Skip to main content

PAITS Features Pricing About Login

Features Cheat Sheets Answers Math & CS Wiki Pricing

Business Blog About How it works Download

Runnable Answer Engine • Interactive CodeSandbox

Interactive Engineering Answers

Direct, zero-fluff engineering answers paired with runnable CodeSandbox environments, automated unit test assertions, and mathematical complexity bounds.

pythonData Structures & Algorithms

Intermediate • 7 min

How to Implement an O(1) LRU Cache Using a Doubly Linked List and Hash Map

An O(1) Least Recently Used (LRU) cache pairs a hash map (mapping keys to node pointers for O(1) lookup) with a doubly linked list bounded by dummy head and tail sentinels (enabling O(1) node splice and insertion at the MRU front). On every get(key) or put(key, val), the accessed node is unlinked and moved immediately after head; when capacity is exceeded, head's opposite sentinel (tail.prev) is evicted in O(1).

Time: O(1) Space: O(C) 3 Unit Tests

LRU CacheHash MapDoubly Linked List

Open CodeSandbox

typescriptFrontend & Concurrency

Intermediate • 6 min

Debouncing vs. Throttling in TypeScript & JavaScript: Implementation, Leading/Trailing Edge & Rate Control

Debouncing groups rapid bursts of events into a single execution after a quiet period of W milliseconds has elapsed since the last invocation (ideal for search-as-you-type). Throttling guarantees at most one execution every W milliseconds during continuous firing (ideal for scroll, resize, and mousemove handlers).

Time: O(1) Space: O(1) 2 Unit Tests

TypeScriptJavaScriptDebounce

Open CodeSandbox

pythonAI & Machine Learning

Advanced • 9 min

How to Implement Scaled Dot-Product & Causal Multi-Head Attention in PyTorch

Scaled Dot-Product Attention computes pairwise token affinity via Q @ K.transpose(-2, -1) / sqrt(d_k), applies an upper-triangular -inf causal mask so token i cannot attend to future tokens j > i, normalizes rows with softmax, and computes a convex combination of Value vectors V.

Time: O(N^2 * d_model) Space: O(H * N^2) 2 Unit Tests

TransformersPyTorchSelf-Attention

Open CodeSandbox

pythonSystem Design

Advanced • 8 min

How Consistent Hashing with Virtual Nodes Minimizes Cache Key Remapping

Consistent hashing maps both storage nodes and keys onto a circular integer ring [0, 2^32 - 1]. A key routes to the first node whose hash position is >= the key's hash (wrapping around to index 0). Adding or removing a node among N nodes remaps only K/N keys on average instead of ~100% under hash(key) % N.

Time: O(log(N * V)) Space: O(N * V) 2 Unit Tests

Consistent HashingDistributed SystemsSystem Design

Open CodeSandbox

pythonData Structures & Algorithms

Intermediate • 8 min

Dijkstra's Algorithm vs. A* Search: Priority Queue Implementation & Admissible Heuristics

Dijkstra's algorithm expands nodes ordered by path cost from the source g(u), guaranteeing optimal shortest paths on any graph with non-negative edge weights in O((V + E) log V). A* search orders the priority queue by f(u) = g(u) + h(u), using an admissible heuristic h(u) <= h*(u) to direct search toward the target while preserving optimality.

Time: O((V + E) log V) Space: O(V + E) 1 Unit Tests

GraphsDijkstraA* Search

Open CodeSandbox

pythonSystem Design

Intermediate • 7 min

How to Build an Atomic Token-Bucket Rate Limiter in Python & Redis

A Token Bucket rate limiter stores two numbers per client: current tokens and last_refill_ts. On each request at timestamp now, it lazily computes replenished tokens (now - last_refill_ts) * refill_rate capped at burst capacity B, and consumes 1 token if tokens >= 1—requiring zero background cron threads.

Time: O(1) Space: O(1) 1 Unit Tests

Rate LimitingSystem DesignRedis

Open CodeSandbox

pythonData Structures & Algorithms

Intermediate • 7 min

How to Implement Topological Sort & Cycle Detection in a DAG (Kahn's Algorithm)

Kahn's algorithm computes the in-degree of every vertex, enqueues all vertices with in-degree 0, and repeatedly pops a vertex u, appends u to the topological order, and decrements the in-degree of its outgoing neighbors v—enqueuing v when in_degree[v] reaches 0. If the resulting order contains fewer than V vertices, the directed graph contains a cycle.

Time: O(V + E) Space: O(V + E) 1 Unit Tests

GraphsTopological SortDAG

Open CodeSandbox

pythonAI & Machine Learning

Intermediate • 6 min

How to Compute Batch Cosine Similarity & Top-K Vector Retrieval for RAG

Cosine similarity measures the cosine of the angle between two vectors: (u . v) / (||u||_2 ||v||_2). In production RAG systems, pre-normalizing all document embeddings to unit L2 norm (||d_i||_2 = 1) at ingestion time reduces cosine similarity to a pure inner product (q_hat @ D_hat.T) with zero division overhead.

Time: O(N * D + N log K) Space: O(N) 1 Unit Tests

Vector SearchEmbeddingsRAG

Open CodeSandbox

© 2026 PAITS. By using our AI-powered learning platform, you agree to our Terms and have read our Privacy Policy · Cheat Sheets · Runnable Answers · Math & CS Wiki · llms.txt.