LeetCode Problem 1300 Critical Connections In A Network Missing Test Case Analysis And Solution

July 6, 2025 by StackCamp Team 96 views

Missing Test Case - 1300. Critical Connections in a Network

Introduction

This article addresses a missing test case issue identified in the LeetCode problem 1300, titled "Critical Connections in a Network." This problem falls under the category of graph theory and requires identifying the critical connections within a network represented as a graph. A critical connection is defined as an edge that, if removed, would disconnect the graph. The absence of certain test cases can lead to incorrect or inefficient code being accepted, which this article aims to highlight and rectify. We will delve into the specifics of the bug, provide a detailed description, and offer a code example demonstrating the issue. Understanding and addressing such missing test cases is crucial for ensuring the robustness and accuracy of solutions in algorithmic problem-solving. By examining the problem, the reported bug, and the provided code, we can gain a deeper insight into graph algorithms and the importance of comprehensive testing. Through this discussion, we aim to improve the quality of problem-solving on platforms like LeetCode and contribute to a more reliable coding experience. The goal is to ensure that solutions accepted are genuinely correct and efficient, thus fostering a better understanding of algorithmic concepts and their practical applications.

Problem Overview: Critical Connections in a Network

The problem, "Critical Connections in a Network," presents a scenario where we are given a network of servers (nodes) and their connections (edges). The task is to identify the critical connections, which are the connections that, if removed, would disconnect the network. In graph theory terms, these are also known as bridges. Understanding the core concept of graph connectivity is crucial here. A graph is said to be connected if there is a path between every pair of vertices. A critical connection, therefore, is an edge whose removal increases the number of connected components in the graph. This problem typically requires the application of graph traversal algorithms, such as Depth-First Search (DFS), and concepts like Tarjan's algorithm, which is specifically designed for finding bridges in a graph. The efficiency of a solution often depends on how effectively these algorithms are implemented and optimized. Furthermore, it's essential to consider various edge cases and scenarios to ensure the robustness of the solution. These may include graphs with no edges, graphs that are already disconnected, or graphs with multiple critical connections. A comprehensive solution should be able to handle these cases correctly and efficiently. The challenge lies in not only identifying a working solution but also in crafting one that performs optimally under different conditions. This involves careful consideration of time and space complexity, as well as the suitability of the chosen algorithm for the problem's constraints.

Bug Report Analysis: Missing Test Case

The bug report highlights a crucial issue: the absence of a specific test case that exposes a flaw in the provided code. The LeetCode username, rohitkgpt, identified this deficiency in problem 1300. The bug category is classified as a missing test case, indicating that the reported code incorrectly passes the existing test suite due to the absence of a particular scenario. The description elaborates on the issue by presenting three test cases that the code fails to handle correctly. These test cases are designed to expose situations where the code's logic breaks down, leading to an incorrect result. Specifically, the expected behavior for these test cases is an empty list [], signifying that there are no critical connections in the given networks. The provided code, written in Python/Python3, attempts to solve the problem by iteratively removing each connection and checking if the graph remains connected. This approach, while conceptually straightforward, is inefficient and prone to errors, especially in larger graphs. The missing test case likely involves a graph structure that the code's connectivity check fails to handle accurately. This could be due to various factors, such as the order in which nodes are visited during the graph traversal or the way the code handles disconnected components. The significance of this bug report lies in its demonstration of the importance of comprehensive test suites. A well-designed test suite should cover a wide range of scenarios, including edge cases and boundary conditions, to ensure that the submitted code is truly correct.

Code Examination: Python Implementation

The Python code provided in the bug report attempts to find critical connections by iteratively removing each edge and then checking if the remaining graph is still connected. Let's break down the code step by step.

class Solution:
    def criticalConnections(self, n: int, connections: List[List[int]]) -> List[List[int]]:
        # as mentioned in question "Any server can reach other servers directly or indirectly through the network." it means all nodes are connected in some way or the other so there is only 1 component in graph
        cnt,ans,leng=0,[],len(connections)
        for a in range(leng):
            edges=connections[0:a]+connections[a+1:]
            if edges==[]:
                ans.append(connections[a])
                continue
            al,va=[],[]
            for _ in range(n):
                al.append([])
                va.append(0)

            for b,c in edges:
                al[b].append(c)
                al[c].append(b)

            def dfs(node):
                va[node]=1
                for d in al[node]:
                    if va[d]==0:
                        dfs(d)

            print(edges)
            src=edges[0][0]
            dfs(src)
            
            if 0 in va:
                cnt+=1
                ans.append(connections[a])

        return ans

The code iterates through each connection in the input list. In each iteration, it removes the current connection and constructs a new graph with the remaining connections. It then performs a Depth-First Search (DFS) to check if the graph remains connected. If the DFS traversal doesn't visit all nodes (indicated by the presence of 0 in the va list, which tracks visited nodes), the removed connection is considered critical and added to the result list. The main flaw in this approach is its inefficiency and incorrectness. The time complexity is high due to the repeated graph construction and DFS traversals. More importantly, the connectivity check is flawed. It only checks if all nodes are reachable from the first node in the edges list. If the graph is disconnected and the first node happens to be in a connected component, the DFS will not explore other components, leading to a false conclusion that the removed edge is critical. This is precisely the kind of scenario that the missing test case likely targets. A more efficient and correct approach would involve using Tarjan's algorithm, which can find critical connections in linear time.

