Blog

Information Theory Basics: Entropy, Compression, and Shannon's Theorem

Shannon entropy, the source coding theorem, compression limits, Huffman coding, and why you can never compress truly random data.

2025-10-24

Randomized Algorithms and Probabilistic Analysis

Las Vegas vs Monte Carlo algorithms, expected running time, randomized quicksort, skip list analysis, and why adding randomness can make algorithms simpler and faster.

2025-10-23

Amortized Analysis: The True Cost of Operations

Understanding amortized analysis through the aggregate method, accounting method, and potential method, with dynamic array and splay tree examples showing why worst-case per-operation analysis can be misleading.

2025-10-22

Turing Machines and Computability

The universal Turing machine, the Church-Turing thesis, the Halting Problem, undecidability, and why some problems can never be solved by any computer.

2025-10-21

Context-Free Grammars and Parsing

Context-free grammars, pushdown automata, why programming languages are context-free, the Chomsky hierarchy, and how parsers turn source code into syntax trees.

2025-10-20

Finite Automata and Regular Languages

DFAs, NFAs, regular expressions under the hood, the equivalence between them, and the Pumping Lemma for proving what finite automata cannot do.

2025-10-19

Ethics and Bias in Language Models: What Training Data Teaches Machines About Us

How training data bias, toxicity, and hallucination emerge in language models, what debiasing techniques exist, and why responsible deployment requires more than better algorithms.

2025-10-19

NP-Hard and Approximation Algorithms

When exact solutions are impossible at scale, approximation algorithms provide provable guarantees. Covering NP-hardness, the traveling salesman problem, knapsack, and greedy/DP approximation strategies.

2025-10-18

Evaluation Metrics for NLP: Why Measuring Language Quality Is Hard

From BLEU and ROUGE to METEOR and perplexity, exploring the automatic metrics that score NLP systems, their blind spots, and why human evaluation remains the gold standard.

2025-10-18

NP-Complete Problems: The Hardest Problems in NP

Understanding NP-completeness, polynomial-time reductions, the Cook-Levin theorem, SAT, 3-SAT, and why proving any single NP-complete problem is in P would solve them all.

2025-10-17

Text Generation and Decoding Strategies: How ChatGPT Picks the Next Token

From greedy decoding to top-k, top-p nucleus sampling, and temperature scaling, exploring the algorithms that turn a probability distribution over tokens into coherent, creative text.

2025-10-17

P vs NP: The Million-Dollar Question in Computer Science

Understanding the P vs NP problem, decision problems, polynomial-time verification, and why resolving this question would reshape cryptography, optimization, and all of engineering.

2025-10-16

Machine Translation: From Encoder-Decoder to Why Google Translate Got Good Overnight

How sequence-to-sequence models, attention, beam search, and the BLEU score transformed machine translation from a rule-based curiosity into a system that handles 100+ languages.

2025-10-16

Recurrence Relations and the Master Theorem

How to solve recurrence relations using the Master Theorem, recursion tree method, and substitution method, with a detailed walkthrough of T(n) = 2T(n/2) + n.

2025-10-15

Attention and Self-Attention Through the Linguistics Lens

How the attention mechanism lets transformers learn which words relate to which, why positional encoding replaces word order, and what self-attention looks like from a linguist's perspective.

2025-10-15

Big-O, Big-Theta, Big-Omega: Formal Asymptotic Analysis

Formal definitions of Big-O, Big-Theta, and Big-Omega notation with limit-based proofs, common growth rates, and why Big-O dominates interviews while Big-Theta is more precise.

2025-10-14

Semantic Similarity and Sentence Embeddings: From Cosine Distance to RAG Pipelines

How BERT embeddings and sentence transformers measure meaning similarity between texts, enabling semantic search, duplicate detection, and retrieval-augmented generation.

