Solved DP_2

CoinChange_II.py

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+'''
+To solve this problem, I chose exhaustive approach by building a DP table
+- Initially I stored first row and column with dummy values
+- As long as the coin denomination is greater than the amount the case would be invalid
+- If its not less, we calculate the number of ways, by taking its repeated sub problems which calculate the number of ways and add them
+'''
+class Solution:
+    def change(self, amount: int, coins: List[int]) -> int:
+
+        n = len(coins) # rows
+        m = amount # columns
+
+        # creating a DP table of n+1 rows(coins) and m+1 columns(amount)
+        dp = [[0] * (m+1) for _ in range(n+1)]
+        dp[0][0] = 1 # making first element as 1/ there is only 1 way to make zero amount, by not choosing any coin
+
+        for i in range(1, n+1):
+            for j in range (m+1):
+                # till denomination > amount, case would be invalid
+                if coins[i-1] > j:
+                    # so we copy the result from above row
+                    dp[i][j] = dp[i-1][j]
+                else:
+                    # filling current element with addition of:
+                    # above row elemet (sub problem calculating part of no of ways)
+                    # and element in the denominations steps back element(sub problem calculating part of no of ways)
+                    dp[i][j] = dp[i-1][j] + dp[i][j-coins[i-1]]
+        # returning the last elemet which has no of ways to calculate the amount
+        return dp[n][m]
+
+'''
+Time Complexity: O(n*m)
+Since we are iterating on the DP table of size (n+1) * (m+1), we ignore the constants so O(n*m)
+Space Complexity: O(m*n)
+Since DP tables stores the results of all the coins count and amount
+'''

HousePaint.py

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+'''
+For solving the House Paint problem, I considered using exahustive approach by optimizing the sub problems.
+- Initially I kept the costs of last row constant, and stored in the DP matrix in the last row.
+- Then by iterating in the houses and colors matrix from bottom elements to top, I calculated the sum of current element and minimum cost of either of the colors.
+- After filling the DP matrix, we return the minimum of top most row elements, because by the time the elements are 
+'''
+class Solution:
+    def minCost(self, costs):
+        # no of rows/houses
+        n = len(costs)
+        # n houses and 3 columns
+        dp = [[0] * 3 for _ in range(n)]
+        # keeping the last row as it is
+        dp[n-1][0] = costs[n-1][0]
+        dp[n-1][1] = costs[n-1][1]
+        dp[n-1][2] = costs[n-1][2]
+
+        # filling the dp matrix from second last index, since the last row already has same values from costs matrix
+        for i in range(n-2, -1, -1):
+            # filling the elements with cost of that color with minimum of other two colors from below row
+            # eg:taking the cost of painting the houses with red and cost of painting with minimum of blue/green
+            dp[i][0] = costs[i][0] + min(dp[i+1][1], dp[i+1][2])
+            dp[i][1] = costs[i][1] + min(dp[i+1][0], dp[i+1][2])
+            dp[i][2] = costs[i][2] + min(dp[i+1][0], dp[i+1][1])
+
+        # minimum of top most row
+        return min(dp[0][0], dp[0][1], dp[0][2])
+
+'''
+Time Complexity: O(n)
+Here I am creating a DP matrix whose size is n*3, where 3 is constant, so we itearte on n elements which takes O(n)
+Space Complexity: O(n)
+Similarly for space the size of DP matrix would be n*3, where 3 is constant, so the average space taken is O(n)
+'''