给定一个由N 个整数组成的数组arr[] ,任务是通过将结果子序列的每个元素乘以其索引(基于 1 的索引)来找到最大子序列总和。
例子:
Input: arr[] = {-1, 2, -10, 4, -20}
Output: 15
Explanation:
For the subsequence {-1, 2, 4}, sum of the given subsequence after performing the given operations = 1*(-1) + 2*2 + 3*4 = 15, which is maximum possible.
Input: arr[] = {1, 0, -1, 2}
Output: 7
Explanation:
For the subsequence {1, 0, 2}, sum of the given subsequence after performing the given operations = 1*1 + 2*0 + 3*2 = 7, which is maximum possible.
朴素的方法:这个想法是使用递归从数组中生成所有可能的子序列,并计算每个子序列所需的总和并打印最大总和。
时间复杂度: O(2 N )
辅助空间: O(N)
高效方法:上述方法可以使用动态规划进行优化,因为问题包含许多重叠的子问题。以下是步骤:
- 初始化一个辅助矩阵dp[][] ,其中dp[i][j]存储长度为j的子序列的最大值,直到索引i 。
- 遍历给定的数组,对于每个元素,有两种可能性:
- 要么将当前元素包含在子序列总和中并增加子序列中的元素数。
- 或者从子序列中排除当前元素并继续下一个元素。
- 因此,递推关系由以下等式给出:
dp[i][j] = max(a[i] * j + maximumSum(j + 1, i + 1), maximumSum(j, i + 1))
- 通过对每个索引使用上述递推关系不断更新dp[][]表,并返回整个数组可能的最大和。
- 在上述步骤后打印最大值。
下面是上述方法的实现:
C++
// C++ program for the above approach
#include
using namespace std;
// Initialize dp array
int dp[1005][1005];
// Function to find the maximum
// sum of the subsequence formed
int maximumSumUtil(int a[], int index,
int count, int n)
{
// Base Case
if (index > n || count > n + 1) {
return 0;
}
// If already calculated
// state occurs
if (dp[index][count] != -1)
return dp[index][count];
// Include the current element
int ans1 = maximumSumUtil(a, index + 1,
count + 1, n)
+ a[index] * count;
// Exclude the current element
int ans2 = maximumSumUtil(a, index + 1,
count, n);
// Update the maximum ans
return (dp[index][count]
= max(ans1, ans2));
}
// Function to calculate maximum sum
// of the subsequence obtained
int maximumSum(int arr[], int N)
{
// Initialise the dp array with -1
memset(dp, -1, sizeof(dp));
// Print the maximum sum possible
cout << maximumSumUtil(arr, 0, 1,
N - 1);
}
// Driver Code
int main()
{
// Given array
int arr[] = { -1, 2, -10, 4, -20 };
// Size of the array
int N = sizeof(arr) / sizeof(arr[0]);
// Function Call
maximumSum(arr, N);
return 0;
}
Java
// Java program for
// the above approach
import java.util.*;
class GFG{
// Initialize dp array
static int [][]dp = new int[1005][1005];
// Function to find the maximum
// sum of the subsequence formed
static int maximumSumUtil(int a[], int index,
int count, int n)
{
// Base Case
if (index > n || count > n + 1)
{
return 0;
}
// If already calculated
// state occurs
if (dp[index][count] != -1)
return dp[index][count];
// Include the current element
int ans1 = maximumSumUtil(a, index + 1,
count + 1, n) +
a[index] * count;
// Exclude the current element
int ans2 = maximumSumUtil(a, index + 1,
count, n);
// Update the maximum ans
return (dp[index][count] =
Math.max(ans1, ans2));
}
// Function to calculate maximum sum
// of the subsequence obtained
static void maximumSum(int arr[], int N)
{
// Initialise the dp array with -1
for(int i = 0; i < 1005; i++)
{
for (int j = 0; j < 1005; j++)
{
dp[i][j] = -1;
}
}
// Print the maximum sum possible
System.out.print(maximumSumUtil(arr, 0,
1, N - 1));
}
// Driver Code
public static void main(String[] args)
{
// Given array
int arr[] = {-1, 2, -10, 4, -20};
// Size of the array
int N = arr.length;
// Function Call
maximumSum(arr, N);
}
}
// This code is contributed by 29AjayKumar
Python3
# Python3 program for the above approach
# Initialize dp array
dp = [[-1 for x in range(1005)]
for y in range(1005)]
# Function to find the maximum
# sum of the subsequence formed
def maximumSumUtil(a, index, count, n):
# Base Case
if (index > n or count > n + 1):
return 0
# If already calculated
# state occurs
if (dp[index][count] != -1):
return dp[index][count]
# Include the current element
ans1 = (maximumSumUtil(a, index + 1,
count + 1, n) +
a[index] * count)
# Exclude the current element
ans2 = maximumSumUtil(a, index + 1,
count, n)
# Update the maximum ans
dp[index][count] = max(ans1, ans2)
return dp[index][count]
# Function to calculate maximum sum
# of the subsequence obtained
def maximumSum(arr, N):
# Print the maximum sum possible
print(maximumSumUtil(arr, 0, 1,
N - 1))
# Driver Code
# Given array
arr = [ -1, 2, -10, 4, -20 ]
# Size of the array
N = len(arr)
# Function call
maximumSum(arr, N)
# This code is contributed by Shivam Singh
C#
// C# program for
// the above approach
using System;
class GFG{
// Initialize dp array
static int [,]dp = new int[1005, 1005];
// Function to find the maximum
// sum of the subsequence formed
static int maximumSumUtil(int []a, int index,
int count, int n)
{
// Base Case
if (index > n || count > n + 1)
{
return 0;
}
// If already calculated
// state occurs
if (dp[index, count] != -1)
return dp[index, count];
// Include the current element
int ans1 = maximumSumUtil(a, index + 1,
count + 1, n) +
a[index] * count;
// Exclude the current element
int ans2 = maximumSumUtil(a, index + 1,
count, n);
// Update the maximum ans
return (dp[index, count] =
Math.Max(ans1, ans2));
}
// Function to calculate maximum sum
// of the subsequence obtained
static void maximumSum(int []arr, int N)
{
// Initialise the dp array with -1
for(int i = 0; i < 1005; i++)
{
for (int j = 0; j < 1005; j++)
{
dp[i, j] = -1;
}
}
// Print the maximum sum possible
Console.Write(maximumSumUtil(arr, 0,
1, N - 1));
}
// Driver Code
public static void Main(String[] args)
{
// Given array
int []arr = {-1, 2, -10, 4, -20};
// Size of the array
int N = arr.Length;
// Function Call
maximumSum(arr, N);
}
}
// This code is contributed by Rajput-Ji
Javascript
15
时间复杂度: O(N 2 )
辅助空间: O(N 2 )
如果您希望与专家一起参加现场课程,请参阅DSA 现场工作专业课程和学生竞争性编程现场课程。