2025-10-13

Parsing and Syntax Trees: Where Compilers and NLP Share Roots

From constituency parsing to dependency parsing and the CYK algorithm, exploring how machines discover the grammatical structure of sentences using the same ideas that power programming language compilers.

2025-10-12

Sequence Labeling: POS Tagging, NER, and the Viterbi Algorithm

How Hidden Markov Models and BIO tagging turn the problem of labeling every word in a sentence into a tractable dynamic programming problem, powering part-of-speech tagging and named entity recognition.

2025-10-11

Word Embeddings: How King Minus Man Plus Woman Equals Queen

From one-hot vectors to Word2Vec and GloVe, exploring how the distributional hypothesis lets us learn dense vector representations where arithmetic on words captures meaning.

2025-10-10

TF-IDF and Bag of Words: Why Keyword Search Still Works

How term frequency and inverse document frequency turn documents into weighted vectors, powering search engines and text classification long before deep learning arrived.

2025-10-09

N-Grams and Language Models: Predicting the Next Word with Counting

From unigrams to trigrams, exploring how the Markov assumption turns word prediction into a counting problem, and why smoothing techniques like Kneser-Ney keep probabilities from collapsing to zero.

2025-10-08

Tokenization and Text Normalization: Why GPT Does Not See Words

From whitespace splitting to Byte Pair Encoding and SentencePiece, exploring how machines break text into processable units and why subword tokenization changed NLP forever.

2025-09-28

Graph Coloring and Bipartite Checking

Understand graph coloring, 2-coloring via BFS/DFS for bipartite checking, possible bipartition problems, and an introduction to the chromatic number.

2025-09-27

Binary Search on the Answer

Apply binary search to optimization problems by searching over the answer space: Koko eating bananas, split array largest sum, and capacity to ship packages.

2025-09-26

Trie-Based String Problems

Use tries for autocomplete, word search, longest common prefix, and wildcard matching, with efficient prefix lookups and character-by-character traversal.

2025-09-25

Top-K and Heap Patterns

Solve Top-K problems efficiently using heaps: Kth largest element, merge K sorted lists, median from data stream, and min-heap vs max-heap selection strategies.

2025-09-24

Open-Ended Evolution: The Quest for Systems That Never Stop Surprising

Why most simulations plateau, how novelty search and MAP-Elites push past convergence, and the grand challenge of building systems that generate unbounded complexity.

2025-09-24

Bit Manipulation Patterns

Master essential bit manipulation techniques including XOR tricks, single number detection, counting bits, power of two checks, and bitmask subset enumeration.

2025-09-23

Artificial Life vs Artificial Intelligence: Bottom-Up Emergence vs Top-Down Optimization

Why ALife researchers think modern AI is missing something, how embodied intelligence differs from statistical learning, and where the two fields might converge.

2025-09-23

Interval Merging and Sweep Line

Master interval problems including merging overlapping intervals, meeting rooms scheduling, insert interval, and the event-based sweep line technique.

2025-09-22

Artificial Chemistry and Self-Replication: From Von Neumann to the Origin of Life

How computational models of chemistry explore self-reproducing automata, autocatalytic sets, and the fundamental question of how life bootstraps itself.

2025-09-22

Union-Find (Disjoint Set Union)

Learn the Union-Find data structure with path compression and union by rank for connected components, dynamic connectivity, and Kruskal's MST algorithm.

2025-09-21

Agent-Based Modeling: Local Rules, Global Patterns

How autonomous agents with simple local rules produce ant trails, predator-prey cycles, and Schelling's segregation, and how to build and analyze these simulations.

2025-09-21

BFS and DFS on Grids

Apply BFS and DFS to 2D grid problems including island counting, flood fill, shortest path in a maze, and 4-directional vs 8-directional traversal.

2025-09-20

L-Systems: Growing Fractal Plants with Rewriting Rules

How Lindenmayer systems use string rewriting and turtle graphics to model biological branching, fractal geometry, and procedural generation.

