Ural 1029 Ministry 题解

题意

给定一个\(n\times m(1\le n \le10,1\le m \le500)\)的矩阵，矩阵中的每个值都是一个小于等于\(10^9\)的正整数。

现在从第\(1\)行的任意位置开始，在第\(n\)行的任意位置结束。每次有\(3\)种移动选择(不能移动到矩阵外)。

设当前位置为\((i,j)\)

  • 移动到\((i+1,j)\)

  • 移动到\((i,j-1)\)

  • 移动到\((i,j+1)\)

每条路径的价值是路径走过所有的位置上的值的和(小于等于\(10^9\))。

问在所有路径中，路径价值最小的，输出这条路径所有位置的列号。

题解

考虑记忆化搜索(DP也可以)。

对于每个点，记忆化搜索可以移动到它的\(3\)种位置，取最小值即可，顺便记录一下路径。

也可以无脑最短路。

Tips:

  1. WA\(1\)的同学不要着急，Test\(1\)并不是样例，仔细找找有没有错误。

  2. 设\((i,j)\)的答案为\(dp(i,j)\)，矩阵中的值为\(a(i,j)\)，状态转移方程如果是\(dp(i,j)=\min\{dp(i-1,j),dp(i,j-1),dp(i,j+1) \}+a(i,j)\)可能会WA\(1\)，改为\(dp(i,j)=\min\{dp(i-1,j)+a(i,j),dp(i,j-1)+a(i,j),dp(i,j+1)+a(i,j) \}\)即可AC。目前并不知道原因(我太弱了)，如果有知道的可以在评论区留言，谢谢！

程序

AC

// #pragma GCC optimize(2)

// #pragma G++ optimize(2)

// #pragma comment(linker,"/STACK:102400000,102400000")

// #include <bits/stdc++.h>

#include <map>

#include <set>

#include <list>

#include <array>

#include <cfenv>

#include <cmath>

#include <ctime>

#include <deque>

#include <mutex>

#include <queue>

#include <ratio>

#include <regex>

#include <stack>

#include <tuple>

#include <atomic>

#include <bitset>

#include <cctype>

#include <cerrno>

#include <cfloat>

#include <chrono>

#include <cstdio>

#include <cwchar>

#include <future>

#include <limits>

#include <locale>

#include <memory>

#include <random>

#include <string>

#include <thread>

#include <vector>

#include <cassert>

#include <climits>

#include <clocale>

#include <complex>

#include <csetjmp>

#include <csignal>

#include <cstdarg>

#include <cstddef>

#include <cstdint>

#include <cstdlib>

#include <cstring>

#include <ctgmath>

#include <cwctype>

#include <fstream>

#include <iomanip>

#include <numeric>

#include <sstream>

#include <ccomplex>

#include <cstdbool>

#include <iostream>

#include <typeinfo>

#include <valarray>

#include <algorithm>

#include <cinttypes>

#include <cstdalign>

#include <stdexcept>

#include <typeindex>

#include <functional>

#include <forward_list>

#include <system_error>

#include <unordered_map>

#include <unordered_set>

#include <scoped_allocator>

#include <condition_variable>

// #include <conio.h>

// #include <windows.h>

using namespace std;

typedef long long LL;

typedef unsigned int ui;

typedef unsigned long long ull;

typedef float fl;

typedef double ld;

typedef long double LD;

typedef pair<int,int> pii;

#if (WIN32) || (WIN64) || (__WIN32) || (__WIN64) || (_WIN32) || (_WIN64) || (WINDOWS)

#define lld "%I64d"

#define llu "%I64u"

#else

#define lld "%lld"

#define llu "%llu"

#endif

#define ui(n) ((unsigned int)(n))

#define LL(n) ((long long)(n))

#define ull(n) ((unsigned long long)(n))

#define fl(n) ((float)(n))

#define ld(n) ((double)(n))

#define LD(n) ((long double)(n))

#define char(n) ((char)(n))

#define Bool(n) ((bool)(n))

#define fixpoint(n) fixed<<setprecision(n)

const int INF=1061109567;

const int NINF=-1044266559;

const LL LINF=4557430888798830399;

const ld eps=1e-15;

#define MOD (1000000007)

#define PI (3.1415926535897932384626433832795028841971)

