poj 1077 Eight
2009-08-30 14:58
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Eight
Description
The 15-puzzle has been around for over 100 years; even if you don't know it by that name, you've seen it. It is constructed with 15 sliding tiles, each with a number from 1 to 15 on it, and all packed into a 4 by 4 frame with one
tile missing. Let's call the missing tile 'x'; the object of the puzzle is to arrange the tiles so that they are ordered as:
where the only legal operation is to exchange 'x' with one of the tiles with which it shares an edge. As an example, the following sequence of moves solves a slightly scrambled puzzle:
The letters in the previous row indicate which neighbor of the 'x' tile is swapped with the 'x' tile at each step; legal values are 'r','l','u' and 'd', for right, left, up, and down, respectively.
Not all puzzles can be solved; in 1870, a man named Sam Loyd was famous for distributing an unsolvable version of the puzzle, and
frustrating many people. In fact, all you have to do to make a regular puzzle into an unsolvable one is to swap two tiles (not counting the missing 'x' tile, of course).
In this problem, you will write a program for solving the less well-known 8-puzzle, composed of tiles on a three by three
arrangement.
Input
You will receive a description of a configuration of the 8 puzzle. The description is just a list of the tiles in their initial positions, with the rows listed from top to bottom, and the tiles listed from left to right within
a row, where the tiles are represented by numbers 1 to 8, plus 'x'. For example, this puzzle
is described by this list:
Output
You will print to standard output either the word ``unsolvable'', if the puzzle has no solution, or a string consisting entirely of the letters 'r', 'l', 'u' and 'd' that describes a series of moves that produce a solution. The
string should include no spaces and start at the beginning of the line.
Sample Input
Sample Output
Source
South Central USA 1998
Time Limit: 1000MS | Memory Limit: 65536K | |||
Total Submissions: 9915 | Accepted: 4303 | Special Judge |
The 15-puzzle has been around for over 100 years; even if you don't know it by that name, you've seen it. It is constructed with 15 sliding tiles, each with a number from 1 to 15 on it, and all packed into a 4 by 4 frame with one
tile missing. Let's call the missing tile 'x'; the object of the puzzle is to arrange the tiles so that they are ordered as:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 x
where the only legal operation is to exchange 'x' with one of the tiles with which it shares an edge. As an example, the following sequence of moves solves a slightly scrambled puzzle:
1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 5 6 7 8 5 6 7 8 5 6 7 8 5 6 7 8 9 x 10 12 9 10 x 12 9 10 11 12 9 10 11 12 13 14 11 15 13 14 11 15 13 14 x 15 13 14 15 x r-> d-> r->
The letters in the previous row indicate which neighbor of the 'x' tile is swapped with the 'x' tile at each step; legal values are 'r','l','u' and 'd', for right, left, up, and down, respectively.
Not all puzzles can be solved; in 1870, a man named Sam Loyd was famous for distributing an unsolvable version of the puzzle, and
frustrating many people. In fact, all you have to do to make a regular puzzle into an unsolvable one is to swap two tiles (not counting the missing 'x' tile, of course).
In this problem, you will write a program for solving the less well-known 8-puzzle, composed of tiles on a three by three
arrangement.
Input
You will receive a description of a configuration of the 8 puzzle. The description is just a list of the tiles in their initial positions, with the rows listed from top to bottom, and the tiles listed from left to right within
a row, where the tiles are represented by numbers 1 to 8, plus 'x'. For example, this puzzle
1 2 3 x 4 6 7 5 8
is described by this list:
1 2 3 x 4 6 7 5 8
Output
You will print to standard output either the word ``unsolvable'', if the puzzle has no solution, or a string consisting entirely of the letters 'r', 'l', 'u' and 'd' that describes a series of moves that produce a solution. The
string should include no spaces and start at the beginning of the line.
