第41棵 - 图的存储结构
1. 邻接矩阵法
用一维数组存储顶点--描述顶点相关的数据。
用二维数组存储边--描述顶点的边。
设图A = (V,E)是一个有n个顶点的图，图的邻接矩阵为Edge[n][n]，则：Edge[i][j] = W，W>0，i和j连接；Edge[i][j] = 0，i == j 或者i和j不链接。
注：W为权值，当需要权值时，取W为1表示结点间连接。
无向图的邻接矩阵是对称的。
有向图的邻接矩阵可能是不对称的。
2. 邻接矩阵法的头结点
记录定点的个数。
记录与顶点相关的数据描述。
记录描述边集的二维数组。
typedef struct _tag_MGraph
{
int count;
MVertex** v;
int** matrix;
}TMGraph;
问题：如何根据顶点数目，动态创建二维数组？
3. 动态申请二维数组的原理
通过二级指针动态申请一位数组。
通过一级指针申请数据空间。
将一维指针数组中的指针连接到数据空间。
int** malloc2d(int row, int col)
{
int** ret = (int**)malloc(sizeof(int*) * row);
int* p = (int*)malloc(sizeof(int) * row *col);
int i = 0;
if(p && ret)
{
for(i=0;i<row;i++)
{
ret[i] = p + i * col;
}
}
else
{
free(ret);
fre(p);
ret = NULL;
}
return ret;
}
4. 程序——邻接矩阵法实现图结构
main.c
#include <stdio.h>
#include <stdlib.h>
#include "LGraph.h"
/* run this program using the console pauser or add your own getch, system("pause") or input loop */
void print_data(LVertex* v)
{
printf("%s", (char*)v);
}
int main(int argc, char *argv[])
{
LVertex* v[] = {"A", "B", "C", "D", "E", "F"};
LGraph* graph = LGraph_Create(v, 6);
LGraph_AddEdge(graph, 0, 1, 1);
LGraph_AddEdge(graph, 0, 2, 1);
LGraph_AddEdge(graph, 0, 3, 1);
LGraph_AddEdge(graph, 1, 5, 1);
LGraph_AddEdge(graph, 1, 4, 1);
LGraph_AddEdge(graph, 2, 1, 1);
LGraph_AddEdge(graph, 3, 4, 1);
LGraph_AddEdge(graph, 4, 2, 1);
LGraph_Display(graph, print_data);
LGraph_DFS(graph, 0, print_data);
LGraph_BFS(graph, 0, print_data);
LGraph_Destroy(graph);
return 0;
}
LGraph.h
#ifndef _LGRAPH_H_
#define _LGRAPH_H_
typedef void LGraph;
typedef void LVertex;
typedef void (LGraph_Printf)(LVertex*);
LGraph* LGraph_Create(LVertex** v, int n);
void LGraph_Destroy(LGraph* graph);
void LGraph_Clear(LGraph* graph);
int LGraph_AddEdge(LGraph* graph, int v1, int v2, int w);
int LGraph_RemoveEdge(LGraph* graph, int v1, int v2);
int LGraph_GetEdge(LGraph* graph, int v1, int v2);
int LGraph_TD(LGraph* graph, int v);
int LGraph_VertexCount(LGraph* graph);
int LGraph_EdgeCount(LGraph* graph);
void LGraph_DFS(LGraph* graph, int v, LGraph_Printf* pFunc);
void LGraph_BFS(LGraph* graph, int v, LGraph_Printf* pFunc);
void LGraph_Display(LGraph* graph, LGraph_Printf* pFunc);
#endif
LGraph.c
#include <malloc.h>
#include <stdio.h>
#include "LGraph.h"
#include "LinkList.h"
#include "LinkQueue.h"
typedef struct _tag_LGraph
{
int count;
LVertex** v;
LinkList** la;
} TLGraph;
typedef struct _tag_ListNode
{
LinkListNode header;
int v;
int w;
} TListNode;
static void recursive_dfs(TLGraph* graph, int v, int visited[], LGraph_Printf* pFunc)
{
int i = 0;
pFunc(graph->v[v]);
visited[v] = 1;
printf(", ");
