题目大意
给出一个由2*S*(S+1)构成的S*S大小的火柴格。火柴可以构成1x1,2x2...SxS大小的方格。其中已经拿走了几个火柴,问最少再拿走几个火柴可以使得这些火柴无法构成任何一个方格。
题目分析
本题,采用的是搜索+剪枝来实现。需要做的是保存每个搜索节点的状态,以及通过合理的记录数据,对状态进行推演。
这里状态为:当前需要被拆除的火柴序号(match_index,可以拆除或者不拆除)+当前剩余的完整的方格的数目(left_square_num)+
当前已经拆除的火柴数目(taken_num,可以用于最优化剪枝)。
而记录数据可以为:火柴i是否位于方块j中 gMatchInSquare[i][j]. 方块s中最大的火柴序号 gMaxMatchInSquare[s](用于剪枝)。
这样,使用最优化剪枝,DFS搜索。剪枝:
(1)对于当前节点,若taken_num > gMinTakenNum,则剪枝返回;
(2)如果火柴 match_index 不存在任何一个剩余的完整的方块中,则不必拆除match_index,即剪枝拆除match_index的情况;
(3)如果火柴 match_index 是当前剩余的某个完整方块的构成火柴的最大的序号,则必须进行拆除(因为,对于火柴是按照序号从小到大进行递归搜索,如果match_index为某个方格的最大序号,则若不删除,之后的任何火柴都不在该方格中,无法破坏该方格),即剪枝不拆除的情况;
单纯使用以上剪枝,仍然会超时,则考虑使用估计函数来进行深度剪枝:考虑当前剩余的所有完整方格中不相交的方格的个数K,则从当前状态开始,至少还需要拆除K个火柴,才可能达到没有完整方格的状态。因此 taken_num >= gMinTakenNum
改为 taken_num + SeperateCompleteSquareNum() > gMinTakenNum
,进行剪枝。
实现方法
可以采用单纯的剪枝,或者采用IDA算法。
实现(c++)
#define _CRT_SECURE_NO_WARNINGS #include<stdio.h> #include<vector> #include<algorithm> #define INFINITE 1 << 30 #define MAX_MATCH_NUM 2*5*6 #define MAX_SQUARE_NUM MAX_MATCH_NUM*5 using namespace std; bool gMatchInSquare[MAX_MATCH_NUM][MAX_SQUARE_NUM]; //判断火柴i是否位于方块j中 bool gSquareComplete[MAX_SQUARE_NUM]; //方块s是否完整 int gMaxMatchInSquare[MAX_SQUARE_NUM]; //方块s中最大的火柴序号 int gMinTakenNum; //最少需要拿走的火柴数目 int gTotalSquareNum; //没有任何火柴被拿走的情况下,总的方格数目 int gTotalMatchNum; //没有任何火柴被拿走的情况下,总的火柴数 vector<int> gNotMissedMatch; //没有被拿走的火柴集合,从中选择拿走的火柴 //初始化,主要是对于S*S的网格,判断 每个火柴位于那些方格中,以及每个方格中的最大的火柴序号 void Init(int size){ memset(gMatchInSquare, false, sizeof(gMatchInSquare)); memset(gSquareComplete, true, sizeof(gSquareComplete)); gTotalMatchNum = 2 * (size + 1)*size; int s = size; gTotalSquareNum = 0; while (s > 0){ gTotalSquareNum += s*s; s--; } s = 1; int total_square_index = 0; while (s <= size){ for (int square_index = 0; square_index < (size - s + 1)*(size - s + 1); square_index++){ int match_index = (square_index / (size - s + 1))*(2 * size + 1) + (square_index % (size - s + 1)); int up_beg = match_index; int left_beg = match_index + size; int right_beg = left_beg + s; int down_beg = up_beg + s*(1 + size*2); for (int i = 0; i < s; i++){ gMatchInSquare[up_beg + i][total_square_index] = true; gMatchInSquare[down_beg + i][total_square_index] = true; gMatchInSquare[left_beg + i*(2 * size + 1)][total_square_index] = true; gMatchInSquare[right_beg + i*(2 * size + 1)][total_square_index] = true; } gMaxMatchInSquare[total_square_index] = down_beg + s - 1; total_square_index++; } s++; } } //判断火柴m位于那些完整的方格中,以及m是否是某些网格的最大序号火柴 void MatchInCompleteSquare(int m, vector<int>& complete_square_contain_match, bool* match_is_max){ *match_is_max = false; for (int s = 0; s < gTotalSquareNum; s++){ if (gMatchInSquare[m][s] && gSquareComplete[s]){ complete_square_contain_match.push_back(s); if (gMaxMatchInSquare[s] == m){ *match_is_max = true; } } } } //获得当前剩余的完整网格中,不相交的网格的数目 int SeperateCompleteSquareNum(int n){ int result = 0; typedef