2025-09-20

Backtracking: Permutations, Combinations, and Pruning

Master the backtracking pattern for generating permutations, combinations, and subsets, with decision tree modeling, pruning strategies, N-Queens, and Sudoku solving.

2025-09-19

Neuroevolution: When Evolution Builds the Brain

Evolving neural network weights and topologies with NEAT, and why evolutionary strategies sometimes beat gradient descent.

2025-09-19

Monotonic Stack and Monotonic Queue

Learn the monotonic stack and monotonic queue patterns for next greater element, daily temperatures, and sliding window maximum problems.

2025-09-18

Genetic Algorithms: Evolving Solutions Through Selection, Crossover, and Mutation

How Darwinian evolution can be turned into an optimization algorithm, and when genetic algorithms outperform gradient-based methods.

2025-09-18

Prefix Sums: Range Queries in Constant Time

Master the prefix sum technique for O(1) range sum queries, subarray sum equals K with hash maps, and extending the pattern to 2D grids.

2025-09-17

Boids and Flocking: Emergent Swarm Behavior from Three Simple Rules

How Reynolds' separation, alignment, and cohesion rules produce realistic flocking, and how spatial hashing makes it scale.

2025-09-17

Fast and Slow Pointers: Floyd's Cycle Detection

Understand the fast and slow pointer technique for cycle detection in linked lists, finding the cycle start node, and solving the happy number problem.

2025-09-16

Cellular Automata Beyond Conway: Wolfram's Elementary Rules and the Edge of Chaos

From 1D binary rules to Wolfram's four classes of behavior, exploring how the simplest possible automata reveal deep truths about computation and complexity.

2025-09-16

Two Pointers: Same Direction and Opposite Direction

Learn the two-pointer technique for sorted arrays and sequences, covering opposite-direction pair sum, same-direction duplicate removal, and container with most water.

2025-09-15

Conway's Game of Life: Complexity from Simplicity

How four simple rules on a grid produce gliders, oscillators, and Turing-complete computation, launching the field of artificial life.

2025-09-15

Sliding Window: Fixed and Variable Width Patterns

Master the sliding window technique for subarray and substring problems, covering fixed-width windows, variable-width expand/shrink, max subarray sum, and longest substring without repeats.

2025-08-23

GPU Memory Optimization: Shared Memory, Coalescing, Bank Conflicts, and Occupancy

How to write GPU code that makes efficient use of the memory hierarchy through coalesced global memory access, shared memory tiling, bank conflict avoidance, and occupancy tuning.

2025-08-22

Parallel Computing Patterns: Map, Reduce, Scan, and Gather/Scatter

The fundamental parallel algorithmic patterns that form the building blocks of GPU programming, from embarrassingly parallel map operations to the surprisingly tricky parallel prefix sum.

2025-08-21

CUDA Basics: Kernels, Threads, Blocks, and Grids

How the CUDA programming model maps parallel work onto GPU hardware through a hierarchy of threads, blocks, and grids, and how kernels launch thousands of threads to execute in parallel.

2025-08-20

GPU Architecture: Streaming Multiprocessors, Warps, and the SIMT Model

How GPUs achieve massive parallelism through thousands of lightweight cores organized into streaming multiprocessors, executing warps of threads in lockstep under the SIMT execution model.

2025-07-29

P, NP, NP-Hard, and NP-Complete: The Complexity Zoo

Understanding the major complexity classes, what makes a problem polynomial or non-polynomial, the P vs NP question, NP-hardness, NP-completeness, and how they all relate.

2025-07-28

Shannon's Source Coding Theorem: The Fundamental Limit of Compression

Understanding Shannon's source coding theorem, entropy as the absolute limit of lossless compression, optimal code lengths, and Huffman coding as the practical realization.

2025-07-27

Arrow's Impossibility Theorem: Why No Perfect Voting System Exists

Understanding Arrow's impossibility theorem, the five fairness axioms for voting systems, why democracy is mathematically hard, and what this means for real-world elections.