Sample Input
2 3 4 1 5 x 7 6 8
Sample Output
ullddrurdllurdruldr
Source
South Central USA 1998
/*好久没有遇到这么大快人心的题了,虽然做了一天多,但是这次一改以往遇到陌生题比较浮躁的态度,认真去学习了相关知识。 通过这道题学习了A*算法,以及排列数一种常见的hash算法,并实际使用和操作了最大堆,可以说是收获颇多 */ /*注意程序中的两个优化: 如果两个优化都没有,则会TLE 如果只用优化2,而不用优化1,所用时间为,同样会TLE 如果只用优化1,而不用优化2,所用时间为: 5782538 bobten2008 1077 Accepted 8752K 47MS C++ 7345B 2009-08-30 14:52:15 如果两个都用,则时间为: 5782475 bobten2008 1077 Accepted 9128K 16MS C++ 7329B 2009-08-30 14:41:28 所以可见优化二的作用是决定性的 */ #include <iostream> #define MAX_N 362880 //利用阶乘逆序数来计算Hash值,最大hash值是9! - 1 using namespace std; //0-9的阶乘,用来计算哈希值 int fac[10] = {1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880}; int temp[10]; //记录1 - 9 九个位置可以移动的方向的个数 int dirN[10] = {0, 2, 3, 2, 3, 4, 3, 2, 3, 2}; //记录1 - 9 九个位置可以移动的方向的种类 char dirT[10][6] = { {' '}, {' ', 'r', 'd'}, {' ', 'l', 'r', 'd'}, {' ', 'l', 'd'}, {' ', 'u', 'r', 'd'}, {' ', 'u', 'l', 'r', 'd'}, {' ', 'u', 'l', 'd'}, {' ', 'u', 'r'}, {' ', 'u', 'l', 'r'}, {' ', 'u', 'l'}}; int heap[MAX_N + 5]; int heapSize = 0; int initState, finalState = 123456789, curPos; struct node { int type; //0 尚未访问, 1 已经在待考察队列中 2 被访问过并从待访问队列中删除了 int heapPos; //当前状态在堆中的位置 int state; //当前状态,如123456789 char dType; //上一个状态到当前状态所走的方向 int diff; //A*算法中的h()用当前状态和终止状态之间的different * 10 表示(不计空格) int step; //A*算法中的g(),表示当前遍历的深度 int fVal; //diff和step的和,堆算法主要基于这个指标 node() { step = 0; type = 0; } }hashT[MAX_N + 5]; //哈希表 int eP(int l) //计算10的l次方 { int res = 1; for(int i = 1; i <= l; i++) res *= 10; return res; } int getNum(int which, int num) //通过当前状态取3 × 3方格中的第which个数 { if(which > num % 10) return num / eP(9 - which + 1) % 10; else return num / eP(9 - which) % 10; } //以下均是移动空格的位置 int moveL(int state) //由当前状态左移得到新的状态 { return state - 1; } int moveR(int state) //由当前状态右移得到新的状态 { return state + 1; } int moveU(int state) //由当前状态下移得到新的状态 { int t0 = 9 - state % 10 + 1; int t1 = state / eP(t0); int t2 = t1 % 1000; t1 = t1 - t2 + (t2 % 100) * 10 + t2 / 100; t1 = t1 * eP(t0); return t1 + state % eP(t0) - 3; } int moveD(int state) //由当前状态上移得到新的状态 { int t0 = 7 - state % 10; int t1 = state / eP(t0); int t2 = t1 % 1000; t1 = t1 - t2 + (t2 % 10) * 100 + t2 / 10; t1 = t1 * eP(t0); return t1 + state % eP(t0) + 3; } int getRevNum(int state) //得到当前状态的逆序数,最后一位表示空格,不计 { int blank = state % 10; int total = 0; int p = 0; for(int id = 1; id <= 9; id++) { if(id == blank) continue; int n = getNum(id, state); temp[p++] = n; for(int k = p - 2; k >= 0; k--) if(temp[k] >= n) total++; } return total; } //通过当前状态得到其对应的哈希值 //这里计算哈希值采用1-9每位的逆序数乘以(位数减一的阶乘)并取和的方法,这样 //可以保证最大的哈希值为9! - 1且没个状态的哈希值都唯一 int getHashNum(int state) { int blank = state % 10; int hashVal = 0; int p = 0, total; int pp = 