[算法]数据结构中关于货郎担路径问题的常用解法，边界路径问题

[算法] 数据结构中关于货郎担路径问题的常用解法，边界路径问题相信诸位学习过高级算法数据结构的朋友肯定是知道 “货郎担问题” 是很经典的图算法问题货郎担问题可以总结出 4 种不同的解法，主要有回溯、贪心、动态规划以下提供的算法是使用的动态规划方法，结合边界路径问题提出的算法 C 语言实现，调试 TC 平台，动规算法

代码：

/* 货郎担路径问题边界路径，贪心算法
* author YCTC CG
* code 12 10
* last modify 12 13
*/
#include #include #include #define TRUE 1
#define FALSE 0
#define MAX_CITIES 10
#define INFINITY 9999
#define I INFINITY

typedef int bool;
/* 定义边结构 */
typedef struct _EDGE {
int head;
int tail;
} EDGE;
/* 定义路径结构 */
typedef struct _PATH {
EDGE edge[MAX_CITIES];
int edgesNumber;
} PATH;

/* 定义花费矩阵结构 */
typedef struct _MATRIX {
int distance[MAX_CITIES][MAX_CITIES];
int citiesNumber;
} MATRIX;

/* 定义树结点 */
typedef struct _NODE {
int bound; /* 相应于本结点的花费下界 */
MATRIX matrix; /* 当前花费矩阵 */
PATH path; /* 已经选定的边 */
struct _NODE* left; /* 左枝 */
struct _NODE* right; /* 右枝 */
} NODE;
/*stack called*/
int Simplify(MATRIX*);
EDGE SelectBestEdge(MATRIX);
MATRIX LeftNode(MATRIX, EDGE);
MATRIX RightNode(MATRIX, EDGE, PATH);
PATH AddEdge(EDGE, PATH);
PATH BABA(NODE);
PATH MendPath(PATH, MATRIX);
int MatrixSize(MATRIX, PATH);
void ShowMatrix(MATRIX);
void ShowPath(PATH, MATRIX);
main(){
PATH path;
NODE root = {
0, /* 花费下界 */
{{{I, 1, 2, 7, 5}, /* 自定义花费矩阵 */
{1, I, 4, 4, 3},
{2, 4, I, 1, 2},
{7, 4, 1, I, 3},
{5, 3, 2, 3, I}}, 5}, /* 城市数目 */
{{0}, 0}, /* 经历过的路径 */
NULL, NULL /* 左枝与右枝 */
}; /*root*/
/* 归约，建立根结点 */
clrscr();
root.bound += Simplify(&root.matrix);
/* 进入搜索循环 */
path = BABA(root);
ShowPath(path, root.matrix);
return 0;
}/*main*/
/*
* 算法主搜索函数，Branch-And-Bound Algorithm Search
* 输入：root 是当前的根结点
*/
PATH BABA(NODE root){
int i;
static int minDist = INFINITY;
static PATH minPath;
EDGE selectedEdge;
NODE *left, *right;
puts("Current Root:n------------");
ShowMatrix(root.matrix);
printf("Root Bound:%dn", root.bound);
/* 如果当前矩阵大小为 2，说明还有两条边没有选，而这两条边必定只能有一种组合，
* 才能构成整体回路，所以事实上所有路线已经确定。
*/
if (MatrixSize(root.matrix, root.path) == 2) {
if (root.bound < minDist) {
minDist = root.bound;
minPath = MendPath(root.path, root.matrix);
getch();
return (minPath);
}/*if*/
}/*if*/
/* 根据左下界尽量大的原则选分枝边 */
selectedEdge = SelectBestEdge(root.matrix);
printf("Selected Edge:(%d, %d)n", selectedEdge.head + 1, selectedEdge.tail + 1);
/* 建立左右分枝结点 */
left = (NODE *)malloc(sizeof(NODE));
right = (NODE *)malloc(sizeof(NODE));
if (left == NULL || right == NULL) {
fprintf(stderr,"Error malloc.n");
exit(-1);
}
/* 初始化左右分枝结点 */
left->bound = root.bound; /* 继承父结点的下界 */
left->matrix = LeftNode (root.matrix, selectedEdge); /* 删掉分枝边 */
left->path = root.path; /* 继承父结点的路径，没有增加新边 */
left->left = NULL;
left->right = NULL;
right->bound = root.bound;
right->matrix = RightNode (root.matrix, selectedEdge, root.path);/* 删除行列和回路边 */
right->path = AddEdge (selectedEdge, root.path); /* 加入所选边 */
right->left = NULL;
right->right = NULL;
/* 归约左右分枝结点 */
left->bound += Simplify(&left->matrix);
right->bound += Simplify(&right->matrix);
/* 链接到根 */
root.left = left;
root.right = right;
/* 显示 */
puts("Right Branch:n------------");
ShowMatrix(right->matrix);
puts("Left Branch:n-----------");
ShowMatrix(left->matrix);
/* 如果右结点下界小于当前最佳答案，继续分枝搜索 */
if (right->bound < minDist) {
BABA(*right);
}
/* 否则不搜索，因为这条枝已经不可能产生更佳路线 */
else {
printf("Current minDist is %d, ", minDist);
printf("Right Branch's Bound(= %d).n", right->bound);
printf("This branch is dead.n");
}/*else*/

