Advanced Algorithms and Problem Solving Questions

Comprehensive assessment of advanced algorithmic reasoning, design, and optimization for hard and composite problems. Covers advanced dynamic programming techniques including state compression and bitmask dynamic programming, combinatorial generation and backtracking, recursion and divide and conquer strategies, greedy algorithms with correctness proofs, and advanced graph algorithms such as breadth first search, depth first search, shortest path algorithms including Dijkstra and Bellman Ford, minimum spanning tree, network flow, strongly connected components, and topological sort. Also includes advanced tree and string algorithms such as suffix arrays and advanced hashing, bit manipulation and low level optimizations, algorithmic reductions and heuristics, and complexity analysis including amortized reasoning. Candidates should recognize applicable patterns, combine multiple data structures in a single solution, transform brute force approaches into optimized solutions, prove correctness and derive time and space complexity bounds, handle edge cases and invariants, and articulate trade offs and incremental optimization strategies. At senior levels expect mentoring on algorithmic choices, designing for tight constraints, and explaining engineering implications of algorithm selection.

HardSystem Design

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Design a scalable clustering pipeline for streaming high-dimensional embeddings (e.g., user-behavior vectors) that produces around 1k clusters. Use mini-batch k-means with k-means++ seeding, ANN for assignment, and drift detection. Describe heuristics for memory, latency, incremental updates, and re-seeding strategies.

MediumTechnical

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Implement the Knuth–Morris–Pratt (KMP) prefix-function (pi array) and use it to find all occurrences of pattern P in text T in O(|P| + |T|) time. Provide code in Python, explain how the prefix function avoids backtracking, and highlight off-by-one pitfalls.

MediumTechnical

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Solve the Traveling Salesman Problem (TSP) for N ≤ 15 using DP with bitmask: implement the DP[state][last] Held–Karp approach in Python or C++ to compute minimal tour length. Explain time and space complexity O(N^2 * 2^N) and discuss practical optimizations or pruning strategies.

EasyTechnical

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Explain time and space complexity of the naive k-nearest-neighbors (kNN) search (brute-force linear scan) for high-dimensional vectors. Describe kd-tree and its limitations in high dimensions; explain when you would choose approximate methods such as LSH or HNSW instead of exact kd-tree.

EasyTechnical

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Compare recursive DFS to iterative DFS (explicit stack). Provide a Python iterative DFS implementation that can handle a graph with up to 10^5 nodes without recursion limit issues. When is iterative DFS preferable in production code for ML pipelines?

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