Algorithm Design and Dynamic Programming Questions

Comprehensive topic covering algorithm design with a strong emphasis on dynamic programming across beginner to advanced levels. Candidates should be able to recognize overlapping subproblems and optimal substructure, define states and derive recurrence relations, and implement correct top down memoization or bottom up tabulation. Core problem types include Fibonacci and climbing stairs for basics, coin change and basic knapsack, intermediate patterns such as longest increasing subsequence, longest common subsequence, edit distance, and matrix chain multiplication, and advanced domains including bitmask dynamic programming, dynamic programming on trees, digit dynamic programming, game theoretic dynamic programming, and multi dimensional state spaces. Evaluation includes space and time optimization techniques such as rolling arrays, state compression, reducing dimensionality, and other algorithmic optimizations including divide and conquer optimization, monotone queue optimization, and convex hull trick when applicable. Candidates are expected to refactor brute force solutions into efficient dynamic programming implementations, reason about correctness and complexity, discuss trade offs between clarity and performance, and leverage related algorithmic building blocks such as binary search, common sorting algorithms, greedy strategies, and appropriate data structures to improve solutions.

EasyTechnical

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Compare top-down memoization and bottom-up tabulation approaches for dynamic programming. Discuss production concerns relevant to SREs: recursion depth and stack overflow, memory footprint and cache eviction, determinism, reproducibility, and testability. Give concrete cases where you would prefer one approach over the other in a long-running service.

MediumTechnical

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Implement a bitmask DP in Python to solve the following coverage minimization: given N <= 20 services and coverage mask for each service (bitmask of features it covers), find the minimal set of services that covers all features. Return minimal count and one example set of services. Explain time/space complexity and how this approach maps to small-scale rollout planning in SRE.

EasyTechnical

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Implement a function in Python: def count_ways(n: int) -> int that returns the number of distinct ways to climb n stairs when you can take 1, 2, or 3 steps at a time. Assume n <= 40. Use dynamic programming (top-down memoization or bottom-up tabulation). Provide sample inputs/outputs and explain time and space complexity. Discuss production considerations if this function runs in a metrics aggregation service called frequently.

HardTechnical

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Model a cluster as a Markov chain where each server can transition between up and down states with known probabilities. Using DP over time (or matrix exponentiation), compute expected downtime over T steps and steady-state probabilities. Discuss state explosion for n servers, product-form approximations for independence, and practical methods SREs use to estimate reliability metrics.

HardTechnical

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Explain impartial combinatorial games and the Sprague-Grundy theorem. Given a game where each component is an integer health and a move reduces one component's health by a chosen positive amount (with per-component rules), describe how to compute Grundy numbers using DP/memoization for up to n <= 20 components, detect winning positions, and discuss applicability to adversarial rollback planning in SRE.

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