Dynamic Programming (DP) is an algorithm design technique that solves complex problems by breaking them down into smaller subproblems and storing the results of already computed values to avoid redundant calculations. It is mainly used for optimization problems, reducing time complexity and providing efficient solutions.<\/p>\n
This problem involves finding the maximum sum of a contiguous subarray within an array of integers. For example, the maximum subarray sum of the array Print the maximum subarray sum.<\/p>\n To solve this problem using dynamic programming, the following steps can be taken:<\/p>\n The above code first takes the size of the array The time complexity of this algorithm is The space complexity is also Dynamic programming is a powerful technique for efficiently solving complex problems. The maximum subarray sum problem helps understand the basic concepts of dynamic programming and often appears in coding tests, making sufficient practice necessary.<\/p>\n{-2, 1, -3, 4, -1, 2, 1, -5, 4}<\/code> is 6<\/code>, which is the sum of the subarray {4, -1, 2, 1}<\/code>.<\/p>\nInput<\/h3>\n
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N<\/code> (1 \u2264 N \u2264 100,000).<\/li>\nOutput<\/h3>\n
3. Problem Solving Process<\/h2>\n
3.1. Approach<\/h3>\n
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dp[i]<\/code> as the maximum subarray sum up to the i<\/code>th element.<\/li>\ndp[i] = max(arr[i], dp[i-1] + arr[i])<\/code>. That is, decide whether to include the i<\/code>th element or not.<\/li>\ndp[0] = arr[0]<\/code>.<\/li>\n3.2. C++ Code Implementation<\/h3>\n
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\n#include <iostream>\n#include <vector>\n#include <algorithm>\n\nusing namespace std;\n\nint main() {\n int N;\n cin >> N;\n vector4. Code Explanation<\/h2>\n
N<\/code> and the elements of the array as input. Then, it declares a dp<\/code> array to store the maximum subarray sum up to each index. After that, it updates the dp<\/code> values inside a loop while finding the maximum value in the process.<\/p>\n4.1. Time Complexity<\/h3>\n
O(N)<\/code>. It has linear time complexity because it iterates through the array once.<\/p>\n4.2. Space Complexity<\/h3>\n
O(N)<\/code>, as it stores up to N<\/code> values. However, there is a method to find the maximum sum using just two variables without using the dp<\/code> array.<\/p>\n5. Conclusion<\/h2>\n