This problem requires efficiently checking if paths exist between nodes with constraints on edge distances. A good approach involves two-pointer scanning with invariant tracking, combined with union-find to manage connectivity. The challenge lies in minimizing redundant computations, especially since queries are pre-given. Using sorting and graph traversal optimizations ensures a more efficient solution.
Problem Statement
You are given an undirected graph with n nodes and an edge list where each edge is represented as [u, v, dis] indicating an edge between nodes u and v with distance dis. Note that there may be multiple edges between the same nodes.
Your task is to determine, for each query [p, q, limit], if there exists a path between nodes p and q such that every edge on the path has a distance strictly less than limit. Return an array where each element corresponds to the result of the respective query, either true or false.
Examples
Example 1
Input: n = 3, edgeList = [[0,1,2],[1,2,4],[2,0,8],[1,0,16]], queries = [[0,1,2],[0,2,5]]
Output: [false,true]
The above figure shows the given graph. Note that there are two overlapping edges between 0 and 1 with distances 2 and 16. For the first query, between 0 and 1 there is no path where each distance is less than 2, thus we return false for this query. For the second query, there is a path (0 -> 1 -> 2) of two edges with distances less than 5, thus we return true for this query.
Example 2
Input: n = 5, edgeList = [[0,1,10],[1,2,5],[2,3,9],[3,4,13]], queries = [[0,4,14],[1,4,13]]
Output: [true,false]
The above figure shows the given graph.
Constraints
- 2 <= n <= 105
- 1 <= edgeList.length, queries.length <= 105
- edgeList[i].length == 3
- queries[j].length == 3
- 0 <= ui, vi, pj, qj <= n - 1
- ui != vi
- pj != qj
- 1 <= disi, limitj <= 109
- There may be multiple edges between two nodes.
Solution Approach
Sorting Edges and Queries
Begin by sorting both the edges by their distance and the queries by the path length limit. This allows processing edges progressively, using union-find to determine connectivity as the edges' distances increase.
Union-Find for Connectivity
Union-find is used to track which nodes are connected as we process edges. For each query, only consider edges with distances less than the specified limit. This minimizes unnecessary checks.
Two-Pointer Scanning
Use a two-pointer approach to scan through both sorted edge list and sorted queries simultaneously. This ensures that queries are answered as efficiently as possible, avoiding re-evaluation of the same edges for different queries.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
The time complexity is mainly determined by the sorting steps: sorting the edges takes O(E log E), and sorting the queries takes O(Q log Q). Processing the edges and answering the queries both take linear time. Thus, the overall time complexity is O((E + Q) log Q). The space complexity is dominated by the union-find structure, which is O(n).
What Interviewers Usually Probe
- Check the candidate's ability to optimize repeated computations across multiple queries.
- Look for a solid understanding of union-find and graph traversal algorithms.
- Assess how efficiently the candidate integrates sorting and two-pointer techniques into their solution.
Common Pitfalls or Variants
Common pitfalls
- Failing to optimize for multiple queries by not sorting or reordering them properly.
- Mismanaging the union-find data structure, leading to unnecessary recomputation of connected components.
- Overcomplicating the solution by attempting brute force for each query without leveraging sorted edges.
Follow-up variants
- Implementing the solution with a depth-first search (DFS) approach instead of union-find.
- Handling cases where the edge distances are extremely large, and the graph is sparse.
- Optimizing for very large graphs with a focus on minimizing space complexity.
How GhostInterview Helps
- GhostInterview helps by suggesting efficient sorting techniques and ensuring the solution avoids unnecessary recomputation, improving performance for large datasets.
- It provides tips on using union-find effectively, preventing costly rechecks of already processed edges.
- GhostInterview guides the candidate through problem-specific optimization strategies, offering clear hints for leveraging two-pointer scanning and edge sorting.
Topic Pages
Related GhostInterview Pages
- LeetCode Interview Copilot - Use GhostInterview as a live solver when you want direct help with LeetCode-style coding questions.
- Coding Interview Assistant - See how GhostInterview supports array, string, linked list, graph, and tree interview workflows.
- How GhostInterview Works - Review the screenshot, reasoning, and answer flow before using the solver in a live interview.
FAQ
How does two-pointer scanning help with the Checking Existence of Edge Length Limited Paths problem?
Two-pointer scanning allows us to process sorted edges and queries in tandem, efficiently answering multiple queries while avoiding redundant checks for edges already evaluated.
What is the role of union-find in this problem?
Union-find is used to track connectivity between nodes as edges are processed, ensuring we only check edges with distances that meet the query constraints.
What makes the Checking Existence of Edge Length Limited Paths problem hard?
The challenge lies in optimizing for multiple queries and efficiently managing the graph traversal without redundant computations, especially given large input sizes.
What does it mean that the problem involves 'multiple edges between nodes'?
This means that there could be more than one edge between any pair of nodes with different distances. Handling this requires careful edge selection and traversal during query processing.
How can GhostInterview help me optimize the solution for large datasets?
GhostInterview suggests techniques like sorting edges and queries and using efficient data structures such as union-find to ensure the solution remains scalable for large datasets.
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