1. ��본 ��념

2. ��붓 ��리��

3. ��

3.1. 배��
3.2. 리��
3.3. 배�� Graph�� 리�� Graph��

4. ��본 ��

5. �� 리(Minimum Cost Spanning Trees)

6. Shortest Paths

6.1. Single Source, All Destination(�� 모�� Shortest Path를 �� 말��)
6.2. All Pairs Shortest Path

��만�� 백�� ! ��료 �� 리를 �� .^^;

1. ��본 ��념 ¶

Vertex( �� )
Edge( �� )
Directed Graph - Edge�� 방�� 는 ��
Undirected Graph - Edge�� 방�� 는 ��

2. ��붓 ��리�� ¶

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

3. �� ¶

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

3.1. 배�� ¶

2�� 배�� . �� Adjacency Matrix(맞��?--;)
먼�� 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�� 1��
LeftUpper -> RightLower 대�� 대��
�� Directed Graph

~cpp 
         2 
       / |
 0 <- 1   |
      | /
      3  
 * .... ��������� ��������������� ���������������..--; �������� < Vertex1, Vertex2 > ���런 ��������� ���������.
 * 만������ Vertex1 -> Vertex2��� 방�������� Edge��� ������면 < Vertex1, Vertex2 > ��������� ������ 말���������.
 * ���������를 ���릴��� ���는 �������� �������������� ���������면
 * <1,0>, <2,1>, <3,1>, <3,2>

�� 리��면

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

| - 방�� Edge�� 1��
LeftUpper -> RightLower 대�� 대��

3.2. 리�� ¶

Adjacency List(Linked List�� )

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

3.3. 배�� Graph�� 리�� Graph�� ¶

�� Vertex �� Edge�� ?
- 배�� : θ(1) - 2�� 배�� arxy ��번�� 능! �� 배�� 말��.
- 리�� : θ(n)
��는 메모리�� ( n�� Vertices, e�� Edges )
- 배�� : θ(n^2) - n X n �� 배�� n^2 --;
- 리�� : �� 리�� 보면 Head Node�� 는 n�� Head Node��부�� 뻗��는 Node�� 는 2*e �� . θ(n+e) �� 리�� 리�� 말��.
��더 : Weighted Graph��
- 배�� : 1대�� Weight를 ��.
- 리�� : ��를 �� . (��)
�� : 뭐�� 더 낳�� 런�� 만.. �� 리�� 부�� 따��볼�� 배�� 는�� 말��!

4. ��본 �� ¶

Traversal(��)
- Depth First Search(��리말�� ) : ��물�� 는 말��. �� 막��면 빽. �� (��는 ��). �� 돌��면 ��.
- Breadth First Search(��리말�� ) : �� Vertex를 �� . 뺍��. �� Vertex를 �� . ��부�� 빼면�� 노�� Vertex를 �� . Queue�� 면 ��는 ��랍��. �� 리�� 면 �� 바�� Level Order�� 는 ��란 말��.

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

Weighted Graph�� , �� Vertex�� 마��막 Vertex�� 끊�� 번�� 린��.
Kruskal's Algorithm
- Edge들�� Cost �� Sort(��부��)
- Edge들�� 따�� . �� Cycle�� 면 �� 말�� . �� 면 ��만��.
Prim's Algorithm
- Vertex를 �� .
- �� Edge�� .
- �� Edge�� Vertex�� Vertex�� Edge�� .
- �� 복��.

6. Shortest Paths ¶

6.1. Single Source, All Destination(�� 모�� Shortest Path를 �� 말��) ¶

Dijkstra's Algorithm
- �� (Adjancey(??) Matrix)�� (��러�� 2�� 배��)
- costi, j : i -> j �� 는 Cost
- �� : S = { V0 } V0 : Source,
- �� S : { �� Shortest Path�� Vertex들 } : ��러�� 면 S�� 는��는 말��.
- distw : v0�� w�� Shortest Path�� . �� w를 ��는 S��내�� 는 Vertex들만 �� .
- ��를 들��면

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

    ....
   
    ��복������.

�� 리�� 면,
- for n-1 번 ��복
  - dist �� 는 Vertex u ��
  - S = S U {u}
  - dist�� : distw = min { distw, distu + costu, w }

6.2. All Pairs Shortest Path ¶

Floyd - Warshall Algorithm
- �� 2�� 배�� . ��런데 �� 리�� (-) Weight �� .(��리�� 면 �� 는 말��.) ��만 Negative Cycle�� .
  - Negative Cycle? �� 돌면 - �� 는�� 말��.
- �� A(-1)i, j �� . ��복�� . �� n-1�� 복��.
- A(k)i, j �� 면 i�� j�� 는 Shortest Path �� .�� 모�� 0 ~ k�� vertex만�� .
- 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. ��본 ��념 ¶

2. ��붓 ��리�� ¶

3. �� ¶

3.1. 배�� ¶

3.2. 리�� ¶

3.3. 배�� Graph�� 리�� Graph�� ¶

4. ��본 �� ¶

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

6. Shortest Paths ¶

6.1. Single Source, All Destination(�� 모�� Shortest Path를 �� 말��) ¶

6.2. All Pairs Shortest Path ¶

Data Structure/Graph

Contents

1. ���본 ���념 ¶

2. ���붓 ���리��� ¶

3. ������ ¶

3.1. 배��� ¶

3.2. 리������ ¶

3.3. 배����� ��������� Graph��� 리�������� ��������� Graph��� ����� ¶

4. ���본 ������ ¶

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

6. Shortest Paths ¶

6.1. Single Source, All Destination(�������� ��������������� 모�� ����������� Shortest Path를 ��������� ������말������) ¶

6.2. All Pairs Shortest Path ¶

1. ��본 ��념 ¶

2. ��붓 ��리�� ¶

3. �� ¶

3.1. 배�� ¶

3.2. 리�� ¶

3.3. 배�� Graph�� 리�� Graph�� ¶

4. ��본 �� ¶

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

6.1. Single Source, All Destination(�� 모�� Shortest Path를 �� 말��) ¶