add 53 maximum subarray

Author: teckoo

_posts/coding/leetcode/2021-01-01-53-maximum-subarray.md

@@ -0,0 +1,67 @@
+---
+layout: leetcode
+title: "53. Maximum Subarray"
+categories: [leetcode]
+---
+
+[Leetcode Link](https://leetcode.com/problems/maximum-subarray/)
+
+Given an integer array nums, find the contiguous subarray (containing at least one number) which has the largest sum and return its sum.
+
+ 
+```
+Example 1:
+
+Input: nums = [-2,1,-3,4,-1,2,1,-5,4]
+Output: 6
+Explanation: [4,-1,2,1] has the largest sum = 6.
+
+Example 2:
+
+Input: nums = [1]
+Output: 1
+
+Example 3:
+
+Input: nums = [5,4,-1,7,8]
+Output: 23
+ 
+
+Constraints:
+
+1 <= nums.length <= 3 * 104
+-105 <= nums[i] <= 105
+```
+
+# Solution
+
+## prefix sum
+
+T: O(n), S:O(n)
+
+```python
+def maxSubArray(self, nums: List[int]) -> int:
+  for i in range(1, len(nums)):
+    if nums[i-1] > 0:
+      nums[i] += nums[i-1]
+  return max(nums)
+```
+
+## Kadane's algorithm
+
+T: O(n), S:O(n)
+
+```python
+def maxSubArray(self, nums: List[int]) -> int:
+  # Initialize our variables using the first element.
+  current_subarray = max_subarray = nums[0]
+  
+  # Start with the 2nd element since we already used the first one.
+  for num in nums[1:]:
+      # If current_subarray is negative, throw it away. 
+      # Otherwise, keep adding to it.
+      current_subarray = max(num, current_subarray + num)
+      max_subarray = max(max_subarray, current_subarray)
+  
+  return max_subarray
+```

media/js/tag-search-data.js

@@ -135,6 +135,13 @@ var problems = {
         templates: ["backtrack"],
         notes: "Backtrack template. bt(grid, 0). End condition: row==len(grid). diag_l[row+col], diag_r[col-row+n-1], grid[row][col]='Q';bt();grid[row][col]='.'. Time: O(N!), Space: O(N!)",
     },
+    "53": {
+        name: "53. maximum-subarray",
+        tags: ["array", "prefix_sum"],
+        keywords: ["subarray", "sum", "lv_1"],
+        url: "/leetcode/53-maximum-subarray",
+        notes: "Kadane's algorithm. Or get prefix sum for array, convert to buy/sell stock problem",
+    },
     "75": {
         name: "75. Sort Colors",
         tags: ["array", "two_pointers"],