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Google

Given n pairs of parentheses, what is the total number of different, valid combinations of those parenthesis? For example: For n = 1, the only valid combination is (), so the answer is 1. For n = 2, the valid combinations are ()() and (()), so the answer is 2. For n = 3, the valid combinations are ((())), (()()), (())(), ()(()), and ()()(), so the answer is 5. Calculate the number of valid parenthesis combinations for a given n. Explain the formula or method used to arrive at the answer. What is the underlying mathematical concept at play here, and how does it ensure that we only count valid combinations, avoiding invalid ones such as )()(? Detail the relationship between this problem and Catalan numbers, and discuss how this relationship can be utilized to efficiently compute the number of valid combinations, even for relatively large values of n.

Google

Let's design a rate limiter. This is a crucial component in many systems to prevent abuse and ensure fair usage. Your rate limiter should meet these requirements: Functionality: It should limit the number of requests a user can make within a specific time window. For example, allow a user to make 10 requests per minute. Scalability: The rate limiter needs to handle a large number of users and requests concurrently. Imagine millions of users accessing the system. Accuracy: It should accurately track and enforce the rate limits. A small degree of error is acceptable, but significant deviations are not. Low Latency: The rate limiter must not introduce significant delays in request processing. The overhead should be minimal. Fault Tolerance: The system should continue to function correctly even if some components fail. It should be resilient to outages. Cost-Effectiveness: The solution should be cost-effective in terms of resources used (e.g., memory, CPU, network bandwidth). Consider these scenarios and constraints: Users are identified by a unique ID. The time window is configurable (e.g., seconds, minutes, hours). The rate limit is also configurable per user or group of users. You can use any data structures and algorithms you deem appropriate. Assume you have access to a distributed cache (e.g., Redis, Memcached). Walk me through your design. Discuss different approaches, their trade-offs, and how you would address the requirements. Specifically, consider the following: Data structures for storing request counts. Algorithms for incrementing and checking request counts. Handling concurrency and race conditions. Strategies for distributing the rate limiter across multiple servers. How to handle exceeding the rate limit (e.g., returning an error code). Monitoring and alerting. For example, if a user with ID user123 tries to make 11 requests within a minute when their limit is 10, the rate limiter should reject the 11th request. How would your system achieve this efficiently and reliably at scale?

Google

Describe a time you faced a tough technical problem. What made it so difficult, and how did you approach solving it? What resources did you consult, and what was the final outcome? What did you learn from this experience, and how has it shaped your problem-solving skills since then? For example, perhaps you were tasked with optimizing a slow-running query in a large database. The initial attempts to improve performance, such as adding indexes, didn't yield the desired results. You then had to dive deep into the query execution plan, identify bottlenecks, and rewrite the query using more efficient techniques, like leveraging window functions or temporary tables. You could also describe the challenges faced while integrating a new feature into a legacy system with poorly documented code. You might have had to reverse engineer parts of the system, deal with unexpected dependencies, and carefully test the integration to avoid breaking existing functionality. Elaborate on the specific tools and methodologies you employed during these challenges, such as debuggers, profilers, or version control systems. Also, explain how you collaborated with team members or sought assistance from external resources like online forums or documentation to overcome the obstacles.