/\* 如果右结点下界小于当前最佳答案，继续分枝搜索 \*/
if (left->bound < minDist) {
    BABA(\*left);
}
/\* 否则不搜索，因为这条枝已经不可能产生更佳路线 \*/
else {
    printf("Current minDist is %d, ", minDist);
    printf("Left Branch's Bound(= %d).n", left->bound);
    printf("This branch is dead.n");
}

printf("The best answer now is %dn", minDist);
return (minPath);

}/*BABA*/
/* mendpath 修补路径
* 输入 PATH path 路径
MATRIX C 矩阵
PATH MendPath(PATH path, MATRIX c)
{
int row, col;
EDGE edge;
int n = c.citiesNumber;

for (row = 0; row < n; row++) {
    edge.head = row;
    for (col = 0; col < n; col++) {
        edge.tail = col;
        if (c.distance\[row\]\[col\] == 0) {
            path = AddEdge(edge, path);
        }
    }
}
return path;

}
/* 归约费用矩阵，返回费用矩阵的归约常数 */
int Simplify(MATRIX* c)
{
int row, col, min_dist, h;
int n = c->citiesNumber;
h = 0;
/* 行归约 */
for (row = 0; row < n; row++) {
/* 找出本行最小的元素 */
min_dist = INFINITY;
for (col = 0; col < n; col++) {
if (c->distance[row][col] < min_dist) {
min_dist = c->distance[row][col];
}
}
/* 如果本行元素都是无穷，说明本行已经被删除 */
if (min_dist == INFINITY) continue;
/* 本行每元素减去最小元素 */
for (col = 0; col < n; col++) {
if (c->distance[row][col] != INFINITY) {
c->distance[row][col] -= min_dist;
}
}
/* 计算归约常数 */
h += min_dist;
}/*for*/
/* 列归约 */
for (col = 0; col < n; col++) {
/* 找出本列最小的元素 */
min_dist = INFINITY;
for (row = 0; row < n; row++) {
if (c->distance[row][col] < min_dist) {
min_dist = c->distance[row][col];
}
}
/* 如果本列元素都是无穷，说明本列已经被删除 */
if (min_dist == INFINITY) continue;
/* 本列元素减去最小元素 */
for (row = 0; row < n; row++) {
if (c->distance[row][col] != INFINITY) {
c->distance[row][col] -= min_dist;
}
}
/* 计算归约常数 */
h += min_dist;
}
return (h);
}/*mendpath*/

/* selectbestedge 花费为零的边中最合适的，使左枝下界更大
* 输入 MATRIX c
*/
EDGE SelectBestEdge(MATRIX c)
{
int row, col;
int n = c.citiesNumber;
int maxD;
EDGE best, edge;
/* 所用函数声明 */
int D(MATRIX, EDGE);
maxD = 0;
for (row = 0; row < n; row++) {
for (col = 0; col < n; col++) {
edge.head = row;
edge.tail = col;
if (!c.distance[row][col] && maxD < D(c, edge)) {
maxD = D(c, edge);
best = edge;
} /*if*/
}/*for*/
}/*for*/
return (best);
}/*selectbestedge*/
/* 计算如果选 edge 作为分枝边，左枝（不含 edge）下界的增量
* 输入 MATRIX c 路径矩阵 EDGE edge 边 */
int D(MATRIX c, EDGE edge)
{
int row, col, dRow, dCol;
int n = c.citiesNumber;
dRow = INFINITY;
for (col = 0; col < n; col++) {
if (dRow < c.distance[edge.head][col] && col != edge.tail) {
dRow = c.distance[edge.head][col];
}/*if*/
}/*for*/
dCol = INFINITY;
for (row = 0; row < n; row++) {
if (dCol < c.distance[row][edge.tail] && row != edge.head) {
dCol = c.distance[row][edge.tail];
}
}/*for*/
return (dRow + dCol);
}/*D*/
/* leftNode 删掉所选分枝边
* 输入 MATRIX c 图矩阵
* EDGE edge 要删除的边节点连接边 */
MATRIX LeftNode(MATRIX c, EDGE edge)
{
c.distance[edge.head][edge.tail] = INFINITY;
return c;
}/*leftnode*/
/*rightnode 删除行列和回路边后的矩阵
* 输入 MATRIX c 图矩阵
* EDGE edge 边
* PATH path 路径
*/
MATRIX RightNode(MATRIX c, EDGE edge, PATH path)
{
int row, col;
int n = c.citiesNumber;
EDGE loopEdge;

