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+"""
+0/1 Knapsack Problem using 2D Dynamic Programming
+-------------------------------------------------
+
+Objective:
+This file explains the **2D DP approach** to the classic 0/1 Knapsack problem
+in the simplest and most educational way possible.
+
+Problem Statement:
+Given N items, each with a certain weight and value, and a bag with a maximum weight
+capacity (W), we must select items to maximize total value without exceeding W.
+
+This is a perfect problem to understand **2D Dynamic Programming**.
+
+-------------------------------------------------
+Approach Explanation
+-------------------------------------------------
+We build a DP table (2D array) where:
+    dp[i][w] = Maximum value we can achieve
+               using the first i items (0..i-1) and
+               capacity w.
+
+Transition:
+    If we do NOT take the ith item → dp[i][w] = dp[i-1][w]
+    If we take the ith item (if it fits) → dp[i][w] = value[i-1] + dp[i-1][w - weight[i-1]]
+
+Final Answer → dp[n][W]
+
+-------------------------------------------------
+Complexity Analysis:
+-------------------------------------------------
+Time Complexity:  O(N * W)
+Space Complexity: O(N * W)
+Where:
+    N = number of items
+    W = maximum capacity of the bag
+
+"""
+
+# -------------------------------------------------
+# Step 1: Define the knapsack solver function
+# -------------------------------------------------
+def knapsack_2d(weights, values, capacity):
+    """
+    Solves the 0/1 Knapsack problem using 2D Dynamic Programming.
+
+    Parameters:
+        weights  : list of item weights
+        values   : list of item values
+        capacity : maximum capacity of the bag
+
+    Returns:
+        max_value       : Maximum value that can be obtained
+        chosen_items    : List of item indices chosen
+        dp              : 2D DP table
+    """
+
+    n = len(weights)
+
+    # Create DP table with all 0s
+    dp = [[0 for _ in range(capacity + 1)] for _ in range(n + 1)]
+
+    # -------------------------------------------------
+    # Step 2: Fill DP table
+    # -------------------------------------------------
+    for i in range(1, n + 1):
+        for w in range(1, capacity + 1):
+
+            # If current item's weight is more than current capacity, skip it
+            if weights[i - 1] > w:
+                dp[i][w] = dp[i - 1][w]
+
+            else:
+                # Either we take the item or we don’t — choose max
+                include = values[i - 1] + dp[i - 1][w - weights[i - 1]]
+                exclude = dp[i - 1][w]
+                dp[i][w] = max(include, exclude)
+
+    # -------------------------------------------------
+    # Step 3: Backtrack to find chosen items
+    # -------------------------------------------------
+    chosen_items = []
+    w = capacity
+
+    for i in range(n, 0, -1):
+        # If value comes from including the current item
+        if dp[i][w] != dp[i - 1][w]:
+            chosen_items.append(i - 1)
+            w -= weights[i - 1]
+
+    chosen_items.reverse()  # maintain original order
+
+    # -------------------------------------------------
+    # Step 4: Return final results
+    # -------------------------------------------------
+    max_value = dp[n][capacity]
+    return max_value, chosen_items, dp
+
+
+# -------------------------------------------------
+# Step 5: Utility function to print DP table
+# -------------------------------------------------
+def print_dp_table(dp, weights, values):
+    """
+    Prints the DP table in a readable format with item info.
+    """
+    n = len(weights)
+    cap = len(dp[0]) - 1
+
+    print("\n2D DP Table (rows = items, columns = capacity):\n")
+    header = ["w\\i"] + [str(i) for i in range(cap + 1)]
+    print("\t".join(header))
+
+    for i in range(n + 1):
+        row_label = "0 (no items)" if i == 0 else f"{i} (wt={weights[i-1]}, val={values[i-1]})"
+        row_values = [str(dp[i][j]) for j in range(cap + 1)]
+        print(row_label + "\t" + "\t".join(row_values))
+
+
+# -------------------------------------------------
+# Step 6: Run example test cases
+# -------------------------------------------------
+if __name__ == "__main__":
+
+    # Example 1 (Basic)
+    weights = [2, 1, 3, 2]
+    values = [12, 10, 20, 15]
+    capacity = 5
+
+    print("Example 1: Understanding 2D DP through 0/1 Knapsack\n")
+    print(f"Weights : {weights}")
+    print(f"Values  : {values}")
+    print(f"Capacity: {capacity}\n")
+
+    max_value, chosen, dp = knapsack_2d(weights, values, capacity)
+
+    # Print DP table
+    print_dp_table(dp, weights, values)
+
+    print("\nFinal Results:")
+    print(f"Maximum Value: {max_value}")
+    print(f"Chosen Item Indices: {chosen}")
+    print("Chosen Items (index → weight, value):")
+    for idx in chosen:
+        print(f"  Item {idx}: (w={weights[idx]}, v={values[idx]})")
+
+    # Example 2 (Edge Case: small capacity)
+    print("\nExample 2: Edge Case with Small Capacity")
+    weights2 = [4, 5, 1]
+    values2 = [1, 2, 3]
+    cap2 = 4
+    max_val2, chosen2, dp2 = knapsack_2d(weights2, values2, cap2)
+    print(f"Weights: {weights2}, Values: {values2}, Capacity: {cap2}")
+    print(f"Max Value = {max_val2}, Chosen = {chosen2}")