Chapter 16: Graphs and Digraphs

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1 Chapter 16: Graphs and Digraphs Exercises adj: 2. adj: 3. adj: 4. adj: 5. E A B 1 2 data: data: B BE data: BEA BEAR BEARD cat zebra horse data: rat bear dog D C 308

2 6. CAT DOG RAT BAT Alpha Tho Beta Pi Gamma Mu Delta 309

8 A 123 D B C cat cat dog rat dog rat cat Al dog rat Rob Bob Nan Dot Mo 315

9 Exercises Step v w Visit 2. Step v w Visit Step v Visit Queue 4. Step v Visit Queue 1, , a 1 3a 2 3b b a 2 4 3a 1 3 3b b a 4 3 3a 3 4 3b 4 3 3b 3 4 3a 3 3a 4 3b 3 3b 4 5. Step v distance Queue Labels for: ,2, p[2] = 4, p[1] = 1, p[0] = 2 6. Step v w Visit 7. Step v w Visit 1 cat cat 1 zebra zebra 2 cat zebra 2 zebra horse 1 zebra zebra 1 horse horse 2 zebra horse 2 horse 1 horse horse 2 horse 1 rat rat 2 rat bear 1 bear bear 2 bear 1 dog dog 2 dog 316

10 8. Step v Visit Queue 1,2 cat cat 3a cat 3b cat zebra, rat, dog zebra rat dog 3a zebra rat dog 3b zebra horse rat dog horse 3a rat dog horse 3b rat bear dog horse bear 3a dog horse bear 3b dog horse bear 3a horse bear 3b horse bear 3a bear 3b bear 9. Step v Visit Queue 1,2 zebra zebra 3a zebra 3b zebra horse horse 3a horse 3b horse 10. Step v distance Queue Labels for: cat zebra horse rat bear dog 1,2,3 0 rat 0 4 rat 0 cat horse bear cat 1 zebra dog p[2] = dog, p[1] = cat, p[0] = rat 317

11 11. Step v w Visit 12. Step v w Visit 1 A A 1 F F 2 A B 2 F J 1 B B 1 J J 2 B C 2 J K 1 C C 1 K K 2 C E 2 K I 1 E E 1 I I 2 E F 2 I 1 F F 2 F J 1 J J 2 J K 1 K K 2 K I 1 I I 2 I 1 G G 2 G H 1 H H 2 H 1 D D 2 D 13. Step v Visit Queue 14. Step v Visit Queue 1,2 A A 1,2 F F 3a A 3a F 3b A B D B D 3b F J J 3a B D 3a J 3b B C D C 3b J K K 3a D C 3a K 3b D C 3b K I I 3a C 3a I 3b C E F E F 3b I 3a E F 3b E G I F G I 3a F G I 3b F J G I J 3a G I J 3b G H I J H 3a I J H 3b I J H 3a J H 3b J K H K 3a H K 3b H K 3a K 3b K 318

12 15. Step v distance Queue Labels for: A B C D E F G H I J K 1,2,3 0 A 0 4 A 0 B D B 1 D C D 1 C D 2 EF p[3] = E, p[2] = C, p[1] = B, p[0] = A 16. Step v distance Queue Labels for: A B C D E F G H I J K 1,2,3 0 A 0 4 A 0 B D B 1 D C D 1 C C 2 E F E 3 F G I F 3 G I J G 4 I J H p[5] = H, p[4] = G, p[3] = E, p[2] = C, p[1] = B, p[0] = A 17. Step v distance Queue Labels for: A B C D E F G H I J K 1,2,3 0 G 0 4 G 0 H I H 1 I A E I 1 A E J A 2 E J B D E 2 J B D F J 2 B D F K B 3 D F K C p[4] = C, p[3] = B, p[2] = A, p[1] = H, p[0] = G 18. Step v distance Queue Labels for: A B C D E F G H I J K 1,2,3 0 J 0 4 J 0 K K 1 I p[2] = I, p[1] = K, p[0] = J 319

13 19. Step v distance Queue Labels for: A B C D E F G H I J K 1,2,3 0 J 0 4 J 0 K K 1 I I 2 empty 5 No such path 20. #include <queue> // Put prototype in public section of class template Digraph void breadthfirstsearch(int start); /* Breadth-first search of digraph, starting at vertex start. Uses a Queue class template to store vertices. Precondition: start is a vertex. Postcondition: Digraph has been breadth-first searched from start */ // Put definition in Digraph.h below the class declaration template <typename DataType> void Digraph<DataType>::breadthFirstSearch(int start) { // Process v[start].data here cout << myadjacencylists[start].data << endl; queue<vertexinfo> q; q.enqueue(myadjacencylists[start]); vector<bool> unvisited(myadjacencylists.size(), true); while (!q.empty()) { VertexInfo v = q.front(); q.dequeue(); for (list<int>::iterator it = v[start].adjacencylist.begin(); it!= v[start].adjacencylist.end(); it++) if (unvisited[*it]) { // process *it here cout << (*it).data << endl; } } } unvisited[(*it).data] = false; q.enqueue(*it); 320

