem: Find the shortest path between two vertices v and w in a graph with positive weighted edges. One Attack: Let x 1,x 2,,x j be the neighbors of w. Let d(v,x i ) be the weight of the shortest path from v to x i for i=1,2,,j. Then the shortest path from v to w will go through the neighbor of w with the minimal value of d(v,x i ) + wt(x i,w).

13 Dijkstra s Algorithm Goal: Find shortest path in G from v to w. Initially, consider the distance to x 1,x 2,,x j the neighbors of v. Let d(v,x i ) =wt(v,x i ) for each i=1..j. If x k has the smallest weight among the neighbors, set T to be the graph with one edge (v,x k ), abreviated vx k. Consider the distance from v to all of the neighbors of x k, call them y 1,, y p. Set d(v,y i )=d(v,x k )+wt(x k,y i ) if y i is not adjacent to v or min(wt(v,y i ) vs d(v,x k )+wt(x k,y i ) ) if y i is adjacent to both v and x k.

14 Dijkstra s Algorithm Among all of the vertices y not yet in T for which d(v,y) has been set, find the smallest one, say z. Add z to T along with the edge connecting z to v via the shortest path. Update the distance table by (re)considering the distance from v to all neighbors of z via the path in T to z. Continue until w is added to T. Answer: The shortest path from v to w in G will be the unique path in T between them and d(v,w) will be the total weight of the shortest possible path in G.

15 Dijkstra s Algorithm Goal: Find shortest path in G from 1 to 5.

16 Dijkstra s Tree Dijkstra s tree is the final tree T in the algorithm.

17 Traveling Salesman Problem TSP: Find a minimal cost tour visiting n cities and returning home. The band is considering starting in Seattle, visiting 7 other cities and returning to Seattle. Portland, Reno, Yellowstone, Denver, Salt Lake City, San Francisco, Flagstaff How many ways can they do that?

18 Traveling Salesman Problem How many ways can they do that? (9!= ) (10!= ) (11!= ) (12!= ) (13!= ) (14!= ) (15!= ) (16!= ) (17!= )

19 Biggest TSP s Solved 1977 Tour of Germany: 177 cities 1998 Tour of US: 13,509 cites 2001 Tour of Germany:15,112 cities Tour of Sweden: 24,978 cites VLSI Chip: 85,900 locations.

20 Branch and Bound First, find a tour that looks efficient and compute its total distance D. Start building up tours of the cities and compute the required distance C for each initial segment of tour. If C> D, then the best tour will never begin this way. Don t extend this initial segment any further. If a complete tour is ever found with shorter total distance than D, then update D and save the new route as best seen so far.