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Number of Orders in the Backlog

You are given a 2D integer array orders, where each orders[i] = [pricei, amounti, orderTypei] denotes that amounti orders have been placed of type orderTypei at the price pricei. The orderTypei is:

  • 0 if it is a batch of buy orders, or
  • 1 if it is a batch of sell orders.

Note that orders[i] represents a batch of amounti independent orders with the same price and order type. All orders represented by orders[i] will be placed before all orders represented by orders[i+1] for all valid i.

There is a backlog that consists of orders that have not been executed. The backlog is initially empty. When an order is placed, the following happens:

  • If the order is a buy order, you look at the sell order with the smallest price in the backlog. If that sell order's price is smaller than or equal to the current buy order's price, they will match and be executed, and that sell order will be removed from the backlog. Else, the buy order is added to the backlog.
  • Vice versa, if the order is a sell order, you look at the buy order with the largest price in the backlog. If that buy order's price is larger than or equal to the current sell order's price, they will match and be executed, and that buy order will be removed from the backlog. Else, the sell order is added to the backlog.

Return the total amount of orders in the backlog after placing all the orders from the input. Since this number can be large, return it modulo 109 + 7.

 

Example 1:

Input: orders = [[10,5,0],[15,2,1],[25,1,1],[30,4,0]]
Output: 6
Explanation: Here is what happens with the orders:
- 5 orders of type buy with price 10 are placed. There are no sell orders, so the 5 orders are added to the backlog.
- 2 orders of type sell with price 15 are placed. There are no buy orders with prices larger than or equal to 15, so the 2 orders are added to the backlog.
- 1 order of type sell with price 25 is placed. There are no buy orders with prices larger than or equal to 25 in the backlog, so this order is added to the backlog.
- 4 orders of type buy with price 30 are placed. The first 2 orders are matched with the 2 sell orders of the least price, which is 15 and these 2 sell orders are removed from the backlog. The 3rd order is matched with the sell order of the least price, which is 25 and this sell order is removed from the backlog. Then, there are no more sell orders in the backlog, so the 4th order is added to the backlog.
Finally, the backlog has 5 buy orders with price 10, and 1 buy order with price 30. So the total number of orders in the backlog is 6.

Example 2:

Input: orders = [[7,1000000000,1],[15,3,0],[5,999999995,0],[5,1,1]]
Output: 999999984
Explanation: Here is what happens with the orders:
- 109 orders of type sell with price 7 are placed. There are no buy orders, so the 109 orders are added to the backlog.
- 3 orders of type buy with price 15 are placed. They are matched with the 3 sell orders with the least price which is 7, and those 3 sell orders are removed from the backlog.
- 999999995 orders of type buy with price 5 are placed. The least price of a sell order is 7, so the 999999995 orders are added to the backlog.
- 1 order of type sell with price 5 is placed. It is matched with the buy order of the highest price, which is 5, and that buy order is removed from the backlog.
Finally, the backlog has (1000000000-3) sell orders with price 7, and (999999995-1) buy orders with price 5. So the total number of orders = 1999999991, which is equal to 999999984 % (109 + 7).

 

Constraints:

  • 1 <= orders.length <= 105
  • orders[i].length == 3
  • 1 <= pricei, amounti <= 109
  • orderTypei is either 0 or 1.

Solution Explanation: Number of Orders in the Backlog

This problem involves managing a backlog of buy and sell orders. The key to an efficient solution lies in using priority queues (heaps) to efficiently track the best buy and sell orders.

Approach:

  1. Data Structures: We use two priority queues:

    • buy: A max-heap to store buy orders. We use a max-heap because we need to quickly access the buy order with the highest price (to match with sell orders). Elements are stored as (price, amount). The negative price ensures a max-heap behavior.
    • sell: A min-heap to store sell orders. We need quick access to the sell order with the lowest price (to match with buy orders). Elements are stored as (price, amount).

  2. Processing Orders: We iterate through the orders array. For each order:

    • Buy Order (orderType == 0):
      • We repeatedly match the buy order with the lowest priced sell orders until either the buy order is fully matched or there are no more suitable sell orders.
      • If the buy order is not fully matched, we add the remaining amount to the buy heap.
    • Sell Order (orderType == 1):
      • We repeatedly match the sell order with the highest priced buy orders until either the sell order is fully matched or there are no more suitable buy orders.
      • If the sell order is not fully matched, we add the remaining amount to the sell heap.

  3. Calculating the Result: After processing all orders, we sum the amounts of all remaining buy and sell orders in their respective heaps. This sum is the total amount of orders in the backlog. We take the modulo with 10^9 + 7 to handle potential integer overflow.

Time Complexity Analysis:

  • Each order is added to or removed from a heap at most once. Heap operations (insertion, deletion, peeking) take O(log n) time, where n is the maximum number of orders in either heap. In the worst case, n is proportional to the length of the orders array.
  • The matching process involves iterating and removing elements from the heap. The total number of elements processed in the matching phase is bounded by the total number of orders.
  • Therefore, the overall time complexity is O(m log m), where m is the length of the orders array.

Space Complexity Analysis:

  • The space used is dominated by the priority queues, which can store up to m elements in the worst case (where all orders end up in the backlog).
  • Hence, the space complexity is O(m).

Code Examples:

The code examples in Python, Java, C++, and Go provided previously accurately reflect this approach and complexity analysis. They efficiently utilize priority queues to manage the buy and sell order backlogs and accurately calculate the final result while handling potential integer overflow.