[LeetCode] 53. Maximum Subarray 最大子数组

Given an integer array nums, find the contiguous subarray (containing at least one number) which has the largest sum and return its sum.

Example:

Input: [-2,1,-3,4,-1,2,1,-5,4],
Output: 6
Explanation: [4,-1,2,1] has the largest sum = 6.

Follow up:

If you have figured out the O(n) solution, try coding another solution using the divide and conquer approach, which is more subtle.

 

给定一个数组，求元素和最大的子数组。

解法1： 动态规划DP，dp[i] 表示到元素i时，末尾为i的数组的最大和。dp[i-1] 的最大元素和如果为正，那么有用就和当前元素相加。如果为负那就无用，干脆舍弃，只用dp[i] 自己来组成。

解法2：题中让用divide and conquer的方法来做。

Java:

public class Solution {
    public int maxSubArray(int[] nums) {
        int res = Integer.MIN_VALUE, curSum = 0;
        for (int num : nums) {
            curSum = Math.max(curSum + num, num);
            res = Math.max(res, curSum);
        }
        return res;
    }
}

Java: Divide and conquer

public class Solution {
    public int maxSubArray(int[] nums) {
        if (nums.length == 0) return 0;
        return helper(nums, 0, nums.length - 1);
    }
    public int helper(int[] nums, int left, int right) {
        if (left >= right) return nums[left];
        int mid = left + (right - left) / 2;
        int lmax = helper(nums, left, mid - 1);
        int rmax = helper(nums, mid + 1, right);
        int mmax = nums[mid], t = mmax;
        for (int i = mid - 1; i >= left; --i) {
            t += nums[i];
            mmax = Math.max(mmax, t);
        }
        t = mmax;
        for (int i = mid + 1; i <= right; ++i) {
            t += nums[i];
            mmax = Math.max(mmax, t);
        }
        return Math.max(mmax, Math.max(lmax, rmax));
    }
}

Python:

class Solution(object):
    def maxSubArray(self, nums):
        if max(nums) < 0:
            return max(nums)
        global_max, local_max = 0, 0
        for x in nums:
            local_max = max(0, local_max + x)
            global_max = max(global_max, local_max)
        return global_max

Python: wo

class Solution(object):
    def maxSubArray(self, nums):
        """
        :type nums: List[int]
        :rtype: int
        """      
        dp = [0] * (len(nums) + 1)
        max_sum = float('-inf')  # cannot set max_sum = 0, error when input is [-1]
        for i in xrange(1, len(nums) + 1):
            if dp[i-1] >= 0:                
                dp[i] = dp[i-1] + nums[i-1]
            else:
                dp[i] = nums[i-1]
            max_sum = max(max_sum, dp[i])    
                
        return max_sum 　　

Python: wo

class Solution(object):
    def maxSubArray(self, nums):
        """
        :type nums: List[int]
        :rtype: int
        """      
        dp = [0] * len(nums)
        # res = float('-inf') # error [1]
        dp[0] = nums[0]
        res = dp[0]
        for i in xrange(1, len(nums)):
            dp[i] = nums[i] + max(0, dp[i-1])
            res = max(res, dp[i])  
            
        return res 　　

C++:

class Solution {
public:
    int maxSubArray(vector<int>& nums) {
        int res = INT_MIN, curSum = 0;
        for (int num : nums) {
            curSum = max(curSum + num, num);
            res = max(res, curSum);
        }
        return res;
    }
};

 C++: Divide and conquer

class Solution {
public:
    int maxSubArray(vector<int>& nums) {
        if (nums.empty()) return 0;
        return helper(nums, 0, (int)nums.size() - 1);
    }
    int helper(vector<int>& nums, int left, int right) {
        if (left >= right) return nums[left];
        int mid = left + (right - left) / 2;
        int lmax = helper(nums, left, mid - 1);
        int rmax = helper(nums, mid + 1, right);
        int mmax = nums[mid], t = mmax;
        for (int i = mid - 1; i >= left; --i) {
            t += nums[i];
            mmax = max(mmax, t);
        }
        t = mmax;
        for (int i = mid + 1; i <= right; ++i) {
            t += nums[i];
            mmax = max(mmax, t);
        }
        return max(mmax, max(lmax, rmax));
    }
};

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