Topics Covered. Introduction to Graphs Euler s Theorem Hamiltonian Circuits The Traveling Salesman Problem Trees and Kruskal s Algorithm

Transcription

1 Graph Theory

2 Topics Covered Introduction to Graphs Euler s Theorem Hamiltonian Circuits The Traveling Salesman Problem Trees and Kruskal s Algorithm

3 What is a graph? A collection of points, called vertices (or vertex if singular), together with a set of lines, called edges, connecting pairs of vertices. The degree of a vertex is the number of edges that use that vertex as an endpoint, with loops contributing twice to the degree. A vertex is odd if. A vertex is even if.

4 What is a graph? A graph is completely described by its vertices and edges. Example: Sketch the graph with vertices A, B, C, D, E, F, G and edges {AB, AC, BC, CD, DE, DF, EG, FG}. *Both pictures represent the graph described.

5 What is a graph? We can also represent the same graph with different labels on the vertices. We say two graphs are the same graph represented differently if we can find a one-to-one correspondence between the vertices and edges of the two graphs. *Both pictures represent the graph described.

6 These graphs represent the same graph under the correspondence: Vertices A! 1 B! 2 C! 3 D! 4 E! 5 F! 6 G! 7 Edges AB AC BC CD DE DF EG FG " " " " " " " "

7 Graphs We can use graphs to model real-life situations. A C B C A D B D Let each land mass be a vertex, and the bridges be edges. image taken from wikipedia.com

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9 A graph which has no loops and satisfies the condition that no pair of vertices is joined by more than one edge is called a simple graph. A Terminology A C B C B Two edges connecting vertices C and D. D Not simple D Simple

10 Terminology A path in a graph is a sequence (an ordered list) of distinct edges of the graph such that the ending vertex of one edge in the sequence is the beginning vertex of the next edge in the sequence. A circuit is a path that begins and ends at the same vertex. The length of a path or circuit is the number of edges traveled in the path. We usually describe a path by listing the sequence of vertices traveled by the path. A B The path ABCDE is highlighted in red. This path has length. A B D E C D E C

11 Terminology A graph is connected if there exists some path between every pair of vertices. A subgraph of a graph is a set of vertices and edges chosen from the original graph. A component of a graph is a maximally connected subgraph of the original graph. disconnected graph A subgraph of the graph to the left. Find another subgraph of the graph above. How many components does the graph have? What are they? *Connected graphs are also called networks.

12 Example: Determine which of the graphs below is a subgraph of the graph to the right. How many components does the graph have? I H K J K I J H (a) (b) (c)

13 Terminology A bridge in a connected graph is an edge such that if it were removed the graph would no longer be connected. You Try: Construct a connected graph with 6 vertices which contains a bridge. Identify the bridge in your graph. connected graph edge CD is a bridge

14 The Seven Bridges of Koenigsberg Problem Starting at some point in the city, can you visit all parts of the city, crossing each bridge once and only once, and return to the place you started? A A C B C B D D image taken from wikipedia.com

15 Terminology The mathematician Leonard Euler ( ) solved the Koenigsberg bridge problem in 1735 using graph theory. An Euler path is a path that uses every edge of the graph exactly once. An Euler circuit is an Euler path that begins and ends at the same vertex. Can you make a Euler path or an Euler circuit on the graphs below? A B A B A B E E D C D C D C What has to happen for a graph to have a Euler path?

16 Euler s Theorem *Look at the degrees of the vertices in the graph. (1)If a graph has more than two vertices of odd degree then it has no Euler paths. (2)If a graph is connected and has zero or exactly two vertices of odd degree, then it has at least one Euler path (probably more). Any such path must start at one of the odd-degree vertices and end at the other if the graph has two odd-degree vertices. Otherwise, it must start and end at the same vertex. (3)If a connected graph has zero vertices of odd degree, then it has at least one Euler circuit. Conversely, if a graph has an Euler circuit, then it has no vertices of odd degree.

17 Which of the graphs have an Euler path? Which have an Euler circuit? A B A E C B D C D A B E D C G F

18 Euler s Theorem Add an edge to the graph below (without repeating an existing edge) so that the resulting graph contains an Euler path. Hint: Look at the degrees of each vertex. How many vertices have odd degree? How many have even degree? 18

19 Euler s Theorem Add an edge to the graph below (without repeating an existing edge) so that the resulting graph contains an Euler path. 19

20 Euler s Theorem Example: Using the graph below, answer the following. (1)Is AED a path in the graph? (2)Is the graph connected? (3)What is the degree of vertex D? (4)What is the length of the path ABED? (5)Does the graph have an Euler path? If so, give an Euler path. (6)Does it have an Euler circuit?

