题目叙述：九宫问题又称“八数码问题”，是说在3×3的方格棋盘上，摆放着1到8这八个数码，有1个方格是空的，从某种初始状态开始，对空格执行空格左移、空格右移、空格上移和空格下移这四个操作使得棋盘从初始状态变换至目标状态。求其最少步数。
/*****************************/
/* EIGHT DIGIT PROBLEM */
/* 唐国峰 2012年4月23日 */
/*****************************/
// 预编译命令
#include "iostream"
#include "stdlib.h"
using namespace std;
//棋盘大小
#define size 3
//定义二维数组来存储数据表示某一个特定状态
typedef int status[size][size];
//定义状态图中的节点数据结构，即节点的状态信息等
typedef struct Node
{
status data; //节点所存储的状态
struct Node *parent; //指向节点的父亲节点
struct SpringLink *child; //指向节点的后继节点
struct Node *next; //指向链表的后一个节点
int f_value; //由初始状态经由当前节点至目标状态的总耗散值
int g_value; //由初始状态经到当前节点实际耗散值
int h_value; //由当前节点到目标状态的预计耗散值
}NNode, *PNode;
//定义表述指向当前节点的扩展节点的链表
typedef struct SpringLink
{
struct Node *pointData; //指向节点的指针
struct SpringLink *next; //指向当前节点的其他扩展节点
}SPLink, *PSPLink;
//声明OPEN表和CLOSED表
PNode open;
PNode closed;
//计算棋盘状态的逆序数
int InverseNumber(status a)
{
int i, j, sum=0;
int data_chang[size*size]={0};
//将二维数组转换成一维数组，以方便求逆序数
for(i=0;i
{
for(j=0;j
{
data_chang[i*size+j]=a[i][j];
}
}
//计算序列中除零外的逆序数
for(i=0;i<=size*size;i++)
{
if(data_chang[i]!=0)
{
//要比较多少次,从最后一个元素开始比较
for(j=i;j>=0;j--)
{
//当后一个数比前一个数小时
if(data_chang[i]
{
sum++;
}
}
}
}
return sum;
}
//判断是否存在解决方案
bool hasSolution(status startStatus,status targetStatus)
{
int startInverseNumber=InverseNumber(startStatus);
int tatgetInverseNumber=InverseNumber(targetStatus);
//判断初始状态和目标状态除零外的序列逆序数奇偶性，相同则可求值，不同则不可求
if( (startInverseNumber%2) != (tatgetInverseNumber%2) )
{
return false;
}
else
{
return true;
}
}
//初始化一个空链表
void initLink(PNode &Head)
{
Head = (PNode)malloc(sizeof(NNode));
Head->next = NULL;
}
//判断链表是否为空
bool isEmpty(PNode Head)
{
if(Head->next == NULL)
{
return true;
}
else
{
return false;
}
}
//从链表中拿出一个数据
void popNode(PNode &Head , PNode &FNode)
{
if(isEmpty(Head))
{
FNode = NULL;
return;
}
FNode = Head->next;
Head->next = Head->next->next;
FNode->next = NULL;
}
//向节点的最终后继节点链表中添加新的子节点
void addSpringNode(PNode &Head , PNode newData)
{
PSPLink newNode = (PSPLink)malloc(sizeof(SPLink));
newNode->pointData = newData;
newNode->next = Head->child;
Head->child = newNode;
}
//释放状态图中存放节点后继节点地址的空间
void freeSpringLink(PSPLink &Head)
{
PSPLink tmm;
while(Head != NULL)
{
tmm = Head;
Head = Head->next;
free(tmm);
}
}
//释放open表与closed表中的资源
void freeLink(PNode &Head)
{
PNode tmn;
tmn = Head;
Head = Head->next;
free(tmn);
while(Head != NULL)
{
//首先释放存放节点后继节点地址的空间
freeSpringLink(Head->child);
tmn = Head;
Head = Head->next;
free(tmn);
}
}
//向普通链表中添加一个节点
void addNode(PNode &Head , PNode &newNode)
{
newNode->next = Head->next;
Head->next = newNode;
}
//向非递减排列的链表中添加一个节点
void addAscNode(PNode &Head , PNode &newNode)
{
PNode P;
PNode Q;
P = Head->next;
Q = Head;
while(P != NULL && P->f_value < newNode->f_value)
{
Q = P;
P = P->next;
}
//上面判断好位置之后，下面就是简单的插入节点了
newNode->next = Q->next;
Q->next = newNode;
}
//计算节点到目标状态的预计耗散值
int computeh_value(PNode theNode,status targetStatus)
{
int num = 0;
for(int i = 0 ; i < 3 ; i++)
{
for(int j = 0 ; j < 3 ; j++)
{
if(theNode->data[i][j] != targetStatus[i][j])
{
num++;
}
}
}
return num;
}
//计算节点的f，g，h值
void computeAllValue(PNode &theNode , PNode parentNode, status targetStatus)
{
if(parentNode == NULL)
{
theNode->g_value = 0;
}
else
{
theNode->g_value = parentNode->g_value + 1;