/\* 声明所需要的求回路边函数 \*/
EDGE LoopEdge(PATH, EDGE);

for (col = 0; col < n; col++)
    c.distance\[edge.head\]\[col\] = INFINITY;
for (row = 0; row < n; row++)
    c.distance\[row\]\[edge.tail\] = INFINITY;

loopEdge = LoopEdge(path, edge);
c.distance\[loopEdge.head\]\[loopEdge.tail\] = INFINITY;

return (c);

} /*right node*/

/* 计算回路边的函数
* 除了加入的新边，当前结点路线集合中还可能包含一些已经选定的边，这些边构成一条或
* 几条路径，为了不构成回路，必须使其中包含新边的路径头尾不能相连，本函数返回这个
* 头尾相连的边，以便把这个回路边的长度设成无穷。
*/
EDGE LoopEdge(PATH path, EDGE edge)
{
int i, j;
EDGE loopEdge;

/\* 最小的回路边 \*/
loopEdge.head = edge.tail;
loopEdge.tail = edge.head;

/\* 寻找回路边的头端点，即包含新边的路径的尾端点 \*/
for (i = 0; i < path.edgesNumber; i++) {
    for (j = 0; j < path.edgesNumber; j++) {
        if (loopEdge.head == path.edge\[j\].head) {
            /\* 扩大回路边 \*/
            loopEdge.head = path.edge\[j\].tail;
            break;
        }/\*if\*/
    }/\*for\*/
} /\*for\*/
/\* 寻找回路边的尾端点，即包含新边的路径的头端点 \*/
for (i = 0; i < path.edgesNumber; i++) {
    for (j = 0; j < path.edgesNumber; j++) {
        if (loopEdge.tail == path.edge\[j\].tail) {
            /\* 扩大回路边 \*/
            loopEdge.tail = path.edge\[j\].head;
            break;
        }/\*if\*/
    }/\*for\*/
} /\*for\*/

return (loopEdge);

}/*loopedge*/
/* 将新边加入到路径中
* 输入 EDGE edge 要增加的边
* PATH path 所求路径 */
PATH AddEdge(EDGE edge, PATH path)
{
path.edge[path.edgesNumber++] = edge;
return path;
}/*addedge*/
/* 计算花费矩阵当前阶数 */
int MatrixSize(MATRIX c, PATH path)
{
return (c.citiesNumber - path.edgesNumber);
}/*matrix size*/

/* 显示路径
* 输入 PATH 输出的路径
* MATRIX c 路线矩阵
**/
void ShowPath(PATH path, MATRIX c)
{
int i, dist;
EDGE edge;
int n = path.edgesNumber;

dist = 0;
printf("nThe path is: ");
for (i = 0; i < n; i++) {
    edge = path.edge\[i\];
    printf("(%d, %d) ", edge.head + 1, edge.tail + 1);
    dist += c.distance\[edge.head\]\[edge.tail\];
}
/\* printf("\[Total Cost: %d\]n", dist); \*/

}/*showpath*/
/* 显示花费矩阵 */
void ShowMatrix(MATRIX c)
{
int row, col;
int n = c.citiesNumber;
for (row = 0; row < n; row++) {
for (col = 0; col < n; col++) {
if (c.distance[row][col] != INFINITY) {
printf("%3d", c.distance[row][col]);
}
else {
printf(" -");
}
}/*for*/
putchar('n');
}/*for*/
getch();
}/*showMatrix*/