Onsite interview problem Google
Onsite Interview Problem at Google: A Deep Dive
In the realm of software engineering interviews, particularly at tech giants like Google, questions that challenge our understanding of graph theory and probability often arise. One such intriguing problem involves calculating the expected number of steps required to “erase” a city represented as a directed graph. Let’s break down this problem and explore its nuances.
Problem Overview
The City Map
You are given a city map, modeled as a directed graph with number_of_vertices intersections. Each intersection is connected by one-way roads, represented as a series of strings. Each string corresponds to a vertex, and the characters within the string indicate the direction of the edges.
The Task
The objective is to determine the expected number of steps needed to erase the city. In each step, you randomly select an intersection (vertex) and remove it along with all intersections that are reachable from it.
Input/Output Example
Input:
3 000 000 000
Output:
3.00000000000000000000
Key Observations
- Graph Characteristics: The graph is directed, and in the given input example, there are no edges connecting any vertices, making them isolated. Each intersection can be removed individually.
- Random Selection: The problem highlights the importance of randomness in choosing which vertex to remove, thus introducing a probabilistic element to the expected number of steps.
Solution Approach
Understanding the Expected Steps
To find the expected number of steps to erase the city, we can leverage the concept of Markov processes. Each state in the process corresponds to the current configuration of the graph after certain vertices have been removed.
The expected number of steps can be computed using the following recurrence relation:
-
Let
E(n)be the expected number of steps needed withnvertices remaining. The expected value is given by:[ E(n) = 1 + \frac{1}{n} \sum_{k=1}^{n} E(n - k) ]
Where:
- The term
1accounts for the current step. - The summation averages the expected steps over all possible choices of the vertex to remove.
Base Case
For the base case, when there are no vertices left (E(0) = 0), the process terminates since there are no more intersections to erase.
Time Complexity
The time complexity of this approach is O(n^2) due to the nested recurrence, where n is the number of vertices. Each expected value relies on previous values, leading to a quadratic growth in computation.
Space Complexity
The space complexity is O(n) to store the computed expected values for each state.
Clever Trick
One clever approach to optimize this problem is to recognize the structure of the graph. If the graph has separate components (disconnected subgraphs), the expected value can be calculated independently for each component and then combined. This reduces redundant calculations and can significantly enhance efficiency.
Conclusion
The problem of calculating the expected number of steps to erase a city modeled as a directed graph is a fascinating exercise in probability and graph theory. By understanding the underlying mechanics and utilizing recurrence relations, we can arrive at a solution that highlights the interplay between random processes and structured data.
As we continue to explore such complex problems, let’s keep the conversation going! Have you encountered similar challenges in your interviews or projects? What strategies did you find effective in solving them?