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3532. Path Existence Queries in a Graph I

You are given an integer n representing the number of nodes in a graph, labeled from 0 to n - 1.

You are also given an integer array nums of length n sorted in non-decreasing order, and an integer maxDiff.

An undirected edge exists between nodes i and j if the absolute difference between nums[i] and nums[j] is at most maxDiff (i.e., |nums[i] - nums[j]| <= maxDiff).

You are also given a 2D integer array queries. For each queries[i] = [ui, vi], determine whether there exists a path between nodes ui and vi.

Return a boolean array answer, where answer[i] is true if there exists a path between ui and vi in the ith query and false otherwise.

Example 1

Input: n = 2, nums = [1,3], maxDiff = 1, queries = [[0,0],[0,1]]
Output: [true,false]
Explanation:
- Query [0,0]: Node 0 has a trivial path to itself.
- Query [0,1]: There is no edge between Node 0 and Node 1 because |nums[0] - nums[1]| = |1 - 3| = 2, which is greater than maxDiff.
- Thus, the final answer after processing all the queries is [true, false].

Example 2

Input: n = 4, nums = [2,5,6,8], maxDiff = 2, queries = [[0,1],[0,2],[1,3],[2,3]]
Output: [false,false,true,true]
Explanation:
- Query [0,1]: No edge; |2 - 5| = 3 > maxDiff.
- Query [0,2]: No path; |2 - 6| = 4 > maxDiff.
- Query [1,3]: Path 1 -> 2 -> 3 via edges with gaps 1 and 2.
- Query [2,3]: Direct edge; |6 - 8| = 2 <= maxDiff.
- Thus, the final answer is [false, false, true, true].

Constraints

  • 1 <= n == nums.length <= 10^5
  • 0 <= nums[i] <= 10^5
  • nums is sorted in non-decreasing order.
  • 0 <= maxDiff <= 10^5
  • 1 <= queries.length <= 10^5
  • queries[i] == [ui, vi]
  • 0 <= ui, vi < n