14 21. Matrix multiplication and addition can be used to find the reachability matrix from a digraph's adjacency matrix. In the following, the type Matrix is defined by statements like the following (or preferably, a class with built-in operations): const int MAX_SIZE =... // max # rows/columns + 1 typedef int Matrix[MAX_SIZE][MAX_SIZE]; /* Function to find reachability matrix of a digraph from its n x n adjacency matrix adj. Receive: Adjacency matrix adj Return: Reachability matrix */ findreach(matrix adj, int n, Matrix reach) { Matrix ident; makeidentity(n, ident); } Matrix sum, // cumulative sum of powers of adj power, // a power of adj prod, // used to find adj X power reach; // reachability matrix copymatrix(ident, prod); for (int count = 1; count < n; count++) { findproduct(a, prod, power); findsum(reach, power, sum); copy(power, prod); copy(sum, reach); } The following functions can be used for the matrix operations: // Create n x n identity matrix ident void makeidentity(int n, Matrix ident) { for (int i = 1; i <= n; i++) for (int j = 1; j <= n; j++) ident[i][j] = 0; for (int i = 1; i <= n; i++) ident[i][i] = 1; } // Copy n x n matrix mat into matrix copy void copymatrix(int n, Matrix mat, Matrix copy) { for (int i = 1; i <= n; i++) for (int j = 1; j <= n; j++) copy[i][j] = mat[i][j]; } 321

15 // Find sum of n x nmatrices mat1 and mat2. Matrix findsum(matrix mat1, Matrix mat2, int n, Matrix sum) { for (int i = 1; i <= n; i++) for (int j = 1; j <= n; j++) sum[i][j] = mat1[i][j] + mat2[i][j]; } // Find product of n x n matrices mat1 and mat2. Matrix findproduct(matrix mat1, Matrix mat2, int n, Matrix prod) { for (int i = 1; i <= n; i++) for (int j = 1; j <= n; j++) { int sum = 1; for (int k = 1; k <= n; k++) sum += mat1[i][k] * mat2[k][j]; prod[i][j] = sum; } } 22. This is a simple modification of the solution in Exercise 21. Use && instead of * and instead of multiply (*) in the matrix functions /* Function to find reachability matrix of a digraph from its n x n adjacency matrix adj using Warshall's algorithm Receive: Adjacency matrix adj Return: Reachability matrix */ findreach(matrix adj, int n, Matrix reach) { Matrix reach; copymatrix(adj, reach); int k = 1; 322

16 } while (k <= n &&!allones(reach)) { for (int i = 1; i <= n && i!= k; i++) if (reach[i][k]!= q for (int j = 1; j <= n, j++) reach[i][j] = reach[k][j]; k++ } //-- Check if matrix mat contains all 1s bool allones(matrix mat) { for (int i = 1; i <= n; i++) for (int j = 1; j <= n; j++) if (mat[i][j]!= 1) return false; return true; } Exercises adj: 2. adj: 3. adj: data: data: B BE data: BEA BEAR BEARD 323

17 4. adj: moose zebra horse data: rat bear dog cow

23 e e4 e3 e1 2 e2 3 1 e3 e2 e1 4 e5 2 e4 e e e11 e7 1 e5 e2 e1 e3 2 e4 3 e8 e6 5 e

24 A1 cat 1 A1 4D 2B cow 5 dog 2 C2 3B C3 bat 4 rat A1 ) A1 (f) Al C2 3B C2 3B Rob Bob Nan Dot Mo

26 32. Note: The following algorithms assume the adjacency list representation of a graph. 33. Algorithm to insert a new vertex and its connecting edges into a graph: 1. Get a new head node. 2. Search the array v to find the first v[k] that is null, and set v[k] to point to this new head node. //-- Assume all elements of v are initialized to null pointers. 3. Set the data field of the head node to the given item. 4. For each vertex adjacent to the new vertex (as entered by the user): a. Add a new node for the adjacent vertex to the new vertex list,x. b. Add a new node for the new vertex to the list for the adjacent vertex. End for. 333

27 34. Algorithm to search for a vertex containing a given data item: Simply search the array v for a head node containing the data item. 35. Algorithm to delete a given data item and all edges incident to it: 1. Find the location k in the array v of the head node containing the data item; if it is not found, terminate this algorithm and display an appropriate message. 2. Set ptr = v[k]->next. 3. While (ptr!= 0), do the following: a. Search the list pointed to by v[ptr->vertex] for the node that contains the given data item and delete this node from the list. b. Set ptr = ptr ->next. 4. Delete the entire list at v[k] and make v[k] a null pointer. 36. Algorithm to find all vertices adjacent to a given vertex: 1. Find the location k in the array v of the head node containing the data item; if it is not found, terminate this algorithm and display an appropriate message. 2.Set ptr = v[k]->next. 3.While (ptr!= 0), do the following: a. Process v[ptr ->vertex], which is one of the adjacent vertices. b. Set ptr = ptr ->next. End while. 37. One method of determining if a graph (with adjacency matrix A) has a cycle follows: 1. Multiply A times itself (producing matrix A 2 ). 2. Set i = 2 and done = false. 3. While (! done), do the following: a. Multiply A times itself. b. If the diagonal of A has a non-zero entry, or i = number of vertices in the graph, Set done = true. Else Increment i by 1. End if End while. 334

28 38. One way to find a spanning tree is to do a depth-first traversal of the graph. This is a simple modification of the depthfirstsearch() function member of the class Graph in Figure 16.4 of the text. Simply add another parameter vector<bool> & edgelist and on each recursive call, inside the while loop, add the edge pointed to by ptr to edgelist. 39. Kruskal's algorithm is a simple modification of the spanning-tree function in Exercise 38 where one selects at each step the cheapest candidate for an edge. To do the selections most efficiently, store the edge nodes in a priority queue with the highest priority edges being those with minimal cost. 335

Graphs. directed and undirected graphs weighted graphs adjacency matrices. abstract data type adjacency list adjacency matrix

Graphs. directed and undirected graphs weighted graphs adjacency matrices. abstract data type adjacency list adjacency matrix Graphs 1 Graphs directed and undirected graphs weighted graphs adjacency matrices 2 Graph Representations abstract data type adjacency list adjacency matrix 3 Graph Implementations adjacency matrix adjacency