Google

PART 1: ALGORITHMIC PROBLEM - GENERAL GRAPH MATCHING & TUTTE MATRIXBASE PROBLEMYou are building a matchmaking engine for a massive multiplayer game where players can team up in pairs. You have $N$ players and $M$ potential compatibility links between them. Any player can match with any compatible player (the graph is strictly non-bipartite, as there are no distinct "two halves" of the player base).Task: Design an algorithm to find the absolute maximum number of valid pairs.Standard Maximum Bipartite Matching algorithms (like Hopcroft-Karp or Ford-Fulkerson) fail completely on general graphs because odd-length cycles trap the augmenting paths. Explain the mathematical mechanics of Edmonds' Blossom Algorithm. How does it dynamically detect odd-length cycles, contract them into a single "blossom" super-node, and recursively expand them to find the maximum matching in $O(V^3)$ time?FOLLOW-UP 1The matchmaking engine scales to thousands of players, and $O(V^3)$ is too slow for the real-time lobby system. However, you only need to know the size of the maximum matching, not the exact pairs.How do you utilize algebraic graph theory to solve this? Specifically, explain how to construct the Tutte Matrix (a skew-symmetric symbolic adjacency matrix). How does the Schwartz-Zippel Lemma guarantee that evaluating the determinant of this matrix with random integers modulo a large prime accurately identifies the existence of a perfect matching? How does computing the matrix rank yield the size of the maximum matching in $O(V^\omega)$ time (where $\omega \approx 2.37$ is the matrix multiplication exponent)?FOLLOW-UP 2To hit strict microsecond latency budgets, this algebraic algorithm must be implemented in bare-metal C++. Standard Gaussian Elimination to find the matrix rank still requires heavy modulo arithmetic and nested loops, causing branching penalties and cache misses.How do you optimize the Tutte Matrix evaluation over the Galois Field of 2, $GF(2)$? Detail how mapping the matrix to an array of std::bitset allows you to entirely replace inner-loop modulo arithmetic with highly parallel bitwise XOR operations, computing the matrix rank in strictly $O(V^3 / 64)$ time while perfectly utilizing the L1 CPU cache and SIMD registers.PART 2: SYSTEM DESIGN - DISTRIBUTED GRAPH DATABASE (NEO4J / TIGERGRAPH)BASE PROBLEMYou are designing a distributed Graph Database to store and query a massive social network with 2 billion nodes (Users) and 50 billion edges (Friendships, Likes).Design the core storage engine architecture. Explain the concept of "Index-Free Adjacency." Unlike a Relational Database that relies on $O(\log N)$ B-Tree index lookups for every JOIN, how does a native graph database physically store memory pointers (or localized IDs) directly inside the node records to allow $O(1)$ edge traversal per hop, enabling complex 5-hop recursive queries to execute in milliseconds?FOLLOW-UP 1As the graph exceeds the RAM of a single bare-metal server, it must be partitioned across a 50-node cluster.If you use a standard "Edge-Cut" partitioning strategy (assigning nodes to shards via consistent hashing), traversing edges frequently requires expensive cross-network RPC calls. Furthermore, "Supernodes" (e.g., a celebrity with 100 million followers) will create catastrophic network hotspots that crash individual shards. How do you design a "Vertex-Cut" (or Bipartite) partitioning strategy? Detail how splitting the Supernode itself into smaller logical replicas across multiple machines distributes the query load and minimizes cross-shard network chatter during graph traversals.FOLLOW-UP 2The data science team needs to run global graph analytics, such as computing the PageRank of all 2 billion users simultaneously.Running a standard centralized BFS traversal over the distributed database is impossible due to network latency. How do you design an offline Graph Processing Engine using the Bulk Synchronous Parallel (BSP) model (like Apache Giraph or Google Pregel)? Explain the execution of "Supersteps": how nodes independently compute their new state, securely pass messages to their neighbors across the network boundary, and synchronize at global barriers completely lock-free until the PageRank algorithm converges.PART 3: AI / LLM DISCUSSION QUESTIONSIn hardware optimization for LLM inference, the "Roofline Model" maps theoretical peak performance against operational intensity. Why are the Prefill Phase (processing the prompt) and the Decode Phase (generating tokens) located on completely opposite slopes of the Roofline? Specifically, why is batch-size-1 decoding mathematically bottlenecked entirely by High Bandwidth Memory (HBM) speeds rather than GPU TFLOPS compute limits?What is Rank-One Model Editing (ROME)? When a foundational model outputs an outdated fact (e.g., "The Prime Minister of the UK is Boris Johnson"), how does ROME mathematically locate the specific multi-layer perceptron (MLP) weights acting as the "knowledge store," and how does it compute a localized, rank-one algebraic update to directly overwrite that single association without causing catastrophic forgetting of other facts?Explain the architectural advantages of Hybrid SSM-Transformer models (like Jamba). How does interleaving recurrent State Space Model layers (like Mamba) with standard Attention layers mathematically compress the KV-cache footprint while retaining the exact in-context recall and long-range dependency tracking that pure SSMs historically struggle with?In post-training alignment, what is Self-Play Fine-Tuning (SPIN)? How does this algorithm fundamentally differ from RLHF or DPO by forcing the LLM to play a two-player game against previous iterations of itself, effectively pushing the model's generated data distribution to match the human target distribution without ever requiring an external reward model or human preference annotations?