Gold

Divisibility

Rare

Using the information that one integer evenly divides another.
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Modular Arithmetic

Not Frequent

Working with remainders from division.
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Introduction to DP

Very Frequent

Speeding up naive recursive solutions with memoization.
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Knapsack DP

Not Frequent

Problems that can be modeled as filling a limited-size container with a subset of items.
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Paths on Grids

Not Frequent

Counting the number of "special" paths on a grid, and how some string problems can be solved using grids.
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Longest Increasing Subsequence

Has Not Appeared

Finding and using the longest increasing subsequence of an array.
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Bitmask DP

Rare

DP problems that require iterating over subsets.
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Range DP

Rare

Solving the problem on every contiguous subarray of the original array.
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Breadth First Search (BFS)

Not Frequent

Traversing a graph in a way such that vertices closer to the starting vertex are processed first.
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Disjoint Set Union

Somewhat Frequent

The Disjoint Set Union (DSU) data structure allows you to add edges to an initially empty graph and test whether two vertices of the graph are connected.
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Topological Sort

Rare

An ordering of vertices in a directed acyclic graph that ensures that a node is visited before every node it has a directed edge to.
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Shortest Paths with Non-Negative Edge Weights

Somewhat Frequent

Introduces Bellman-Ford, Floyd-Warshall, Dijkstra.
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Minimum Spanning Trees

Not Frequent

A subset of the edges of a connected, undirected, edge-weighted graph that connects all the vertices to each other of minimum total weight, where no cycles are allowed.
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Stacks

Rare

A data structures that only allows insertion and deletion at one end.
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Sliding Window

Not Frequent

Maintaining data over consecutive subarrays.
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Point Update Range Sum

Somewhat Frequent

Introduces Segment Tree, Binary Indexed Tree, and C++ Order Statistic Tree.
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Euler Tour Technique

Not Frequent

Flattening a tree into an array to easily query and update subtrees.
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DP on Trees - Introduction

Not Frequent

Using subtrees as subproblems.
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DP on Trees - Solving For All Roots

Rare

Tree DP that uses the subtree from excluding each node's subtree.
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String Hashing

Rare

Quickly test equality of substrings with a small probability of failure.
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(Optional) More on Unordered Sets & Maps

Rare

Maintaining collections of distinct elements with hashing.
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(Optional) A Faster Hash Table in C++

Rare

Introduces gp_hash_table.
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Intro to Bitwise Operators

Has Not Appeared

Goes over the six bitwise operators and the common ways they are used.
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Additional Practice for USACO Gold

Congratulations on making it this far! Here are some final tips for Gold, as well as additional practice problems you may find useful.
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