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The AOI Roadmap ​

This roadmap is a recommended learning path for AOI students who want to gradually build their competitive programming and problem-solving skills.

⚠️ Important:

  • Focus on problem-solving skills more than rushing to advance in the roadmap.
  • Make sure your basics are strong before moving on.
  • You are not obliged to complete the whole roadmap — it is just a suggestion of what to do next.
  • Practice a lot — solving problems consistently is the most effective way to improve.
  • Learn to write clean solutions and explain your reasoning — clarity is as important as correctness.
  • Always review your mistakes and analyze why a solution failed — this is where the real learning happens.

Phase 1 – Foundations ​

Phase 2 – First Data Structures & Problem-Solving Tools ​

Phase 3 – Brute Force, Recursion & Intro to Graphs/DP ​

  • Brute force & complexity limits
  • Sorting techniques: sorts, Radix sort, Bucket sort
  • Recursion
  • Backtracking (N-queens, permutations)
  • Bitmasks for subsets
  • Divisibility, Sieve of Eratosthenes, factorization in
  • Binary search on answer
  • Binary exponentiation
  • Interactive problems
  • Intro to DP: Fibonacci, Coin Change
  • Graph representation: adjacency list vs. matrix
  • Connected components

Phase 4 – Intermediate DS & DP ​

  • Sets, multisets, hashmaps, policy-based DS
  • Coordinate compression
  • Pointers & Iterators
  • Trees (DFS, BFS, properties, DP)
  • Tries
  • Intermediate DP:
    • Grid paths
    • Prefix DP
    • 0/1 Knapsack
    • Interval DP (matrix chain, merging stones)
    • LIS (Longest Increasing Subsequence)
  • Counting & combinatorics basics (nCr, Pascal’s triangle, factorials mod, Binomial Theorem)
  • Modular Inverses & Fermat’s Little Theorem
  • Game Theory Basics & Minimax
  • Line sweeping

Phase 5 – Graph Algorithms (Part I) ​

  • BFS, DFS in depth
  • Planar Graphs, Eulerian Graphs, Hamiltonian Graphs
  • Topological Sort & Euler Tour
  • Shortest paths:
    • Dijkstra
    • Bellman–Ford
    • Floyd–Warshall

Phase 6 – Divide & Conquer, Trees & DS ​

  • Fast sorting (merge sort, quick sort)
  • Divide & conquer on problems (inversion count, Karatsuba multiplication)
  • Sparse tables, Segment Trees, Fenwick Tree, Merge Sort Tree
  • Binary heaps
  • Union-Find (DSU)

Phase 7 – Advanced Graph Algorithms ​

  • Minimum Spanning Tree (Kruskal, Prim)
  • Binary Lifting (LCA)
  • Strongly Connected Components & Condensation graph
  • Flows (Ford–Fulkerson)
  • Bipartite Matching, Hungarian Algorithm
  • More interactive problems
  • Bridges & Articulation Points

Phase 8 – Advanced Topics ​

  • Persistent Data Structures
  • Advanced Number Theory
    • Euler’s Totient Function
    • Bézout’s Identity
  • Ternary search (unimodal functions)
  • Sqrt decomposition & Mo’s algorithm
  • Heavy-light decomposition
  • Centroid decomposition
  • Knuth–Yao optimization
  • Randomized Algorithms
  • Lazy propagation
  • Geometry:
    • Convex hull
    • Rotations & angle problems
    • Polygon area

Final Tips ​

  • Consistency beats intensity: Solve a few problems every day rather than cramming.
  • Mix topics: Don’t stay stuck on only one topic for months; revisit earlier phases to reinforce them.
  • Balance theory and practice: Understanding algorithms is important, but solving problems under time pressure matters more.
  • Compete often: Join Codeforces, AtCoder, and AOI contests regularly to measure your progress.
  • Discuss with peers: Teaching or explaining a solution to a friend helps you deeply understand it.
  • Don’t fear hard problems: Even if you can’t solve them, trying will push your thinking forward.

🚀 Remember: becoming strong at problem-solving is not about how fast you progress through the roadmap, but how deeply you understand and apply each step.