1. a c b c a d e f c f b e c a e d f c e f Adjacent lists Graph Algorithms Data structure for graph algorithms: Adjacent list, Adjacent matrix Adjacent matrix

2. Adjacent list: Each vertex u has an adjacent list Adj[u] (1) For a connected graph G, if G is directed graph the total size of all the adjacent lists is |E|, and if G is undirected graph then the total size of all the adjacent lists is 2|E|. Generally, the total size of adjacent lists is O(V+E). (2)For a weighted graph G, weight w(u,v) of edge (u,v) is kept in Adj[u] with vertex v. Adjacent matrix: Each vertex is given a number from 1,2,…,|V|. (1) For a undirected graph, its adjacent matrix is symmetric. (2) For a weighted graph, weight w(u,v) is kept in its adjacent matrix at row i and column j.

3. １ ２ 5 1 2 2 5 3 1 4 ３ 3 4 2 ５ ４ 3 4 5 2 5 1 2 4 4 1 2 １ 3 ２ 2 5 3 5 6 4 2 4 6 5 5 4 6 6 • Comparison between adjacent list and adjacent matrix • If |E| is much smaller than then adjacent list is better (using less memory). • It costs time using adjacent lists to find if v is adjacent to u. 1 2 3 4 5 1 0 1 0 0 1 2 1 0 1 1 1 3 0 1 0 1 0 4 0 1 1 0 1 5 1 1 0 1 0 1 2 3 4 5 6 1 0 1 0 1 0 0 2 0 0 0 0 1 0 3 0 0 0 0 1 1 4 0 1 0 0 0 0 5 0 0 0 1 0 0 6 0 0 0 0 0 1

4. Given G = (V,E) and vertex s, search all the vertices that s can arrive. Breadth-first search (BFS): Searching the vertices whose distance from s is k ealier than visiting those whose distance from s is k+1. Ｑ ８ ８ ０ １ Ｑ ｗ ｒ (2) (1) ｓ r r s s t t u u r r s s t t u u ８ ８ ８ １　１ １ ０ ０ ８ ８ ８ Ｑ Ｑ ８ ８ 2 ０ １ 2 ０ １ ８ ８ ８ ８ (3) t x v (4) r t x v w x y v v w w x x y y ８ v w x y ８ 2 2 2 2 2 ８ １ 2 １ １ 2 2 u r r s s t u t Ｑ Ｑ 3 2 ０ １ 3 2 ０ １ (6) v u y (5) x u v 2 3 3 2 2 １ 3 ８ 2 2 2 2 3 １ v w x y v w x y u s t r s u r t Ｑ Ｑ 3 3 2 2 ０ ０ １ １ (8) (7) y u y 2 2 3 2 2 １ 3 １ 3 3 3 v w x v w x y y r s t u 3 2 ０ １ (8) Ｑ: 2 2 １ 3 v w x y

5. Analysis of the algorithm • Each vertex is put into queue Q at most once. Therefore, the number of operation for the queue is O(|V|). • Each adjacent list is at most scanned once. Therefore, the total running time for scanning adjacent lists is O(|E|). • The running time for initiation is O(|V|). • Therefore, the total running time of the algorithm is O(|V|+|E|).

6. Find the path from s to v in BSF

7. Depth-first search：Search deeper in the graph whenever possible. (1) Each vertex has two timestamps: d[v] is the first timestamp when v is first discovered, and f[v] is the second timestamps when the search finishes examining v’s adjacent list. (2) It generates a number of depth-first search trees. u v w u v w u v w u v w 1/ 1/ 2/ 1/ 2/ 1/ 2/ 4/ 3/ 3/ x y z x y z x y z x y z (a) (d) (c) (b) u v w u v w u v w u v w 1/ 2/ 1/ 2/7 1/ 2/ 1/ 2/ B B B B 4/ 3/ 4/5 3/6 4/5 3/6 4/5 3/ x y z x y z x y z x y z (e) (h) (g) (f) u v w u v w u v w u v w 1/8 2/7 9/ 1/8 2/7 1/8 2/7 9/ 1/8 2/7 C B B B B F F F F 4/5 3/6 4/5 3/6 4/5 3/6 4/5 3/6 x y z x y z x y z x y z (l) (i) (k) (j) u v w u v w u v w u v w 1/8 2/7 9/ 1/8 2/7 9/ 1/8 2/7 9/12 1/8 2/7 9/ C C C C B B B B F F F F B B B 4/5 3/6 10/ 4/5 3/6 10/11 4/5 3/6 10/11 4/5 3/6 10/ x y z x y z x y z x y z (n) (o) (p) (m)

8. Running time (1)The running time except DFS-VIST is O(|V|). (2) Each vertex is called by DFS-VISIT only once, because only white vertices will be called by DES-VISIT and when they are called their color is changed to gray immediately. (3) The loop in DFS-VISIT is executed only |Adj[v]| times. Therefore, the total running time of the algorithm is O(|V|+|E|).