1 問題描述
拼圖遊戲，通常簡單的也叫做八數位問題，就是解決如何通過移動有限的步數，從圖的一個狀态移動到另外一個狀态，如下面的圖所示：
2 問題解決的思路
一個簡單的思路就是采取廣度優先搜尋算法，算法思路如下所示：
把起始狀态圖存儲在哈希表中；
把起始狀态加入隊列當中；
循環判斷目前狀态是否等于結束狀态
從隊列中取出元素
尋找所有可以移動的操作，産生新狀态X
若新位置狀态X不在哈希表中
将X加入隊列和哈希表
3 算法的實作
Puzzle.h
/*************************************************************************************/
/*"NITE and DAY" Puzzle */
/*************************************************************************************/
/*Special symbols.*/
#define SymbolBlank '.' /*Symbol that denotes a blank square.*/
#define SymbolFixed '$' /*Symbol that cannot move.*/
/*Board dimensiona.*/
#define BoardHeight 3
#define BoardWidth 4
#define BoardSize 12
/*Start position.*/
/* N I T E */
/* $ . $ . */
/* D A Y . */
#define StartBoard "NITE$.$.DAY."
/*Goal position.*/
/* D A Y . */
/* $ . $ . */
/* N I T E */
#define GoalBoard "DAY.$.$.NITE"
/*Number of moves in a minimal length solution for this puzzle,*/
/* where one move is defined as s moving a piece vertically or horizontally*/
/* to an adjacent unoccupied square (and the SymbolFixed piece cannot move).*/
#define MinSolution 66
/*Data structure parameters.*/
#define QueueArraySize 50000 /*Queue goes from 0 to QueueArraySize-1.*/
#define HashArraySize 1000003 /*Hash table goes from 0 to HashArraySize-1.*/
#define HashStatsMAX 50 /*Max number of hash bucket size statistics to keep.*/
Output.h
/*Output a position; takes 4 arguments:
*POS: String that lists pieces in row major order.
step: The step number (ignore next two args when step=0).
piece: Piece that moved.
dir: Direction that the piece moved; 0=north, 1=east, 2=south, 3=west.
*/
void OutputPosition(char *POS, int step, char piece, int dir) {
static char *DirectionString[4] = {"north", "east", "south", "west"};
int i;
printf("\nStep %d",step);
if (step>0) printf(", move %c %s",piece,DirectionString[dir]);
printf(":\n");
for (i=0; i<BoardSize; i++) {
printf("%c",POS[i]);
if ( (i%BoardWidth) == (BoardWidth-1) ) printf("\n"); else printf(" ");
}
}
/*Utility function to make a positive integer into a string with commas.*/
char *MakeString(int n) {
static int block[100];
static char s[100], *x;
int i=-1, j;
do {block[++i] = n%1000; n /= 1000;} while (n>0);
sprintf(s,"%d",block[i]);
if (block[i]<10) x=s+1; else if (block[i]<100) x=s+2; else x=s+3;
for (j=(i-1); j>=0; j--) {
if (block[j]<10) sprintf(x,",00%d",block[j]);
else if (block[j]<100) sprintf(x,",0%d",block[j]);
else sprintf(x,",%d",block[j]);
x += 4;
}
sprintf(x,"%c",'\0');
return(s);
}
/*Print queue statistics.*/
void QueueStats(int Qarray, int Qmax, int Qfront, int Qrear) {
printf("\nQueue statistics:\n");
printf(" Queue array size: %s\n", MakeString(Qarray));
printf(" Max positions ever in queue: %s\n", MakeString(Qmax));
printf(" Index of Qfront: %s\n", MakeString(Qfront));
printf(" Index of Qrear: %s\n", MakeString(Qrear));
printf(" |Qfront-Qrear| = %s\n", MakeString(abs(Qfront-Qrear)));
}
/*Print hash statistics.*/
void HashStats(int Harray, int Hcount, int Hmax) {
printf("\nHash table statistics:\n");
printf(" Hash array size: %s\n", MakeString(Harray));
printf(" No. positions in hash table: %s\n", MakeString(Hcount));
printf(" Max bucket size: %s\n", MakeString(Hmax));
printf("\nHash buckets at the end:\n");
}
/*Print number of buckets of given size.*/
/*Call this function once for each bucket size, startiing at 0 and going*/
/* up to the maximum bucket size (the Hmax argument to HashStats).*/
