Introduction
The USA Computing Olympiad (USACO) is a prestigious programming competition divided into four divisions: Bronze, Silver, Gold, and Platinum. Among these divisions, passing USACO Silver stands out as one of the most challenging tasks, arguably second only to qualifying for the USACO Camp. This is because the gap in terms of difficulty and algorithmic maturity between Bronze and Silver is very steep compared to other divisions. While past coding or math competition experience might suffice for advancing from Bronze to Silver, Silver demands a significantly more extensive approach. Not only does it require learning a handful of new algorithmic concepts and data structures, but it also requires competitors to have a thorough understanding of exactly when and how to apply each one.
Especially with the increasing difficulty of contests, which particularly impacts the Silver division, participants must dedicate a substantial amount of time to learn new algorithms and hone their problem-solving skills. In this comprehensive blog post, we aim to provide an in-depth understanding of the USACO Silver division, covering everything necessary to advance to USACO Gold in your next competition!
Note that the rest of the blog assumes that you have already passed Bronze (or are already very familiar with the division), and thus will not cover prerequisite information from Bronze.
Concepts You Need to Learn For USACO Silver
To begin preparing for Silver, we recommend starting by learning each concept individually before practicing on past contest problems. Jumping straight into practicing on past problems before learning the concepts can often be counter productive, because it’s very difficult to figure out these concepts on your own (especially while trying to solve a problem) and will spoil the problem for future practice. Instead, the recommended approach is to learn each topic individually, followed by practicing with a curated list of problems (e.g., the USACO Guide or Codeforces) that require that concept.
Beyond the topics covered in USACO Bronze (programming fundamentals, recursion, basic data structures, and sorting), USACO Silver introduces a wide variety of new concepts.
Prefix Sums
Prefix sums have a wide range of applications: most notably, they enable computing range sums over arrays in constant time. Generally, you should think about prefix sums whenever you want to optimize some sort of operation or query over a range. In addition to prefix sums, it’s important to learn the difference array technique and computing 2D prefix sums. In modern USACO Silver problems, prefix sums are generally not the crux of the problem solution (i.e. the question will never just be to compute a certain operation over a range). However, they still serve as an important component in optimizing efficiency in a more complex solution.
Binary Search
Binary search is an efficient algorithm to search for values over a sorted array, reducing time complexity from O(N) with linear search down to O(log N). Lower and upper bound binary search are variations of the binary search algorithm, which can efficiently find the first and last occurrence of an element in a sorted array respectively.
It’s also important to learn the technique of “binary searching for the answer,” which can typically be used when the problem asks you to find the maximum or minimum value of some variable that satisfies a certain condition. Many of the past and recent USACO Silver problems utilize this technique.
Data Structures
In addition to a solid grasp of foundational data structures such as static and dynamic arrays, unordered sets, and maps, Silver level proficiency necessitates a thorough command of several additional data structures. These include stacks, queues, priority queues, ordered sets, ordered maps, and linked lists. While these data structures are crucial tools for optimizing the time complexity of solutions, they are not intended to be the sole focus of problem-solving. Problems at this level in USACO will demand the adept integration of these structures within more complex solutions, highlighting their supportive role rather than being the sole solution component. As such, simply knowing how each data structure works is not at all enough to master this general concept, instead competitors need to know the ins and outs of the tradeoffs between each data structure and how to apply each one effectively.
Graphs, Trees, and Traversals
Graphs is by far the most important topic in USACO Silver. The concept is simple: representing a network of relationships via a collection of “nodes” connected by “edges.” Every Silver contest will have at least one graph problem. Graphs also happen to be one of the most diverse types of problems. Some important concepts within graph theory are as follows:
General terminology (nodes, edges, directed edges, connected components, cycles, etc.)
Representing graphs with adjacency list vs. matrix
Depth-first and breadth-first traversal
Applying graph traversals to the flood fill algorithm on grids
Special types of graphs, most notably trees (connected acyclic graphs) and functional graphs (directed graphs in which all vertices have an outdegree of 1)
Typically, whenever a problem mentions a network of objects that are connected in some way, it’s a clear indication that the solution involves graphs (i.e. barns connected by paths, friendships between cows, etc.).
