Roadmap

The algorithms roadmap

13 stages in the order they should be learned. Every stage depends only on the ones before it. ★ marks topics built on Zaza Gamezardashvili's lectures.

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Stage 0

Foundations

Measure an algorithm before you write one

✓Big-O & Growth RatesPredict how running time grows with input size12 min›✓Recursion & the Call StackSee what really happens when a function calls itself12 min›

Stage 1

Arrays & Hashing

The two structures behind most real code

✓Arrays & Dynamic ArraysWhy indexing is O(1) and push_back is O(1) on average10 min›✓Hash Tables: Sets & MapsLook anything up in O(1) on average14 min›✓Prefix Sums, Two Pointers & Sliding WindowAnswer range questions without rescanning14 min›

Stage 2

Sorting & Searching

Order the data, then find anything in O(log n)

✓Elementary Sorts★▶Bubble, selection, insertion: the O(n²) baseline10 min›✓Merge Sort: Divide & ConquerSplit, sort halves, merge: O(n log n) guaranteed12 min›✓Quicksort & PartitioningPartition around a pivot; fast in practice12 min›✓Counting Sort★▶Beat n log n when keys are small integers8 min›✓Comparators & StabilitySort by any rule with std::sort8 min›✓Binary Search & Search on the AnswerHalve the search space every step14 min›

Stage 3

Linear Structures

Stacks, queues and lists: discipline about who goes next

✓Linked ListsInsert anywhere in O(1) by rewiring pointers8 min›✓Stack (LIFO) & Expression Evaluation★▶Last in, first out, and how calculators use it15 min›✓Monotonic StackNext greater element for every item in O(n)10 min›✓Queue (FIFO) & DequeFirst in, first out: the engine of BFS8 min›

Stage 4

Complete Search

Try everything, cleverly: recursion that explores

✓Generating Subsets & PermutationsWalk the decision tree of include / exclude12 min›✓Backtracking & Pruning: N-Queens★▶Abandon a branch the moment it cannot work14 min›

Stage 5

Trees

Hierarchies that make search logarithmic

✓Trees: Concepts, Properties, Traversals★▶Root, depth, height and the three traversals20 min›✓Binary Search Tree★▶Search, insert and delete in O(h)20 min›✓Balanced Trees & set / mapWhy balance keeps h = O(log n)10 min›✓Binary Heap & Priority Queue★▶Always know the minimum in O(1)12 min›✓Fenwick Tree (Binary Indexed Tree)★▶Change an element and get any range sum, both in O(log n)20 min›✓Trie & AutocompleteStore words by shared prefixes10 min›

Stage 6

Graphs I: Traversal

Explore any network level by level or deep first

✓Graph RepresentationsAdjacency matrix, list and edge list8 min›✓Breadth-First Search (BFS)★▶Shortest paths in unweighted graphs, wave by wave20 min›✓Depth-First Search (DFS)Go deep, backtrack, timestamp everything12 min›✓Cycles, Topological Sort, Bipartite CheckOrder tasks with dependencies14 min›✓Tree Diameter & Center★▶Two BFS runs find the longest path12 min›

Stage 7

Greedy

Take the best move now, and prove it is safe

✓The Greedy Idea & When It FailsActivity selection, and a coin system that breaks greedy12 min›✓Fractional Knapsack★▶Take the best value per kilogram first10 min›

Stage 8

Graphs II: Weighted

Shortest paths and cheapest networks

✓Dijkstra's Algorithm★▶Shortest paths when no weight is negative20 min›✓Bellman–Ford & Negative EdgesHandle negative weights, detect negative cycles12 min›✓Union-Find (DSU)Merge groups and ask "same group?" in near O(1)12 min›✓Minimum Spanning Tree: Kruskal & PrimConnect everything at the lowest total cost14 min›

Stage 9

Math Toolkit

Bits, primes and fast powers you will keep reaching for

✓Bit ManipulationAND, OR, XOR, shifts and subsets as numbers10 min›✓GCD & the Sieve of EratosthenesEuclid in O(log n), all primes up to n fast12 min›✓Fast Exponentiation & Modular Arithmeticaⁿ mod m in O(log n) by squaring10 min›

Stage 10

Dynamic Programming

Solve each subproblem once, reuse it forever

✓DP Fundamentals: Memo & TableTurn exponential recursion into linear work14 min›✓1-D DP: Coins & LISOne array of answers, filled left to right14 min›✓DP on GridsCount and optimise paths through a grid10 min›✓0-1 Knapsack★▶Pick items under a weight limit, optimally20 min›✓Longest Common Subsequence★▶Compare two sequences with a 2-D table20 min›✓Edit DistanceHow many edits turn one word into another10 min›✓Floyd–Warshall: DP on GraphsShortest paths between every pair, in three loops12 min›✓DP on TreesAnswers bubble up from the leaves12 min›✓Bitmask DPDP over subsets: the travelling salesman12 min›

Stage 11

Strings

Find a pattern in a text without wasted comparisons

✓Naive Pattern MatchingThe O(n·m) baseline, and where it wastes work8 min›✓Rolling Hash & Rabin–KarpCompare strings by fingerprints12 min›✓Prefix Function & KMPNever reread a character of the text15 min›

Stage 12

Capstone: Hard Problems

Know when no fast algorithm is likely to exist

✓P vs NP & Choosing an AlgorithmRecognise hard problems and pick the right tool15 min›

Electives

Advanced tools for contests and specialised work

✓Segment TreeAny range query, even min and max, with updates in O(log n)15 min›✓Z-FunctionAn alternative to KMP10 min›✓Suffix ArrayAll suffixes, sorted12 min›✓Manacher's AlgorithmEvery palindrome in linear time10 min›