2025-07-26

Kolmogorov Complexity: Randomness as Incompressibility

Understanding Kolmogorov complexity, the shortest program that produces a string, Berry's paradox, randomness defined as incompressibility, and connections to Gödel and Turing.

2025-07-25

The Four Color Theorem: The First Computer-Assisted Proof

Understanding the four color theorem, why it was controversial as the first major computer-assisted proof, the techniques of reducibility and discharging, and what it means for the nature of proof.

2025-07-24

Supercomputer Design: TOP500 Rankings, Interconnect Topologies, and Scientific Workloads

How the world's fastest supercomputers are designed, from interconnect topologies like fat trees and dragonfly networks to cooling systems and the scientific workloads that drive their architecture.

2025-07-24

Fixed-Point Theorems: Why Your GPS Converges and Nash Equilibria Exist

Understanding Brouwer's fixed-point theorem, Banach's contraction mapping, and Knaster-Tarski, with applications from GPS convergence to game theory equilibria.

2025-07-23

Parallel File Systems: Lustre, GPFS, and High-Speed I/O for Supercomputers

How parallel file systems like Lustre and GPFS stripe data across hundreds of storage servers to deliver the bandwidth that thousands of compute nodes demand.

2025-07-23

The P vs NP Barrier Proofs: Why the Hardest Problem Stays Hard

Understanding why proving P != NP is so difficult: the three known barriers of relativization, natural proofs, and algebrization that block every known proof technique.

2025-07-22

MPI and Distributed Computing: Message Passing Across Thousands of Nodes

How the Message Passing Interface enables parallel programs to communicate across distributed-memory clusters using point-to-point and collective operations.

2025-07-22

Proof Techniques Toolkit: The Methods Behind Every Theorem

A comprehensive guide to proof techniques: direct proof, contradiction, contrapositive, weak and strong induction, well-ordering, and constructive vs non-constructive proofs.

2025-07-21

Job Scheduling and Resource Management: SLURM, PBS, and the Art of Sharing a Supercomputer

How job schedulers like SLURM and PBS allocate compute resources across hundreds of users, balancing fairness, throughput, and utilization on shared HPC clusters.

2025-07-21

The Pigeonhole Principle and Ramsey Theory: Guaranteed Structure in Chaos

Understanding the pigeonhole principle, its surprisingly deep consequences, Ramsey theory's guarantee that complete disorder is impossible, and the party problem.

2025-07-20

Cluster Architecture: Nodes, Interconnects, and the Anatomy of a Supercomputing Cluster

Understanding how compute clusters are built from nodes, interconnects, and memory architectures, from Beowulf clusters to modern blade servers.

2025-07-20

Euler's Identity: Where Five Constants Meet in One Equation

Understanding Euler's identity e^(i*pi) + 1 = 0, the Taylor series that make it work, complex exponentials, and why five fundamental constants converge in a single equation.

2025-07-19

The Irrationality of sqrt(2): The Pythagoreans' Crisis

Understanding the classic proof by contradiction that the square root of 2 is irrational, the historical crisis it caused, and why rational numbers have gaps.

2025-07-18

Euclid's Proof of Infinite Primes: The Oldest Beautiful Proof

Understanding Euclid's elegant proof by contradiction that there are infinitely many prime numbers, why you can always find one more prime, and the enduring power of simple reasoning.

2025-07-17

The Halting Problem: Turing's Proof That Some Questions Are Unanswerable

Understanding Turing's 1936 proof by contradiction that no program can decide whether all programs halt, the self-referential paradox at its core, and its connection to Gödel's incompleteness.

2025-07-16

Cantor's Diagonalization: Why Some Infinities Are Bigger Than Others

Understanding Cantor's diagonal argument, the proof that the real numbers are uncountable, different sizes of infinity, and the Continuum Hypothesis.