/*

#define MB_LEN_MAX 5

#define SHRT_MIN (-32768)

#define SHRT_MAX 32767

#define USHRT_MAX 0xffffU

#define INT_MIN (-2147483647 - 1)

#define INT_MAX 2147483647

#define UINT_MAX 0xffffffffU

#define LONG_MIN (-2147483647L - 1)

#define LONG_MAX 2147483647L

#define ULONG_MAX 0xffffffffUL

#define LLONG_MAX 9223372036854775807ll

#define LLONG_MIN (-9223372036854775807ll - 1)

#define ULLONG_MAX 0xffffffffffffffffull

*/

#define MP make_pair

#define MT make_tuple

#define All(a) (a).begin(),(a).end()

#define pall(a) (a).rbegin(),(a).rend()

#define log2(x) log(x)/log(2)

#define Log(x,y) log(x)/log(y)

#define SZ(a) ((int)(a).size())

#define rep(i,n) for(int i=0;i<((int)(n));i++)

#define rep1(i,n) for(int i=1;i<=((int)(n));i++)

#define repa(i,a,n) for(int i=((int)(a));i<((int)(n));i++)

#define repa1(i,a,n) for(int i=((int)(a));i<=((int)(n));i++)

#define repd(i,n) for(int i=((int)(n))-1;i>=0;i--)

#define repd1(i,n) for(int i=((int)(n));i>=1;i--)

#define repda(i,n,a) for(int i=((int)(n));i>((int)(a));i--)

#define repda1(i,n,a) for(int i=((int)(n));i>=((int)(a));i--)

#define FOR(i,a,n,step) for(int i=((int)(a));i<((int)(n));i+=((int)(step)))

#define repv(itr,v) for(__typeof((v).begin()) itr=(v).begin();itr!=(v).end();itr++)

#define repV(i,v) for(auto i:v)

#define repE(i,v) for(auto &i:v)

#define MS(x,y) memset(x,y,sizeof(x))

#define MC(x) MS(x,0)

#define MINF(x) MS(x,63)

#define MCP(x,y) memcpy(x,y,sizeof(y))

#define sqr(x) ((x)*(x))

#define UN(v) sort(All(v)),v.erase(unique(All(v)),v.end())

#define filein(x) freopen(x,"r",stdin)

#define fileout(x) freopen(x,"w",stdout)

#define fileio(x)\

	freopen(x".in","r",stdin);\

	freopen(x".out","w",stdout)

#define filein2(filename,name) ifstream name(filename,ios::in)

#define fileout2(filename,name) ofstream name(filename,ios::out)

#define file(filename,name) fstream name(filename,ios::in|ios::out)

#define Pause system("pause")

#define Cls system("cls")

#define fs first

#define sc second

#define PC(x) putchar(x)

#define GC(x) x=getchar()

#define Endl PC('\n')

#define SF scanf

#define PF printf

inline int Read()

{

    int X=0,w=0;char ch=0;while(!isdigit(ch)){w|=ch=='-';ch=getchar();}while(isdigit(ch))X=(X<<3)+(X<<1)+(ch^48),ch=getchar();

	return w?-X:X;

}

inline void Write(int x){if(x<0)putchar('-'),x=-x;if(x>9)Write(x/10);putchar(x%10+'0');}

inline LL powmod(LL a,LL b){LL RES=1;a%=MOD;assert(b>=0);for(;b;b>>=1){if(b&1)RES=RES*a%MOD;a=a*a%MOD;}return RES%MOD;}

inline LL gcdll(LL a,LL b){return b?gcdll(b,a%b):a;}

const int dx[]={0,1,0,-1,1,-1,-1,1};

const int dy[]={1,0,-1,0,-1,-1,1,1};

/************************************************************Begin************************************************************/

const int maxn=110,maxm=510;

int n,m,a[maxn][maxm],dp[maxn][maxm];

pair<int,int> pre[maxn][maxm];

inline int sol(int x,int y)

{

	if(y<1||y>m) return dp[x][y]=INF;

	if(dp[x][y]!=-1) return dp[x][y]; else dp[x][y]=0;

	int res=sol(x-1,y)+a[x][y];

	dp[x][y]=res;

	pre[x][y]={x-1,y};

	res=sol(x,y-1)+a[x][y];

	if(res<dp[x][y])