}
theNode->h_value = computeh_value(theNode,targetStatus);
theNode->f_value = theNode->g_value + theNode->h_value;
}
//初始化函数，进行算法初始条件的设置
void initial(status startStatus,status targetStatus)
{
//初始化open以及closed表
initLink(open);
initLink(closed);
//初始化起始节点，令初始节点的父节点为空节点
PNode NULLNode = NULL;
PNode StartNode = (PNode)malloc(sizeof(NNode));
for(int i = 0 ; i < 3 ; i++)
{
for(int j = 0 ; j < 3 ; j++)
{
StartNode->data[i][j] = startStatus[i][j];
}
}
StartNode->parent = NULL;
StartNode->child = NULL;
StartNode->next = NULL;
computeAllValue(StartNode, NULLNode,targetStatus);
//起始节点进入OPEN表
addAscNode(open , StartNode);
}
//将B节点的状态赋值给A节点
void statusAEB(PNode &ANode , PNode BNode)
{
for(int i = 0 ; i < 3 ; i++)
{
for(int j = 0 ; j < 3 ; j++)
{
ANode->data[i][j] = BNode->data[i][j];
}
}
}
//两个节点是否有相同的状态
bool hasSameStatus(PNode ANode , PNode BNode)
{
for(int i = 0 ; i < size ; i++)
{
for(int j = 0 ; j < size ; j++)
{
if(ANode->data[i][j] != BNode->data[i][j])
return false;
}
}
return true;
}
//节点与其祖先节点是否有相同的状态
bool hasAnceSameStatus(PNode OrigiNode , PNode AnceNode)
{
while(AnceNode != NULL)
{
if(hasSameStatus(OrigiNode , AnceNode))
return true;
AnceNode = AnceNode->parent;
}
return false;
}
//取得方格中空的格子的位置
void getPosition(PNode theNode , int &row , int &col)
{
for(int i = 0 ; i < size ; i++)
{
for(int j = 0 ; j < size ; j++)
{
if(theNode->data[i][j] == 0)
{
row = i;
col = j;
return;
}
}
}
}
//交换两个数字的值
void changeAB(int &a , int &b)
{
int c;
c = b;
b = a;
a = c;
}
//检查相应的状态是否在某一个链表中
bool inLink(PNode spciNode , PNode theLink , PNode &theNodeLink , PNode &preNode)
{
preNode = theLink;
theLink = theLink->next;
while(theLink != NULL)
{
if(hasSameStatus(spciNode , theLink))
{
theNodeLink = theLink;
return true;
}
preNode = theLink;
theLink = theLink->next;
}
return false;
}
//产生节点的后继节点链表
void SpringLink(PNode theNode , PNode &spring, status targetStatus)
{
int row;
int col;
getPosition(theNode , row , col);
//空的格子右边的格子向左移动
if(col != 2)
{
PNode rlNewNode = (PNode)malloc(sizeof(NNode));
statusAEB(rlNewNode, theNode);
changeAB(rlNewNode->data[row][col], rlNewNode->data[row][col + 1]);
if(hasAnceSameStatus(rlNewNode, theNode->parent))
{
free(rlNewNode);//与父辈相同，丢弃本节点
}
else
{
rlNewNode->parent = theNode;
rlNewNode->child = NULL;
rlNewNode->next = NULL;
computeAllValue(rlNewNode, theNode, targetStatus);
//将本节点加入后继节点链表
addNode(spring, rlNewNode);
}
}
//空的格子左边的格子向右移动
if(col != 0)
{
PNode lrNewNode = (PNode)malloc(sizeof(NNode));
statusAEB(lrNewNode, theNode);
changeAB(lrNewNode->data[row][col], lrNewNode->data[row][col - 1]);
if(hasAnceSameStatus(lrNewNode, theNode->parent))
{
free(lrNewNode);//与父辈相同，丢弃本节点
}
else
{
lrNewNode->parent = theNode;
lrNewNode->child = NULL;
lrNewNode->next = NULL;
computeAllValue(lrNewNode, theNode, targetStatus);
//将本节点加入后继节点链表
addNode(spring , lrNewNode);
}
}
//空的格子上边的格子向下移动
if(row != 0)
{
PNode udNewNode = (PNode)malloc(sizeof(NNode));
statusAEB(udNewNode , theNode);
changeAB(udNewNode->data[row][col], udNewNode->data[row - 1][col]);
if(hasAnceSameStatus(udNewNode, theNode->parent))
{
free(udNewNode);//与父辈相同，丢弃本节点
}
else
{
udNewNode->parent = theNode;
udNewNode->child = NULL;
udNewNode->next = NULL;
computeAllValue(udNewNode, theNode, targetStatus);