Test Cases and Expected Behavior

The bug report includes three specific test cases that the provided code fails to handle correctly. These test cases are crucial for understanding the limitations of the code and the importance of comprehensive testing. Let's examine each test case in detail:

Test Case 1

4
[[0,1],[1,2],[2,0],[1,3]]

In this test case, we have a graph with 4 nodes (0 to 3) and 4 connections. The connections form a cycle (0-1-2-0) and a separate edge (1-3). This graph is connected, and there are no critical connections because removing any single edge will not disconnect the graph. The expected behavior is an empty list [].

Test Case 2

2
[[0,1]]

This test case represents a simple graph with 2 nodes and 1 connection. Removing the only connection will disconnect the graph. Therefore, this connection is critical. However, the provided code incorrectly identifies this as non-critical due to its flawed logic. The expected behavior should be [[0, 1]].

Test Case 3

1
[]

This test case represents a graph with a single node and no connections. There are no edges to remove, and thus no critical connections. The expected behavior is an empty list []. The provided code's handling of this case is also incorrect, as it may produce an unexpected output due to the way it initializes and processes the graph data. These test cases highlight the importance of considering various graph structures and edge cases when designing test suites. A robust solution should be able to correctly identify critical connections in all these scenarios, demonstrating the need for a more sophisticated algorithm like Tarjan's algorithm.

Corrected Approach: Tarjan's Algorithm

To address the limitations of the provided code and accurately identify critical connections, a more efficient and robust algorithm is required. Tarjan's algorithm is a well-established method for finding bridges (critical connections) in a graph. It leverages Depth-First Search (DFS) and maintains two key arrays: discovery_time and low_link. The discovery_time array stores the time at which each node is first visited during the DFS traversal. The low_link array, on the other hand, stores the lowest discovery time reachable from a node through its subtree and back edges. A critical connection (u, v) is identified when low_link[v] > discovery_time[u], where 'u' is the parent of 'v' in the DFS tree. This condition indicates that there is no back edge from the subtree rooted at 'v' to any ancestor of 'u', meaning that the edge (u, v) is a bridge. Let's outline the steps involved in Tarjan's algorithm:

Initialize discovery_time and low_link arrays with -1 and a timer to 0.
Perform a DFS traversal of the graph.
During the DFS, update discovery_time and low_link for each node.
For each edge (u, v), check if low_link[v] > discovery_time[u]. If true, (u, v) is a critical connection.

The advantages of Tarjan's algorithm are its efficiency, with a time complexity of O(V + E), where V is the number of vertices and E is the number of edges, and its correctness in identifying all critical connections in a graph. This approach avoids the inefficient iterative edge removal and connectivity checks present in the original code. Furthermore, Tarjan's algorithm provides a systematic way to explore the graph and identify bridges based on the topological relationships between nodes and edges. By implementing Tarjan's algorithm, we can ensure that the solution accurately handles various graph structures and edge cases, leading to a more reliable and robust solution for the "Critical Connections in a Network" problem. This highlights the importance of selecting the appropriate algorithm for the task and understanding its underlying principles.

Conclusion

In conclusion, the analysis of the missing test case in LeetCode problem 1300, "Critical Connections in a Network," underscores the critical role of comprehensive testing in algorithmic problem-solving. The reported bug, identified by LeetCode user rohitkgpt, highlights how the absence of specific test scenarios can lead to the acceptance of incorrect or inefficient code. The provided Python code, while attempting to solve the problem, suffers from both performance and correctness issues due to its iterative edge removal and flawed connectivity check. The test cases presented in the bug report effectively expose these limitations, demonstrating the need for a more robust and efficient approach. The discussion of Tarjan's algorithm as a corrected approach emphasizes the importance of selecting the appropriate algorithm for the task. Tarjan's algorithm, with its linear time complexity and systematic bridge identification, offers a significant improvement over the original code's approach. This case study serves as a valuable lesson for both problem solvers and problem setters. For problem solvers, it highlights the importance of thoroughly testing their code with a variety of inputs, including edge cases and boundary conditions. For problem setters, it underscores the need to create comprehensive test suites that cover a wide range of scenarios to ensure the accuracy and reliability of solutions. Ultimately, addressing missing test cases and promoting the use of efficient algorithms contribute to a more robust and effective problem-solving environment on platforms like LeetCode. This, in turn, fosters a deeper understanding of algorithmic concepts and their practical applications, leading to better code and more confident programmers.