This problem requires finding the minimum time at which removing edges will split the graph into at least k connected components. A binary search over the time values can be used to determine the optimal time t. The solution involves applying union-find to check the number of components at each potential removal time.
Problem Statement
You are given an integer n representing the number of nodes in an undirected graph, with edges given in a 2D array. Each edge is defined by three values: ui, vi, and timei, indicating an undirected edge between nodes ui and vi that can be removed at time timei.
You are also provided with an integer k, and your goal is to find the minimum time t such that after removing all edges with time <= t, the graph will be split into at least k connected components.
Examples
Example 1
Input: n = 2, edges = [[0,1,3]], k = 2
Output: 3
Example 2
Input: n = 3, edges = [[0,1,2],[1,2,4]], k = 3
Output: 4
Example 3
Input: n = 3, edges = [[0,2,5]], k = 2
Output: 0
Constraints
- 1 <= n <= 105
- 0 <= edges.length <= 105
- edges[i] = [ui, vi, timei]
- 0 <= ui, vi < n
- ui != vi
- 1 <= timei <= 109
- 1 <= k <= n
- There are no duplicate edges.
Solution Approach
Binary Search over Time
The problem can be approached by performing a binary search over the possible edge removal times. The smallest time t that results in the graph splitting into at least k connected components is the answer.
Union-Find for Component Tracking
Union-Find (Disjoint Set Union) will be used to efficiently track connected components as edges are removed. After each binary search step, union-find helps check if the graph is split into the desired number of components.
Edge Removal Simulation
For each candidate time t found by the binary search, simulate the removal of all edges with time <= t. Union-find operations will help check if the number of connected components is >= k at that point.
Complexity Analysis
| Metric | Value |
|---|---|
| Time | Depends on the final approach |
| Space | Depends on the final approach |
The time complexity is dominated by the binary search, which runs in O(log(max_time)), and the union-find operations for each step, resulting in a total time complexity of O(log(max_time) * n). The space complexity is O(n) due to the union-find data structure.
What Interviewers Usually Probe
- The candidate must demonstrate a clear understanding of binary search principles and how they apply to this problem.
- The candidate should effectively use union-find to manage connected components during the binary search.
- A solid solution will include reasoning on time complexity and justify the approach with optimal space utilization.
Common Pitfalls or Variants
Common pitfalls
- Candidates might attempt to use brute force methods instead of binary search, leading to inefficient solutions.
- Misunderstanding the problem constraints and removing edges prematurely may cause incorrect results.
- Failing to optimize union-find operations with path compression and union by rank could lead to slower performance.
Follow-up variants
- What if the graph starts already with k connected components?
- How can this approach be modified for directed graphs?
- What happens if there are more edges with the same time value?
How GhostInterview Helps
- GhostInterview helps by guiding you through the critical steps of binary search and union-find with relevant examples and hints.
- GhostInterview offers specific feedback on optimizing union-find techniques to handle large inputs effectively.
- GhostInterview assists in analyzing the time complexity and optimizing your solution to meet problem constraints.
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
What is the binary search approach for the 'Minimum Time for K Connected Components' problem?
The binary search approach involves testing different removal times and checking if the graph splits into k connected components using union-find.
How does union-find help in solving the problem?
Union-find helps track the number of connected components as edges are removed, enabling efficient checks of graph connectivity at each binary search step.
What is the minimum time for the graph to have 3 components in this problem?
The minimum time is determined by performing binary search to find the smallest time t where removing all edges with time <= t results in at least 3 connected components.
Why does binary search work for this problem?
Binary search works by narrowing down the valid answer space for time t, testing each time to find the smallest t that creates at least k components.
What are the main challenges in solving the 'Minimum Time for K Connected Components' problem?
The main challenges are implementing an efficient binary search and optimizing union-find operations to handle large input sizes and edge cases.
Need direct help with Minimum Time for K Connected Components instead of spending more time grinding it?
Download GhostInterview when you want a LeetCode solver, not another long practice loop. Capture Minimum Time for K Connected Components from a screenshot, get the answer path and complexity, and use supported stealth workflows that stay outside captured layers.
Capture the prompt fast instead of rewriting the problem by hand.
Get the solution path, trade-offs, and complexity summary in one pass.
Stay outside captured layers on supported screen-share workflows.
Stay in the same pattern family
Minimize Maximum Component Cost leverages binary search over edge weights to optimize the cost of graph components after edge removal.
Open problem page#3600 Maximize Spanning Tree Stability with UpgradesMaximizing the stability of a spanning tree with upgrades requires careful optimization of edge strengths using binary search and graph algorithms.
Open problem page#3534 Path Existence Queries in a Graph IISolve path existence queries in a graph using binary search on the answer space, focusing on sorted nodes and maximum difference.
Open problem page