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

id	title	author	prerequisites
data-structures-gold	Data Structures (Gold)	Michael Cao	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)
  • Extremely difficult.

(add more once codeforces comes up)