Generally, all silver graph problems will require you to perform some sort of graph traversal, which typically boils down to using depth-first search. Many silver problems in USACO will also contain special types of graphs, which as mentioned earlier include trees and functional graphs. You’ll know you’re facing a tree problem whenever the problem statement mentions a connected graph with N vertices and N - 1 edges, a graph where every pair of nodes are connected by a unique path, and/or a connected graph that does not contain any cycles. Functional graph problems are generally not as obvious (it may not even be obvious that it’s a graph problem in the first place), however will generally boil down to exploiting cycles within a directed graph.
Two Pointers and Sliding Window
The two pointers technique involves moving two “pointers” rightwards across an array, typically to search for a pair of indices that satisfy some condition. Since each pointer only passes through the array once, this technique is used to optimize a solution that brute forces over all pairs of indices in an array, which would take quadratic time, down to an approach that only takes linear time.
The sliding window technique is a special case of the two pointers technique, where a fixed distance is maintained between the two pointers. This approach considers all constant size subarrays of an array, maintaining and updating data between each subarray.
It will almost never be obvious when either of these techniques should be applied, as problems generally become much simpler once you know that one of the two techniques can solve the problem. As such, you should always have these floating around in the back of your mind whenever you’re solving a USACO Silver problem.
Greedy Algorithms
The Greedy approach is an algorithmic strategy that constructs a solution incrementally, selecting each subsequent piece based on the most apparent and immediate advantage. In other words, it finds the globally optimal solution by taking locally optimal choices at each step. Greedy algorithms are different from all the other concepts listed in the sense that they can vary a lot from case to case, and are solely defined based on the principle of choosing the local optimum.
It’s important to note that a greedy approach will not always work, as there are many instances where locally optimal steps do not lead to a globally optimal solution. However, greedy solutions are generally simple to implement, so if you are ever unsure of whether a problem can be solved using greedy, just implement it and see if it passes (this may not be a good idea for more complex problems). If it does, great – you’re done, and you didn’t even have to waste time proving to yourself that your solution works! Otherwise, back to the drawing board!
Coordinate Compression
Within USACO Silver, coordinate compression is a rare, but nonetheless important, technique in which you compress values from a large range into a smaller range. The process involves assigning each value within a list to its corresponding index if the list were sorted.
This technique is useful whenever you do not care about absolute values, but rather the relative ordering of the values. Note that you will not always need to use coordinate compression – only use it when you need the relative values to be small for whatever reason (i.e. you want to use the values as array indices, which would require compressing the values down to a smaller range). Coordinate compression can also be applied to the 2-dimensional case, and can often be paired with 2D prefix sums.
Ad Hoc
Ad hoc is the only concept that you do not have to learn – because it isn’t a concept! Ad hoc problems are those that do not fall in a well-defined category (i.e. the main solution does not involve any of the specified concepts in Bronze or Silver). Instead, they involve some clever insight or observation about the problem itself. As a result, these types of problems require very strong problem solving skills.
The only way to really prepare for ad hoc problems is to practice a bunch of them! Ad hoc problems can only really be solved by having strong problem solving, and the only way to get better at problem solving is to practice.
Resources and How to Prepare For USACO Silver
A comprehensive list of resources available can be found on the USACO Resources Page. Below, we’ve listed resources that competitors have historically found to be most useful for each stage in preparing for silver: learning each concept, practicing each concept, and preparing for contests.
Learning USACO Silver Concepts
There are a ton of free resources available online to learn and master each concept required in USACO Silver. In particular, the USACO Guide does a great job at providing links to videos, articles, and textbooks that explain each concept in detail.
If you want to follow a textbook specifically geared for competitive programming, some great resources include the Competitive Programmer’s Handbook by Antti Laaksonen (note: many of the topics listed are Gold and Platinum topics, out of the scope of Silver) and An Introduction to the USA Computing Olympiad by Darren Yao.
