How do you solve Minimum Number of Vertices to Reach All Nodes on LeetCode?

Minimum Number of Vertices to Reach All Nodes (LeetCode #1557) can be solved using Brute Force or Optimal Solution. The optimal approach is Optimal Solution with O(n + m) time complexity and O(n) space complexity. Key insight: Nodes with no incoming edges must be included in the result as they cannot be reached from any other node.

What is the time complexity of Minimum Number of Vertices to Reach All Nodes?

The optimal solution for Minimum Number of Vertices to Reach All Nodes runs in O(n + m) time and uses O(n) space. The time complexity is linear with respect to the number of vertices (n) and edges (m) because we traverse each edge once to build the in-degree array.

What is the brute force solution for Minimum Number of Vertices to Reach All Nodes?

The brute force approach for Minimum Number of Vertices to Reach All Nodes is Brute Force, with O(n²) time complexity and O(1) space complexity. The brute-force approach involves checking each vertex to see if it can reach all other vertices. If a vertex cannot reach all nodes, we need to include it in our result set. This is straightforward but inefficient for larger graphs. The optimal Optimal Solution approach improves this to O(n + m).

What algorithm or data structure is used to solve Minimum Number of Vertices to Reach All Nodes?

Minimum Number of Vertices to Reach All Nodes uses the following concepts: Graph Theory. The recommended patterns to study are: Graph Traversal, Counting In-Degrees.

What are the interview tips for Minimum Number of Vertices to Reach All Nodes?

Always think about the properties of the graph (like acyclic nature) to simplify your approach. Be clear about the definitions of incoming and outgoing edges when discussing graph traversal. Practice explaining your thought process while coding, as communication is key in interviews.

What are common mistakes when solving Minimum Number of Vertices to Reach All Nodes?

Overlooking nodes with zero incoming edges and assuming all nodes can be reached from any starting point. Not considering the properties of directed acyclic graphs, which can simplify the problem.

#1557

Minimum Number of Vertices to Reach All Nodes

Medium

Graph TheoryGraph TraversalCounting In-Degrees

LeetCode ↗

Approaches

Brute ForceOptimal

Complexity Comparison

	Brute Force	Optimal Solution★
Time	O(n²)	O(n + m)
Space	O(1)	O(n)

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Intuition

Time O(n + m)Space O(n)

The optimal solution leverages the property of directed acyclic graphs (DAGs) that nodes with no incoming edges are essential to reach other nodes. Thus, we can simply identify nodes with no incoming edges to find the minimum set of vertices.

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Algorithm

3 steps

1Step 1: Initialize an array to count incoming edges for each vertex.
2Step 2: Traverse the edges and populate the incoming edge count for each vertex.
3Step 3: Collect all vertices with zero incoming edges into the result list.

solution.py8 lines

1# Full working Python code
2from collections import defaultdict
3
4def minVertices(n, edges):
5    in_degree = [0] * n
6    for u, v in edges:
7        in_degree[v] += 1
8    return [i for i in range(n) if in_degree[i] == 0]

ℹ

Complexity note: The time complexity is linear with respect to the number of vertices (n) and edges (m) because we traverse each edge once to build the in-degree array.

1Nodes with no incoming edges must be included in the result as they cannot be reached from any other node.
2In a directed acyclic graph, the minimum set of vertices to reach all nodes corresponds to the sources of the graph.

Solutions and explanations are original Tejav content. Problem titles © LeetCode — use the LeetCode button above for the full problem statement.