Reflections on the ACMer's Growth Path
Where is Your North Star?
Many people plunge into the world of competitive programming, clutching a long “list of knowledge points” as if they’ve found a treasure map, believing they can unearth riches by simply following it. But they often overlook the most crucial question: Why are you embarking on this journey?
The destination on this map differs vastly for everyone.
If your goal is to join a desired tech company, your course is relatively clear. You’ll need to master the most core and classic data structures and algorithms, such as trees, graphs, and basic dynamic programming. Interviewers value the robustness of your code, the clarity of your logic, and how you communicate and handle edge cases. In this scenario, those “dragon-slaying techniques” on the list—like Pólya’s theorem or LCT—might truly be just that for you, offering low cost-effectiveness.
However, if your journey aims for the stars and the stage of the ICPC World Finals, then you cannot overlook any corner of this map. You’ll need extreme speed, absolute precision, and a strong heart that can perform consistently under pressure. Those seemingly obscure knowledge points might be the key factors determining victory or defeat in a competition.
Of course, there’s also a group who are more like pure travelers, driven by curiosity and a love for knowledge itself. They might linger in the deep valleys of number theory or revel in the stellar beauty outlined by computational geometry. For them, learning is the goal, and the process is the best reward.
Therefore, before you set off, be sure to find your own North Star. It will determine your direction, your equipment, and everything you encounter on this path.
A Not-So-”Standard” Map
With your direction clear, let’s look at the map. Traditional knowledge checklists always try to lay everything out neatly, but I prefer to see it as a continent full of opportunities and challenges, primarily composed of several sections.
Core Methodologies: Your Craft
These are the fundamental skills you rely on, the three essential axes you must wield in this arena.
Dynamic Programming (DP): The Weaver of Time
In my view, DP is the art of conversing with time. Its core lies in solving problems that exhibit “optimal substructure” and “overlapping subproblems.” You elegantly predict the future (solve the final problem) by “remembering” the past (storing solutions to subproblems).
State design is the soul of DP, and often the most challenging part. From basic linear DP and knapsack DP, to more complex tree DP (recursive state transitions on trees, often used for problems like maximum independent set or tree diameter), bitmask DP (using binary bits to represent set states, a powerful tool for small-scale combinatorial optimization problems), and further to digit DP and probability DP, each type is a significant test of your problem modeling ability.
It’s perfectly normal to feel like you understand a model, only to be stumped by a slightly different problem statement. What’s more critical is that naive DP often leads to Time Limit Exceeded, which calls for DP optimization. Monotonic queue and convex hull trick (slope optimization) are like turbochargers for your weaving machine; they leverage the monotonicity or geometric properties of decisions to optimize O(N^2) complexity down to O(N) or O(N log N). These are key differentiators between a proficient DP user and a master. The gap between “knowing about” DP and “being able to use” DP is bridged by hundreds, if not thousands, of practice problems and a deep insight into the essence of the problem.
Graph Theory: The Architect of Relationships
Graph theory is the philosophy of “connections.” Everything in the world, from social networks to transportation hubs, can be abstracted into nodes and edges. Learning graph theory is like learning to be an urban planner in an abstract world.
Your toolbox will contain a variety of powerful tools. For instance, shortest path algorithms (Dijkstra/SPFA) help you plan the quickest routes; connectivity algorithms (strongly/biconnected components, cut vertices/bridges) help you analyze network stability and critical nodes; and bipartite matching (Hungarian algorithm/Hopcroft-Karp algorithm) can solve various pairing problems.
And network flow is the culmination of graph theory. Maximum flow not only solves transportation problems but can also be cleverly transformed into models like bipartite matching and project assignment. Its dual problem—minimum cut—is a powerful tool for solving minimum cost problems related to “partitions,” with applications in areas like image segmentation. When you also need to consider costs, min-cost max-flow can help you find the lowest-cost solution.
But remember, the essence of graph theory often lies not in whether you can recite the Dinic algorithm template, but in graph construction. How to transform a problem that seemingly has no relation to graphs into a graph theory model—that’s where your true intelligence is tested.
Data Structures: The Sculptor of Information
If data is uncarved jade, then data structures are the tools and techniques for sculpting it. They bring order to chaotic data and make slow queries swift.
Segment trees and Fenwick trees (BIT) are your right and left hands for handling range problems; one is powerful, the other lightweight and fast. For more complex tree problems, Heavy-Light Decomposition can transform tree path operations into familiar sequence range operations, a truly brilliant technique. When you need to dynamically maintain the connectivity of a forest, LCT (Link-Cut Tree) can make a dazzling appearance.
