1. Data structure for graph algorithms: Adjacent list, Adjacent matrixb c d e f
Adjacent lists a d e c b f
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. Comparison between adjacent list and adjacent matrix １ ２ ５ ４ ３ 3 4 2 1 5 １ ２ 4 5 3 6 2 1
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.

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. ０ １ ８ ｗ Ｑ ｒ
１　１ u r s t v w x y
(2) ｓ Ｑ ０ ８ u r s t v w x y
(1) ０ １ ８ 2 u r s t v w x y １ Ｑ
(3) ０ １ 2 ８ u r s t v w x y Ｑ
(4) r ０ １ 2 ８ 3 u s t v w x y Ｑ
(5) ０ １ 2 3 u r s t v w x y Ｑ
(6) ０ １ 2 3 u r s t v w x y Ｑ
3 3
(7) t s u ０ １ 2 3 r v w x y Ｑ
(8) ０ １ 2 3 u r s t v w x y Ｑ:
(8)

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 1/
x y z
(a)
u v w 1/ 2/
x y z
(b)
u v w 1/ 2/ 3/
x y z
(c)
u v w 1/ 2/ 4/ 3/
x y z
(d)
u v w 1/ 2/ 4/ 3/
x y z
(e) B
u v w 1/ 2/
4/5 3/
x y z
(f) B
u v w 1/ 2/
4/5
3/6
x y z
(g) B
u v w 1/
2/7
4/5
3/6
x y z
(h) B
u v w
1/8
2/7
4/5
3/6
x y z
(i) B F
u v w
1/8
2/7
4/5
3/6
x y z
(j) B F
(k)
u v w
1/8
2/7 9/
4/5
3/6
x y z B F
(l)
u v w
1/8
2/7 9/
4/5
3/6
x y z B F C
(m)
u v w
1/8
2/7 9/
4/5
3/6
10/
x y z B F C
(n)
u v w
1/8
2/7 9/
4/5
3/6
10/
x y z B F C
(o)
u v w
1/8
2/7 9/
4/5
3/6
10/11
x y z B F C
(p)
u v w
1/8
2/7
9/12
4/5
3/6
10/11
x y z B F C

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|).