pair<int, int> MatchNumSquarePair; vector<MatchNumSquarePair> ms_vec; for (int s = 0; s < gTotalSquareNum; s++){ if (!gSquareComplete[s]) continue; int num = 0; for (int m = 0; m < gTotalMatchNum; m++){ if (gMatchInSquare[m][s]) num++; } ms_vec.push_back(MatchNumSquarePair(num, s)); } sort(ms_vec.begin(), ms_vec.end()); vector<bool> match_used(gTotalMatchNum, false); for (int i = 0; i < ms_vec.size(); i++){ MatchNumSquarePair ms_pair = ms_vec[i]; bool ok = true; for (int m = n; m < gTotalMatchNum; m++){ if (match_used[m] && gMatchInSquare[m][ms_pair.second]){ ok = false; } } if (ok){ for (int m = n; m < gTotalMatchNum; m++){ if (gMatchInSquare[m][ms_pair.second]){ match_used[m] = true; } } result++; } } return result; } /* //单纯的估计函数进行剪枝,不适用IDA算法 void Destroy(int n, int taken_num, int left_complete_square){ if (n == gNotMissedMatch.size()){ return; } if (left_complete_square == 0){ gMinTakenNum = gMinTakenNum < taken_num ? gMinTakenNum : taken_num; return; } //估价函数剪枝 if (taken_num + SeperateCompleteSquareNum(gNotMissedMatch[n]) >= gMinTakenNum){ return; } int match = gNotMissedMatch[n]; vector<int> complete_square_contain_match; bool match_is_max_in_square; MatchInCompleteSquare(match, complete_square_contain_match, &match_is_max_in_square); //如果火柴 match_index 不存在任何一个剩余的完整的方块中,则不必拆除match_index,剪枝1 if (complete_square_contain_match.empty()){ Destroy(n + 1, taken_num, left_complete_square); } else{ //如果火柴 match_index 是当前剩余的某个完整方块的构成火柴的最大的序号,则必须进行拆除,即剪枝不拆除的情况;剪枝2 if (!match_is_max_in_square){ Destroy(n + 1, taken_num, left_complete_square); } for (int i = 0; i < complete_square_contain_match.size(); i++){ int s = complete_square_contain_match[i]; gSquareComplete[s] = false; } Destroy(n + 1, taken_num + 1, left_complete_square - complete_square_contain_match.size()); for (int i = 0; i < complete_square_contain_match.size(); i++){ int s = complete_square_contain_match[i]; gSquareComplete[s] = true; } } }*/ /* //IDA 迭代加深,每次只增加1个深度 void Destroy(int n, int taken_num, int left_complete_square, bool* destroy_over){ if (*destroy_over) return; if (n == gNotMissedMatch.size()){ return; } if (left_complete_square == 0){ *destroy_over = true; return; } int seperate_complete_square_num = SeperateCompleteSquareNum(gNotMissedMatch[n]); if (taken_num + seperate_complete_square_num > gMinTakenNum){ return; } int match = gNotMissedMatch[n]; vector<int> complete_square_contain_match; bool match_is_max_in_square; MatchInCompleteSquare(match, complete_square_contain_match, &match_is_max_in_square); if (complete_square_contain_match.empty()){ Destroy(n + 1, taken_num, left_complete_square, destroy_over); } else{ if (!match_is_max_in_square){ Destroy(n + 1, taken_num, left_complete_square, destroy_over); } for (int i = 0; i < complete_square_contain_match.size(); i++){ int s = complete_square_contain_match[i]; gSquareComplete[s] = false; } Destroy(n + 1, taken_num + 1, left_complete_square - complete_square_contain_match.size(), destroy_over); for (int i = 0; i < complete_square_contain_match.size(); i++){ int s = complete_square_contain_match[i]; gSquareComplete[s] = true; } } } */ //IDA迭代加深,每次可能增加多个深度,由next_min_taken_num指定 void Destroy(int n, int taken_num, int left_complete_square, int & next_min_taken_num){ if (next_min_taken_num <= gMinTakenNum){ return; } if (n == gNotMissedMatch.size()){ return; } if (left_complete_square == 0){ next_min_taken_num = next_min_taken_num < taken_num ? next_min_taken_num : taken_num; return; } int seperate_complete_square_num = SeperateCompleteSquareNum(gNotMissedMatch[n]); if (taken_num + seperate_complete_square_num > gMinTakenNum){ next_min_taken_num = next_min_taken_num < taken_num + seperate_complete_square_num ? next_min_taken_num : seperate_complete_square_num + taken_num; return; } int match = gNotMissedMatch[n]; vector<int> complete_square_contain_match; bool match_is_max_in_square; MatchInCompleteSquare(match, complete_square_contain_match, &match_is_max_in_square); if (complete_square_contain_match.empty()){ Destroy(n + 1, taken_num, left_complete_square, next_min_taken_num); } else{ if (!match_is_max_in_square){ Destroy(n + 1, taken_num, left_complete_square, next_min_taken_num); } for (int i = 0; i < complete_square_contain_match.size(); i++){ int s = complete_square_contain_match[i]; gSquareComplete[s] = false; } Destroy(n + 1, taken_num + 1, left_complete_square - complete_square_contain_match.size(), next_min_taken_num); for (int i = 0; i < complete_square_contain_match.size(); i++){ int s = complete_square_contain_match[i]; gSquareComplete[s] = true; } } } //IDA方法 void Resolve(int left_complete_square){ gMinTakenNum = SeperateCompleteSquareNum(gNotMissedMatch[0]); int next_min_taken_num; bool destroy_over; while (true){ //IDA2 next_min_taken_num = INFINITE; Destroy(0, 0, left_complete_square, next_min_taken_num); if (next_min_taken_num <= gMinTakenNum){ gMinTakenNum = next_min_taken_num; return; } gMinTakenNum = next_min_taken_num; /* IDA1 destroy_over = false; Destroy(0, 0, left_complete_square, &destroy_over); if (destroy_over){ return; } gMinTakenNum++; */ } } int main(){ int T; scanf("%d", &T); while (T--){ int size, k; scanf("%d %d", &size, &k); Init(size); gNotMissedMatch.clear(); for (int i = 0; i < gTotalMatchNum; i++){ gNotMissedMatch.push_back(i); } gMinTakenNum = INFINITE; int missed_match_index, left_complete_square = gTotalSquareNum; for (int i = 0; i < k; i++){ scanf("%d", &missed_match_index); missed_match_index--; gNotMissedMatch.erase(find(gNotMissedMatch.begin(), gNotMissedMatch.end(), missed_match_index)); for (int j = 0; j < gTotalSquareNum; j++){ if (gMatchInSquare[missed_match_index][j] && gSquareComplete[j]){ gSquareComplete[j] = false; left_complete_square--; } } } //普通的 估价剪枝 //Destroy(0, 0, left_complete_square); //IDA 1或者2 Resolve(left_complete_square); printf("%d ", gMinTakenNum); } return 0; }