Another way to learn Silver concepts is to take an online algorithms course provided by a college. Coursera hosts many good courses, most notably Algorithms Specialization by Stanford University and Algorithms Part I and II by Princeton University. While they cost money, Coursera allows you to audit them for free, or to apply for a fee waiver (it’s fairly easy to get approved as a high school student). In addition to covering relevant topics for USACO, they provide interesting projects and exercises to apply what you’ve learned, which can help place these concepts in a broader context and give you a taste for what college-level computer science classes are like. If your only goal is to prepare for USACO, however, these courses won’t be as efficient of a way to learn the concepts you need.
If you prefer learning in a more structured manner, or don’t learn on your own effectively, you may want to consider signing up for a USACO class. Classes follow a set curriculum and pace, and provide more personalized help through teachers. However, classes do cost money, and quality may vary between different organizations – do your research before signing up for one!
Practicing USACO Silver Concepts
After learning each concept and how to implement them, the next step is to practice using that concept in past problems. Often, it’s wise to organize your practice by topic – for example focusing on practicing DFS one week and then practicing two pointers another week.
The USACO Guide, once again, is an amazing resource to practice concepts once you’ve learned them. It provides a curated list of problems for each concept, enabling targeted practice once you have learned each one.
Another useful resource for targeted practice is Codeforces, which is the biggest online site that hosts competitive programming contests. It’s also a great resource if you want to practice, but want to save past USACO Silver problems for when you feel more comfortable. To organize your practice by concept, navigate to the problem set tab and filter the problems by a specific tag (i.e. the concept you want to practice) and difficulty range (1200 - 1900 is generally a good range for Silver). For a comprehensive guide on how to utilize Codeforces, check out our blog here.
How to Prepare For USACO Silver
Once you feel comfortable with all the concepts tested in USACO’s Silver division, as well as how to implement them in past problems, it’s time to prepare for upcoming contests. Do not continue practicing with concept lists! While they are great for learning concepts initially, relying on topic lists to prepare for contests can be counterproductive, as it’s much easier to solve a problem if you already know what algorithm/data structure you are supposed to use beforehand.
The best way to prepare for upcoming contests is simply to practice on past problems! A great way to do this is to go through problems in the past contests page on USACO, starting from the 2016-2017 contest season and working your way through to the more recent contests. Newer contests are typically harder than older ones, so by practicing in chronological order, you’ll gradually face harder problems as you improve through more practice.
If you have run out of USACO Silver problems to practice, or want to save the remaining ones for later, Codeforces is a great resource to find a ton more problems and contests.
What to Do if You’re Stuck In USACO Silver
Many competitors often run into some scenario where they do really well in Bronze and pass, but find themselves stuck in Silver for several contests—maybe even a year or two. If this is happening to you, it’s time to re-evaluate how “ready” you think you are for upcoming contests, and why your expectations of passing are not matching up with your past results.
Maybe you’ve learned all the Silver concepts and have practiced them on a bunch of problems already, or you’ve been able to solve 50% of all past Silver problems already. In these cases, would it be reasonable to expect to be promoted in the next contest? In many cases (unless you’re naturally super talented), no! Our blog post about Closing the Gap Between Expectations and Reality goes into this topic in depth. Essentially, even if you’ve solved 50% of all past Silver problems already, the problems you’ve solved are likely to be roughly the easiest 50%. However, modern contest problems are all typically on the harder end of the difficulty spectrum, meaning even though you could solve half of the Silver problems, you won’t be able to solve any problems in a contest because the level of difficulty you can handle does not encompass the contest difficulty.
If you truly want to anticipate success for an upcoming contest, you should reach a point where you can solve more than 90% of all USACO Silver problems. At this point, it is a lot more likely that all three contest problems will fall within the range of what you can handle — and thus you should expect success in these cases.
What if you’ve reached a point where you can solve all past Silver problems, but you still find yourself struggling in contests? This brings us to the next topic: contest strategy.
Silver Contest Strategy
Having a strong contest strategy can make or break your contest performance. In fact, during USACO camp each year, the coaches host a special lecture focused solely on developing a strong contest strategy. And every time, the campers see their scores shoot up after implementing the contest strategy.
So, what makes a good contest strategy? The key distinction between practicing on your own time and competing in contests is the time limit. While 4 or 5 hours may seem like a lot of time – it really isn’t. Thus, being able to allocate your time well is absolutely crucial for succeeding in contests.
