Street-Routing Problems Lecture 26 Sections 5.1-5.2 Robb T. Koether Hampden-Sydney College Wed, Oct 25, 2017 Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Wed, Oct 25, 2017 1 / 21
1 Street-Routing Problems 2 Definitions 3 Examples 4 Assignment Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Wed, Oct 25, 2017 2 / 21
Outline 1 Street-Routing Problems 2 Definitions 3 Examples 4 Assignment Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Wed, Oct 25, 2017 3 / 21
Street-Routing Problems There are many problems that fall under the heading Street-Routing Problems. We will consider five of them. Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Wed, Oct 25, 2017 4 / 21
Street-Routing Problems There are many problems that fall under the heading Street-Routing Problems. We will consider five of them. The Security Guard Problem Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Wed, Oct 25, 2017 4 / 21
Street-Routing Problems There are many problems that fall under the heading Street-Routing Problems. We will consider five of them. The Security Guard Problem The Mail Carrier Problem Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Wed, Oct 25, 2017 4 / 21
Street-Routing Problems There are many problems that fall under the heading Street-Routing Problems. We will consider five of them. The Security Guard Problem The Mail Carrier Problem The Königsberg Bridge Problem (famous) Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Wed, Oct 25, 2017 4 / 21
Street-Routing Problems There are many problems that fall under the heading Street-Routing Problems. We will consider five of them. The Security Guard Problem The Mail Carrier Problem The Königsberg Bridge Problem (famous) The Bridges of Madison County Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Wed, Oct 25, 2017 4 / 21
Street-Routing Problems There are many problems that fall under the heading Street-Routing Problems. We will consider five of them. The Security Guard Problem The Mail Carrier Problem The Königsberg Bridge Problem (famous) The Bridges of Madison County The Traveling Salesman Problem (famous) Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Wed, Oct 25, 2017 4 / 21
The Security Guard Problem Example (The Security Guard Problem) A security guard must patrol every street of the neighborhood. What route should he follow to minimize the total distance? Must he walk some streets twice? Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Wed, Oct 25, 2017 5 / 21
The Mail Carrier Problem Example (The Mail Carrier Problem) A mail carrier must deliver mail to both sides of every street of the neighborhood, except the boundary. What route should he follow to minimize the total distance? Must he walk some streets (same side) twice? Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Wed, Oct 25, 2017 6 / 21
The Königsberg Problem Example (The Königsberg Problem) A Königsberger would like to take a stroll across the seven bridges of Königsberg. Can it be done without ever crossing the same bridge twice? Does it matter where the stroller starts? Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Wed, Oct 25, 2017 7 / 21
The Bridges of Madison County Problem Example (The Bridges of Madison County Problem) A photographer wants to photograph each of the 11 bridges of Madison County. He must cross the bridge to photograph it and each bridge has a $5 toll. What route will minimize the total cost? Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Wed, Oct 25, 2017 8 / 21
The Bridges of Madison County Problem Example (The Bridges of Madison County Problem) 1 3 2 5 4 6 7 8 9 10 11 A photographer wants to photograph each of the 11 bridges of Madison County. He must cross the bridge to photograph it and each bridge has a $5 toll. What route will minimize the total cost? Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Wed, Oct 25, 2017 9 / 21
The Traveling Salesman Problem Example (The Traveling Salesman Problem) 8 B I 9 A 7 6 2 1 3 3 2 4 J 9 7 C 3 4 8 F 10 6 E 7 D 10 10 11 H 9 G A salesman is located in a city. He must make a trip during which he visits each of a number of other cities and return to his home city. He knows the distance from every city to every other city. What route will minimize the total distance traveled? Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Wed, Oct 25, 2017 10 / 21
Outline 1 Street-Routing Problems 2 Definitions 3 Examples 4 Assignment Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Wed, Oct 25, 2017 11 / 21
Definitions Definition A graph is a collection of vertices and edges. We normally draw the vertices as dots and the edges as lines. Each edge connects a pair of vertices. Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Wed, Oct 25, 2017 12 / 21
Definitions We can give the vertices labels, e.g., A, B, C, etc. Then use those labels to identify the edges, e.g., AB, AC, etc. A B D E C Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Wed, Oct 25, 2017 12 / 21
Definitions Definition (Path) A path is a sequence of edges, each edge adjacent to the next edge. We may denote a path by listing the vertices through which it passes. (E.g., DABC.) A B D E C Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Wed, Oct 25, 2017 12 / 21
Definitions Definition (Circuit) A circuit is a path that begins and ends at the same vertex. (E.g., ABCA.) A B D E C Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Wed, Oct 25, 2017 12 / 21
Definitions Definition (Weighted Graph) A weighted graph is a graph in which every edge is assigned a value (its weight). A 5 B 8 15 7 3 C D 12 E Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Wed, Oct 25, 2017 12 / 21
Outline 1 Street-Routing Problems 2 Definitions 3 Examples 4 Assignment Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Wed, Oct 25, 2017 13 / 21
Example Example (The Security Guard Problem) In the Security Guard Problem, we want a path that traverses every edge at least once and has the minimal total length. Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Wed, Oct 25, 2017 14 / 21
Example Example (The Mail Carrier Problem) In the Mail Carrier Problem, we want a path that traverses every edge at least twice (except the boundary) and has the minimal total length. With the duplicated edges, the Mail Carrier Problem is the same as the Security Guard Problem. Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Wed, Oct 25, 2017 15 / 21
Example Example (The Bridges of Königsberg Problem) In the Bridges of Königsberg Problem, we want a circuit that traverses each edge exactly once. Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Wed, Oct 25, 2017 16 / 21
Example Example (The Bridges of Königsberg Problem) We draw a graph that shows only the relevant parts. Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Wed, Oct 25, 2017 17 / 21
Example Example (The Bridges of Madison County Problem) In the Bridges of Madison County Problem, we want a circuit that traverses each edge at least once and has the minimal total length. This is the same as the Security Guard Problem. Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Wed, Oct 25, 2017 18 / 21
Example Example (The Bridges of Madison County Problem) In the Bridges of Madison County Problem, we want a circuit that traverses each edge at least once and has the minimal total length. This is the same as the Security Guard Problem. Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Wed, Oct 25, 2017 18 / 21
Example Example (The Traveling Salesman Problem) 8 B I 9 A 7 6 2 1 3 3 2 4 J 9 7 C 3 4 8 F 10 6 E 7 D 10 10 11 H 9 G In the Traveling Salesman Problem, we want a circuit that visits each vertex at least once and has the minimal total length. Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Wed, Oct 25, 2017 19 / 21
Example Example (The Traveling Salesman Problem) A 8 11 B 6 7 8 C 10 D 9 I 10 11 15 4 J 15 17 11 11 F 9 14 E 7 H 9 G In the Traveling Salesman Problem, we want a circuit that visits each vertex at least once and has the minimal total length. Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Wed, Oct 25, 2017 19 / 21
Outline 1 Street-Routing Problems 2 Definitions 3 Examples 4 Assignment Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Wed, Oct 25, 2017 20 / 21
Assignment Assignment Chapter 5: Exercises 2, 3, 7, 13, 15, 19, 20, 21, 22, 27. Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Wed, Oct 25, 2017 21 / 21