21 Fleury s Algorithm - used to find Euler Circuits Given a connected graph with all even vertices, choose any vertex with which to begin. Travel along edges in the graph according to the following rules. 1. After traveling an edge, erase it. Once all the edges for a vertex have been used, erase the vertex, too. 2. You may only travel over a bridge if there is no other option. Example: The graph to the right represents the streets in a neighborhood. Starting at vertex A, use Fleury s algorithm to find an Euler circuit to provide a mail route for this neighborhood that does not repeat streets.

22 erase erase erase The Euler circuit is ACBFGJKHIEDHGCDA.

23 Eulerizing a Graph Eulerizing a graph converts a non-eulerian graph into an Eulerian graph by making copies of existing edges in the graph so that all vertices have even degree. Example: A newspaper boy has been assigned the neighborhood depicted below for his newspaper route. Find a route that uses every road, begins and ends at the same point, and minimizes repetition of streets. The odd degree vertices are labeled. We duplicate edges to make these vertices have even degrees. *Note we can t add edges like an edge from D to E, that aren t in the original graph.

24 Solution: Eulerize You try: Duplicate edges in the graph below to make it Eulerian. A D B H I E C J G F

25 The Traveling Salesman Problem Kristen lives in Kansas City and must make visits next week to give sales pitches in Denver, Minneapolis, Chicago, and Nashville. She has found prices of flights between each pair of cities. What sequence of visits minimizes the cost? We ll use a graph to model the problem.

26 Hamiltonian Paths A path that passes through all the vertices of a graph exactly once is called a Hamiltonian path. If a Hamiltonian path begins and ends at the same vertex, it is called a Hamiltonian circuit. If a graph has a Hamiltonian circuit we say it is Hamiltonian. What is the difference between a Hamiltonian circuit and an Euler circuit? Find a Hamiltonian path in the graph to the right starting at vertex A and ending at vertex G.

27 Complete Graphs A complete graph is one in which every pair of vertices is joined by an edge. A complete graph with n vertices is denoted by K n. Often, we want to find all the Hamiltonian circuits in a graph. This is easy to accomplish for complete graphs.

28 Example: Find all the Hamiltonian Circuits in K 4. *Note that the circuit ADCBA is the same as the circuit DCBAD, which is the same as CBADC, etc. So we can assume all circuits begin and end at vertex A. If we start at vertex A, what vertex could we visit next? Then which vertex could we visit? We can display this in a tree diagram.

29 Example: Find all the Hamiltonian Circuits in K 4. start A choices for the second vertex B C D choices for the third vertex C D B D B C choices for the last vertex D C D B C B Tracing through the tree diagram we see that K 4 has the following 6 circuits: ABCDA, ABDCA, ACBDA, ACDBA, ADBCA, ADCBA

30 You try: How many Hamiltonian circuits does K 3 have? What about K 5? K n?

31 Solving the Traveling Salesman Problem (TSP) - find all Hamiltonian circuits using weighted graphs A weighted graph is a graph with numbers, called weights, assigned to the edges. The weight of a path is the sum of the weights of the edges of the path.

32 Example: Kristen lives in Kansas City and must make visits next week to give sales pitches in Denver, Minneapolis, Chicago, and Nashville. To determine which would be her cheapest trip, she has obtained prices of flights between each pair of cities. What sequence of visits minimizes the cost? Find all the Hamiltonian circuits and their weights in the graph. Which Hamiltonian circuit has the minimum weight?

33 The Brute Force Algorithm for solving the TSP Step 1: List all Hamiltonian circuits in the graph. Step 2: Find the weight of each circuit found in step 1. Step 3: The circuits with the smallest weights give the solution to the TSP. This algorithm is time consuming. There are other less timeconsuming algorithms that give good approximations to the TSP.

34 The Nearest Neighbor Algorithm (NNA) Step 1: Start at any vertex X. Step 2: Choose any one edge connected to X that has the smallest weight. (There may be several with smallest weight.) Select the vertex at the end of this edge as the next vertex in your circuit. This vertex is called the nearest neighbor of X. Step 3: Choose subsequent new vertices as you did in step 2. (i.e., at every step, choose the next nearest neighbor that hasn t already been chosen). Step 4: After all vertices have been chosen, close the circuit by returning to the starting vertex.