for(i=0; i<LinkList_Length(graph->la[v]); i++)
{
TListNode* node = (TListNode*)LinkList_Get(graph->la[v], i);
if( !visited[node->v] )
{
recursive_dfs(graph, node->v, visited, pFunc);
}
}
}
static void bfs(TLGraph* graph, int v, int visited[], LGraph_Printf* pFunc)
{
LinkQueue* queue = LinkQueue_Create();
if( queue != NULL )
{
LinkQueue_Append(queue, graph->v + v);
visited[v] = 1;
while( LinkQueue_Length(queue) > 0 )
{
int i = 0;
v = (LVertex**)LinkQueue_Retrieve(queue) - graph->v;
pFunc(graph->v[v]);
printf(", ");
for(i=0; i<LinkList_Length(graph->la[v]); i++)
{
TListNode* node = (TListNode*)LinkList_Get(graph->la[v], i);
if( !visited[node->v] )
{
LinkQueue_Append(queue, graph->v + node->v);
visited[node->v] = 1;
}
}
}
}
LinkQueue_Destroy(queue);
}
LGraph* LGraph_Create(LVertex** v, int n) // O(n)
{
TLGraph* ret = NULL;
int ok = 1;
if( (v != NULL ) && (n > 0) )
{
ret = (TLGraph*)malloc(sizeof(TLGraph));
if( ret != NULL )
{
ret->count = n;
ret->v = (LVertex**)calloc(n, sizeof(LVertex*));
ret->la = (LinkList**)calloc(n, sizeof(LinkList*));
ok = (ret->v != NULL) && (ret->la != NULL);
if( ok )
{
int i = 0;
for(i=0; i<n; i++)
{
ret->v[i] = v[i];
}
for(i=0; (i<n) && ok; i++)
{
ok = ok && ((ret->la[i] = LinkList_Create()) != NULL);
}
}
if( !ok )
{
if( ret->la != NULL )
{
int i = 0;
for(i=0; i<n; i++)
{
LinkList_Destroy(ret->la[i]);
}
}
free(ret->la);
free(ret->v);
free(ret);
ret = NULL;
}
}
}
return ret;
}
void LGraph_Destroy(LGraph* graph) // O(n*n)
{
TLGraph* tGraph = (TLGraph*)graph;
LGraph_Clear(tGraph);
if( tGraph != NULL )
{
int i = 0;
for(i=0; i<tGraph->count; i++)
{
LinkList_Destroy(tGraph->la[i]);
}
free(tGraph->la);
free(tGraph->v);
free(tGraph);
}
}
void LGraph_Clear(LGraph* graph) // O(n*n)
{
TLGraph* tGraph = (TLGraph*)graph;
if( tGraph != NULL )
{
int i = 0;
for(i=0; i<tGraph->count; i++)
{
while( LinkList_Length(tGraph->la[i]) > 0 )
{
free(LinkList_Delete(tGraph->la[i], 0));
}
}
}
}
int LGraph_AddEdge(LGraph* graph, int v1, int v2, int w) // O(1)
{
TLGraph* tGraph = (TLGraph*)graph;
TListNode* node = NULL;
int ret = (tGraph != NULL);
ret = ret && (0 <= v1) && (v1 < tGraph->count);
ret = ret && (0 <= v2) && (v2 < tGraph->count);
ret = ret && (0 < w) && ((node = (TListNode*)malloc(sizeof(TListNode))) != NULL);
if( ret )
{
node->v = v2;
node->w = w;
LinkList_Insert(tGraph->la[v1], (LinkListNode*)node, 0);
}
return ret;
}
int LGraph_RemoveEdge(LGraph* graph, int v1, int v2) // O(n*n)
{
TLGraph* tGraph = (TLGraph*)graph;
int condition = (tGraph != NULL);
int ret = 0;
condition = condition && (0 <= v1) && (v1 < tGraph->count);
