Theory

• Fair Division: Cut-and-Choose, Selfridge-Conway, Brams-Taylor, and the Mathematics of Envy-Freeness · 2020-01-15

  A rigorous exploration of fair division—from cake-cutting protocols to envy-free allocations—and their application to cloud resource allocation and beyond.

• Mechanism Design: VCG Auctions, the Revelation Principle, and the Architecture of Truthfulness · 2019-12-23

  A deep exploration of mechanism design—the VCG mechanism, Myerson optimal auction, incentive compatibility, and how to design games where truth-telling is a dominant strategy.

• Algorithmic Game Theory: Nash Equilibrium Computation, PPAD-Completeness, and the Computational Lens on Strategy · 2019-11-23

  A rigorous look at algorithmic game theory—computing Nash equilibria, the PPAD complexity class, and how computational constraints reshape strategic reasoning.

• Smoothed Analysis: Why Simplex Works in Practice and the Spielman-Teng Framework · 2019-11-19

  An exploration of smoothed analysis—Spielman and Teng's framework that explains why the simplex method and other algorithms transcend their worst-case bounds.

• Average-Case Complexity: Levin's Distributional Problems, AvgP, and Cryptographic Implications · 2019-10-20

  A deep examination of average-case complexity—Levin's theory of distributional NP-completeness, the class AvgP, and why cryptography needs hard-on-average problems.

• Descriptive Complexity: Fagin's Theorem, Logic, and an Alternative to Turing Machines · 2019-10-12

  An exploration of descriptive complexity—where computational classes are characterized by logical definability—and Fagin's theorem that NP equals existential second-order logic.

• Circuit Complexity: AC0, NC, P/poly, and the PARITY ∉ AC0 Proof · 2019-09-29

  A rigorous journey through circuit complexity classes—AC0, NC, P/poly—and the landmark result that PARITY cannot be computed by constant-depth polynomial-size circuits.

• Communication Complexity: Yao's Two-Party Model, the Rectangle Method, and Lower Bounds Galore · 2019-08-18

  A deep investigation of communication complexity—the mathematics of information exchange between parties—and its far-reaching implications for circuits, data structures, and streaming.

• Sublinear Algorithms: Property Testing, Query Complexity, and the Power of Random Sampling · 2019-07-07

  An exploration of sublinear-time algorithms—property testing, the regularity lemma connection, and how random sampling reveals global structure without reading the whole input.

• Streaming Algorithms: Misra-Gries, Count-Min Sketch, AMS, and the Power of Small Space · 2019-06-27

  A comprehensive tour of streaming algorithms—from frequency estimation sketches to frequency moments—and the space lower bounds that define what's possible.

• Online Algorithms: Competitive Analysis, Ski Rental, Paging, and the Primal-Dual Method · 2019-05-12

  A thorough examination of online algorithms—decisions without foresight—through the lens of competitive analysis, from ski rental and paging to the k-server problem.

• Parameterized Complexity: FPT, the W-Hierarchy, Kernelization, and Bounded Search Trees · 2019-05-11

  An in-depth exploration of parameterized complexity theory—how structural parameters beyond input size can tame NP-hardness through FPT algorithms, kernelization, and the W-hierarchy.

• NP-Completeness: The Cook-Levin Theorem, Polynomial Reductions, and the Hardest Problems in NP · 2019-05-03

  A deep dive into the theory of NP-completeness—from Turing machines and the Cook-Levin theorem to the taxonomy of NP-complete problems and the P versus NP question.

• Linear Programming: Simplex Geometry, Duality, and the Interior-Point Revolution · 2019-04-19

  An exploration of linear programming from Dantzig's simplex method through von Neumann's duality to Karmarkar's interior-point breakthrough that reshaped optimization theory.

• Network Flow: From Ford-Fulkerson to Push-Relabel and the Max-Flow Min-Cut Theorem · 2019-04-13

  A rigorous journey through the algorithms that solve maximum flow—Ford-Fulkerson, Edmonds-Karp, Dinic, and Push-Relabel—together with the duality that binds flows to cuts.

• Implementing A Lock Free Concurrent Hash Map: From Theoretical Foundations To Practical Performance Optimization In C++20 · 2019-03-16

  A comprehensive technical exploration of implementing a lock free concurrent hash map: from theoretical foundations to practical performance optimization in c++20, covering key concepts, practical implementations, and real-world applications.

• Dynamic Programming: Bellman's Principle of Optimality and the Art of Reusing Computation · 2019-01-27

  A deep exploration of how Bellman's recursive insight transforms exponential despair into polynomial hope across knapsack, shortest paths, sequence alignment, and reinforcement learning.

• Designing A Conflict Free Replicated Data Type (Crdt) For Collaborative Text Editing: A Deep Dive Into Rope Structures, Vector Clocks, And Operational Transformation Alternatives · 2019-01-01

  A comprehensive technical exploration of designing a conflict free replicated data type (crdt) for collaborative text editing: a deep dive into rope structures, vector clocks, and operational transformation alternatives, covering key concepts, practical implementations, and real-world applications.

• Implementing A Distributed Consensus Protocol From Scratch: Raft With Leader Election, Log Replication, Membership Changes, And Cluster Reconfiguration In Go · 2019-01-01

  A comprehensive technical exploration of implementing a distributed consensus protocol from scratch: raft with leader election, log replication, membership changes, and cluster reconfiguration in go, covering key concepts, practical implementations, and real-world applications.