Riya filmed a great video detailing contest strategy in depth, and we also have a blog post regarding contest strategy. The key takeaway from the video is to take practice contests with the same time constraints as a real contest. During these contests (and actual contests too), keep a contest log where you note down what you are currently working on every 15 minutes. Maintaining this log can help you identify time leaks during contests: if you notice that you’ve been working on the same problem for one and a half hours without significant progress, you should choose to cut your losses and move on to a different problem, coming back to the current problem later.
After the contest is over, analyze your performance. Did you allocate too much time on a single problem? Were you too caught up in debugging because you didn’t plan out implementation well enough? Could you have solved a problem with better time management? Asking questions like these can help you identify mistakes you’ve made and learn from them, making sure that you do not repeat these mistakes in the future.
How Much You Should Practice
A lot of competitors often ask the question, “how many hours a week should I put into USACO to pass X division in Y time?” The answer to this question is different for everyone. It could take some competitors 2 months to pass USACO Silver practicing 2 hours a week, and other competitors 10 months practicing 5 hours a week.
Instead of comparing yourself to how much others practice, or frantically searching online to find a routine that worked for someone else, you need to figure out what works for you. To achieve this, create some goal in which you hope to promote by, and allocate time throughout your schedule so that you can consistently make progress every week. If you find that you’re not progressing as fast as you need to, either put more hours into practicing every week or reconsider if your goal is realistic. Another option is to find a USACO coach, which can often make your practice and improvement a lot faster, giving you time to focus on strengthening the rest of your college application.
At VPlanet, we have private coaching programs starting from USACO Bronze all the way up to USACO Platinum.
Coaching from VPlanet Coding
If you’re super busy, then you need our program, so that you can advance faster and spend your time more efficiently. If you can get half your time back, then all those remaining hours can be spent on something else to make your college apps that much more competitive.
Here are 5 key benefits that we provide over other alternatives:
Comprehensive curriculum for each division
VPlanet provides dozens of hours of video content with in-depth algorithms and problem walkthroughs, covering all the knowledge a student needs to know to become successful. Additionally, our curriculum teaches the essential problem-solving skills that are required beyond the knowledge of algorithms.
Carefully curated problems
Practicing on your own can be frustrating and inefficient. We know what it’s like to spend 2 hours thinking through a problem just to give up and read a solution you can barely understand. VPlanet has handpicked 100+ problems for each division, ensuring that the problems students work on are never too easy or too hard. We guarantee that each problem has just the right level of difficulty to optimize students’ improvement trajectory.
Daily support from coaches
If students are ever stuck or confused, VPlanet provides daily support in the form of Facebook group discussions, Q&A, and one-on-one sessions with highly-experienced coaches. These sessions maintain a very small ratio between students and coaches to ensure that everyone receives as much personalized help as possible.
In contrast, a typical USACO class can have as many as 30 students assigned to one teacher, with only 1-2 class sessions per week to ask for help. This class environment makes it difficult to ask for help and risks leaving many students behind.
Work at your own pace
An inevitable issue with USACO classes is that they fail to account for each student’s individual skill level. This issue guarantees that either a significant number of students are falling behind in class, or the class is too slow for many high-achieving students. VPlanet overcomes this issue, enabling students to go at the pace they want with regular support from coaches.
Advance faster
We find that VPlanet students tend to advance at a much faster pace than others. Our carefully crafted curriculum and sessions with coaches provides just the right amount of content and personalized support for students to improve as fast as possible. This leaves our students with more time to focus on other activities to strengthen their college applications.
Conclusion
Mastering the USACO Silver division requires a multifaceted approach, delving into an array of advanced concepts and problem-solving strategies. This comprehensive guide serves as a roadmap for aspiring participants, emphasizing the importance of deeply understanding concepts, targeted practice, and realistic expectations. Transitioning from theoretical comprehension to practical implementation is pivotal, coupled with an emphasis on effective contest strategies to manage time and learning from mistakes. By embracing these principles, individuals can navigate the challenges of Silver, enhancing their competitive programming skills and setting a solid foundation for future success in higher USACO divisions.