0; //pp记录阶乘的位置 for(int id = 1; id <= 9; id++) { pp++; if(id == blank) continue; int n = getNum(id, state); temp[p++] = n; total = 0; for(int k = p - 2; k >= 0; k--) { if(temp[k] >= n) total++; } hashVal += total * fac[p - 1]; } // 空格作为最小值计算 return hashVal + (blank - 1) * fac[blank - 1]; } //计算两个状态的距离 int getDiff(int state1, int state2) { int blank1 = state1 % 10, blank2 = state2 % 10, i, diff = 0; if(blank1 != blank2) diff += 2; for(i = 1; i <= 9; i++) { if(i != blank1 && i != blank2 && getNum(i, state1) != getNum(i, state2)) diff++; } return diff; } void swapHeapV(int i, int j) { int temp = heap[i]; heap[i] = heap[j]; heap[j] = temp; hashT[heap[i]].heapPos = i; hashT[heap[j]].heapPos = j; } //从i结点保证堆的性质 void maxHeapify(int i) { int l = 2 * i, r = 2 * i + 1, largest; if(l <= heapSize && hashT[heap[l]].fVal < hashT[heap[i]].fVal) largest = l; else largest = i; if(r <= heapSize && hashT[heap[r]].fVal < hashT[heap[largest]].fVal) largest = r; if(largest != i) { swapHeapV(i, largest); maxHeapify(largest); } } //取堆中的最大值 int getHeapMax() { return heap[1]; } //取堆中的最大值,并删除这个值,需要用maxHeapify(1)来保持最大堆的性质 int extracMaxHeap() { if(heapSize < 1) return -1; int maxv = heap[1]; heap[1] = heap[heapSize]; hashT[heap[1]].heapPos = 1; heapSize--; maxHeapify(1); return maxv; } //向对中插入结点i void heapInsert(int i) { heapSize++; heap[heapSize] = i; hashT[i].heapPos = heapSize; int pos = heapSize; while(pos > 1 && hashT[heap[int(pos / 2)]].fVal > hashT[heap[pos]].fVal) { swapHeapV(pos, int(pos / 2)); pos = pos / 2; } } //当结点i的权值发生变化时,需要更新堆的结构以保证最大堆的性质 void upMoveHeap(int i) { int pos = i; while(pos > 1 && hashT[heap[int(pos / 2)]].fVal > hashT[heap[pos]].fVal) { swapHeapV(pos, int(pos / 2)); pos = pos / 2; } } //递归打印结果 void printRes(int curState) { int hashVal = getHashNum(curState); char dType = hashT[hashVal].dType; if(dType != 's') { if(dType == 'l') { printRes(moveR(curState)); cout<<'l'; } else if(dType == 'r') { printRes(moveL(curState)); cout<<'r'; } else if(dType == 'u') { printRes(moveD(curState)); cout<<'u'; } else if(dType == 'd') { printRes(moveU(curState)); cout<<'d'; } } } //计算的核心控制函数 void getBest() { //计算并初始化初始结点的相关性质 int h = getHashNum(initState), d, curHashVal, curState, xPos, nextState, nextHashVal, nextHeapPos; char type; hashT[h].state = initState; hashT[h].type = 1; hashT[h].dType = 's'; hashT[h].step = 0; //将初始节点插入堆中 heapInsert(h); while(true) { //取当前堆中最大的结点 curHashVal = extracMaxHeap(); //设置当前状态为从待考察队列中删除 hashT[curHashVal].type = 2; curState = hashT[curHashVal].state; //到达终止结点 if(curState == finalState) { printRes(curState); cout<<endl; break; } xPos = curState % 10; //遍历空格可以走的方向 for(d = 1; d <= dirN[xPos]; d++) { type = dirT[xPos][d]; if(type == 'r') { //从当前节点向右走到达新的状态 nextState = moveR(curState); //计算新状态的哈希值 nextHashVal = getHashNum(nextState); nextHeapPos = hashT[nextHashVal].heapPos; //如果此状态已经被访问过 if(hashT[nextHashVal].type != 0) { //优化1:更新此状态的g函数,即遍历的深度 if(hashT[curHashVal].step + 1 < hashT[nextHashVal].step) { hashT[nextHashVal].step = hashT[curHashVal].step + 1; hashT[nextHashVal].fVal = hashT[nextHashVal].step + hashT[nextHashVal].diff; //更新堆的结构以保证最大堆的性质 upMoveHeap(nextHeapPos); } continue; } //记录方向 hashT[nextHashVal].dType = 'r'; } else if(type == 'l') { nextState = moveL(curState); nextHashVal = getHashNum(nextState); nextHeapPos = hashT[nextHashVal].heapPos; if(hashT[nextHashVal].type != 0) { if(hashT[curHashVal].step + 1 < hashT[nextHashVal].step) { hashT[nextHashVal].step = hashT[curHashVal].step + 1; hashT[nextHashVal].fVal = hashT[nextHashVal].step + hashT[nextHashVal].diff; upMoveHeap(nextHeapPos); } continue; } hashT[nextHashVal].dType = 'l'; } else if(type == 'u') { nextState = moveU(curState); nextHashVal = getHashNum(nextState); nextHeapPos = hashT[nextHashVal].heapPos; if(hashT[nextHashVal].type != 0) { if(hashT[curHashVal].step + 1 < hashT[nextHashVal].step) { hashT[nextHashVal].step = hashT[curHashVal].step + 1; hashT[nextHashVal].fVal = hashT[nextHashVal].step + hashT[nextHashVal].diff; upMoveHeap(nextHeapPos); } continue; } hashT[nextHashVal].dType = 'u'; } else if(type == 'd') { nextState = moveD(curState); nextHashVal = getHashNum(nextState); nextHeapPos = hashT[nextHashVal].heapPos; if(hashT[nextHashVal].type != 0) { if(hashT[curHashVal].step + 1 < hashT[nextHashVal].step) { hashT[nextHashVal].step = hashT[curHashVal].step + 1; hashT[nextHashVal].fVal = hashT[nextHashVal].step + hashT[nextHashVal].diff; upMoveHeap(nextHeapPos); } continue; } hashT[nextHashVal].dType = 'd'; } //存储并计算新状态的相关信息 hashT[nextHashVal].state = nextState; //g函数:遍历的深度加一 hashT[nextHashVal].step = hashT[curHashVal].step + 1; //h函数:优化2,将h函数乘以10可以降低深度的影响,从而提高A*算法的效率,但是结果一般不是最优的了 hashT[nextHashVal].diff = getDiff(nextState, finalState) * 10; //计算:f函数即g函数和h函数之和 hashT[nextHashVal].fVal = hashT[nextHashVal].diff + hashT[nextHashVal].step; //标记此状态已经加入待考察队列 hashT[nextHashVal].type = 1; //将新状态加入堆(待考察队列) heapInsert(nextHashVal); } } } int main() { char ch; int i; initState = 0; for(i = 1; i <= 9; i++) { cin>>ch; if(ch == 'x') curPos = i; else initState = initState * 10 + int(ch - '0'); } initState = initState * 10 + curPos; int revInit = getRevNum(initState); if(revInit % 2 != 0) cout<<"unsolvable"<<endl; else if(initState == finalState) cout<<"lr"<<endl; else getBest(); return 0; }
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