Solution Explanation: Shopping Offers

This problem involves finding the minimum cost to buy a set of items given individual item prices and special offers. The solution presented uses a combination of bit manipulation, memoization (dynamic programming), and a depth-first search (DFS) approach.

Approach

1. Bit Manipulation for State Representation: Since the number of item types (n) is limited (n ≤ 6), we can represent the current shopping needs efficiently using bit manipulation. Each item's quantity is encoded within a 4-bit segment of an integer (mask). This allows us to represent the entire shopping list's state compactly.

2. Memoization (Dynamic Programming): To avoid redundant calculations, a memoization table (f in the code) stores the minimum cost for each state (mask). If a state is already computed, the stored value is retrieved directly.

3. Depth-First Search (DFS): A recursive DFS function (dfs) explores different combinations of using special offers. It starts with the complete shopping list (needs) and iteratively considers applying each special offer.

   • Base Case: If no special offers can be applied, the cost is calculated using individual item prices.
   • Recursive Step: For each special offer, the function checks if it can be applied (enough items are available). If so, it applies the offer, recursively calculates the cost for the remaining needs, and updates the minimum cost.

4. State Transition: When an offer is applied, the corresponding bits in the mask are updated to reflect the reduced needs. The recursive call is made with the updated mask.

Time and Space Complexity Analysis

• Time Complexity: The time complexity is difficult to express precisely, but it's bounded by O(m * 24n), where:

  • m is the number of special offers.
  • n is the number of item types (at most 6).
  • 24n represents the possible states (masks) – each of the n items has at most 16 possible quantities (4 bits). The actual number of states visited will be significantly less due to memoization and the constraints on item quantities in the problem statement.

• Space Complexity: The space complexity is dominated by the memoization table, which can store up to O(24n) entries in the worst case. Again, this is an upper bound, and the actual space used will likely be much less due to memoization.

Code Explanation (Python as Example)

The Python code provided demonstrates the solution elegantly:

class Solution:
    def shoppingOffers(self, price, special, needs):
        @cache # Python's @cache decorator for memoization.
        def dfs(cur):
            ans = sum(p * (cur >> (i * bits) & 0xF) for i, p in enumerate(price))
            for offer in special:
                nxt = cur
                for j in range(len(needs)):
                    if (cur >> (j * bits) & 0xF) < offer[j]:
                        break  # Offer cannot be applied.
                    nxt -= offer[j] << (j * bits) # Update mask after applying offer
                else:
                    ans = min(ans, offer[-1] + dfs(nxt))  # Recursive call
            return ans
 
        bits, mask = 4, 0
        for i, need in enumerate(needs):
            mask |= need << i * bits
        return dfs(mask)

The Java, C++, Go, and Typescript codes follow a similar structure, adapting the memoization and bit manipulation techniques to their respective languages.

The core idea is to efficiently explore the search space using bit manipulation and avoid redundant calculations using memoization. The algorithm cleverly handles the state transitions and constraints of the problem.