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1 p a. Find the shortest path from A to all other vertices for the graph in Figure 9.8. b. Find the shortest unweighted path from B to all other vertices for the graph in Figure 9.8. A 5 B C D G F E a. source destination path cost A B AB 5 C AC D ABGED 9 E ABGE 7 F ABGEF 8 G ABG 6 b. Source Destination path cost B A BEDA C BC D BCD E BE F BEF G BG p. 9. (ZOJ 56) a. Explain how to modify Dijkstra's algorithm to produce a count of the number of different minimum paths from v to w. b. Explain how to modify Dijkstra's algorithm so that if there is more than one minimum path from v to w, a path with the fewest number of edges is chosen.

2 Answer a: void Dijkstra( Table T ) /* T[ ].Count is initialized to be. T[start].Count = */ vertex v, w; for ( ; ; ) v = smallest unknown distance vertex; if ( v == NotAVertex ) break; T[v].Known = True; for ( each w adjacent to v ) if(!t[w].known ) if( T[v].Dist + Cvw < T[w].Dist ) Decrease( T[w].Dist to T[v]+Cvw ) T[w].Path = v; T[w].Count = T[v].Count; /* NOT T[w].Count = */ else if( T[v].Dist + Cvw == T[w].Dist ) T[w].Count += T[v].Count; /* NOT T[w].Count += */ Answer b: void Dijkstra( Table T ) /* T[ ].Count is initialized to be */ vertex v, w; for ( ; ; ) v = smallest unknown distance vertex; if ( v == NotAVertex ) break; T[v].Known = True; for ( each w adjacent to v ) if(!t[w].known ) if( T[v].Dist + Cvw < T[w].Dist ) Decrease( T[w].Dist to T[v]+Cvw ) T[w].Path = v; T[w].Count = T[v].Count + ; else if( ( T[v].Dist + Cvw == T[w].Dist ) && ( T[v].Count + < T[w].Count ) ) T[w].Count = T[v].Count + ; T[w].Path = v; /* DO NOT forget this */

3 p. 9. Find the maximum flow in the network of Figure Answer: A B C s D E F 6 G H I t p. 9.5 a. Find a minimum spanning tree for the graph in Figure 9.8 using both Prim's and Kruskal's algorithms. b. Is this minimum spanning tree unique? Why? Answer a: They are the same A B C D E F G H I 7 J Answer b: This minimum spanning tree is not unique. For example, another minimum spanning tree obtained from Prim s algorithm is: A B C D E F G H I 7 J

4 p Write a program to find the strongly connected components in a digraph. #define MaxVertices /* maximum number of vertices */ typedef int Vertex; /* vertices are numbered from to MaxVertices */ typedef enum FALSE, TRUE boolean; /* declarations for a graph with adjacency list representation */ #ifndef _Graph_h struct VNode; typedef struct VNode *PtrToVNode; struct GNode; typedef struct GNode *PtrToGNode; typedef PtrToGNode Graph; /* create a graph with NumOfVertices vertices and no edge */ Graph CreateGraph( int NumOfVertices ); /* insert edge V->W to G */ boolean InsertEdge( Vertex V, Vertex W, Graph G ); /* reverse all the edges in G and return the resulting graph Gr */ Graph ReverseGraph( Graph G ); /* free spaces taken by G */ void DeleteGraph( Graph G ); /* implementations of the above functions are omitted */ #endif /*_Graph_h */ struct VNode Vertex Vert; PtrToVNode Next; ; struct GNode int NumOfVertices; int NumOfEdges; PtrToVNode *Array; ; boolean Visited[MaxVertices]; /* global mark for visited vertices */ Vertex DfsOrder[MaxVertices]; /* store vertices in dfs order */ int DfsNum; /* dfs number for vertices */

5 void PostOrder( Vertex V ) /* store vertices during postorder dfs */ DfsOrder[DfsNum++] = V; void PrintV( Vertex V ) /* print V during postorder dfs */ printf("%d ", V); void PostorderDfs( Vertex V, Graph G, void (*f)(vertex V) ) /* postorder dfs template with visiting function f */ PtrToVNode W; Visited[V] = TRUE; for ( W=G->Array[V]->Next; W; W=W->Next ) if (!Visited[W->Vert] ) PostorderDfs( W->Vert, G, (*f) ); (*f)(v); void StronglyConnectedComponents( Graph G ) /* print the strongly connected components in G */ /* output format: V, V,... V, V, */ Graph Gr; Vertex V; /* Step : mark the postorder dfs number for each vertex in G */ InitializeVisited( G->NumOfVertices ); /* Visited[ ] is initialized to be FALSE */ DfsNum = ; for ( V=; V<G->NumOfVertices; V++ ) if (!Visited[V] ) PostorderDfs( V, G, PostOrder ); /* end for */ /* Step : reverse edges in G and save the resulting graph in Gr */ Gr = ReverseGraph( G ); if (!Gr )

6 printf("program failed: cannot reverse graph.\n"); else /* Step : print components by postorder dfs on Gr */ InitializeVisited( Gr->NumOfVertices ); while ( DfsNum ) /* always start at the vertex with the largest dfs number */ V = DfsOrder[DfsNum]; if (!Visited[V] ) /* print this component in a line*/ printf( " " ); PostorderDfs( V, Gr, PrintV ); printf( "\n" ); /* end - if */ /* end - while */ DeleteGraph( Gr ); /* free space */ /* end - else */ Sketch of the proof of correctness:. V, W Comp(G) Path(V->W) and Path(W->V) in both G and Gr;. V, W Comp(G) V, W DfsT(G) and DfsT(Gr);. For V DfsT(Gr) with X as the root, Path(X->V) in Gr, and hence Path(V->X) in G.. Step V, X DfsT(G); 5. DfsNum(X) > DfsNum(V) V must be numbered before X in DfsT(G); 6. Step 5 V is a descendant of X in DfsT(G) since it was a postorder visit; 7. Step 6 Path(X->V) in G. G/G r DfsT(G) DfsT(G r )

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