8. String to Integer (atoi) - Detailed Explanation
Problem Statement
You are given an undirected weighted graph with n nodes labeled from 0 to n-1 and an array of edges, where each edge is represented as [u, v, w]. Here, u and v are the two endpoints of the edge and w is its weight. The graph is guaranteed to be connected. Your task is to choose a subset of edges that connects all nodes (i.e., forms a spanning tree) such that the maximum edge weight among those chosen is as small as possible. In other words, you need to minimize the maximum edge weight used to connect all nodes.
Examples
Example 1
- Input:
- n = 4
- edges = [[0, 1, 3], [1, 2, 4], [2, 3, 5], [0, 3, 10]]
- Output: 5
- Explanation:
- One possible spanning tree is obtained by selecting the edges [0,1,3], [1,2,4], and [2,3,5].
- The maximum edge weight in this tree is 5, which is the minimum possible among all spanning trees.
Example 2
- Input:
- n = 3
- edges = [[0, 1, 2], [1, 2, 2], [0, 2, 3]]
- Output: 2
- Explanation:
- The spanning tree consisting of edges [0,1,2] and [1,2,2] connects all nodes with the maximum edge weight equal to 2.
Example 3
- Input:
- n = 5
- edges = [[0, 1, 4], [0, 2, 3], [1, 2, 1], [1, 3, 2], [2, 4, 6], [3, 4, 5]]
- Output: 5
- Explanation:
- A minimum spanning tree for this graph is given by the edges [1,2,1], [1,3,2], [0,2,3], and [3,4,5].
- The maximum edge weight among these is 5, which is the answer.
Constraints
- The graph has n nodes labeled from 0 to n-1.
- The input list of edges represents a connected graph.
- Edge weights are non-negative integers.
Hints
-
Think MST: Notice that the problem is asking for a spanning tree that minimizes the maximum edge weight. This is related to finding a Minimum Spanning Tree (MST), as an MST minimizes the overall cost and, by extension, the weight of the heaviest edge in many cases.
-
Binary Search Possibility: Alternatively, you can binary search on the maximum allowed edge weight and test connectivity using a DFS/BFS, which can also lead you to the optimal answer.
Approach 1: Minimum Spanning Tree (Kruskal’s Algorithm)
Explanation
The key observation is that among all spanning trees, the Minimum Spanning Tree (MST) minimizes the maximum edge weight. We can use Kruskal’s algorithm to build the MST:
-
Sort the Edges:
Sort all edges in ascending order based on their weights. -
Union-Find Structure:
Initialize a union-find (disjoint-set) data structure to efficiently check and connect components. Initially, each node is its own component. -
Iterate Through Edges:
Process each edge in sorted order. For an edge [u, v, w]:- If u and v belong to different components (i.e., not connected), include this edge in your spanning tree by unioning their components.
- Update the current maximum edge weight with w.
- Continue until all nodes are connected (i.e., when you have used n-1 edges).
-
Answer:
The maximum edge weight among the selected edges in the MST is the minimized maximum edge weight for the graph.
Visual Walkthrough (Example 1)
- Sorted Edges: [[0, 1, 3], [1, 2, 4], [2, 3, 5], [0, 3, 10]]
- Step 1: Add edge [0, 1, 3]. Maximum so far = 3.
- Step 2: Add edge [1, 2, 4]. Maximum becomes 4.
- Step 3: Add edge [2, 3, 5]. Maximum becomes 5. Now all nodes are connected.
- Result: 5
Code in Python
Code in Java
Approach 2: Binary Search on Maximum Edge Weight
Explanation
Another way to solve the problem is to use binary search over the range of edge weights to find the minimum threshold that still allows the graph to be connected.
-
Determine Search Range:
- Set the lower bound as the minimum edge weight and the upper bound as the maximum edge weight in the graph.
-
Binary Search Iteration:
- For a candidate maximum edge weight (mid), build a subgraph using only the edges with weight ≤ mid.
- Use DFS or BFS to check if this subgraph is connected (i.e., all nodes are reachable).
- If the subgraph is connected, try a lower threshold; otherwise, increase the threshold.
-
Answer:
- The smallest threshold for which the subgraph is connected is the answer.
Visual Walkthrough (Example 1)
- Search Range: 3 to 10
- Mid = 6:
- Consider edges with weight ≤ 6: [0,1,3], [1,2,4], [2,3,5].
- The graph is connected → try lowering the threshold.
- Adjust and Repeat:
- Continue the binary search until you determine that 5 is the minimum threshold that still connects the graph.
Complexity Analysis
- Time Complexity:
- Each binary search iteration involves checking connectivity, which takes O(n + m).
- The overall time complexity is O((n + m) × log(maxEdge)).
- Space Complexity: O(n) for the DFS/BFS.
Complexity Analysis (Approach 1)
- Time Complexity:
- Sorting the edges takes O(m log m), where m is the number of edges.
- The union-find operations are nearly O(1) each (amortized), so overall time is O(m log m).
- Space Complexity: O(n) for the union-find structure.
Step-by-Step Walkthrough (Kruskal’s Approach)
- Sort the Edges: Arrange the edges in increasing order by weight.
- Initialize Union-Find: Each node is its own component initially.
- Process Each Edge:
- For each edge in sorted order, if the endpoints belong to different components, union them and update the maximum edge weight.
- Stop once all nodes are connected (n - 1 edges have been added).
- Return the Maximum Edge Weight: This value is the minimized maximum edge weight for the spanning tree.
Common Mistakes
- Skipping the Sorting Step: Not sorting the edges correctly will result in an incorrect MST.
- Union-Find Pitfalls: Incorrect implementation of union-find (e.g., forgetting path compression) can lead to inefficient or incorrect behavior.
- Ignoring Graph Connectivity: Ensure that the graph is connected; if not, a spanning tree does not exist.
Edge Cases
-
Single Node Graph: When n = 1, there are no edges. Depending on the problem definition, the answer might be 0 or undefined.
-
Graph Already Optimally Connected: The graph might already be structured such that the maximum edge weight is as low as possible.
-
Disconnected Graph: If the graph isn’t connected (which should not occur under the given constraints), the problem must specify how to handle it.
Alternative Variations
-
Minimizing the Maximum Edge Weight on a Path:
- Instead of connecting all nodes, you might be asked to find a path between two specific nodes that minimizes the maximum edge weight along that path.
-
Multiple Queries:
- Given multiple pairs of nodes, determine the minimized maximum edge weight for a path connecting each pair. This variation may require offline processing with union-find or other advanced techniques.
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