1. ��

2. 한��

3. 표��

3.1. ��
3.2. ��트
3.3. �� 표��한 Graph�� 트�� 표��한 Graph��

4. ��

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

6. Shortest Paths

6.1. Single Source, All Destination(하�� Shortest Path�� 할�� )
6.2. All Pairs Shortest Path

�� ! �� 하�� 하��.^^;

1. �� ¶

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

2. 한�� ¶

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

3. 표�� ¶

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

3.1. �� ¶

2�� 표��합��. �� 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�� 할�� 1�� 팅
LeftUpper -> RightLower ��
�� Directed Graph

~cpp 
         2 
       / |
 0 <- 1   |
      | /
      3  
 * .... 화���표 표��하������ ������하������..--; ��������� < Vertex1, Vertex2 > ������ 표����� ���������.
 * ��������� Vertex1 -> Vertex2��� ���향������ Edge��� ��������� < Vertex1, Vertex2 > ��������� ������ ������������.
 * 화���표��� ��������� ������ ��������� ��������������� 표��하���
 * <1,0>, <2,1>, <3,1>, <3,2>

��표��

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

| - ��향�� Edge�� 할�� 1�� 팅
LeftUpper -> RightLower ��

3.2. ��트 ¶

Adjacency List(Linked List�� 표��)

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

3.3. �� 표��한 Graph�� 트�� 표��한 Graph�� ¶

�� Vertex �� Edge�� ?
- �� : θ(1) - 2�� arxy �� ! �� .
- ��트 : θ(n)
��하�� ( n�� Vertices, e�� Edges )
- �� : θ(n^2) - n X n �� 히 n^2 --;
- ��트 : �� 트�� 표�� Head Node�� n�� 하�� Head Node�� Node�� 2*e �� 하��. θ(n+e) �� 트�� 하�� .
한�� : Weighted Graph�� 표��
- �� : 1�� Weight�� .
- ��트 : �� 하�� 하��. (��)
�� : �� 할 �� .. �� 편�� 하�� !

4. �� ¶

Traversal(탐��)
- Depth First Search(�� 탐��) : 한�� 파�� . �� 히�� . ��택 ��(�� ). �� .
- Breadth First Search(�� 탐��) : �� Vertex�� 큐�� . ��. �� Vertex�� 큐�� . �� Vertex�� . Queue�� . �� 탐�� 트�� 키�� Level Order�� .

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

Weighted Graph�� , �� Vertex�� Vertex�� 한�� .
Kruskal's Algorithm
- Edge�� Cost �� Sort(��)
- Edge�� 하�� 한��. ��하�� Cycle�� 한��. �� .
Prim's Algorithm
- Vertex�� 하�� 택한��.
- �� Edge�� 택한��.
- �� Edge�� Vertex�� Vertex�� Edge�� 택한��.
- �� 한��.

6. Shortest Paths ¶

6.1. Single Source, All Destination(하�� Shortest Path�� 할�� ) ¶

Dijkstra's Algorithm
- 표�� 행��(Adjancey(??) Matrix)�� 표��(�� 2�� )
- costi, j : i -> j �� Cost
- �� : S = { V0 } V0 : Source,
- �� S : { �� Shortest Path�� Vertex�� } : �� S�� .
- distw : v0�� 하�� w�� Shortest Path�� . �� w�� 하�� S��합�� Vertex�� 한��.
- ��

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

    ....
   
    ������한���.

�� 화 하��,
- for n-1 ��
  - dist �� Vertex u ��
  - S = S U {u}
  - dist�� : distw = min { distw, distu + costu, w }

6.2. All Pairs Shortest Path ¶

Floyd - Warshall Algorithm
- �� 표�� 2�� 한��. �� (-) Weight �� 허��한��.(�� .) 하�� Negative Cycle�� .
  - Negative Cycle? �� 클�� - �� 한��.
- �� 행�� A(-1)i, j �� 한��. ��할�� 호 �� . �� n-1�� 한��.
- A(k)i, j �� 하�� i�� j�� Shortest Path �� .�� 0 ~ k�� vertex�� .
- 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. �� ¶

2. 한�� ¶

3. 표�� ¶

3.1. �� ¶

3.2. ��트 ¶

3.3. �� 표��한 Graph�� 트�� 표��한 Graph�� ¶

4. �� ¶

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

6. Shortest Paths ¶

6.1. Single Source, All Destination(하�� Shortest Path�� 할�� ) ¶

6.2. All Pairs Shortest Path ¶

Data Structure/Graph

Contents

1. ������ ������ ¶

2. 한��� ��������� ¶

3. 표�� ¶

3.1. ������ ¶

3.2. ������트 ¶

3.3. ��������� 표��한 Graph��� ������트��� 표��한 Graph��� ������ ¶

4. ������ ������ ¶

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

6. Shortest Paths ¶

6.1. Single Source, All Destination(하������ ��������������� ������ ������������ Shortest Path��� ���할��� ���������������) ¶

6.2. All Pairs Shortest Path ¶

1. �� ¶

2. 한�� ¶

3.1. �� ¶

3.2. ��트 ¶

3.3. �� 표��한 Graph�� 트�� 표��한 Graph�� ¶

4. �� ¶

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

6.1. Single Source, All Destination(하�� Shortest Path�� 할�� ) ¶