Problem Description

The problem presents a scenario in which we're managing the orders in a restaurant. We have an array called orders, where each entry is a list with three elements: the name of the customer, the table number they're sitting at, and the food item they've ordered. Our task is to organize this data into a "display table".

The "display table" is effectively a two-dimensional array where each row represents a table in the restaurant, and each column represents a different food item. The top-left cell is labeled "Table", and the first row lists all the unique food items in alphabetical order, excluding this top-left cell. The first column lists the table numbers in ascending order. The rest of the cells in the table display the count of each food item ordered at each table.

The final output should not include customer names, and should be sorted by table number first, then by food item names alphabetically.

Intuition

To solve this problem, we must aggregate the orders for each table, and then count how many times each food item appears for each table.

Here's the step-by-step approach to do this:

1. Identify Unique Elements: We need to collect all unique table numbers and food items. The table numbers will form the rows of our display table, and the food items will form the columns.

2. Count Ordered Items: For each order, we count the number of times a food item is ordered per table. This is best done using a data structure like a dictionary or, in the case of Python, a Counter which is efficient for this kind of tallying.

3. Initialize the Display Table: Create the display table's header by adding "Table" followed by the sorted list of unique food items. This becomes the first row of our final result.

4. Populate the Display Table: Iterate through the sorted list of table numbers. For each table, create a new row starting with the table number. Then for each food item, add the count for that item at the current table. If the item hasn't been ordered at the table, the count is 0.

5. Sort the Data: Ensure that the table numbers and food items are sorted before creating the display table. Table numbers should be sorted numerically and food items alphabetically.

This approach guarantees that the final data representation matches the required format: a table sorted by table number with each type of food item ordered listed in alphabetical order, showing the quantity ordered at each table.

Learn more about Sorting patterns.

Solution Approach

The solution utilizes Python's standard library to effectively manage and compute the required output. The process includes:

1. Data Storage: Two Python set data structures are used to store unique table numbers (tables) and unique food items (foods) encountered in the list of orders. As sets, they will ignore any duplicate entries automatically.

2. Counting with Counter: The Counter from the collections module in Python is a subclass of the dictionary specifically designed for counting hashable objects. It is used here to tally the occurrences of food item per table number. The key is designed as a string with the format 'table.food', concatenating the table number and food item with a delimiter.

3. Sorting and Conversion to List: Once we have all unique table numbers and food items, these are sorted using Python's built-in sorted function. They are also converted to lists to prepare for iterating through them to produce the final display table format.

4. Building the Display Table: The display table is initialized with its header — a list beginning with Table followed by the sorted list of food items. For every table number in the sorted tables list, a new row is created starting with the table number in string format. Subsequently, for each food item in the sorted foods list, the number of times the food item was ordered at the table is retrieved from the Counter and converted to a string before being appended to the current row.

5. Constructing the Final Output: The final result (res) is a list of lists that mimics the tabular form. It starts with the header row and is followed by one row for each table displaying the counts for each food item.

The pattern followed here is straightforward:

• Use sets to derive uniqueness.
• Use Counter for efficient counting of items.
• Convert and sort the sets to list data structures for ordered access.
• Build the desired output in the format of a two-dimensional list, iterating through all table numbers and food items and using the tallies from the Counter.

This algorithm is effective as it breaks the problem down into simpler steps, each clearly executed with appropriate Python data structures and library functions. The use of the Counter particularly simplifies the problem of tracking and counting the number of food items per table number. The resulting implementation is clean, easy to understand, and performs the required task efficiently.

Example Walkthrough

Let's consider a small set of orders given to us in the following format: [["David", "3", "Cesar Salad"], ["Alice", "3", "Chicken Sandwich"], ["Alice", "3", "Cesar Salad"], ["David", "2", "Pasta"]]. Here's how the solution approach will handle this data:

1. Identify Unique Elements:

   • Unique table numbers: {"3", "2"}
   • Unique food items: {"Cesar Salad", "Chicken Sandwich", "Pasta"}

2. Count Ordered Items:

   • We count each food item ordered at each table. For example, "Cesar Salad" is ordered twice at table "3". We use a Counter to track this, resulting in a dictionary that may look something like {"3.Cesar Salad": 2, "3.Chicken Sandwich": 1, "2.Pasta": 1}.

3. Initialize the Display Table:

   • The header row is created as ["Table", "Cesar Salad", "Chicken Sandwich", "Pasta"], based on the sorted list of unique food items.

4. Populate the Display Table:

   • Create a row for table "2": ["2", "0", "0", "1"] which means table "2" didn't order "Cesar Salad" or "Chicken Sandwich", but ordered "Pasta" once.
   • Create a row for table "3": ["3", "2", "1", "0"] indicating the counts of "Cesar Salad", "Chicken Sandwich", and "Pasta" respectively.

5. Constructing the Final Output:

   • The resulting display table is:

     [ ["Table", "Cesar Salad", "Chicken Sandwich", "Pasta"], 
       ["2", "0", "0", "1"], 
       ["3", "2", "1", "0"] ]

   • This table is in the correct format, showing the counts of food items ordered at each table.

In summary, the example provided highlights the efficiency of the approach in organizing and presenting the data for a restaurant's order management system. Using Python's Counter, set, and sorting capabilities, we've transformed the list of orders into a display table that's easy to read and sorted accordingly.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the code can be broken down into several parts:

1. Iterating through the list of orders: This takes O(N) time, where N is the total number of orders.
2. Adding items to and creating the foods and tables sets: Insertions take O(1) on average, so for N orders, the complexity is O(N).
3. The Counter updates (mp[f'{table}.{food}'] += 1) also occur N times, and they take O(1) time each, thus O(N) in total.
4. Sorting the foods list takes O(F log F) time, where F is the number of unique foods.
5. Sorting the tables list takes O(T log T) time, where T is the number of unique tables.
6. Building the res list involves a double loop which iterates T times outside and F times inside, leading to O(T * F).

Adding these up, the total time complexity is O(N) + O(N) + O(N) + O(F log F) + O(T log T) + O(T * F), which simplifies to O(N + F log F + T log T + T * F).

Space Complexity

The space complexity can also be dissected into:

1. The tables and foods sets, which take O(T + F) space.
2. The mp counter, which will store at most N key-value pairs, hence O(N) space.
3. The res list, which contains a T+1 by F matrix, thus taking O(T * F) space.

Combining these, the overall space complexity is O(T + F + N + T * F). Since N can be at most T * F if every table orders every type of food once, the space complexity simplifies to O(T * F).

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