2070. Most Beautiful Item for Each Query - Detailed Explanation

Problem Statement

You are given a 2D integer array items where each element is of the form [price, beauty], meaning that an item has a given price and a beauty value. You are also given an integer array queries. For each query, you need to determine the maximum beauty among all items whose price is less than or equal to the query value. If no item satisfies the condition for a query, the answer for that query is 0.

Return an array of integers where the (j)th element is the answer for the (j)th query in the original order.

Example 1

• Input:
  • items = [[1, 2], [3, 2], [2, 4], [5, 6]]
  • queries = [3, 1, 5]
• Explanation:
  • For query = 3:
    • Items with price ≤ 3: [1,2], [3,2], [2,4]
    • Maximum beauty among these items is 4.
  • For query = 1:
    • Only item with price ≤ 1 is [1,2] → beauty = 2.
  • For query = 5:
    • All items with price ≤ 5: [1,2], [3,2], [2,4], [5,6]
    • Maximum beauty is 6.
• Output: [4, 2, 6]

Example 2

• Input:
  • items = [[4, 5], [1, 2], [3, 7]]
  • queries = [2, 4, 1]
• Explanation:
  • For query = 2:
    • Items with price ≤ 2: [1,2] → maximum beauty = 2.
  • For query = 4:
    • Items with price ≤ 4: [4,5], [1,2], [3,7] → maximum beauty = 7.
  • For query = 1:
    • Items with price ≤ 1: [1,2] → maximum beauty = 2.
• Output: [2, 7, 2]

Constraints

• (1 \leq \text{items.length}, \, \text{queries.length} \leq 10^5)
• (1 \leq \text{price}, \text{beauty}, \text{queries}[i] \leq 10^5)
• The queries can be in any order and the items array is not necessarily sorted.

Hints

1. Naive Approach Pitfall:

   • Iterating through all items for every query results in an (O(n \times q)) solution, which is too slow given the constraints.

2. Sorting & Offline Query Processing:

   • If you sort the items by price and also process the queries in increasing order, you can “sweep” through the items only once while maintaining the best (maximum) beauty seen so far.

3. Maintaining Original Order:

   • Since the answer must be in the original query order, you can store the query’s index along with its value when sorting.

Approach 1: Sorting with Two-Pointers (Offline Query Processing)

Detailed Explanation

1. Sort the Items:
   Sort the array items by the price (first element of each subarray). This allows you to efficiently consider all items that meet a given price threshold.

2. Process Queries Offline:

   • Create a list of tuples for the queries where each tuple is (query_value, original_index).
   • Sort this list by the query value in ascending order.

3. Iterate with a Two-Pointer Technique:

   • Use a pointer to traverse through the sorted items array.
   • For each query (in sorted order), move the pointer forward until the item’s price is greater than the query value.
   • During this iteration, update a variable max_beauty to be the maximum beauty seen so far among items with price ≤ current query.
   • Assign max_beauty as the answer for the current query.

4. Output the Answers:

   • Since you stored the original index for each query, you can put each answer in its corresponding position in the result array.

Why This Works

• Efficiency:
  Sorting items and queries takes (O(n \log n)) and (O(q \log q)), respectively. Then a single pass over items for all queries gives an (O(n + q)) sweep.
• Correctness:
  Since items are processed in increasing order of price, when you reach a query with threshold (q), you have already considered all items with price ≤ (q). The maximum beauty among these items is correctly maintained.

Code Solutions

Python Code