2025-07-15

Gödel's Incompleteness Theorems: The Limits of Formal Systems

Understanding Gödel's two incompleteness theorems, self-reference through Gödel numbering, and why no consistent formal system can prove all truths about arithmetic.

2025-06-24

Monoids and Semigroups in Programming: Why MapReduce Works

How the humble monoid, a set with an associative operation and identity, explains why MapReduce parallelizes, why folds compose, and why functional programming patterns work.

2025-06-23

Lattices and Order Theory: The Algebra of Static Analysis

How lattices provide the mathematical foundation for fixed-point computation in program analysis, type systems, and abstract interpretation.

2025-06-22

Polynomials over Finite Fields: Why Your Network Card Does Algebra

How polynomial division over GF(2) powers CRC checksums, and how irreducible polynomials enable BCH codes and error detection in every network packet.

2025-06-21

Modular Arithmetic as Algebra: The Ring Behind RSA

Viewing modular arithmetic through the lens of ring theory, from Euler's totient to the Chinese Remainder Theorem, and why RSA works.

2025-06-20

Finite Fields (Galois Fields): The Algebra Inside AES and Reed-Solomon

How GF(2^n) arithmetic works, why AES uses GF(2^8), and how Reed-Solomon codes protect your data from QR codes to deep-space probes.

2025-06-19

Rings and Fields: Two Operations, Infinite Power

How adding a second operation to a group creates rings and fields, the algebraic structures behind modular arithmetic, polynomial math, and finite field cryptography.

2025-06-18

Group Homomorphisms and Isomorphisms: Structure-Preserving Maps

Understanding maps that preserve algebraic structure, and why 'same shape' enables code reuse, type safety, and database schema transformations.

2025-06-17

Permutation Groups: Rearrangements, Cycles, and Shuffles

How the symmetric group S_n captures every possible rearrangement, and why cycle notation connects to sorting networks, shuffle algorithms, and puzzle solvers.

2025-06-16

Cyclic Groups and Generators: The Engine Behind Diffie-Hellman

How a single element can generate an entire group, and why this property is the algebraic foundation of modern key exchange.

2025-06-15

Groups: The Algebra of Symmetry

Understanding groups, their four axioms, and why this single algebraic structure underpins error detection, cryptography, and algorithm design.

2021-04-09

Topological Sort, Strongly Connected Components, and Tarjan's Algorithm

Advanced graph algorithms for directed graphs: ordering dependencies with topological sort and finding strongly connected components with Tarjan's algorithm.

2021-04-08

Minimum Spanning Trees: Kruskal's, Prim's, and Union-Find

Connecting all vertices at minimum total cost with Kruskal's and Prim's algorithms, powered by the union-find data structure.

2021-04-07

Shortest Path Algorithms: Dijkstra's and Bellman-Ford

Finding the shortest path in weighted graphs with Dijkstra's algorithm for non-negative weights and Bellman-Ford for graphs with negative edges.

2021-04-06

Greedy Algorithms: Activity Selection, Huffman Coding, and Greedy vs DP

When making the locally optimal choice at each step leads to a globally optimal solution, and when it does not.

2021-04-05

Advanced DP Patterns: Knapsack, LCS, Matrix Chain, and Bitmask DP

Four essential dynamic programming patterns that appear everywhere: 0/1 Knapsack, Longest Common Subsequence, Matrix Chain Multiplication, and Bitmask DP.

2021-04-04

Dynamic Programming Fundamentals

Understanding overlapping subproblems, optimal substructure, memoization, and tabulation: the core ideas behind dynamic programming.

2021-04-03

Binary Search and Its Variants

Mastering the classic binary search algorithm and its powerful variants: lower bound, upper bound, and searching rotated arrays.

2021-04-02

Non-Comparison Sorting: Counting Sort, Radix Sort, and Bucket Sort

Breaking the n log n barrier by exploiting structure in the data, with Counting Sort, Radix Sort, and Bucket Sort.

