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Math & Number Theory Master Recap: Interview Cheatsheet
Complete reference for math and number theory DSA patterns: algorithm selection guide, complexity table, and top 25 interview problems.
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Number-theory
20 articles
dsa4 min read
Inclusion-Exclusion Principle: Counting with Overlaps
Apply the inclusion-exclusion principle to count elements satisfying union of conditions. Solves divisibility, coverage, and derangement problems.
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Randomized Algorithms: Reservoir Sampling and Quickselect
Master randomized DSA algorithms: reservoir sampling for streams, quickselect for O(n) kth element, and random shuffling.
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Game Theory: Nim, Grundy Numbers, and Sprague-Grundy
Master combinatorial game theory: Nim XOR strategy, Grundy (nimber) values, and Sprague-Grundy theorem for composite games.
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Computational Geometry: Points, Lines, and Convex Hull
Tackle geometry problems in coding interviews: cross product, point-in-polygon, line intersection, and Graham scan convex hull.
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Probability and Expected Value in Competitive Programming
Apply probability theory and expected value DP to competitive programming: dice problems, random walks, and geometric distribution.
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Integer Overflow, Precision, and Safe Arithmetic
Prevent integer overflow in competitive programming: safe multiplication, __int128, binary search on answers, and floating-point gotchas.
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Square Root Decomposition: Range Queries in O(sqrt n)
Build range query structures in O(sqrt n) per query with block decomposition. Simpler alternative to segment trees.
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Chinese Remainder Theorem: Solving Modular Equations
Solve systems of modular congruences with the Chinese Remainder Theorem. Fundamental for cryptography and competitive math.
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Number Sequences: Fibonacci Variants and Mathematical Patterns
Explore classic number sequences: Fibonacci, Lucas, Pell, tribonacci, with DP and matrix approaches.
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Bit Manipulation: XOR Tricks and Bitmasking
Master bitwise operations for DSA: XOR tricks, Brian Kernighan bit counting, subset enumeration, and bitmask DP.
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Matrix Exponentiation: Fibonacci and Linear Recurrences
Solve linear recurrences like Fibonacci in O(log n) using matrix exponentiation. Essential for DP optimization on large n.
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Number Theory: Perfect Numbers, Abundant, and Digit Problems
Explore digit manipulation, perfect/abundant numbers, Armstrong numbers, and common math interview patterns.
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Combinatorics: nCr, Pascal's Triangle, and Catalan Numbers
Compute combinations nCr efficiently with precomputed factorials, Pascal's triangle for small n, and Catalan numbers for tree/parenthesis counting.
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Euler's Totient Function and Applications
Compute Euler's totient phi(n) for cryptography and modular inverse applications. Sieve variant for all values up to n.
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Prime Factorization: Trial Division and Pollard's Rho
Factor integers efficiently with trial division O(sqrt n), SPF sieve O(log n), and understand when each approach is optimal.
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Modular Arithmetic: Fast Exponentiation and Inverse
Master modular arithmetic for competitive programming: binary exponentiation O(log n), modular inverse, and Chinese Remainder Theorem.
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GCD, LCM, and Extended Euclidean Algorithm
Master GCD/LCM with Euclidean algorithm O(log n) and the Extended Euclidean for modular inverse computation.
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Sieve of Eratosthenes: Find All Primes Efficiently
Generate all primes up to n in O(n log log n) with the Sieve of Eratosthenes. Includes segmented sieve and prime factorization variants.
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Math & Number Theory for DSA: Complete Guide
Master mathematical algorithms for DSA: primes, GCD, modular arithmetic, combinatorics, and fast exponentiation with 5-language implementations.
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