added modified version of Minimum Path Sum

Author: minhyeong-joe

DynamicProgramming/MinPathSum/README.md

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+# Minimum Path Sum (**MODIFIED**)
+
+[Leetcode Link](https://leetcode.com/problems/minimum-path-sum/)
+
+## Problem:
+
+Given a m x n grid filled with non-negative numbers, find a path from top left to bottom right, which minimizes the sum of all numbers along its path.
+
+**modified**: You can move either down or right _or bottom-right (i+1, j+1)_
+
+## Example:
+
+```
+Input: grid = [[1,2,3],
+               [4,8,2],
+               [1,5,3]]
+Output: 8
+Explanation: Because the path 1 → 2 → 2 → 3 minimizes the sum.
+```

DynamicProgramming/MinPathSum/solution.py

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+def minPathSum(grid):
+    # init dynamic programming table
+    dp = [[None for _ in grid[0]] for _ in grid]
+    dp[0][0] = grid[0][0]
+    for i in range(1, len(dp)):
+        dp[i][0] = dp[i-1][0] + grid[i][0]
+    for j in range(1, len(dp[0])):
+        dp[0][j] = dp[0][j-1] + grid[0][j]
+    print("initial dp:", dp)
+    # fill in dp table
+    for i in range(1, len(dp)):
+        for j in range(1, len(dp[i])):
+            minCostSoFar = min(dp[i-1][j], dp[i][j-1], dp[i-1][j-1])
+            dp[i][j] = minCostSoFar + grid[i][j]
+    print("after fill-in:", dp)
+    return dp[-1][-1]
+
+
+grid = [[1, 2, 3],
+        [4, 8, 2],
+        [1, 5, 3]]
+print(minPathSum(grid))