condition = condition && (0 <= v2) && (v2 < tGraph->count);
if( condition )
{
TListNode* node = NULL;
int i = 0;
for(i=0; i<LinkList_Length(tGraph->la[v1]); i++)
{
node = (TListNode*)LinkList_Get(tGraph->la[v1], i);
if( node->v == v2)
{
ret = node->w;
LinkList_Delete(tGraph->la[v1], i);
free(node);
break;
}
}
}
return ret;
}
int LGraph_GetEdge(LGraph* graph, int v1, int v2) // O(n*n)
{
TLGraph* tGraph = (TLGraph*)graph;
int condition = (tGraph != NULL);
int ret = 0;
condition = condition && (0 <= v1) && (v1 < tGraph->count);
condition = condition && (0 <= v2) && (v2 < tGraph->count);
if( condition )
{
TListNode* node = NULL;
int i = 0;
for(i=0; i<LinkList_Length(tGraph->la[v1]); i++)
{
node = (TListNode*)LinkList_Get(tGraph->la[v1], i);
if( node->v == v2)
{
ret = node->w;
break;
}
}
}
return ret;
}
int LGraph_TD(LGraph* graph, int v) // O(n*n*n)
{
TLGraph* tGraph = (TLGraph*)graph;
int condition = (tGraph != NULL);
int ret = 0;
condition = condition && (0 <= v) && (v < tGraph->count);
if( condition )
{
int i = 0;
int j = 0;
for(i=0; i<tGraph->count; i++)
{
for(j=0; j<LinkList_Length(tGraph->la[i]); j++)
{
TListNode* node = (TListNode*)LinkList_Get(tGraph->la[i], j);
if( node->v == v )
{
ret++;
}
}
}
ret += LinkList_Length(tGraph->la[v]);
}
return ret;
}
int LGraph_VertexCount(LGraph* graph) // O(1)
{
TLGraph* tGraph = (TLGraph*)graph;
int ret = 0;
if( tGraph != NULL )
{
ret = tGraph->count;
}
return ret;
}
int LGraph_EdgeCount(LGraph* graph) // O(n)
{
TLGraph* tGraph = (TLGraph*)graph;
int ret = 0;
if( tGraph != NULL )
{
int i = 0;
for(i=0; i<tGraph->count; i++)
{
ret += LinkList_Length(tGraph->la[i]);
}
}
return ret;
}
void LGraph_DFS(LGraph* graph, int v, LGraph_Printf* pFunc)
{
TLGraph* tGraph = (TLGraph*)graph;
int* visited = NULL;
int condition = (tGraph != NULL);
condition = condition && (0 <= v) && (v < tGraph->count);
condition = condition && (pFunc != NULL);
condition = condition && ((visited = (int*)calloc(tGraph->count, sizeof(int))) != NULL);
if( condition )
{
int i = 0;
recursive_dfs(tGraph, v, visited, pFunc);
for(i=0; i<tGraph->count; i++)
{
if( !visited[i] )
{
recursive_dfs(tGraph, i, visited, pFunc);
}
}
printf("\n");
}
free(visited);
}
void LGraph_BFS(LGraph* graph, int v, LGraph_Printf* pFunc)
{
TLGraph* tGraph = (TLGraph*)graph;
int* visited = NULL;
int condition = (tGraph != NULL);
condition = condition && (0 <= v) && (v < tGraph->count);
condition = condition && (pFunc != NULL);
condition = condition && ((visited = (int*)calloc(tGraph->count, sizeof(int))) != NULL);
if( condition )
{
int i = 0;
