1. 기�� 개��

2. �� 그��기

3. ��

3.1. ��
3.2. ��
3.3. �� Graph�� Graph�� 교

4. 기��

5. �� (Minimum Cost Spanning Trees)

6. Shortest Paths

6.1. Single Source, All Destination(�� 곳�� Shortest Path�� 구�� )
6.2. All Pairs Shortest Path

�� 공�� 깨고! �� 구�� 계�� 겠��.^^;

1. 기�� 개�� ¶

Vertex( �� )
Edge( �� )
Directed Graph - Edge�� 그��
Undirected Graph - Edge�� 그��

2. �� 그��기 ¶

�� Vertex�� Edge�� 개 ��
Vertex�� Edge�� 개�� 2개��

3. �� ¶

~cpp 
������ ��������� Undirected Graph�� ������고 ��������������.
         2 
       / |
 0 - 1   |
     | /
     3       
������ ������ ������������..--;
V = { 0, 1, 2, 3 }
E = { (0,1), (1,3), (1,2), (2,3) }

3.1. �� ¶

2�� . �� Adjacency Matrix(��?--;)
�� Undirected Graph

	0	1	2	3
0	0	1	0	0
1	1	0	1	1
2	0	1	0	1
3	0	1	1	0

Vertex �� Edge�� 그 곳�� 값�� 1��
LeftUpper -> RightLower ��각�� 기��
�� Directed Graph

~cpp 
         2 
       / |
 0 <- 1   |
      | /
      3  
 * .... ��������� ���������기�� ���������군���..--; ��������� < Vertex1, Vertex2 > ������ ��������� ���������.
 * ��������� Vertex1 -> Vertex2��� ������������ Edge�� ��������� < Vertex1, Vertex2 > ������게 ������ ������������.
 * ������������ 그������ ������ ���계��� ��������������� ������������
 * <1,0>, <2,1>, <3,1>, <3,2>

�� 그��

	0	1	2	3
0	0	0	0	0
1	1	0	0	0
2	0	1	0	0
3	0	1	1	0

| - �� Edge�� 1��
LeftUpper -> RightLower ��각�� 기��

3.2. �� ¶

Adjacency List(Linked List�� )

0	->	1	X
1	->	0	-	->	2	-	->	3	X
2	->	1	-	->	3	X
3	->	1	-	->	2	X

3.3. �� Graph�� Graph�� 교 ¶

�� Vertex �� Edge�� ?
- �� : θ(1) - 2�� arxy �� 근 ��! 고�� .
- �� : θ(n)
�� ( n개�� Vertices, e개�� Edges )
- �� : θ(n^2) - n X n 개�� n^2 --;
- �� : �� 그�� Head Node�� n개�� 고 Head Node�� Node�� 2*e 개�� . θ(n+e) 고�� .
�� : Weighted Graph��
- �� : 1�� Weight�� .
- �� : �� . (��)
결�� : �� 그�� .. 구�� 과 �� 게 �� !

4. 기�� ¶

Traversal(��)
- Depth First Search(�� 깊�� ) : �� 간�� . �� . �� (�� ). �� .
- Breadth First Search(�� ) : �� Vertex�� . ��. 거기�� Vertex�� . �� 그 �� 결�� Vertex�� 계�� . Queue�� 게 �� 거��. �� 그게 �� Level Order�� .

5. �� (Minimum Cost Spanning Trees) ¶

Weighted Graph�� , �� Vertex�� Vertex까�� 게 �� 그��.
Kruskal's Algorithm
- Edge�� Cost �� Sort(��)
- Edge�� 결��. ��결�� Cycle�� 기�� 그�� 고 ��거��. �� 그��.
Prim's Algorithm
- Vertex�� .
- 거기�� Edge�� .
- 그 Edge�� Vertex�� Vertex�� Edge�� 걸 ��.
- �� .

6. Shortest Paths ¶

6.1. Single Source, All Destination(�� 곳�� Shortest Path�� 구�� ) ¶

Dijkstra's Algorithm
- �� (Adjancey(??) Matrix)�� (그��까 2�� )
- costi, j : i -> j �� Cost
- ��기값 : S = { V0 } V0 : Source,
- ��간 S : { ��금까�� Shortest Path�� 결�� Vertex�� } : 그��까 결�� S�� .
- distw : v0�� w까�� Shortest Path�� 값. �� w�� 고�� S�� Vertex�� 거�� .
- ��

~cpp 
    Start Vertex : 4 -> S = {4}
    dist[3] = 1500         
    dist[5] = 250                3,5 ��� 4��� ���결������ ������
    dist[others] = ���������
           -> dist[5]�� ����� ������. 5��� S��� ���������.
    ...

    ....
   
    ������������.

��걸 ��고�� ,
- for n-1 ��
  - dist 값�� Vertex u ��기
  - S = S U {u}
  - dist값 갱�� : distw = min { distw, distu + costu, w }

6.2. All Pairs Shortest Path ¶

Floyd - Warshall Algorithm
- �� 2�� . 그�� 고�� (-) Weight �� .(그�� .) �� Negative Cycle�� .
  - Negative Cycle? 그 �� - �� 길�� .
- ��기 �� A(-1)i, j �� . �� 값�� . ��걸 n-1까�� .
- A(k)i, j ��고 �� i�� j�� Shortest Path �� 값��.�� 길�� 0 ~ k�� vertex�� 간�� .
- A(k)i, j = min { A(k-1)i,j, A(k-1)i, k + A(k-1)k, j }

DataStructure

Data Structure/Graph

Contents

1. 기�� 개�� ¶

2. �� 그��기 ¶

3. �� ¶

3.1. �� ¶

3.2. �� ¶

3.3. �� Graph�� Graph�� 교 ¶

4. 기�� ¶

5. �� (Minimum Cost Spanning Trees) ¶

6. Shortest Paths ¶

6.1. Single Source, All Destination(�� 곳�� Shortest Path�� 구�� ) ¶

6.2. All Pairs Shortest Path ¶

Data Structure/Graph

Contents

1. 기��� 개��� ¶

2. ������ 그���기 ¶

3. ������ ¶

3.1. ������ ¶

3.2. ��������� ¶

3.3. ��������� ��������� Graph��� ������������ ��������� Graph��� ���교 ¶

4. 기��� ������ ¶

5. ������ ������ ������ ������(Minimum Cost Spanning Trees) ¶

6. Shortest Paths ¶

6.1. Single Source, All Destination(��������� ��������������� ������ 곳��������� Shortest Path��� 구������ ���������������) ¶

6.2. All Pairs Shortest Path ¶

1. 기�� 개�� ¶

2. �� 그��기 ¶

3. �� ¶

3.1. �� ¶

3.2. �� ¶

3.3. �� Graph�� Graph�� 교 ¶

4. 기�� ¶

5. �� (Minimum Cost Spanning Trees) ¶

6.1. Single Source, All Destination(�� 곳�� Shortest Path�� 구�� ) ¶