Merge pull request UTSAVS26#1094 from NK-Works/dp-3

Catalan Algo, Min Steps to One and Knapsack With Constraints Added

Algorithms_and_Data_Structures/Dynamic-Programming-Series/Hard-DP-Problems/catalan-numbers.py

@@ -0,0 +1,16 @@
+def catalan_numbers(n):
+    # Initialize a list to store Catalan numbers
+    catalan = [0] * (n + 1)
+
+    # Base case
+    catalan[0] = 1
+
+    # Calculate the remaining catalan numbers up to nth
+    for i in range(1, n + 1):
+        catalan[i] = sum(catalan[j] * catalan[i - 1 - j] for j in range(i))
+
+    return catalan
+
+# Test the function
+n = 10  # Calculate the first 10 Catalan numbers
+print(f"The first {n} Catalan numbers are: {catalan_numbers(n)}")

Algorithms_and_Data_Structures/Dynamic-Programming-Series/Hard-DP-Problems/knapsack-with-constraints.py

@@ -0,0 +1,22 @@
+def knapsack(weights, values, capacity):
+    n = len(weights)
+    # Create a DP array with (n+1) rows for items and (capacity+1) columns for capacity
+    dp = [[0] * (capacity + 1) for _ in range(n + 1)]
+
+    # Fill the DP table
+    for i in range(1, n + 1):
+        for w in range(1, capacity + 1):
+            if weights[i - 1] <= w:
+                # Include the item or exclude it, choose the better option
+                dp[i][w] = max(dp[i - 1][w], dp[i - 1][w - weights[i - 1]] + values[i - 1])
+            else:
+                # Cannot include the item, so carry forward the value without it
+                dp[i][w] = dp[i - 1][w]
+
+    return dp[n][capacity]
+
+# Test the function
+weights = [1, 2, 3, 5]
+values = [10, 20, 30, 50]
+capacity = 6
+print(f"Maximum value in the knapsack: {knapsack(weights, values, capacity)}")

Algorithms_and_Data_Structures/Dynamic-Programming-Series/Hard-DP-Problems/min-steps-to-one.py

@@ -0,0 +1,21 @@
+def min_steps_to_one(n):
+    # Initialize a DP array where dp[i] will store minimum steps to reach 1 from i
+    dp = [float('inf')] * (n + 1)
+    dp[1] = 0  # Base case: it takes 0 steps to reach 1 from 1
+
+    # Fill the DP table for each number from 2 up to n
+    for i in range(2, n + 1):
+        # Subtract 1
+        dp[i] = dp[i - 1] + 1
+        # Divide by 2 if applicable
+        if i % 2 == 0:
+            dp[i] = min(dp[i], dp[i // 2] + 1)
+        # Divide by 3 if applicable
+        if i % 3 == 0:
+            dp[i] = min(dp[i], dp[i // 3] + 1)
+
+    return dp[n]
+
+# Test the function
+n = 10
+print(f"Minimum steps to reduce {n} to 1: {min_steps_to_one(n)}")

Project-Structure.md

@@ -95,8 +95,11 @@
       * [Nth Tribonacci Num](Algorithms_and_Data_Structures/Dynamic-Programming-Series/Basic-DP-Problems/nth_tribonacci_num.py)
       * [Zero One Knapsack](Algorithms_and_Data_Structures/Dynamic-Programming-Series/Basic-DP-Problems/zero_one_knapsack.py)
     * Hard-Dp-Problems
+      * [Catalan-Numbers](Algorithms_and_Data_Structures/Dynamic-Programming-Series/Hard-DP-Problems/catalan-numbers.py)
       * [Cherry-Pick-Algo](Algorithms_and_Data_Structures/Dynamic-Programming-Series/Hard-DP-Problems/cherry-pick-algo.py)
+      * [Knapsack-With-Constraints](Algorithms_and_Data_Structures/Dynamic-Programming-Series/Hard-DP-Problems/knapsack-with-constraints.py)
       * [Levenshtein-Distance](Algorithms_and_Data_Structures/Dynamic-Programming-Series/Hard-DP-Problems/levenshtein-distance.py)
+      * [Min-Steps-To-One](Algorithms_and_Data_Structures/Dynamic-Programming-Series/Hard-DP-Problems/min-steps-to-one.py)
       * [Regular-Expression-Matching](Algorithms_and_Data_Structures/Dynamic-Programming-Series/Hard-DP-Problems/regular-expression-matching.py)
     * Medium-Dp-Problems
       * [Coin-Change](Algorithms_and_Data_Structures/Dynamic-Programming-Series/Medium-DP-Problems/coin-change.py)