> ## Documentation Index
> Fetch the complete documentation index at: https://leetcode-py.wisl.dev/llms.txt
> Use this file to discover all available pages before exploring further.

> ## Agent Instructions
> leetcode-py is a Python LeetCode practice environment generator with one CLI: lcpy. It is not a service or platform.
> Each problem is a directory under leetcode/ with README.md, solution.py, test_solution.py, helpers.py, and playground.ipynb. lcpy gen creates them from JSON templates bundled with the package.
> Examples are backed by tests; copy them verbatim.

# Shopping Offers Python Solution with Tests

> Tested Python solution for LeetCode 638 with 22 pytest cases. Generate a practice environment with lcpy.

LeetCode 638, [Medium](/catalog/medium). Topics: [Array](/catalog/topics/array), [Dynamic Programming](/catalog/topics/dynamic-programming), [Backtracking](/catalog/topics/backtracking), [Bit Manipulation](/catalog/topics/bit-manipulation), [Memoization](/catalog/topics/memoization), [Bitmask](/catalog/topics/bitmask), Knapsack Problem, Complete Knapsack. [View on LeetCode](https://leetcode.com/problems/shopping-offers/description/).

Generate this problem as a practice environment: tested reference solution, 22 [parametrized pytest cases](/practice/testing), and a playground notebook:

```bash theme={"theme":{"light":"github-light","dark":"github-dark"}}
lcpy gen -n 638   # by problem number
lcpy gen -s shopping_offers   # by problem name
```

## Problem

In LeetCode Store, there are `n` items to sell. Each item has a price. However, there are some special offers, and a special offer consists of one or more different kinds of items with a sale price.

You are given an integer array `price` where `price[i]` is the price of the `i_th` item, and an integer array `needs` where `needs[i]` is the number of pieces of the `i_th` item you want to buy.

You are also given an array `special` where `special[i]` is of size `n + 1` where `special[i][j]` is the number of pieces of the `j_th` item in the `i_th` offer and `special[i][n]` (i.e., the last integer in the array) is the price of the `i_th` offer.

Return *the lowest price you have to pay for exactly certain items as given, where you could make optimal use of the special offers*. You are not allowed to buy more items than you want, even if that would lower the overall price. You could use any of the special offers as many times as you want.

### Examples

```
Input: price = [2,5], special = [[3,0,5],[1,2,10]], needs = [3,2]
Output: 14
```

**Explanation:** There are two kinds of items, A and B. Their prices are $2 and $5 respectively. In special offer 1, you can pay $5 for 3A and 0B. In special offer 2, you can pay $10 for 1A and 2B. You need to buy 3A and 2B, so you may pay $10 for 1A and 2B (special offer #2), and $4 for 2A.

```
Input: price = [2,3,4], special = [[1,1,0,4],[2,2,1,9]], needs = [1,2,1]
Output: 11
```

**Explanation:** The price of A is $2, and $3 for B, $4 for C. You may pay $4 for 1A and 1B, and $9 for 2A, 2B and 1C. You need to buy 1A, 2B and 1C, so you may pay $4 for 1A and 1B (special offer #1), and $3 for 1B, $4 for 1C. You cannot add more items, though only \$9 for 2A, 2B and 1C.

### Constraints

* `n == price.length == needs.length`
* `1 <= n <= 6`
* `0 <= price[i], needs[i] <= 10`
* `1 <= special.length <= 100`
* `special[i].length == n + 1`
* `0 <= special[i][j] <= 50`
* The input is generated that at least one of `special[i][j]` is non-zero for `0 <= j <= n - 1`.

## Solution

Reference implementation from [solution.py on GitHub](https://github.com/wislertt/leetcode-py/blob/main/leetcode/shopping_offers/solution.py), full suite in [test\_solution.py](https://github.com/wislertt/leetcode-py/blob/main/leetcode/shopping_offers/test_solution.py):

```python theme={"theme":{"light":"github-light","dark":"github-dark"}}
from functools import cache


class Solution:
    # Time: O(len(special) * product(needs[i] + 1))
    # Space: O(product(needs[i] + 1)) for the memo
    def shopping_offers(self, price: list[int], special: list[list[int]], needs: list[int]) -> int:
        offers = [
            (tuple(offer[:-1]), offer[-1])
            for offer in special
            if sum(a * b for a, b in zip(offer[:-1], price, strict=True)) > offer[-1]
        ]

        @cache
        def dfs(need: tuple[int, ...]) -> int:
            best = sum(p * c for p, c in zip(price, need, strict=True))
            for items, cost in offers:
                if all(have >= take for have, take in zip(need, items, strict=True)):
                    rest = tuple(have - take for have, take in zip(need, items, strict=True))
                    best = min(best, cost + dfs(rest))
            return best

        return dfs(tuple(needs))
```

## Complexity

| Time | Space |
| - | - |
| O(len(special) \* product(needs\[i] + 1)) | O(product(needs\[i] + 1)) for the memo |

## Tags


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