//将本节点加入后继节点链表
addNode(spring, udNewNode);
}
}
//空的格子下边的格子向上移动
if(row != 2)
{
PNode duNewNode = (PNode)malloc(sizeof(NNode));
statusAEB(duNewNode, theNode);
changeAB(duNewNode->data[row][col], duNewNode->data[row + 1][col]);
if(hasAnceSameStatus(duNewNode, theNode->parent))
{
free(duNewNode);//与父辈相同，丢弃本节点
}
else
{
duNewNode->parent = theNode;
duNewNode->child = NULL;
duNewNode->next = NULL;
computeAllValue(duNewNode, theNode, targetStatus);
//将本节点加入后继节点链表
addNode(spring , duNewNode);
}
}
}
//输出给定节点的状态
void outputStatus(PNode stat)
{
for(int i = 0 ; i < 3 ; i++)
{
for(int j = 0 ; j < 3 ; j++)
{
cout << stat->data[i][j] << " ";
}
cout << endl;
}
}
//输出最佳的路径
void outputBestRoad(PNode goal)
{
int deepnum = goal->g_value;
if(goal->parent != NULL)
{
outputBestRoad(goal->parent);
}
cout << "第" << deepnum-- << "步的状态：" << endl;
outputStatus(goal);
}
void AStar(status startStatus,status targetStatus)
{
PNode tmpNode; //指向从open表中拿出并放到closed表中的节点的指针
PNode spring; //tmpNode的后继节点链
PNode tmpLNode; //tmpNode的某一个后继节点
PNode tmpChartNode;
PNode thePreNode; //指向将要从closed表中移到open表中的节点的前一个节点的指针
bool getGoal = false; //标识是否达到目标状态
long numcount = 1; //记录从open表中拿出节点的序号
initial(startStatus,targetStatus); //对函数进行初始化
initLink(spring); //对后继链表的初始化
tmpChartNode = NULL;
cout << "从OPEN表中拿出的节点的状态及相应的值" << endl;
while(!isEmpty(open))
{
//从OPEN表中拿出f值最小的元素,并将拿出的元素放入CLOSED表中
popNode(open, tmpNode);
addNode(closed, tmpNode);
cout << "第" << numcount++ << "个状态是：" << endl;
outputStatus(tmpNode);
cout << "其f值为：" << tmpNode->f_value << endl;
cout << "其g值为：" << tmpNode->g_value << endl;
cout << "其h值为：" << tmpNode->h_value << endl;
//如果拿出的元素是目标状态则跳出循环
if(computeh_value(tmpNode, targetStatus) == 0)
{
getGoal = true;
break;
}
//产生当前检测节点的后继(与祖先不同)节点列表，产生的后继节点的parent属性指向当前检测的节点
SpringLink(tmpNode, spring, targetStatus);
//遍历检测节点的后继节点链表
while(!isEmpty(spring))
{
popNode(spring , tmpLNode);
//状态在OPEN表中已经存在，thePreNode参数在这里并不起作用
if(inLink(tmpLNode , open , tmpChartNode , thePreNode))
{
addSpringNode(tmpNode , tmpChartNode);
if(tmpLNode->g_value < tmpChartNode->g_value)
{
tmpChartNode->parent = tmpLNode->parent;
tmpChartNode->g_value = tmpLNode->g_value;
tmpChartNode->f_value = tmpLNode->f_value;
}
free(tmpLNode);
}
//状态在CLOSED表中已经存在
else if(inLink(tmpLNode, closed, tmpChartNode, thePreNode))
{
addSpringNode(tmpNode, tmpChartNode);
if(tmpLNode->g_value < tmpChartNode->g_value)
{
PNode commu;
tmpChartNode->parent = tmpLNode->parent;
tmpChartNode->g_value = tmpLNode->g_value;
tmpChartNode->f_value = tmpLNode->f_value;
freeSpringLink(tmpChartNode->child);
tmpChartNode->child = NULL;
popNode(thePreNode, commu);
addAscNode(open, commu);
}
free(tmpLNode);
}
//新的状态即此状态既不在OPEN表中也不在CLOSED表中
else
{
addSpringNode(tmpNode, tmpLNode);
addAscNode(open, tmpLNode);
}
}
}
//目标可达的话，输出最佳的路径
if(getGoal)
{
cout << endl;
cout << "路径长度为：" << tmpNode->g_value << endl;
outputBestRoad(tmpNode);
}
//释放节点所占的内存
freeLink(open);
freeLink(closed);
}
void main()
{
//开始状态和目标状态
status startStatus = {2, 8, 3, 1, 6, 4, 7, 0, 5};
status targetStatus = {1, 2, 3, 8, 0, 4, 7, 6, 5};
if(hasSolution(startStatus,targetStatus))
{
AStar(startStatus,targetStatus);
}
else
{
cout << "从当前输入的初始状态无法经过有限步数变换至您期望的目标状态！" << endl;
}
system("pause");
}
