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【动态规划】Part1

發布時間：2023/11/27 生活经验 39 豆豆

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1. 硬幣找零

題目描述：假設有幾種硬幣，如1、3、5，并且數量無限。請找出能夠組成某個數目的找零所使用最少的硬幣數。

分析： ? dp [0] = 0
? ? ? ? ? ?dp [1] = 1 + dp [1-1]
? ? ? ? ? ?dp [2] = 1 + dp [2-1]
? ? ? ? ? ?dp [3] = min (dp [3 - 1] + 1, dp [3 - 3] + 1)

 1 #include<iostream>
 2 #include<algorithm>
 3 #define INF 32767
 4 using namespace std;
 5 
 6 int dp[100];
 7 int coin[3] = { 1, 2, 3 };
 8 
 9 int main()
10 {
11     int sum;
12     cin >> sum;
13     dp[0] = 0;
14     for (int i = 1; i <= 100; ++i)
15         dp[i] = INF;
16     for (int i = 1; i <= sum; ++i)
17         for (int j = 0; j <= 2; ++j)
18             if (coin[j] <= i)
19                 dp[i] = min(dp[i], dp[i - coin[j]] + 1);
20     cout << dp[sum] << endl;
21     return 0;
22 }

2. 最長遞增子序列

? 題目描述：最長遞增子序列（Longest Increasing Subsequence）是指找到一個給定序列的最長子序列的長度，使得子序列中的所有元素單調遞增。

給定一個序列，求解它的最長遞增子序列的長度。比如： arr[] = {3,1,4,1,5,9,2,6,5} ? 的最長遞增子序列長度為4。即為：1,4,5,9

 1 #include<iostream>
 2 #include<algorithm>
 3 using namespace std;
 4 
 5 int arr[9] = { 3, 1, 4, 1, 5, 9, 2, 6, 5 };
 6 int dp[9];
 7 
 8 int main()
 9 {
10     for (int i = 0; i < 9; ++i)
11         dp[i] = 1;
12     for (int i = 1; i < 9; ++i)
13         for (int j = 0; j < i; ++j)
14             if (arr[i] > arr[j])
15                 dp[i] = max(dp[i], dp[j] + 1);
16     int mi = 0;
17     for (int i = 0; i < 6; ++i)
18         mi = max(mi, dp[i]);
19     cout << mi << endl;
20     return 0;
21 }

3. 數字三角形

Problem description

7

3   8

8   1   0

2   7   4   4

4   5   2   6   5


(Figure 1)

Figure 1 shows a number triangle. Write a program that calculates the highest sum of numbers passed on a route that starts at the top and ends somewhere on the base. Each step can go either diagonally down to the left or diagonally down to the right.

Input

Your program is to read from standard input. The first line contains one integer T, the number of test cases, for each test case: the first line contain a integer N: the number of rows in the triangle. The following N lines describe the data of the triangle. The number of rows in the triangle is > 1 but <= 100. The numbers in the triangle, all integers, are between 0 and 99.?

Output

Your program is to write to standard output. The highest sum is written as an integer for each test case one line.

Sample Input

Sample Output

Problem Source

IOI?1994

代碼：

 1 #include<iostream>
 2 #include<algorithm>
 3 using namespace std;
 4 
 5 int dp[100][100];
 6 int arr[100][100];
 7 
 8 int main()
 9 {
10     int N;
11     cin >> N;
12     if (N == 1)
13         cout << N;
14     for (int i = 0; i < N; ++i)
15         for (int j = 0; j <= i; ++j)
16             cin >> arr[i][j];
17     for (int i = 0; i < N; ++i)
18         dp[N - 1][i] = arr[N - 1][i];
19     for (int i = N - 2; i >= 0; --i)
20         for (int j = 0; j <= i; ++j)
21             dp[i][j] = max(arr[i][j] + dp[i + 1][j], arr[i][j] + dp[i + 1][j + 1]);
22     cout << dp[0][0] << endl;
23     return 0;
24 }

?4. 最大最大連續子序列和/積

??求取數組中最大連續子序列和，例如給定數組為A={1， 3， -2， 4， -5}，則最大連續子序列和為6，即1+3+（-2）+ 4 = 6。

??求取數組中最大連續子序列積。

參考資料

? 常見動態規劃問題分析與求解
??關于序列的面試題2------------最大連續子序列和以及積

轉載于:https://www.cnblogs.com/sunbines/p/9011149.html

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