3532 - Path Existence Queries in a Graph I.

Medium

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 <code>queriesi = u_i, v_i</code>, determine whether there exists a path between nodes <code>u_i</code> and <code>v_i</code>.

Return a boolean array answer, where answer[i] is true if there exists a path between <code>u_i</code> and <code>v_i</code> in the <code>i^th</code> 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:

• <code>1 <= n == nums.length <= 10⁵</code>

• <code>0 <= numsi<= 10⁵</code>

• nums is sorted in non-decreasing order.

• <code>0 <= maxDiff <= 10⁵</code>

• <code>1 <= queries.length <= 10⁵</code>

• <code>queriesi == u_i, v_i</code>

• <code>0 <= u_i, v_i< n</code>