	{

		dp[x][y]=res;

		pre[x][y]={x,y-1};

	}

	res=sol(x,y+1)+a[x][y];

	if(res<dp[x][y])

	{

		dp[x][y]=res;

		pre[x][y]={x,y+1};

	}

	return dp[x][y];

}

inline void print(int x,int y)

{

	if(x>1) print(pre[x][y].fs,pre[x][y].sc);

	cout<<y<<' ';

}

int main()

{

	cin>>n>>m;

	rep1(i,n) rep1(j,m) cin>>a[i][j];

	MS(dp,-1);

	rep1(j,m) dp[1][j]=a[1][j];

	int ans=1;

	rep1(j,m) if(sol(n,j)<sol(n,ans)) ans=j;

	print(n,ans);

	return 0;

}

/*************************************************************End**************************************************************/

WA\(1\)程序

// #pragma GCC optimize(2)

// #pragma G++ optimize(2)

// #pragma comment(linker,"/STACK:102400000,102400000")

// #include <bits/stdc++.h>

#include <map>

#include <set>

#include <list>

#include <array>

#include <cfenv>

#include <cmath>

#include <ctime>

#include <deque>

#include <mutex>

#include <queue>

#include <ratio>

#include <regex>

#include <stack>

#include <tuple>

#include <atomic>

#include <bitset>

#include <cctype>

#include <cerrno>

#include <cfloat>

#include <chrono>

#include <cstdio>

#include <cwchar>

#include <future>

#include <limits>

#include <locale>

#include <memory>

#include <random>

#include <string>

#include <thread>

#include <vector>

#include <cassert>

#include <climits>

#include <clocale>

#include <complex>

#include <csetjmp>

#include <csignal>

#include <cstdarg>

#include <cstddef>

#include <cstdint>

#include <cstdlib>

#include <cstring>

#include <ctgmath>

#include <cwctype>

#include <fstream>

#include <iomanip>

#include <numeric>

#include <sstream>

#include <ccomplex>

#include <cstdbool>

#include <iostream>

#include <typeinfo>

#include <valarray>

#include <algorithm>

#include <cinttypes>

#include <cstdalign>

#include <stdexcept>

#include <typeindex>

#include <functional>

#include <forward_list>

#include <system_error>

#include <unordered_map>

#include <unordered_set>

#include <scoped_allocator>

#include <condition_variable>

// #include <conio.h>

// #include <windows.h>

using namespace std;

typedef long long LL;

typedef unsigned int ui;

typedef unsigned long long ull;

typedef float fl;

typedef double ld;

typedef long double LD;

typedef pair<int,int> pii;

#if (WIN32) || (WIN64) || (__WIN32) || (__WIN64) || (_WIN32) || (_WIN64) || (WINDOWS)

#define lld "%I64d"

#define llu "%I64u"

#else

#define lld "%lld"

#define llu "%llu"

#endif

#define ui(n) ((unsigned int)(n))

#define LL(n) ((long long)(n))

#define ull(n) ((unsigned long long)(n))

#define fl(n) ((float)(n))

#define ld(n) ((double)(n))

#define LD(n) ((long double)(n))

#define char(n) ((char)(n))

#define Bool(n) ((bool)(n))

#define fixpoint(n) fixed<<setprecision(n)

const int INF=1061109567;

const int NINF=-1044266559;

const LL LINF=4557430888798830399;

const ld eps=1e-15;

#define MOD (1000000007)

#define PI (3.1415926535897932384626433832795028841971)

/*

#define MB_LEN_MAX 5

#define SHRT_MIN (-32768)

#define SHRT_MAX 32767

#define USHRT_MAX 0xffffU

#define INT_MIN (-2147483647 - 1)

#define INT_MAX 2147483647

#define UINT_MAX 0xffffffffU

#define LONG_MIN (-2147483647L - 1)

#define LONG_MAX 2147483647L

#define ULONG_MAX 0xffffffffUL

#define LLONG_MAX 9223372036854775807ll

#define LLONG_MIN (-9223372036854775807ll - 1)

#define ULLONG_MAX 0xffffffffffffffffull

*/

#define MP make_pair

#define MT make_tuple

#define All(a) (a).begin(),(a).end()

#define pall(a) (a).rbegin(),(a).rend()

#define log2(x) log(x)/log(2)