bfs(tGraph, v, visited, pFunc);
for(i=0; i<tGraph->count; i++)
{
if( !visited[i] )
{
bfs(tGraph, i, visited, pFunc);
}
}
printf("\n");
}
free(visited);
}
void LGraph_Display(LGraph* graph, LGraph_Printf* pFunc) // O(n*n*n)
{
TLGraph* tGraph = (TLGraph*)graph;
if( (tGraph != NULL) && (pFunc != NULL) )
{
int i = 0;
int j = 0;
for(i=0; i<tGraph->count; i++)
{
printf("%d:", i);
pFunc(tGraph->v[i]);
printf(" ");
}
printf("\n");
for(i=0; i<tGraph->count; i++)
{
for(j=0; j<LinkList_Length(tGraph->la[i]); j++)
{
TListNode* node = (TListNode*)LinkList_Get(tGraph->la[i], j);
printf("<");
pFunc(tGraph->v[i]);
printf(", ");
pFunc(tGraph->v[node->v]);
printf(", %d", node->w);
printf(">");
printf(" ");
}
}
printf("\n");
}
}
LinkList.h
LinkList.c
LinkQueue.h
LinkQueue.c
5. 邻接表示法
从一个顶点出发的边连接在同一个链表中。
每一个链表结点代表一条边，结点中保存边的另一个顶点的下标和权值。
6. 邻接链表发的头结点
记录定点个数。
记录与顶点相关的数据描述。
记录描述边集的链表数组。
typedef struct _tag_LGraph
{
int count;
LVertex** v;
LinkList** la;
}TLGraph;
7. 程序——邻接链表发实现图结构
main.c
#include <stdio.h>
#include <stdlib.h>
#include "MGraph.h"
/* run this program using the console pauser or add your own getch, system("pause") or input loop */
void print_data(MVertex* v)
{
printf("%s", (char*)v);
}
int main(int argc, char *argv[])
{
MVertex* v[] = {"A", "B", "C", "D", "E", "F"};
MGraph* graph = MGraph_Create(v, 6);
MGraph_AddEdge(graph, 0, 1, 1);
MGraph_AddEdge(graph, 0, 2, 1);
MGraph_AddEdge(graph, 0, 3, 1);
MGraph_AddEdge(graph, 1, 5, 1);
MGraph_AddEdge(graph, 1, 4, 1);
MGraph_AddEdge(graph, 2, 1, 1);
MGraph_AddEdge(graph, 3, 4, 1);
MGraph_AddEdge(graph, 4, 2, 1);
MGraph_Display(graph, print_data);
MGraph_DFS(graph, 0, print_data);
MGraph_BFS(graph, 0, print_data);
MGraph_Destroy(graph);
return 0;
}
MGraph.h
#ifndef _MGRAPH_H_
#define _MGRAPH_H_
typedef void MGraph;
typedef void MVertex;
typedef void (MGraph_Printf)(MVertex*);
MGraph* MGraph_Create(MVertex** v, int n);
void MGraph_Destroy(MGraph* graph);
void MGraph_Clear(MGraph* graph);
int MGraph_AddEdge(MGraph* graph, int v1, int v2, int w);
int MGraph_RemoveEdge(MGraph* graph, int v1, int v2);
int MGraph_GetEdge(MGraph* graph, int v1, int v2);
int MGraph_TD(MGraph* graph, int v);
int MGraph_VertexCount(MGraph* graph);
int MGraph_EdgeCount(MGraph* graph);
void MGraph_DFS(MGraph* graph, int v, MGraph_Printf* pFunc);
void MGraph_BFS(MGraph* graph, int v, MGraph_Printf* pFunc);
void MGraph_Display(MGraph* graph, MGraph_Printf* pFunc);
#endif
MGraph.c
#include <malloc.h>
#include <stdio.h>
#include "MGraph.h"
#include "LinkQueue.h"
typedef struct _tag_MGraph
{
int count;
MVertex** v;
