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Leetcode 638. Shopping Offers

Problem Statement

A store offers a number of special deals, and they would like to offer customers optimal conditions. The store holds several products, and each product has a fixed price. A customer can buy products in standard quantities at the regular price or via special offers. Each special offer bundles certain quantities of products at a potentially lower price.

You are given:

  1. An integer list price where price[i] is the price of the i-th product.
  2. An integer list needs where needs[i] is the number of units of the i-th product that you need to buy.
  3. A list of special offers, where each special[i] is a list with the last element being the price of the special offer and the rest of the elements represent the quantity of each product included in the offer.

You need to return the minimum cost to satisfy the shopping list needs.

Clarifying Questions

  1. Can the quantity of products required in needs list be zero?
    • Yes, if no quantity of that product is needed.
  2. Are there any constraints on the maximum length of price, needs, or special lists?
    • Usually constrained by reasonable problem settings on Leetcode, often <= 6 for products and may vary accordingly for special offers.
  3. Can the special offers be used more than once?
    • Yes, special offers can be used multiple times as long as they fulfill some part of the needs.
  4. Should we consider cases where applying an offer may exceed the required quantity in needs?
    • No, we should not use offers that exceed the required quantity.

Strategy

To solve this problem, we can use a Depth-First Search (DFS) approach with memoization to reduce the computational overlap. Here are the primary steps:

  1. Recursive Function: Use a recursive function to calculate the minimum cost based on current needs.
  2. Base Case: If all needs are zero, the cost is zero.
  3. Memoization: Store results of subproblems to avoid recalculations.
  4. Offer Check: For each offer, check if the offer can be applied without exceeding the needs.
  5. Recursive Calculation: Calculate costs recursively by applying the special offers and decrementing the needs accordingly.

Code

import java.util.HashMap;
import java.util.List;
import java.util.Map;

public class Solution {

    // Memoization map
    private Map<List<Integer>, Integer> memo = new HashMap<>();

    public int shoppingOffers(List<Integer> price, List<List<Integer>> special, List<Integer> needs) {
        return dfs(price, special, needs);
    }

    private int dfs(List<Integer> price, List<List<Integer>> special, List<Integer> needs) {
        if (memo.containsKey(needs)) {
            return memo.get(needs);
        }

        // Calculate the direct cost without any offers
        int minCost = 0;
        for (int i = 0; i < needs.size(); i++) {
            minCost += needs.get(i) * price.get(i);
        }

        // Try every special offer
        for (List<Integer> offer : special) {
            List<Integer> cloneNeeds = needs.clone();
            boolean valid = true;

            for (int i = 0; i < needs.size(); i++) {
                int diff = cloneNeeds.get(i) - offer.get(i);
                if (diff < 0) {
                    valid = false;
                    break;
                }
                cloneNeeds.set(i, diff);
            }

            if (valid) {
                int offerCost = offer.get(offer.size() - 1) + dfs(price, special, cloneNeeds);
                minCost = Math.min(minCost, offerCost);
            }
        }

        memo.put(needs, minCost);
        return minCost;
    }
}

Time Complexity

The time complexity is challenging to pinpoint exactly due to the recursion and memoization. In the worst-case scenario, each need for each product can range from 0 to some maximum value k. The theoretical upper bound could be exponential in terms of the number of products (O(k^n)) but is significantly mitigated by memoization, which avoids solving the same subproblem multiple times.

With memoization, the practical performance is much better and depends on the number of unique states the needs list can represent and memoize.

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