2021-04-01

Comparison-Based Sorting: Merge Sort, Quick Sort, and Heap Sort

A deep dive into the three most important comparison-based sorting algorithms, their mechanics, and when to reach for each one.

2021-03-10

HTTP/2, TLS, and WebSockets: Modern Application Protocols

How HTTP/2 multiplexes streams over a single connection, how TLS secures communication with encryption and authentication, and how WebSockets enable real-time bidirectional messaging.

2021-03-09

Networking Fundamentals: TCP/IP, UDP, and DNS

How data travels across networks using the TCP/IP stack, the trade-offs between TCP and UDP, how DNS resolves domain names, and the basics of socket programming.

2021-03-08

Virtual Memory: Page Tables, TLBs, and Page Faults

How operating systems use virtual memory to give each process its own address space, translate virtual addresses to physical ones, and handle memory that exceeds physical RAM.

2021-03-07

OS Concepts: Processes, Threads, and Scheduling

How operating systems manage concurrent execution through processes, threads, context switching, and scheduling algorithms that balance fairness, throughput, and responsiveness.

2021-03-06

Memory Hierarchy: From Registers to Disk

How computers organize memory into layers of increasing size and decreasing speed, and why locality of reference makes this hierarchy effective.

2021-03-05

CPU Architecture: Pipelines, Caches, and Branch Prediction

How modern processors execute billions of instructions per second using pipelining, cache hierarchies, branch prediction, and instruction-level parallelism.

2021-02-25

Vector Clocks and Conflict Resolution: Tracking Causality in Distributed Systems

How vector clocks, version vectors, and CRDTs track causality and resolve conflicts in distributed systems where there is no global clock.

2021-02-24

Distributed Transactions: Coordinating Writes Across Services

How distributed systems coordinate multi-node writes using two-phase commit, three-phase commit, and the Saga pattern with compensating transactions.

2021-02-23

Consistency Models: From Strong to Eventual

Understanding the spectrum of consistency models in distributed systems, from linearizability and sequential consistency to causal and eventual consistency, and the trade-offs each model makes.

2021-02-22

Sharding and Partitioning: Splitting Data Across Nodes

How distributed systems split data across multiple nodes using hash-based and range-based partitioning, consistent hashing, and rebalancing strategies.

2021-02-21

Replication: Keeping Copies of Data Across Nodes

How distributed systems replicate data using leader-follower, multi-leader, and leaderless architectures, and how quorum reads and writes ensure correctness.

2021-02-20

Consensus Algorithms: Getting Distributed Nodes to Agree

How Paxos and Raft solve the consensus problem in distributed systems, enabling multiple nodes to agree on a single value even when some nodes fail.

2021-02-19

CAP Theorem: The Fundamental Trade-Off in Distributed Systems

Understanding the CAP theorem and why every distributed system must choose between consistency, availability, and partition tolerance when network failures occur.

2021-02-18

CDNs and Edge Computing

How content delivery networks distribute content to edge servers worldwide, how cache hierarchies reduce origin load, and how edge functions move computation closer to users for lower latency.

2021-02-17

Rate Limiting and Circuit Breakers

How rate limiting algorithms like token bucket and sliding window protect services from overload, and how circuit breakers prevent cascading failures by failing fast when downstream services are unhealthy.

2021-02-16

Microservices and API Gateway Patterns

How microservices decompose monoliths into independently deployable services, how API gateways and service meshes manage cross-cutting concerns, and when the Backend for Frontend pattern simplifies client integration.

2021-02-15

Message Queues and Event-Driven Architecture

How message queues decouple producers from consumers, the differences between pub/sub and point-to-point messaging, and the trade-offs between at-least-once and exactly-once delivery guarantees.

