14 장 . 그래프 알고리즘

PowerPoint
, , , , , ,

, .
, .
, , .
http://cg.myongji.ac.kr
(Object) (Concepts) . (Connection Network Network). (Task) , , . ,
Graph
Vertices
Edges

G‘(Subgraph)
(Path)
(Complete Graph)
V , (V-1)
,
(Forest):
IsAdjacent:
IsEmpty:
. 0 1.
: .
.
.

Vertex
. int AdjacencyMatrix[Max][Max];


int ** InitMatrix(int NRow, int NCol, int Value) Value
{ int **m;
m = malloc(NRow * sizeof(int *)); NRow
for (int i = 0; i < NRow; i++)
m[i] = malloc(NCol * sizeof(int)); NCol
for (int i = 0; i < NRow; i++)
for (int j = 0; j < NCol; j++)
m[i][j] = Value; Value
return m;
}
void FreeMatrix(**m) m
{ for (int i = 0; i < NRow; i++)
free(m[i]);
free(m);
}

int **M;
} graphType;
graphPtr InitGraph(int V)
void InsertEdge(graphPtr g, int V1, int V2)
graphPtr InitGraph(int V) V
{ graphPtr G = malloc(sizeof(graphType));
G->V = V; V
G->E = 0; 0
G->M = InitMatrix(V, V, 0); V 0
return G;
}
void InsertEdge(graphPtr g, int V1, int V2) V1, V2
{ if (g->M[V1][V2] = = 0) 0
{ g->E++;
g->M[V1][V2] = 1; V1->V2 1
}
}
.

E 2

i :
V2 , 2E
,

node* Next;
} node;
} graphType;
graphPtr InitGraph(int V)
void InsertEdge(graphPtr g, int V1, int V2)
graphPtr InitGraph(int V) V
{ graphPtr G = malloc(sizeof(graphType));
G->V = V; V
G->E = 0; 0
G->L = malloc(V * sizeof(node *)); V
for (int i = 0; i < V; i++)
G->L[i] = NULL;
return G;
}
void InsertEdge(graphPtr g, int V1, int V2) V1, V2
{ Nptr temp = malloc(sizeof(node));
temp->Data = V2; V2 ID
temp->Next = g->L[V1];
g->L[V1] = temp; L[V1]
} V2 V1 .

TRUE
FALSE . TRUE . O(2V) .

O(V). O(V2). O(2V + V2) = O(V2)

A , A B O(E). B , B A O(E). O(2V+2E) = O(V+E). (V+E) . V2 .

A->B A B
.

:
.
A, C, D, F, B, E, G . . A-D-E-G A-C-D-B-E-G
: . (DAG: Directed Acyclic Graph) .
(Sink, )
,

(Source, )
,
,
(, Out-Degree):
0 , 0 .

G G . E F. E .

( A, C, D, F, B, E, G)
TopologicalSort(G)
{ Select a Sink T;
L.Insert(1, T);
}
}

(Directed Edge)
.
A-D-B-E-G-F-C
. A-D-B-E-G-F-C G. G, E, B.
F.
TopologicalSort(V)
{ for All Nodes T Adjacent to V
if (T Is Unvisited)
TopologicalSort(T);
}

G G G .
. N , (N-1)
.
A A-B-C-D-G-E-F-H-I
B B-A-F-G-D-C-E-H-I

.
.
.
. .
, . ,
.

“ , .”
{A, B, C, D} {E, F, G, H} . B-E, C-E, D-E, D-F, A-F, A-D 6 . 2 .
. D-F . .

. A
A G, B, F B
A-B .
A B .
B-C, A-G, A-F B-C
B-C
. , .

A A G, B, F .
B , B C, E, D .
(Kruskal's Method)
1 A-B
1 G-E
()
. .
(Greedy Algorithm)
“ ”
“ ”
.
1
,

, . .
. .

.

(Dijkstra Algorithm)
.
i j

, 1 B.
A B C, D, E, F . 1 .

2 C, D, E, F
B , B
[C]: 1 5 . A-B 1 , B-C 2 3 A-B-C . 5 3 .
D, E, F B 1 . F
3 F C, D, E . F C, D, E 2 , 3 C .

4 C . 3 A-C 3 , C-E 1 4. A C . 4 . E . 5 D .
. 5 [D] 5 A-E + E-D = 4 + 1 = 5 . D E E

3 D

(Dynamic Programming)

Y X
X Y

(Transitive Closure)

, .

A ACBDGF
A CBDGF .
H A .

O(V*(V+E)), O(V*V2)
A[B][A] = 1 B A
A B . B A . A C, F A . A[B][C], A[B][F] 1 .
0 , 0 1 . .
O(V3). V2 . V . B C C C
.
:

A[A][B] = 1
A B B A
B B
2 C . A-B + B-C = 1 + 2 = 3. A C A[A][C] 5 .
A[D][B] = 1
D B C D-B-C D-B + B-C = 1 + 2 = 3
A[D][C] 3
.
A
B
C
D
E
F
A
0
1
(Biconnected Graph)
-- . . .
.
(AP: Articulation Point)
. . .

B . G B . B . B , B
C D C C . .
2 . .
{A, B, F, D, E} {C, G}.


D A .
,

A-B A
B-D B D B D
A-E * . E A
B-D D B . D C, B A C A . C A C A .
while (A[i] ≠ Empty) i = A[i];
M N

D-E , C-E , B-E
E . . O(V).
(Union by Weighting)

(a) J-D J J D C .

(Path Compression)
. K L K, A, M
2 : . .
(Halving)

(Network Flow)

5/3 (Capacity) 5 (Flow) 3
S->M->N->T . (Forward Edge) .
(Slack, ) .
S-M 5 - 3 = 2
(Augmenting Path, Alternating Path)
.
S-M-N-T
N-T 0
S-Q-P-T 0
5 + 3 + 5 = 13, 3 + 7 + 3 = 13
. , .
(Backward Edge)
S-M-N-P-T. N-P .
S-M-N 2. 2 S-M 7/5, M-N 4/4 . N . 2 , 2 . P-N . N 4/1 2 N . N P . P 2 2 . P-T 6/3 2 6/5 . 2 .
?

(Cut)
S, A, M , . A-B, A-N, M-N, M-P, S-Q
S, M, Q , . S-A, M-N, M-P, Q-P . .

“ “
. (Capacity) . .

(4 + 3 + 4 + 1 + 6) = 18. (5 + 4 + 1 + 5) = 15. 15 .
.

Documents

14 장 . 그래프 알고리즘