[Learn Dynamic Programming] Maximum Subarray Sum (19)

Table of contents

How to learn dynamic programming?

1. Topic analysis

2. Algorithm principle

1. Status representation

2. State transition equation

3. Initialization

4. Filling order

5. Return value

3. Code writing

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How to learn dynamic programming?

There is no shortcut to learning an algorithm, let alone learning dynamic programming,

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1. Topic analysis

Topic Link: 53. Maximum Subarray Sum - LeetCode

The topic is easy to understand. As the name suggests, it is to find the largest sub-array sum.

2. Algorithm principle

1. Status representation

The position dp[ i ] represents the maximum sum of all subarrays ending with the element at position i.

2. State transition equation

The state transition equation has two cases,

1. When the length of the subarray is 1, the maximum sum is the value at position i

2. If the length of the sub-array is greater than 1, the maximum sum is the maximum sum of the previous position + the value of the current position

So we can get the state transition equation

dp [ i ] = max( nums[ i ]，dp[ i ] + nums[ i ] )

3. Initialization

Initialization is to prevent out-of-bounds, and does not affect subsequent values,

Just initialize it to 0.

4. Filling order

Just go from left to right.

5. Return value

Returns the maximum value in the entire dp table.

3. Code writing

class Solution {
public:
    int maxSubArray(vector<int>& nums) {
        int n = nums.size();
        vector<int> dp(n + 1);
        int ans = INT_MIN;
        for(int i = 1; i <= n ; i++) {
            dp[i] = max(nums[i - 1], dp[i - 1] + nums[i - 1]);
            ans = max(ans, dp[i]);
        }
        return ans;
    }
};

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