for the backtracking part, thanks to the video of stanford cs106b lecture 10 by Julie Zelenski for the nice explanation of recursion and backtracking, highly recommended.
in hdu 2553 cout N-Queens solutions problem, dfs is used.
// please ignore, below analysis is inaccurate, though for inspiration considerations, I decided to let it remain.
and we use a extra occupied[] array to opimize the validation test, early termination when occupied[i]==1, and in function isSafe the test change from 3 to 2 comparisons.
for the improvement, here is an analysis:
noted that total call of dfs for n is less than factorial(n), for convenience we take as factorial(n),
in this appoach, in pow(n,n)-factorial(n) cases, we conduct one comparison, and three for others
otherwise, for all pow(n,n) cases, we must conduct three comparisons,
blow is a list for n from 1 to 10,
n n! n^n n!/n^n
1 1 1 1.00000
2 2 4 0.50000
3 6 27 0.22222
4 24 256 0.09375
5 120 3125 0.03840
6 720 46656 0.01543
7 5040 823543 0.00612
8 40320 16777216 0.00240
9 362880 387420489 0.00094
10 3628800 10000000000 0.00036the table shows that, for almost all of the cases of large n (n>=4), we reduce 3 to 1 comparison in doing validation check.
occupied[i] is initilized to 0, before dfs, mark.
occupied[val]=1;after dfs, unmark,
occupied[val]=0;where val is the value chosen.
inspirations from chapter 3 Decompositions of graphs
in Algorithms(算法概论), Sanjoy Dasgupta University of California, San Diego Christos Papadimitriou University of California at Berkeley Umesh Vazirani University of California at Berkeley.
// leetcode N-Queens(12ms)
class Solution {
vector<vector<string>> ans;
int N;
//int numofsol;
void AddTo_ans(string &conf) {
vector<string> vecstrtmp;
string strtmp(N,'.');
for(auto ind: conf) {
vecstrtmp.push_back(strtmp);
vecstrtmp.back()[(size_t)ind-1]='Q';
}
ans.push_back(vector<string>());
ans.back().swap(vecstrtmp);
}
void RecListNQueens(string soFar, string rest) {
if(rest.empty()) {
//++numofsol;
AddTo_ans(soFar);
}
else {
int i,j,len=soFar.size(), flag;
for(i=0;i<rest.size();++i) {
for(j=0;j<len;++j) {
flag=1;
if(soFar[j]-rest[i]==len-j || soFar[j]-rest[i]==j-len) {
flag=0; break;
}
}
if(flag) RecListNQueens(soFar+rest[i],rest.substr(0,i)+rest.substr(i+1));
}
}
}
public:
vector<vector<string>> solveNQueens(int n) {
ans.clear();
//numofsol=0;
N=n;
string str;
for(int i=1;i<=n;++i) str.push_back(i);
RecListNQueens("", str);
return ans;
}
};// with a tiny modification,
// leetcode N-Queens II (8ms)
class Solution {
//vector<vector<string>> ans;
int N;
int numofsol;
/*void AddTo_ans(string &conf) {
vector<string> vecstrtmp;
string strtmp(N,'.');
for(auto ind: conf) {
vecstrtmp.push_back(strtmp);
vecstrtmp.back()[(size_t)ind-1]='Q';
}
ans.push_back(vector<string>());
ans.back().swap(vecstrtmp);
}*/
void RecListNQueens(string soFar, string rest) {
if(rest.empty()) {
++numofsol;
//AddTo_ans(soFar);
}
else {
int i,j,len=soFar.size(), flag;
for(i=0;i<rest.size();++i) {
for(j=0;j<len;++j) {
flag=1;
if(soFar[j]-rest[i]==len-j || soFar[j]-rest[i]==j-len) {
flag=0; break;
}
}
if(flag) RecListNQueens(soFar+rest[i],rest.substr(0,i)+rest.substr(i+1));
}
}
}
public:
int totalNQueens(int n) {
//ans.clear();
numofsol=0;
N=n;
string str;
for(int i=1;i<=n;++i) str.push_back(i);
RecListNQueens("", str);
return numofsol;
}
};// hdu 2553, 15ms
#include <cstdio>
#include <vector>
#include <algorithm>
class countNQueenSol {
static const int MAXN=12;
static int values[MAXN];
static int occupied[MAXN];
static std::vector<int> ans;
static int cnt, rownum;
static bool isSafe(int val, int row) {
for(int i=0;i<row;++i) {
if(val-values[i]==row-i || val-values[i]==i-row)
return false;
}
return true;
}
static void dfs(int row) {
if(row==rownum) ++cnt;
for(int i=0;i<rownum;++i) {
if(occupied[i]==0 && isSafe(i,row)) {
values[row]=i;
occupied[i]=1;
dfs(row+1);
occupied[i]=0;
}
}
}
public:
static int getMAXN() { return MAXN; }
static int solve(int k) {
if(k<=0 || k>=countNQueenSol::MAXN) return -1;
if(ans[k]<0) {
cnt=0;
rownum=k;
dfs(0);
//if(k==0) cnt=0; // adjust anomaly when k==0
ans[k]=cnt;
}
return ans[k];
}
};
int countNQueenSol::values[countNQueenSol::MAXN]={0};
int countNQueenSol::occupied[countNQueenSol::MAXN]={0};
std::vector<int> countNQueenSol::ans(countNQueenSol::MAXN,-1);
int countNQueenSol::cnt, countNQueenSol::rownum;
int main() {
#ifndef ONLINE_JUDGE
freopen("in.txt","r",stdin);
#endif
int n;
while(scanf("%d",&n)==1 && n>0) {
printf("%d
",countNQueenSol::solve(n));
}
return 0;
}版权声明：本文为博主原创文章，未经博主允许不得转载。// p.s. If in any way improment can be achieved, better performance or whatever, it will be well-appreciated to let me know, thanks in advance.