#define Log(x,y) log(x)/log(y)

#define SZ(a) ((int)(a).size())

#define rep(i,n) for(int i=0;i<((int)(n));i++)

#define rep1(i,n) for(int i=1;i<=((int)(n));i++)

#define repa(i,a,n) for(int i=((int)(a));i<((int)(n));i++)

#define repa1(i,a,n) for(int i=((int)(a));i<=((int)(n));i++)

#define repd(i,n) for(int i=((int)(n))-1;i>=0;i--)

#define repd1(i,n) for(int i=((int)(n));i>=1;i--)

#define repda(i,n,a) for(int i=((int)(n));i>((int)(a));i--)

#define repda1(i,n,a) for(int i=((int)(n));i>=((int)(a));i--)

#define FOR(i,a,n,step) for(int i=((int)(a));i<((int)(n));i+=((int)(step)))

#define repv(itr,v) for(__typeof((v).begin()) itr=(v).begin();itr!=(v).end();itr++)

#define repV(i,v) for(auto i:v)

#define repE(i,v) for(auto &i:v)

#define MS(x,y) memset(x,y,sizeof(x))

#define MC(x) MS(x,0)

#define MINF(x) MS(x,63)

#define MCP(x,y) memcpy(x,y,sizeof(y))

#define sqr(x) ((x)*(x))

#define UN(v) sort(All(v)),v.erase(unique(All(v)),v.end())

#define filein(x) freopen(x,"r",stdin)

#define fileout(x) freopen(x,"w",stdout)

#define fileio(x)\

	freopen(x".in","r",stdin);\

	freopen(x".out","w",stdout)

#define filein2(filename,name) ifstream name(filename,ios::in)

#define fileout2(filename,name) ofstream name(filename,ios::out)

#define file(filename,name) fstream name(filename,ios::in|ios::out)

#define Pause system("pause")

#define Cls system("cls")

#define fs first

#define sc second

#define PC(x) putchar(x)

#define GC(x) x=getchar()

#define Endl PC('\n')

#define SF scanf

#define PF printf

inline int Read()

{

    int X=0,w=0;char ch=0;while(!isdigit(ch)){w|=ch=='-';ch=getchar();}while(isdigit(ch))X=(X<<3)+(X<<1)+(ch^48),ch=getchar();

	return w?-X:X;

}

inline void Write(int x){if(x<0)putchar('-'),x=-x;if(x>9)Write(x/10);putchar(x%10+'0');}

inline LL powmod(LL a,LL b){LL RES=1;a%=MOD;assert(b>=0);for(;b;b>>=1){if(b&1)RES=RES*a%MOD;a=a*a%MOD;}return RES%MOD;}

inline LL gcdll(LL a,LL b){return b?gcdll(b,a%b):a;}

const int dx[]={0,1,0,-1,1,-1,-1,1};

const int dy[]={1,0,-1,0,-1,-1,1,1};

/************************************************************Begin************************************************************/

const int maxn=110,maxm=510;

int n,m,a[maxn][maxm],dp[maxn][maxm];

pair<int,int> pre[maxn][maxm];

inline int sol(int x,int y)

{

	if(y<1||y>m) return dp[x][y]=INF;

	if(dp[x][y]!=-1) return dp[x][y]; else dp[x][y]=0;

	int res=sol(x-1,y);

	dp[x][y]=res;

	pre[x][y]={x-1,y};

	res=sol(x,y-1);

	if(res<dp[x][y])

	{

		dp[x][y]=res;

		pre[x][y]={x,y-1};

	}

	res=sol(x,y+1);

	if(res<dp[x][y])

	{

		dp[x][y]=res;

		pre[x][y]={x,y+1};

	}

	dp[x][y]+=a[x][y];

	return dp[x][y];

}

inline void print(int x,int y)

{

	if(x>1) print(pre[x][y].fs,pre[x][y].sc);

	cout<<y<<' ';

}

int main()

{

	cin>>n>>m;

	rep1(i,n) rep1(j,m) cin>>a[i][j];

	MS(dp,-1);

	rep1(j,m) dp[1][j]=a[1][j];

	int ans=1;

	rep1(j,m) if(sol(n,j)<sol(n,ans)) ans=j;

	print(n,ans);

	return 0;

}

/*************************************************************End**************************************************************/

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