CodeForces 229D Towers

本文探讨了如何通过最少的操作步骤使城市中按照特定顺序排列的塔楼达到非递减高度序列的问题。详细介绍了动态规划方法,通过计算每个塔楼的最低高度并确定合并操作的最优策略。

原题传送门:http://codeforces.com/problemset/problem/229/D

 

 

D. Towers
time limit per test 2 seconds
memory limit per test 256 megabytes
input
standard input
output
standard output

The city of D consists of n towers, built consecutively on a straight line. The height of the tower that goes i-th (from left to right) in the sequence equals hi. The city mayor decided to rebuild the city to make it beautiful. In a beautiful city all towers are are arranged in non-descending order of their height from left to right.

The rebuilding consists of performing several (perhaps zero) operations. An operation constitutes using a crane to take any tower and put it altogether on the top of some other neighboring tower. In other words, we can take the tower that stands i-th and put it on the top of either the (i - 1)-th tower (if it exists), or the (i + 1)-th tower (of it exists). The height of the resulting tower equals the sum of heights of the two towers that were put together. After that the two towers can't be split by any means, but more similar operations can be performed on the resulting tower. Note that after each operation the total number of towers on the straight line decreases by 1.

Help the mayor determine the minimum number of operations required to make the city beautiful.

Input

The first line contains a single integer n (1 ≤ n ≤ 5000) — the number of towers in the city. The next line contains n space-separated integers: the i-th number hi (1 ≤ hi ≤ 105) determines the height of the tower that is i-th (from left to right) in the initial tower sequence.

Output

Print a single integer — the minimum number of operations needed to make the city beautiful.

Sample test(s)
Input
5
8 2 7 3 1
Output
3
Input
3
5 2 1
Output
2

分析:

dp(i)表示对前i座塔进行操作后形成非递减序列所需要的最小操作步数

last(i)表示在dp(i)最小的前提下,第i座塔的最低高度。

易知,将[i,j]区间内的塔合并成一座需要j-i步

于是我们得到dp方程:dp(i)=max{dp(j)+i-j+1| j<i, h(j+1)+h(j+2)+...+h(i-1)+h(i) <= last(j)}

注意,我们在递推的过程中要同时记录、更新last(j)的值来确保得到最优解。

 

#include <cstdio>
using namespace std;
int dp[5010],sum[5010],last[5010];
int main()
{
    int n;
    scanf("%d",&n);
    for(int i = 1;i <= n;i++){
        int a;
        scanf("%d",&a);
         sum[i] = sum[i-1]+a;
         dp[i] = last[i] = 1<<30;
     }
     for(int i = 1;i <= n;i++){
         for(int j = 0;j < i;j++){
             if(sum[i]-sum[j] >= last[j] && dp[i] >= dp[j]+i-j-1){
                 dp[i] = dp[j]+i-j-1;
                 last[i] =sum[i]-sum[j]>last[i]?last[i]:sum[i]-sum[j];
             }
         }
     }
     printf("%d\n",dp[n]);
     return 0;
 }

 

 
### Codeforces 1487D Problem Solution The problem described involves determining the maximum amount of a product that can be created from given quantities of ingredients under an idealized production process. For this specific case on Codeforces with problem number 1487D, while direct details about this exact question are not provided here, similar problems often involve resource allocation or limiting reagent type calculations. For instance, when faced with such constraints-based questions where multiple resources contribute to producing one unit of output but at different ratios, finding the bottleneck becomes crucial. In another context related to crafting items using various materials, it was determined that the formula `min(a[0],a[1],a[2]/2,a[3]/7,a[4]/4)` could represent how these limits interact[^1]. However, applying this directly without knowing specifics like what each array element represents in relation to the actual requirements for creating "philosophical stones" as mentioned would require adjustments based upon the precise conditions outlined within 1487D itself. To solve or discuss solutions effectively regarding Codeforces' challenge numbered 1487D: - Carefully read through all aspects presented by the contest organizers. - Identify which ingredient or component acts as the primary constraint towards achieving full capacity utilization. - Implement logic reflecting those relationships accurately; typically involving loops, conditionals, and possibly dynamic programming depending on complexity level required beyond simple minimum value determination across adjusted inputs. ```cpp #include <iostream> #include <vector> using namespace std; int main() { int n; cin >> n; vector<long long> a(n); for(int i=0;i<n;++i){ cin>>a[i]; } // Assuming indices correspond appropriately per problem statement's ratio requirement cout << min({a[0], a[1], a[2]/2LL, a[3]/7LL, a[4]/4LL}) << endl; } ``` --related questions-- 1. How does identifying bottlenecks help optimize algorithms solving constrained optimization problems? 2. What strategies should contestants adopt when translating mathematical formulas into code during competitive coding events? 3. Can you explain why understanding input-output relations is critical before implementing any algorithmic approach? 4. In what ways do prefix-suffix-middle frameworks enhance model training efficiency outside of just tokenization improvements? 5. Why might adjusting sample proportions specifically benefit models designed for tasks requiring both strong linguistic comprehension alongside logical reasoning skills?
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