Problem Description

In this problem, you are given an array restaurants, where each element in the array represents a restaurant and is itself an array of the form [id_i, rating_i, veganFriendly_i, price_i, distance_i]. You are required to filter these restaurants based on three criteria: whether they are vegan-friendly, their price, and their distance from the user.

The veganFriendly filter is a boolean that, when true, means you should only include restaurants where veganFriendly_i is set to 1. When veganFriendly is false, you can include any restaurant, regardless of its veganFriendly_i value.

Additionally, you are given a maximum price maxPrice and a maximum distance maxDistance. Restaurants exceeding either of these values should not be included in the final list.

Your task is to return an array of restaurant IDs that satisfy all three filter conditions. The IDs should be ordered first by the restaurant's rating in descending order, and in the case of a tie in ratings, by the restaurant's ID in descending order.

Intuition

To solve this problem, we first need to deal with the restaurant ordering. Given that we want the restaurants sorted by rating and ID as secondary criteria, we can achieve this by sorting the restaurants array. In Python, we can use the sort() method with a custom key function that sorts by rating in descending order first and then by ID in descending order as a tiebreaker.

Once we have the sorted array, we need to filter out the restaurants according to the given criteria. We accomplish this by iterating over each restaurant and checking if it satisfies the conditions of being vegan-friendly (if required), within the maximum price, and within the maximum distance. If a restaurant meets all the conditions, we add its ID to our final list of restaurant IDs.

The reason behind sorting the array first before filtering is to ensure that once the filtering is done, we don't need to sort the resulting list again, as it will already be ordered according to the required criteria.

The key aspects of this solution are understanding how to sort the array according to multiple criteria and filtering the elements of the array based on given thresholds.

Learn more about Sorting patterns.

Solution Approach

The solution approach can be detailed through the following steps, which encompass the use of sorting and filtering in Python:

1. Sorting: Firstly, the given list of restaurants is sorted in a descending order based on the rating_i, and in the case where two restaurants have the same rating_i, they're further sorted by id_i in descending order. This is done using the sort() method with a lambda function as the key parameter. The lambda function returns a tuple (-x[1], -x[0]). Here, x represents an element (which is a list) in restaurants. The minus sign (-) is used to sort in descending order since Python's sort functions sort in ascending order by default. The tuple essentially says "sort by the second element (rating) in descending order, and if those are equal, sort by the first element (ID) in descending order".

   restaurants.sort(key=lambda x: (-x[1], -x[0]))

2. Filtering: After sorting, the next step is to filter the restaurants based on the criteria:

   • If veganFriendly is 1, then only include restaurants where veganFriendly_i is also 1.
   • Include restaurants where the price_i is less than or equal to the maxPrice.
   • Include restaurants where the distance_i is less than or equal to the maxDistance.

   We use a for loop to iterate through each restaurant in the sorted restaurants list. For each restaurant (represented as a sublist), we check the three conditions mentioned above.

   for idx, _, vegan, price, dist in restaurants:
       if vegan >= veganFriendly and price <= maxPrice and dist <= maxDistance:
           ans.append(idx)

   We declare an empty list ans to store the IDs of the restaurants that meet all of the filters. For each restaurant, idx represents its ID. We only append idx to ans if all the conditions are met.

3. Return the Result: Once the iteration is complete, the ans list contains the IDs of the restaurants that have passed through the filters, sorted as required. This list is then returned as the final answer.

   return ans

The algorithms used in this implementation include sorting and simple linear iteration for filtering. The data structure used is a list. No complex patterns or data structures like trees, graphs, or dynamic programming are used; the solution is straightforward and only involves array manipulation.

Example Walkthrough

Let's walkthrough an example to illustrate the solution approach.

Suppose we have the following list of restaurants and the filters veganFriendly, maxPrice, and maxDistance given as:

• restaurants = [[1, 4, 1, 40, 10], [2, 8, 0, 50, 5], [3, 8, 1, 30, 4], [4, 10, 0, 10, 3], [5, 10, 1, 15, 1]]
• veganFriendly = 1
• maxPrice = 20
• maxDistance = 10

Initial List of Restaurants

[ID, Rating, VeganFriendly, Price, Distance]

1. [1, 4, 1, 40, 10]
2. [2, 8, 0, 50, 5]
3. [3, 8, 1, 30, 4]
4. [4, 10, 0, 10, 3]
5. [5, 10, 1, 15, 1]

Step 1: Sort Restaurants

First, we sort the list by rating, and for those with the same rating, by ID. In descending order by rating and ID:

• Sort result: [4, 10, 0, 10, 3], [5, 10, 1, 15, 1], [2, 8, 0, 50, 5], [3, 8, 1, 30, 4], [1, 4, 1, 40, 10]

Step 2: Filter Based on Vegan Friendly, Max Price, and Max Distance

Next, we filter by the vegan-friendly requirement, price, and distance. Since veganFriendly is 1, we only select the restaurants where veganFriendly_i is also 1. Then, we ensure price_i is less than or equal to maxPrice and distance_i is less than or equal to maxDistance.

• Restaurant 1: Fails (veganFriendly = 1, but Price = 40 (> 20))
• Restaurant 2: Fails (VeganFriendly = 0)
• Restaurant 3: Fails (Price = 30 (> 20))
• Restaurant 4: Fails (VeganFriendly = 0)
• Restaurant 5: Passes (Rating = 10, VeganFriendly = 1, Price = 15 (<= 20), Distance = 1 (<= 10))

Step 3: Return Result

The final list of restaurant IDs that meet all the criteria is:

• [5]

Thus, after following steps 1 and 2, our filtered and sorted list of restaurant IDs is [5], which is the answer to our problem. This is the array we would return.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the provided solution primarily depends on the sorting operation and the subsequent filtering of the restaurants list based on the specified conditions.

• Sorting: The sort() function is used, which generally has a time complexity of O(n log n), where n is the number of restaurants.
• Filtering: The for loop iterates over each restaurant, performing constant-time checks (if conditions) for each. This gives us O(n).

The overall time complexity combines the above operations, where sorting dominates, resulting in O(n log n) + O(n). Since the O(n log n) term is the dominant factor, the overall time complexity simplifies to O(n log n).

Space Complexity

The space complexity of the solution involves the storage required for the sorted list and the answer list.

• Sorted List: The in-place sort() method is used, so it does not require additional space apart from the input list. Hence, the space required remains O(1) as an auxiliary space.
• Answer List: In the worst case, all restaurants might satisfy the given conditions, so the ans list could contain all restaurant IDs. Therefore, the space complexity for the ans list is O(n).

Taking both into consideration, the overall space complexity of the solution is O(n), where n is the number of restaurants.

Learn more about how to find time and space complexity quickly using problem constraints.