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PART 1: ALGORITHMIC PROBLEM - ADVANCED INCREASING SUBSEQUENCES (DP & SEGMENT TREES) BASE PROBLEM You are analyzing a sequence of stock prices to find the longest continuous trend. Task: Given an integer array `prices`, return the length of the longest strictly increasing subsequence. While a standard Dynamic Programming approach solves this in O(N^2), it is too slow for N = 10^5. How do you implement a Patience Sorting approach (using an auxiliary array and Binary Search) to find the length of the longest increasing subsequence (LIS) in strictly O(N log N) time? FOLLOW-UP 1 The problem expands to two dimensions. You are processing a set of geographical bounding boxes (width and height). You can nest one box inside another if and only if both its width and height are strictly greater than those of the other box. Task: Find the maximum number of boxes you can nest (the "Russian Doll Envelopes" problem). How do you sort the 2D array (e.g., width ascending, height descending) so that the 2D nesting problem collapses perfectly into your 1D LIS Binary Search algorithm? FOLLOW-UP 2 The rules change for the 1D stock prices. You still need to find the longest increasing subsequence, but with a strict volatility constraint: the absolute difference in value between any two adjacent elements in your subsequence cannot exceed a given limit `D`. The Binary Search / Patience Sorting approach breaks down here because the greedy choice of the smallest tail is no longer always optimal under the `D` constraint. Assuming the stock prices are bounded between `0` and `10^5`, how do you use a Segment Tree (where the nodes represent the maximum LIS length ending at a specific price) to solve this in O(N log M) time, where M is the maximum price? ------------------------------------------------ PART 2: SYSTEM DESIGN - DISTRIBUTED OBJECT STORAGE (AMAZON S3) BASE PROBLEM You are designing a globally scalable, highly durable distributed object storage system. Users can upload and download files ranging from small 1 KB text files to massive 5 TB video datasets. Design the overall architecture, detailing how massive files are chunked, uploaded via multi-part upload, and stored across a fleet of storage nodes. FOLLOW-UP 1 Storing the object metadata (object key, size, permissions, and the mapping of chunks to physical storage nodes) in a monolithic relational database will immediately become a bottleneck. How do you design a distributed, highly-available metadata storage layer? If you shard this NoSQL datastore by the `bucket_name`, a single viral bucket will create a massive hot partition. How do you design your partition key and namespace schema to prevent metadata hotspots? FOLLOW-UP 2 Enterprise customers require 99.999999999% (11 nines) of data durability. Standard 3-way replication provides this but incurs a massive 200% storage overhead, which is cost-prohibitive at the exabyte scale. How do you redesign your storage architecture to use Erasure Coding (e.g., Reed-Solomon algorithms)? Explain how splitting a data block into `K` data shards and `M` parity shards allows the system to survive the simultaneous loss of `M` hard drives while drastically reducing the storage footprint compared to full replication. ------------------------------------------------ PART 3: AI / LLM DISCUSSION QUESTIONS In the context of advanced LLM reasoning capabilities, what is "Tree of Thoughts" (ToT) prompting, and how does it improve upon standard Chain-of-Thought (CoT) by utilizing graph traversal algorithms like BFS or DFS? What is the fundamental difference between standard autoregressive decoding and "Classifier-Free Guidance" (CFG) in generative AI models, and how does CFG manipulate the output logits to force stronger alignment with the user's prompt? When performing parameter-efficient fine-tuning on a Large Language Model using LoRA (Low-Rank Adaptation), why does increasing the rank parameter (R) not necessarily yield a linear improvement in model performance, and what are the specific computational and memory trade-offs? What is "KV Cache Offloading" in high-performance inference engines (like DeepSpeed or FlexGen), and how does it orchestrate data movement across the PCIe bus to allow the inference of massive context windows on hardware with heavily constrained GPU VRAM?