#1409

Medium

auto_awesomeArray plus Binary Indexed Tree

LeetCode Problem Workspace

Queries on a Permutation With Key

This problem involves processing queries on a permutation using an array and binary indexed tree for efficient results.

Problem Statement

You are given an array queries consisting of integers, and an integer m. Initially, the permutation P is set to [1, 2, 3, ..., m]. For each element in queries, you must process the following operation: Find the index of the element in the permutation P, move that element to the front, and return the index. Repeat this for all elements in the queries array.

Return an array of integers representing the indices at which the elements of queries were found in the permutation P before they were moved to the front.

Examples

Example 1

Input: queries = [3,1,2,1], m = 5

Output: [2,1,2,1]

The queries are processed as follow: For i=0: queries[i]=3, P=[1,2,3,4,5], position of 3 in P is 2, then we move 3 to the beginning of P resulting in P=[3,1,2,4,5]. For i=1: queries[i]=1, P=[3,1,2,4,5], position of 1 in P is 1, then we move 1 to the beginning of P resulting in P=[1,3,2,4,5]. For i=2: queries[i]=2, P=[1,3,2,4,5], position of 2 in P is 2, then we move 2 to the beginning of P resulting in P=[2,1,3,4,5]. For i=3: queries[i]=1, P=[2,1,3,4,5], position of 1 in P is 1, then we move 1 to the beginning of P resulting in P=[1,2,3,4,5]. Therefore, the array containing the result is [2,1,2,1].

Example 2

Input: queries = [4,1,2,2], m = 4

Output: [3,1,2,0]

Example details omitted.

Example 3

Input: queries = [7,5,5,8,3], m = 8

Output: [6,5,0,7,5]

Example details omitted.

Constraints

1 <= m <= 10^3
1 <= queries.length <= m
1 <= queries[i] <= m

Solution Approach

Initial Permutation Setup

Create the initial permutation P as an array of integers from 1 to m. This array represents the current state of the permutation before processing any queries.

Efficient Index Lookup with Binary Indexed Tree

Using a Binary Indexed Tree (BIT) helps efficiently track the positions of elements in the permutation. It allows us to update and query the position of an element in logarithmic time, reducing the time complexity of the problem.

Process Queries and Adjust Permutation

For each query, find the position of the queried element in the permutation using the BIT, move it to the front of the permutation, and update the BIT accordingly. Repeat this for each element in the query array.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity for each query is O(log m) due to the use of a Binary Indexed Tree for efficient position lookup and updates. Thus, for n queries, the overall time complexity is O(n log m). The space complexity is O(m) for storing the permutation and the BIT data structure.

What Interviewers Usually Probe

Check if the candidate understands how to apply a Binary Indexed Tree for array manipulation and position tracking.
Assess how well the candidate can manage large inputs efficiently through optimized data structures.
Evaluate the candidate's approach to handling queries in constant space or time complexity.

Common Pitfalls or Variants

Common pitfalls

Not using a Binary Indexed Tree or another efficient data structure could lead to a time complexity of O(n*m), which is inefficient for larger inputs.
Failing to update the Binary Indexed Tree properly after each query will lead to incorrect results.
Incorrectly handling edge cases where the queries involve the same element multiple times.

Follow-up variants

Instead of moving elements to the front, the problem could ask to move elements to the back of the array.
Queries could involve different types of operations such as adding or removing elements from the permutation dynamically.
The problem could be extended to handle range queries where multiple elements need to be processed at once.

FAQ

What is the core pattern used in solving the 'Queries on a Permutation With Key' problem?

The core pattern involves using an array and a Binary Indexed Tree to track and update the positions of elements in a permutation efficiently.

How does a Binary Indexed Tree help in this problem?

A Binary Indexed Tree allows us to track the position of elements in the permutation in logarithmic time, improving the efficiency of querying and updating positions.

What are the key challenges when implementing a solution for this problem?

The main challenges involve managing the updates to the permutation after each query and ensuring the solution runs efficiently for large inputs.

Can this problem be solved without using a Binary Indexed Tree?

While it's possible to solve this problem without a Binary Indexed Tree, it would result in inefficient time complexity. A BIT provides a significant optimization.

What is the time complexity of the solution for 'Queries on a Permutation With Key'?

The time complexity is O(n log m), where n is the number of queries and m is the size of the permutation, due to the Binary Indexed Tree operations.

terminal

Solution

Solution 1: Simulation

The problem's data scale is not large, so we can directly simulate it.

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class Solution:
    def processQueries(self, queries: List[int], m: int) -> List[int]:
        p = list(range(1, m + 1))
        ans = []
        for v in queries:
            j = p.index(v)
            ans.append(j)
            p.pop(j)
            p.insert(0, v)
        return ans

Solution 2: Binary Indexed Tree

The Binary Indexed Tree (BIT), also known as the Fenwick Tree, efficiently supports the following two operations:

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class Solution:
    def processQueries(self, queries: List[int], m: int) -> List[int]:
        p = list(range(1, m + 1))
        ans = []
        for v in queries:
            j = p.index(v)
            ans.append(j)
            p.pop(j)
            p.insert(0, v)
        return ans

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