void BucketStat(int Bsize, int Bcount) {
printf(" buckets of size %d = %s\n", Bsize, MakeString(Bcount));
}
main.c
#include <stdio.h>
#include <stdlib.h>
#include "Puzzle.h"
#include "Output.h"
/*Position data type.*/
typedef struct pnode
{
struct pnode *next; /*next pointer for the hash bucket*/
struct pnode *back; /*position from which this one came*/
int cost; /*number of moves to get to this position*/
char piece; /*piece that moved to this position*/
int dir; /*direction moved to enter this position*/
char board[BoardSize]; /*this position's board*/
} PositionBody;
typedef PositionBody *TypePosition;
/*Queue data type.*/
typedef struct Queue_
{
unsigned int head;
unsigned int tail;
TypePosition arr[QueueArraySize];
}Queue;
typedef Queue* TypeQueue;
typedef struct List_
{
TypePosition pos;
struct List_* next;
}List;
enum Move { NORTH = 0, EAST = 1, SOUTH = 2, WEST = 3 };
void initQueue();
void initHash();
void initStartPos();
void enqueue(TypePosition pos);
void runGame();
TypePosition dequeue();
TypePosition newPosition();
List* newList();
unsigned int hash(char* str);
void copyStr(char* str1, char* str2);
int cmpStr(char* str1, char* str2);
TypePosition insertPos(char* str, char piece, int direct, int cost, TypePosition back);
void printResult(TypePosition p);
void statisticQueue();
void statisticHash(char* str);
void clearHash();
List hashTable[HashArraySize];
Queue ques;
char start[BoardSize] = StartBoard;
char goal[BoardSize] = GoalBoard;
unsigned int maxQueueNum = 0;
unsigned int QueueNum = 0;
unsigned int hashNum[HashArraySize];
unsigned int maxBucketSize;
int main() {
initHash();
initQueue();
initStartPos();
runGame();
clearHash();
//system("pause");
return 0;
}
void initQueue()
{
ques.head = 0;
ques.tail = 0;
}
void initHash()
{
for (int i = 0; i < HashArraySize; i++)
{
hashTable[i].pos = NULL;
hashTable[i].next = NULL;
}
}
void initStartPos()
{
TypePosition startPos = insertPos(start, 0, 0, -1, NULL);
enqueue(startPos);
}
void enqueue(TypePosition pos)
{
if (pos == NULL)
return;
unsigned int newTail = (ques.tail + 1) % QueueArraySize;
if (newTail == ques.head)
{
printf("queue overflow!!!");
clearHash();
exit(1);
}
else
{
ques.arr[ques.tail] = pos;
ques.tail = newTail;
QueueNum++;
if (maxQueueNum < QueueNum)
maxQueueNum = QueueNum;
}
}
void runGame()
{
while (1)
{
TypePosition p = dequeue();
if (p == NULL)
return;
if (cmpStr(goal, p->board))
{
//over;
printResult(p);
statisticQueue();
statisticHash(p->board);
return;
}
for (int i = 0; i < BoardSize; i++)
{
int x = i % BoardWidth;
int y = i / BoardWidth;
int dx[4] = { 0,1,0,-1 };
int dy[4] = { -1,0,1,0 };
for (int k = 0; k < 4; k++)
{
int newx = x + dx[k];
int newy = y + dy[k];
if (newx >= 0 && newx < BoardWidth && newy >= 0 && newy < BoardHeight && p->board[i] != SymbolFixed && p->board[i] != SymbolBlank)
{
int newIndex = newy * BoardWidth + newx;
if (p->board[newIndex] == SymbolBlank)
{
char newCs[BoardSize] = { 0 };
copyStr(newCs, p->board);
newCs[i] = SymbolBlank;
newCs[newIndex] = p->board[i];
TypePosition next = insertPos(newCs, p->board[i], k, p->cost, p);
enqueue(next);
}
}
}
}
}
}
TypePosition dequeue()
{
//judge if empty queue
if (ques.head == ques.tail)
return NULL;
//return the head element
unsigned int head = ques.head;
ques.head = (ques.head + 1) % QueueArraySize;
QueueNum--;
return ques.arr[head];
}
unsigned int hash(char* str)
{
unsigned int seed = 131; // 31 131 1313 13131 131313 etc..
