codeforces1215E Marbles

本文详细解析 Codeforces 1215E 的解题思路,介绍了如何通过状态压缩动态规划解决颜色交换问题,避免了贪心算法的陷阱,提供了一个有效的解题模板。

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https://codeforces.com/problemset/problem/1215/E

设f[i][j]为i在houmian,j在后的 颜色 i,j之间需要交换的次数,那么从前枚举到后,当前的颜色a[i],如果a[i]要到某个颜色j前面,那么f[a[i]][j]+=num[j],即当前这个a[i]要和之前已经出现过的num[j]个j颜色的全部交换一次。

最后只要枚举任意两种颜色的相对关系,选择f[i][j]和f[j][i]中较小的作为交换次数就行了。

upd: 上述贪心是错的,这个方法被hack了,不过好像没有unrated。。。

因为如果选择了 (1,2)  (2,3) (3,1)  ,即f[1][2]<f[2][1],f[2][3]<f[3][2],f[3][1]<f[1][3],那么这个先后顺序就矛盾了。需要用状压DP

我们设dp[s]表示把s这个状态中所有数字安排了最少需要多少次交换,然后枚举哪一个放到最前面就行了。

#include<bits/stdc++.h>
#define maxl 400010
using namespace std;

const long long inf=1ll<<61; 

int n,m;
int a[maxl],num[21];
long long ans;
long long f[21][21],dp[1<<21];
char s[maxl];
 
inline void prework()
{
	scanf("%d",&n);
	for(int i=1;i<=n;i++)
	{
		scanf("%d",&a[i]);
		num[a[i]]++;
		for(int j=1;j<=20;j++)
		if(j!=a[i])
			f[a[i]][j]+=num[j];
	} 
}
 
inline void mainwork()
{
	for(int i=0;i<(1<<21);i++)
		dp[i]=inf;
	dp[0]=0;int t;long long tmp=0;
	for(int s=1;s<(1<<20);s++)
	{
		for(int i=1;i<=20;i++)
		if(s&(1<<(i-1)))
		{
			t=s^(1<<(i-1));tmp=0;
			for(int j=1;j<=20;j++)
			if(t&(1<<(j-1)))
				tmp+=f[i][j];
			dp[s]=min(dp[s],dp[t]+tmp);
		}
	}
	ans=dp[(1<<20)-1];
}
 
inline void print()
{
	printf("%lld",ans);
}
 
int main()
{
	int t=1;
	//scanf("%d",&t);
	for(int i=1;i<=t;i++)
	{
		prework();
		mainwork();
		print();
	}
	return 0;
} 

下面这个是之前的错程序。 

#include<bits/stdc++.h>
#define maxl 400010
using namespace std;
 
int n,m;
int a[maxl],num[21];
long long ans;
long long f[21][21];
char s[maxl];
 
inline void prework()
{
	scanf("%d",&n);
	for(int i=1;i<=n;i++)
	{
		scanf("%d",&a[i]);
		num[a[i]]++;
		for(int j=1;j<=20;j++)
		if(j!=a[i])
			f[a[i]][j]+=num[j];
	} 
}
 
inline void mainwork()
{
	for(int i=1;i<=20;i++)
		for(int j=1;j<=20;j++)
		if(i!=j)
			ans+=min(f[i][j],f[j][i]);
}
 
inline void print()
{
	ans=ans/2;
	printf("%lld",ans);
}
 
int main()
{
	int t=1;
	//scanf("%d",&t);
	for(int i=1;i<=t;i++)
	{
		prework();
		mainwork();
		print();
	}
	return 0;
}

 

 

### Codeforces 887E Problem Solution and Discussion The problem **887E - The Great Game** on Codeforces involves a strategic game between two players who take turns to perform operations under specific rules. To tackle this challenge effectively, understanding both dynamic programming (DP) techniques and bitwise manipulation is crucial. #### Dynamic Programming Approach One effective method to approach this problem utilizes DP with memoization. By defining `dp[i][j]` as the optimal result when starting from state `(i,j)` where `i` represents current position and `j` indicates some status flag related to previous moves: ```cpp #include <bits/stdc++.h> using namespace std; const int MAXN = ...; // Define based on constraints int dp[MAXN][2]; // Function to calculate minimum steps using top-down DP int minSteps(int pos, bool prevMoveType) { if (pos >= N) return 0; if (dp[pos][prevMoveType] != -1) return dp[pos][prevMoveType]; int res = INT_MAX; // Try all possible next positions and update 'res' for (...) { /* Logic here */ } dp[pos][prevMoveType] = res; return res; } ``` This code snippet outlines how one might structure a solution involving recursive calls combined with caching results through an array named `dp`. #### Bitwise Operations Insight Another critical aspect lies within efficiently handling large integers via bitwise operators instead of arithmetic ones whenever applicable. This optimization can significantly reduce computation time especially given tight limits often found in competitive coding challenges like those hosted by platforms such as Codeforces[^1]. For detailed discussions about similar problems or more insights into solving strategies specifically tailored towards contest preparation, visiting forums dedicated to algorithmic contests would be beneficial. Websites associated directly with Codeforces offer rich resources including editorials written after each round which provide comprehensive explanations alongside alternative approaches taken by successful contestants during live events. --related questions-- 1. What are common pitfalls encountered while implementing dynamic programming solutions? 2. How does bit manipulation improve performance in algorithms dealing with integer values? 3. Can you recommend any online communities focused on discussing competitive programming tactics? 4. Are there particular patterns that frequently appear across different levels of difficulty within Codeforces contests?
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