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3532. Path Existence Queries in a Graph I

Description

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:

The resulting graph is:

  • Query [0,1]: There is no edge between Node 0 and Node 1 because |nums[0] - nums[1]| = |2 - 5| = 3, which is greater than maxDiff.
  • Query [0,2]: There is no edge between Node 0 and Node 2 because |nums[0] - nums[2]| = |2 - 6| = 4, which is greater than maxDiff.
  • Query [1,3]: There is a path between Node 1 and Node 3 through Node 2 since |nums[1] - nums[2]| = |5 - 6| = 1 and |nums[2] - nums[3]| = |6 - 8| = 2, both of which are within maxDiff.
  • Query [2,3]: There is an edge between Node 2 and Node 3 because |nums[2] - nums[3]| = |6 - 8| = 2, which is equal to maxDiff.
  • Thus, the final answer after processing all the queries is [false, false, true, true].

 

Constraints:

  • 1 <= n == nums.length <= 105
  • 0 <= nums[i] <= 105
  • nums is sorted in non-decreasing order.
  • 0 <= maxDiff <= 105
  • 1 <= queries.length <= 105
  • queries[i] == [ui, vi]
  • 0 <= ui, vi < n

Solutions

Solution 1

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