unsigned int hash = 0;
unsigned int cnt = BoardSize;
while (cnt--)
{
hash = hash * seed + (*str++);
}
return (hash & 0x7FFFFFFF) % HashArraySize;
}
TypePosition insertPos(char* str, char piece, int direct, int cost, TypePosition back)
{
unsigned int key = hash(str);
//the first element in key's bucket
if (hashTable[key].pos == NULL)
{
TypePosition ppos = newPosition();
ppos->back = back;
ppos->next = NULL;
ppos->cost = cost + 1;
ppos->dir = direct;
ppos->piece = piece;
copyStr(ppos->board, str);
hashTable[key].pos = ppos;
hashNum[key]++;
return ppos;
}
//the key's bucket is not empty
List* p = hashTable + key;
List* q = p;
while (p != NULL)
{
if (cmpStr(p->pos->board, str))
return NULL;
q = p;
p = p->next;
}
TypePosition ppos = newPosition();
ppos->back = back;
ppos->next = NULL;
ppos->cost = cost + 1;
ppos->dir = direct;
ppos->piece = piece;
copyStr(ppos->board, str);
List* newNode = newList();
newNode->pos = ppos;
newNode->next = NULL;
q->next = newNode;
//increase the key's biggest bucket size
hashNum[key]++;
if (maxBucketSize < hashNum[key])
maxBucketSize = hashNum[key];
return ppos;
}
void printResult(TypePosition p)
{
if (p->cost == 0)
{
OutputPosition(p->board, p->cost, p->piece, p->dir);
return;
}
else if (p->cost > 0)
printResult(p->back);
OutputPosition(p->board, p->cost, p->piece, p->dir);
}
void copyStr(char* str1, char* str2)
{
for (int i = 0; i < BoardSize; i++)
{
str1[i] = str2[i];
}
}
TypePosition newPosition() {
TypePosition p = (TypePosition)malloc(sizeof(PositionBody));
if (p == NULL) {
printf("Malloc for a new position failed.");
clearHash();
exit(1);
}
return p;
}
List* newList()
{
List* p = (List*)malloc(sizeof(List));
if (p == NULL) {
printf("Malloc for a new List node failed.");
clearHash();
exit(1);
}
return p;
}
int cmpStr(char* str1, char* str2)
{
for (int i = 0; i < BoardSize; i++)
{
if (str1[i] != str2[i])
return 0;
}
return 1;
}
void statisticQueue()
{
QueueStats(QueueArraySize, maxQueueNum, ques.head, ques.tail);
}
void statisticHash(char* str) {
unsigned int key = hash(str);
HashStats(HashArraySize, key, maxBucketSize);
//int* arrSize = (int*)malloc(sizeof(int) * (maxBucketSize + 1));
int* arrSize = (int*)calloc(maxBucketSize + 1, sizeof(int));
for (unsigned int i = 0; i < HashArraySize; i++)
{
arrSize[hashNum[i]]++;
}
for (int i = 0; i <= maxBucketSize; i++)
{
BucketStat(i, arrSize[i]);
}
free(arrSize);
}
void clearHash()
{
for (int i = 0; i < HashArraySize; i++)
{
if (hashTable[i].pos != NULL)
{
List* node = hashTable + i;
free(node->pos);
node = node->next;
while (node != NULL)
{
List* p = node->next;
free(node->pos);
free(node);
node = p;
}
}
}
}
4 輸出
Step 0:
N I T E
$ . $ .
D A Y .
Step 1, move I south:
N . T E
$ I $ .
D A Y .
Step 2, move N east:
. N T E
$ I $ .
D A Y .
Step 3, move Y east:
. N T E
$ I $ .
D A . Y
......
Step 63, move E south:
D A Y .
$ . $ E
N . I T
Step 64, move I west:
D A Y .
$ . $ E
N I . T
Step 65, move T west:
D A Y .
$ . $ E
N I T .
Step 66, move E south:
D A Y .
$ . $ .
N I T E
Queue statistics:
Queue array size: 50,000
Max positions ever in queue: 35,919
Index of Qfront: 4,797
Index of Qrear: 4,800
|Qfront-Qrear| = 3
Hash table statistics:
Hash array size: 1,000,003
No. positions in hash table: 652,885
Max bucket size: 7
Hash buckets at the end:
buckets of size 0 = 546,392
buckets of size 1 = 330,171
buckets of size 2 = 99,613
buckets of size 3 = 20,372
buckets of size 4 = 3,030
buckets of size 5 = 387
buckets of size 6 = 34
buckets of size 7 = 4
![查找算法之二分查找查找算法之二分查找[圖]](data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACwAAAAAAQABAAACAkQBADs=)