Difficulty: Hard
Tags: LeetCode, Hard, Depth-First Search, Breadth-First Search, Graph, Shortest Path, Heap (Priority Queue), leetcode-3123, O(|E| * log|V|), O(|E|), Dijkstra's Algorithm

Problem

You are given an undirected weighted graph of n nodes numbered from 0 to n - 1. The graph consists of m edges represented by a 2D array edges, where edges[i] = [a_i, b_i, w_i] indicates that there is an edge between nodes a_i and b_i with weight w_i.

Consider all the shortest paths from node 0 to node n - 1 in the graph. You need to find a boolean array answer where answer[i] is true if the edge edges[i] is part of at least one shortest path. Otherwise, answer[i] is false.

Return the array answer.

Note that the graph may not be connected.

Example 1:

Input: n = 6, edges = [[0,1,4],[0,2,1],[1,3,2],[1,4,3],[1,5,1],[2,3,1],[3,5,3],[4,5,2]]

Output: [true,true,true,false,true,true,true,false]

Explanation:

The following are all the shortest paths between nodes 0 and 5:

The path 0 -> 1 -> 5: The sum of weights is 4 + 1 = 5.
The path 0 -> 2 -> 3 -> 5: The sum of weights is 1 + 1 + 3 = 5.
The path 0 -> 2 -> 3 -> 1 -> 5: The sum of weights is 1 + 1 + 2 + 1 = 5.

Example 2:

Input: n = 4, edges = [[2,0,1],[0,1,1],[0,3,4],[3,2,2]]

Output: [true,false,false,true]

Explanation:

There is one shortest path between nodes 0 and 3, which is the path 0 -> 2 -> 3 with the sum of weights 1 + 2 = 3.

Constraints:

2 <= n <= 5 * 10⁴
m == edges.length
1 <= m <= min(5 * 10⁴, n * (n - 1) / 2)
0 <= a_i, b_i < n
a_i != b_i
1 <= w_i <= 10⁵
There are no repeated edges.