Problem Description

The task involves identifying the center of an undirected star graph composed of n nodes, which are numbered from 1 to n. A star graph has one central node and exactly n - 1 edges connecting this central node to all other nodes. In simpler terms, a star graph looks like a star, where every point or tip of the star is connected to a central or middle point.

You are given an array edges, where each element edges[i] represents an edge in the graph and consists of two integers [u_i, v_i]. These integers denote an edge connecting the nodes u_i and v_i. The goal is to figure out which node is the center of the star graph.

Intuition

To solve this problem, we can start by noticing that in a star graph, the center node will be the one shared by all the edges. Since we know that this graph is a star graph, we can conclude that the center must be one of the nodes from the first edge described in the edges array.

We can take the first edge edges[0], which is represented by [u1, v1]. This edge must be connected to the center because every edge in a star graph connects directly to the center. To find the center, we simply need to check which node from the first edge is also present in the second edge edges[1].

If the first node u1 of the first edge edges[0] is present in the second edge edges[1], then u1 is the center of the graph. If not, then the second node v1 of the first edge must be the center. Therefore, with just a simple comparison of two edges, we can determine the center of the star graph.

This approach works because of two key properties of the star graph:

1. The center node is connected to all other nodes.
2. There is only one center node in a star graph.

Given these properties, the common node between any two edges in the given star graph must be the center.

Learn more about Graph patterns.

Solution Approach

The implementation of the solution is quite straightforward and doesn't require complex data structures, patterns, or algorithms. The simplicity of the problem stems from the unique structure of the star graph where the center node is the one common node in all edges.

Here is a walk-through of the implementation based on the provided Python code:

class Solution:
    def findCenter(self, edges: List[List[int]]) -> int:
        return edges[0][0] if edges[0][0] in edges[1] else edges[0][1]

1. We define a function findCenter which takes as an input parameter edges, a list of lists, with each sublist representing an edge in the star graph.

2. The function returns either edges[0][0] or edges[0][1]. These are the two nodes in the first edge. One of these must be the center because every edge in the graph must contain the center node due to the definition of a star graph.

3. We use a simple if statement to check if the first node of the first edge edges[0][0] is also in the second edge edges[1]. If edges[0][0] is indeed in the second edge, we return it as it is the center. Otherwise, we return the second node from the first edge edges[0][1].

This implementation essentially leverages the property that the second edge must also contain the center of the star graph. In other words, we are using the fact that the center node will be the only node that repeats in the first two edges.

No additional algorithms or data structures are required, as the star graph's structure ensures that comparing the first two edges is sufficient to identify the center node.

What makes it efficient is that we only need to look at the first two edges. We don't need to check the entire edges list, so the time complexity is constant, O(1), since the number of operations does not depend on the number of edges or nodes in the graph.

Example Walkthrough

Let's use a small example to illustrate the solution approach. Suppose we have a star graph with 4 nodes which results in 3 edges and we are given the edges array representing these edges as:

edges = [[1, 2], [2, 3], [4, 2]]

Based on our solution approach, we know that one of the nodes in edges[0], which is [1, 2], must be the center of the star graph. Now we only need to determine which one it is. To do this, we look at edges[1], which is [2, 3].

Step 1: We identify the nodes in the first edge. They are:

• u1 from edges[0] is 1
• v1 from edges[0] is 2

Step 2: We examine the second edge, which contains:

• u2 from edges[1] is 2
• v2 from edges[1] is 3

Step 3: We determine the common node between the first and second edge:

• 2 is the common node because it is present in both [1, 2] and [2, 3].

Since the node 2 is common to both edges, we can deduce that 2 is the center of our star graph. Hence, the solution function will return 2.

The execution of the code using our example edges array would be as follows:

solution = Solution()
center = solution.findCenter(edges)
print(center)  # This will print 2, as node 2 is the center of the star graph

Our analysis confirms that the first two edges are sufficient to determine the center and that the function findCenter behaves as expected. It returns 2, which is the common node in both edges, and thus the center of the given star graph.

Solution Implementation

Time and Space Complexity

The time complexity of the given code is O(1). This is because the code only checks for the presence of edges[0][0] in edges[1], which is a constant-time operation since it involves two fixed index accesses and one in operation on a list with only two elements.

The space complexity of the code is also O(1). The code does not use any additional space that grows with the input size. It only uses a fixed amount of space for the variables in the function, regardless of the input size.

Learn more about how to find time and space complexity quickly using problem constraints.