35 Example: Use the nearest neighbor algorithm, starting at Kansas City, to solve Kristen s problem.

36 Example: Use the nearest neighbor algorithm, starting at Kansas City, to solve Kristen s problem.

37 Repetitive Nearest Neighbor Algorithm Pick a vertex and apply the Nearest Neighbor Algorithm (NNA) Repeat the NNA for each vertex of the graph Pick the best of all the Hamiltonian circuits you got in the first two steps.

38 The Side Sorted Algorithm (sometimes called the Best Edge or the Cheapest Link Algorithm) Step 1: Choose any edge with the smallest weight. Step 2: Choose any remaining edge in the graph with the smallest weight. Step 3: Keep adding the next smallest weight edge while following the below conditions. - do not form a circuit before all the vertices have been added - do not add an edge that would give a vertex degree three

39 Example: Use the side sorted algorithm to solve Kristen s problem. Edge Chosen Weight MC 137 MK 194 CD 250 DN 462 KN 466

40 Example 1. Does the graph to the left have an Euler circuit? An Euler path? If so, find the path and/or circuit. 2. Find a Hamiltonian circuit in the graph to the left. A 5 D B 1 C 1. Use the Nearest Neighbor Algorithm to find a Hamiltonian circuit in the graph to the left of low weight. 2. Use the Side Sorted Algorithm to find a Hamiltonian circuit in the graph to the left of low weight.

41 Let s add (O)maha and D(A)llas to Kristen s trip. Below is a table with the flight cost between the cities. K C M D N O A K C M D N O A Draw a graph to represent this problem. 2. If we were to use the brute force algorithm to solve the TSP, how many Hamiltonian circuits would we have to consider? 3. Use the Nearest Neighbor Algorithm to find a Hamiltonian circuit beginning at Kansas City.

42 Example: The table below shows the distance (in mi) between five Starbucks stores. A manager from the headquarters must visit each store to check on customer satisfaction. Using any method that we ve discussed, find a route for the manager using as few miles as possible.

43 Some more examples of Hamiltonian Paths Find a Hamiltonian path in each graph below, or explain why you can t.

44 Trees Recall: A subgraph of a given graph is a set of vertices and edges chosen from among those of the original graph. Example: Removing edge HW creates the subgraph shown at right. Original Graph Subgraph

45 Trees Recall that a circuit is a path that begins at a vertex and returns to that vertex without using any edges twice. A connected graph that has no circuits is called a tree. Tree Tree Not a Tree

46 Choose the graph(s) that is a tree. A B C a. F b. A B C D DE F G E c. d. A F C E D B A B C D E F

47 A forest is a collection of one or more trees. Choose the graph(s) that is a forest. C A B a. b. D E c. d. A F C E D B F A B C DE F G A B C D E F

48 Example: The graph below is not a tree because it contains several circuits. Find a subgraph that is a tree.

49 Spanning Trees A spanning tree is a subgraph that Contains all the original vertices in the graph. Is connected Contains no circuits - Every connected graph has at least one spanning tree. Graph Spanning Tree

50 Minimal Spanning Trees In a weighted graph, a spanning tree with the smallest possible total weight is called a minimal spanning tree. - Since a connected weighted graph can have multiple spanning trees, there may be one or more than one of those spanning trees with the smallest total weight. Original Weighted Graph Weight = 76 Spanning Tree Weight = 66 Spanning Tree Weight = 62

51 Kruskal s Algorithm - Used to find a minimal spanning tree of a connected weighted graph. Step 1: Choose any edge with the smallest weight. Step 2: Choose any remaining edge in the graph with the smallest weight. Step 3: Keep adding the next smallest weight edge that does not create a circuit until all vertices have been added to your subgraph.

52 Example: Construct a minimal spanning tree for the weighted graph below.

53 Example: Construct a minimal spanning tree for the weighted graph below. 1 Start with the 5 vertices. Add the edge with the smallest weight. 2 Choose the edge with the next smallest weight.

54 Example: Construct a minimal spanning tree for the weighted graph below. 3 The next lowest edge has weight 4.7, but it is not allowed since its addition would form a circuit. Add edge AC instead. 4 The next lowest edge is BE, so add it. The total weight of the spanning tree is = 18.8

55 Example: The distance between the five most popular animals at a zoo is listed in the table below. What is the minimum distance required to build sidewalks that connect the most popular animals? (A)pe (B)ear (C)heetah (D)ingo (E)lephant (A)pe (B)ear (C)heetah (D)ingo (E)lephant