Algorithms · Prerequisites

Prerequisites for Algorithms

Algorithms combine programming, mathematical reasoning, data structures, proof techniques, complexity analysis, graph theory, probability, and optimization.

Core background

The strongest preparation is practical programming experience together with the ability to reason about runtime, recursion, correctness, graphs, and combinatorial growth.

Essential Background

These topics support complexity analysis, sorting and searching, graph and string algorithms, dynamic programming, approximation, randomization, online algorithms, divide-and-conquer, greedy methods, backtracking, and branch-and-bound.

PF

Programming Fundamentals

Variables · loops · functions

Algorithm design is much easier when basic programming constructs are already automatic.

Technical significance

You should be able to translate pseudocode into working code and trace execution without struggling with syntax.

LoopsFunctionsArrays
Be comfortable with variables, conditionals, loops, functions, arrays/lists, and simple input/output. Trace nested loops and implement pseudocode faithfully.
DS

Data Structures

Arrays · stacks · queues · trees · heaps

Algorithms depend heavily on how input and intermediate state are represented.

Technical significance

A strong algorithm can perform poorly when paired with an unsuitable data structure.

ArraysQueuesTreesHeaps
Know the typical operation costs of arrays, linked lists, stacks, queues, hash tables, trees, heaps, and graph representations.
DM

Discrete Mathematics

Sets · relations · logic

Algorithms are described with sets, functions, relations, sequences, and logical conditions.

Technical significance

Discrete mathematics is the language of correctness and complexity.

Be comfortable with sets, Cartesian products, functions, relations, sequences, predicates, and quantifiers.
PR

Mathematical Proofs

Induction · contradiction · invariants

Correctness is as important as runtime.

Technical significance

Proof techniques explain why an algorithm always returns the right answer.

  • Direct proof and contradiction.
  • Mathematical induction.
  • Loop invariants: initialization, maintenance, termination.
  • Counterexamples.
BO

Asymptotic Analysis

Big-O · Θ · Ω

Algorithm analysis compares how resource use grows with input size.

Recognize O(1), O(log n), O(n), O(n log n), O(n²), O(2ⁿ), and O(n!). Understand upper, lower, and tight asymptotic bounds.
LE

Logarithms & Exponents

log n · powers · growth

Binary search, divide-and-conquer, heaps, and balanced trees produce logarithmic behavior.

Understand powers, log₂n, exponent laws, and why exponential and factorial growth quickly make brute-force algorithms impractical.
RC

Recursion

Base case · recursive case · call stack

Many algorithms are naturally recursive.

Trace recursive calls, identify base cases, understand recursion depth, and relate recursive structure to runtime and memory use.
RR

Recurrence Relations

T(n) · recursive runtime

Recursive algorithms often require recurrence equations for runtime analysis.

Recognize forms such as T(n)=2T(n/2)+n. Use substitution, recursion trees, and the Master Theorem for standard divide-and-conquer recurrences.
GR

Graph Fundamentals

Vertices · edges · paths

Graph algorithms require fluency with graph representations and basic terminology.

Know directed and undirected graphs, weighted graphs, paths, cycles, connectivity, trees, adjacency lists, and adjacency matrices.
CB

Combinatorics

Counting · permutations · combinations

Many search spaces are understood by counting possible configurations.

Know factorials, permutations, combinations, simple counting arguments, and how combinatorial explosion affects exhaustive search.
PB

Probability

Random variables · expectation

Randomized algorithms use probability as part of their design or analysis.

Understand basic probability, conditional probability, random variables, expectation, and the distinction between expected runtime and worst-case runtime.
OP

Optimization Basics

Objectives · constraints · feasible solutions

Greedy, dynamic programming, approximation, and branch-and-bound methods solve optimization problems.

Distinguish decision, search, and optimization problems. Understand feasible solutions, objective functions, lower/upper bounds, and local versus global choices.
NP

Complexity Classes

P · NP · reductions

Later chapters distinguish efficiently solvable problems from problems believed to require super-polynomial time.

Know the high-level meaning of P, NP, NP-hard, NP-complete, polynomial-time verification, and polynomial reductions.
ON

Online Computation

Incremental input · competitive analysis

Online algorithms must make decisions without knowing future requests.

Understand the contrast between online and offline information, irrevocable decisions, adversarial sequences, and competitive ratio.
PA

Parallel Computing Basics

Work · span · synchronization

Parallel algorithms divide work among multiple processing units.

Know the ideas of independent tasks, dependencies, synchronization, communication overhead, load balance, speedup, scalability, work, and critical path/span.
DT

Testing & Debugging

Edge cases · assertions · tracing

A correct idea still needs careful implementation and validation.

Test empty inputs, singleton inputs, duplicates, sorted/reverse-sorted data, disconnected graphs, overflow-sensitive values, and worst-case patterns.
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