int** matrix;
} TMGraph;
static void recursive_dfs(TMGraph* graph, int v, int visited[], MGraph_Printf* pFunc)
{
int i = 0;
pFunc(graph->v[v]);
visited[v] = 1;
printf(", ");
for(i=0; i<graph->count; i++)
{
if( (graph->matrix[v][i] != 0) && !visited[i] )
{
recursive_dfs(graph, i, visited, pFunc);
}
}
}
static void bfs(TMGraph* graph, int v, int visited[], MGraph_Printf* pFunc)
{
LinkQueue* queue = LinkQueue_Create();
if( queue != NULL )
{
LinkQueue_Append(queue, graph->v + v);
visited[v] = 1;
while( LinkQueue_Length(queue) > 0 )
{
int i = 0;
v = (MVertex**)LinkQueue_Retrieve(queue) - graph->v;
pFunc(graph->v[v]);
printf(", ");
for(i=0; i<graph->count; i++)
{
if( (graph->matrix[v][i] != 0) && !visited[i] )
{
LinkQueue_Append(queue, graph->v + i);
visited[i] = 1;
}
}
}
}
LinkQueue_Destroy(queue);
}
MGraph* MGraph_Create(MVertex** v, int n) // O(n)
{
TMGraph* ret = NULL;
if( (v != NULL ) && (n > 0) )
{
ret = (TMGraph*)malloc(sizeof(TMGraph));
if( ret != NULL )
{
int* p = NULL;
ret->count = n;
ret->v = (MVertex**)malloc(sizeof(MVertex*) * n);
ret->matrix = (int**)malloc(sizeof(int*) * n);
p = (int*)calloc(n * n, sizeof(int));
if( (ret->v != NULL) && (ret->matrix != NULL) && (p != NULL) )
{
int i = 0;
for(i=0; i<n; i++)
{
ret->v[i] = v[i];
ret->matrix[i] = p + i * n;
}
}
else
{
free(p);
free(ret->matrix);
free(ret->v);
free(ret);
ret = NULL;
}
}
}
return ret;
}
void MGraph_Destroy(MGraph* graph) // O(1)
{
TMGraph* tGraph = (TMGraph*)graph;
if( tGraph != NULL )
{
free(tGraph->v);
free(tGraph->matrix[0]);
free(tGraph->matrix);
free(tGraph);
}
}
void MGraph_Clear(MGraph* graph) // O(n*n)
{
TMGraph* tGraph = (TMGraph*)graph;
if( tGraph != NULL )
{
int i = 0;
int j = 0;
for(i=0; i<tGraph->count; i++)
{
for(j=0; j<tGraph->count; j++)
{
tGraph->matrix[i][j] = 0;
}
}
}
}
int MGraph_AddEdge(MGraph* graph, int v1, int v2, int w) // O(1)
{
TMGraph* tGraph = (TMGraph*)graph;
int ret = (tGraph != NULL);
ret = ret && (0 <= v1) && (v1 < tGraph->count);
ret = ret && (0 <= v2) && (v2 < tGraph->count);
ret = ret && (0 <= w);
if( ret )
{
tGraph->matrix[v1][v2] = w;
}
return ret;
}
int MGraph_RemoveEdge(MGraph* graph, int v1, int v2) // O(1)
{
int ret = MGraph_GetEdge(graph, v1, v2);
if( ret != 0 )
{
((TMGraph*)graph)->matrix[v1][v2] = 0;
}
return ret;
}
int MGraph_GetEdge(MGraph* graph, int v1, int v2) // O(1)
{
TMGraph* tGraph = (TMGraph*)graph;
int condition = (tGraph != NULL);
int ret = 0;
condition = condition && (0 <= v1) && (v1 < tGraph->count);
condition = condition && (0 <= v2) && (v2 < tGraph->count);
if( condition )
{
ret = tGraph->matrix[v1][v2];