But don’t fall into blind worship of “advanced” data structures. Sometimes, an “elegant brute force,” like square root decomposition (Mo’s algorithm), due to its simple implementation and broad applicability, can instead become a powerful tool for scoring points in competitions. And concepts like persistence (e.g., persistent segment tree, aka Chairman Tree) open up a new world, endowing data with “history” and “memory,” allowing you to query states from any historical version and solve even more incredible problems. Remember, in a competition, the ability to write correct code stably and quickly, and score points, is paramount.
Theoretical Foundations: Inner Cultivation Techniques
If methodologies are your moves, then mathematics is your inner cultivation. Deep internal power can multiply the effectiveness of your techniques.
Number theory and combinatorics are the core of this inner cultivation. From greatest common divisor (GCD/extended Euclidean algorithm) to modular arithmetic (modular inverse, Fermat’s Little Theorem, Chinese Remainder Theorem), and then to various sieve methods and Möbius inversion, these tools may seem abstract, but they can deliver a decisive blow at critical moments. Especially when dealing with counting problems and problems involving modular arithmetic, a solid mathematical foundation allows you to see what others cannot. And combinatorial tools like the Principle of Inclusion-Exclusion, generating functions, and Pólya Enumeration Theorem are the epitome of the “art of counting.”
Linear algebra and polynomials are like higher-level inner techniques. Gaussian elimination for solving linear equations and matrix exponentiation by squaring for accelerating linear recurrences are standard operations. The advent of FFT/NTT has ushered polynomial multiplication into the O(N log N) era, providing powerful computational support for algorithms like generating functions that require polynomial operations, truly a “nuclear weapon” in competitive programming.
Specialized Domains: Dungeon Challenges
Once your craft is refined and your inner power is deep, you can embark on challenges in various “dungeons” with distinct styles.
The world of strings is filled with various ingenious matching and query algorithms. From KMP to Aho-Corasick automaton, and then to the powerful suffix family—Suffix Array (SA) and Suffix Automaton (SAM), as well as Manacher’s algorithm and Palindromic Automaton (PAM) for handling palindromes. But don’t forget, in most cases, the simple and effective string hashing is your most loyal friend; though it might occasionally betray you (get hacked), it’s incredibly fast and highly cost-effective.
Computational geometry is a beautiful yet dangerous place. Its beauty lies in the perfect fusion of logic and form; algorithms like convex hull, half-plane intersection, and rotating calipers all shine with brilliance. But its danger lies in the ubiquitous precision issues. Before entering this domain, be sure to prepare a thoroughly tested and absolutely reliable geometry template, otherwise, you will be tormented by various eps values to the point of questioning your sanity.
Game theory is a pure dance of logic. The SG function and Nim sum are powerful theoretical tools for analyzing impartial games, helping you discern transitions between winning/losing states. However, often, finding the intrinsic patterns of a game or simplifying the model is much more effective than rigidly applying the SG function.
The Invisible Continent: Competitive Programming Engineering
Finally, there’s a crucial continent usually not marked on maps—Competitive Programming Engineering. This includes your debugging skills (double-checking is a divine technique!), your sensitivity to edge cases, your habits for managing and utilizing your code template library, and your time allocation strategies during contests. These “soft skills” determine your baseline, deciding whether you can consistently perform at 80% or more of your 100% knowledge level.
The Journey to Mastery: “Shu, Ha, Ri”
Receiving the map doesn’t mean the journey ends. On the contrary, it’s just the beginning. Mastery in any field is inseparable from an ancient and classic path: “Shu, Ha, Ri”.
Shu: This is the apprentice stage. You strictly follow rules, imitate the code of predecessors, and understand standard solutions to classic problems. This process might be tedious, but it’s indispensable. You must learn to walk before you can run.
Ha: This is the craftsman stage. You begin to understand the principles behind the rules, attempt to break conventions, and integrate knowledge from different domains. You no longer rigidly apply templates but start to think for yourself and adapt.
Ri: This is the master stage. All knowledge and techniques have been internalized into your intuition. You are no longer confined by any fixed moves; you can even start from first principles and create your own solutions.
The fascinating aspect of competitive programming lies precisely here. It’s not just an accumulation of code and logic, but a long journey of learning, growth, and self-transcendence. May you, on this path, be able to gaze at the stars while keeping your feet firmly on the ground.