2021-02-14

Databases: SQL vs NoSQL, Indexing, and Query Optimization

How relational and non-relational databases differ in data modeling and guarantees, how B-tree and hash indexes accelerate lookups, and how query optimizers turn SQL into efficient execution plans.

2021-02-13

Caching: Strategies, Eviction, and Invalidation

How caches accelerate reads with strategies like LRU and LFU, how write-through and write-back policies keep data consistent, and why cache invalidation is one of the hardest problems in computing.

2021-02-12

Load Balancers: Distributing Traffic Across Servers

How load balancers distribute incoming requests across multiple servers using strategies like round robin, least connections, and consistent hashing, and why the choice between L4 and L7 matters.

2021-02-07

Red-Black Trees: Self-Balancing BSTs with Color Invariants

How red-black trees use node coloring and rotations to guarantee O(log n) operations, powering standard library maps and sets across languages.

2021-02-06

LSM Trees: Write-Optimized Storage with Log-Structured Merge

How LSM Trees turn random writes into sequential I/O, powering the storage engines behind Cassandra, RocksDB, and LevelDB.

2021-02-05

Bloom Filters: Probabilistic Membership Testing

How Bloom filters use multiple hash functions and a bit array to answer 'is this element in the set?' with extreme space efficiency, at the cost of occasional false positives.

2021-02-04

Skip Lists: Probabilistic Balancing with Layered Linked Lists

How skip lists use randomized multi-level linked lists to achieve O(log n) search without the complexity of tree rotations.

2021-02-03

Tries: Prefix Trees for Autocomplete and String Matching

How tries store strings character by character in a tree structure, enabling prefix search, autocomplete, and fast string matching.

2021-02-02

B+ Trees: Leaf-Linked Trees for Range Queries and Database Indexing

How B+ Trees extend B-Trees by storing all data in linked leaf nodes, enabling fast range scans that power every major database index.

2021-02-01

B-Trees: Balanced Multi-Way Trees for Disk-Optimized Search

How B-Trees keep data sorted and balanced across multiple children per node, minimizing disk I/O for databases and file systems.

2021-01-25

Matrix Multiplication from Scratch

A dead-simple, step-by-step walkthrough of how matrix multiplication actually works, with every single calculation shown.

2021-01-23

The Dot Product: Multiply, Add, Done

The simplest operation in all of linear algebra, explained like you're five. Two lists of numbers go in, one number comes out.

2021-01-19

BFS and DFS: The Two Ways to Explore a Graph

A deep dive into breadth-first search and depth-first search, their properties, trade-offs, and the problems each one solves best.

2021-01-17

Recursion: Solving Problems by Solving Smaller Problems

How recursive functions work, why they need base cases, and how to think about call stacks, recurrence relations, and memoization.

2021-01-15

Graphs: Representation, BFS, and DFS

How to represent graphs with adjacency lists and matrices, and how breadth-first and depth-first search explore them systematically.

2021-01-13

Heaps and Priority Queues: Always Know the Extreme

How binary heaps maintain the min or max element at the top in O(log n) time, powering priority queues, heap sort, and scheduling systems.

2021-01-11

Trees: Hierarchical Data and Binary Search Trees

Understanding tree structures, binary trees, BSTs, and the three classic traversal orders that unlock recursive thinking.

2021-01-09

Hash Tables: Constant-Time Lookup

How hash functions, collision resolution, and dynamic resizing combine to give hash tables their O(1) average-case performance.

2021-01-07

Queues: First In, First Out

Understanding queues, the FIFO data structure behind task scheduling, breadth-first search, message passing, and buffering systems.

2021-01-05

Stacks: Last In, First Out

Understanding stacks, the LIFO data structure behind call stacks, expression evaluation, undo systems, and backtracking algorithms.

2021-01-03

Linked Lists: Pointers All the Way Down

Singly, doubly, and circular linked lists: how pointer-based data structures trade random access for efficient insertion and deletion.

2021-01-01