}
return ret;
}
int MGraph_TD(MGraph* graph, int v) // O(n)
{
TMGraph* tGraph = (TMGraph*)graph;
int condition = (tGraph != NULL);
int ret = 0;
condition = condition && (0 <= v) && (v < tGraph->count);
if( condition )
{
int i = 0;
for(i=0; i<tGraph->count; i++)
{
if( tGraph->matrix[v][i] != 0 )
{
ret++;
}
if( tGraph->matrix[i][v] != 0 )
{
ret++;
}
}
}
return ret;
}
int MGraph_VertexCount(MGraph* graph) // O(1)
{
TMGraph* tGraph = (TMGraph*)graph;
int ret = 0;
if( tGraph != NULL )
{
ret = tGraph->count;
}
return ret;
}
int MGraph_EdgeCount(MGraph* graph) // O(n*n)
{
TMGraph* tGraph = (TMGraph*)graph;
int ret = 0;
if( tGraph != NULL )
{
int i = 0;
int j = 0;
for(i=0; i<tGraph->count; i++)
{
for(j=0; j<tGraph->count; j++)
{
if( tGraph->matrix[i][j] != 0 )
{
ret++;
}
}
}
}
return ret;
}
void MGraph_DFS(MGraph* graph, int v, MGraph_Printf* pFunc)
{
TMGraph* tGraph = (TMGraph*)graph;
int* visited = NULL;
int condition = (tGraph != NULL);
condition = condition && (0 <= v) && (v < tGraph->count);
condition = condition && (pFunc != NULL);
condition = condition && ((visited = (int*)calloc(tGraph->count, sizeof(int))) != NULL);
if( condition )
{
int i = 0;
recursive_dfs(tGraph, v, visited, pFunc);
for(i=0; i<tGraph->count; i++)
{
if( !visited[i] )
{
recursive_dfs(tGraph, i, visited, pFunc);
}
}
printf("\n");
}
free(visited);
}
void MGraph_BFS(MGraph* graph, int v, MGraph_Printf* pFunc)
{
TMGraph* tGraph = (TMGraph*)graph;
int* visited = NULL;
int condition = (tGraph != NULL);
condition = condition && (0 <= v) && (v < tGraph->count);
condition = condition && (pFunc != NULL);
condition = condition && ((visited = (int*)calloc(tGraph->count, sizeof(int))) != NULL);
if( condition )
{
int i = 0;
bfs(tGraph, v, visited, pFunc);
for(i=0; i<tGraph->count; i++)
{
if( !visited[i] )
{
bfs(tGraph, i, visited, pFunc);
}
}
printf("\n");
}
free(visited);
}
void MGraph_Display(MGraph* graph, MGraph_Printf* pFunc) // O(n*n)
{
TMGraph* tGraph = (TMGraph*)graph;
if( (tGraph != NULL) && (pFunc != NULL) )
{
int i = 0;
int j = 0;
for(i=0; i<tGraph->count; i++)
{
printf("%d:", i);
pFunc(tGraph->v[i]);
printf(" ");
}
printf("\n");
for(i=0; i<tGraph->count; i++)
{
for(j=0; j<tGraph->count; j++)
{
if( tGraph->matrix[i][j] != 0 )
{
printf("<");
pFunc(tGraph->v[i]);
printf(", ");
pFunc(tGraph->v[j]);
printf(", %d", tGraph->matrix[i][j]);
printf(">");
printf(" ");
}
}
}
printf("\n");
}
}
LinkQueue.h
LinkQueue.c
小结：
| 邻接矩阵法 |
邻接链表法 |
| 优点：直观，容易实现。 缺点：当顶点数较多，而边数较少是浪费时间和空间。 |
优点：有效利用空间，非常适合边数较少的图。 缺点：实现相对复杂，不容易查找两个顶点之间的权值。 |
邻接矩阵法和邻接链表法的选择不是绝对的，需要根据实际情况综合考虑。