usaco-guide/content/5_Gold/DS.md

id	title	author	prerequisites
data-structures-gold	Data Structures (Gold)	Michael Cao	Bronze - Data Structures (?) Silver - Data Structures

More advanced applications of data structures introduced in earlier divisions.

Monotonic Stack

Consider the following problem:

Given an array, a, of N (1 \le N \le 10^5) integers, for every index i, find the rightmost index j such that j < i and a_i > a_j.

To solve this problem, let's store a stack of pairs of <value, index> and iterate over the array from left to right. For some index i, we will compute ans_i, the rightmost index for i, as follows:

• Keep popping the top element off the stack as long as value \ge a_i. This is because we know that the pair containing value will never be the solution to any index j > i, since a_i is less than or equal to than value and has an index further to the right.
• If value < a_i, set ans[i] to index, because a stack stores the most recently added values first (or in this case, the rightmost ones), index will contain the rightmost value which is less than a_i. Then, pop the top element off the stack, because index can't be the solution for two elements.

The stack we used is called a "monotonic stack" because we keep popping off the top element of the stack which maintains it's monotonicity (the same property needed for algorithms like binary search) because the elements in the stack are increasing.

Further Reading

• "Nearest Smallest Element" from CPH 8
• CP Algorithms (Min Stack)
• Medium

Problems

• Concurrently Balanced Strings (Old Gold)
• Max Histogram Area (Leetcode)

(add more once codeforces comes up)

Sliding Window

Let's envision a sliding window (or constant size subarray) of size K moving left to right along an array, a. For each position of the window, we want to compute some information.

Let's store a std::set of integers representing the integers inside the window. If the window currently spans the range i \dots j, we observe that moving the range forward to i+1 \dots j+1 only removes a_i and adds a_{j+1} to the window. We can support these two operations and query for the minimum/maximum in the set in O(\log N).

To compute the sum in the range, instead of using a set, we can store a variable s representing the sum. As we move the window forward, we update s by performing the operations s -= a_i and s += a_{j+1}.

Further Reading

• "Sliding Window" from CPH 8
  • Read this article to learn about the "min queue" that CPH describes.
• Medium
• G4G

Problems

• Haybale Feast (Gold)
  • Direct application of sliding window.
• Train Tracking 2 (Plat)
  • Not really sliding window but mentions it.
  • Extremely difficult.

(add more once codeforces comes up)

Merging Sets

Let's consider a tree, rooted at node 1, where each node has a color (see CSES Distinct Colors).

For each node, let's store a set containing only that node, and we want to merge the sets in the nodes subtree together such that each node has a set consisting of all colors in the nodes subtree. Doing this allows us to solve a variety of problems, such as query the number of distinct colors in each subtree. Doing this naively, however, yields a runtime complexity of O(N^2).

However, with just a few lines of code, we can significantly speed this up.

if(a.size() < b.size()){ //for two sets a and b
	swap(a,b);
}

In other words, by merging the smaller set into the larger one, the runtime complexity becomes O(N\log N).

Proof

When merging two sets, you move from the smaller set to the larger set. If the size of the smaller set is X, then the size of the resulting set is at least 2X. Thus, an element that has been moved Y times will be in a set of size 2^Y, and since the maximum size of a set is N (the root), each element will be moved at most O(\log N) times leading to a total complexity of O(N\log N).

Additionally, a set doesn't have to be an std::set. Many data structures can be merged, such as std::map or even adjacency lists.

info | Prove that if you instead merge sets that have size equal to the depths of the subtrees, then small to large merging does O(N) insert calls.

Further Reading

• "Merging Data Structures" from CPH 18

Problems

• Favorite Colors (USACO Gold)
  • Merge Adjacency Lists