Algorithms
Advanced Greedy & Data Structures
Difficulty: Advanced
"For each element in the array, find the next greater element" is a brute-force O(n^2) problem until you notice that a stack maintained in decreasing order lets you answer every query as a side effect of a single left-to-right scan. The...
Advanced Greedy & Data Structures
"For each element in the array, find the next greater element" is a brute-force O(n^2) problem until you notice that a stack maintained in decreasing order lets you answer every query as a side effect of a single left-to-right scan. The whole algorithm runs in O(n), and the same monotonic-stack pattern solves trapping rain water, largest rectangle in a histogram, stock spans, and a long tail of related problems with the same template.
Advanced Greedy & Data Structures is where greedy thinking meets specialized scaffolding. You will implement the monotonic stack pattern for next-greater and next-smaller variants, the monotonic deque for sliding-window maximum and minimum (also a key DP optimization), and sweep-line algorithms for interval and event problems (meeting rooms, interval merge, rectangle area union). The lesson closes with advanced greedy problems that need a heap or priority queue: full Huffman encoding, task scheduling with cooldown, the gas station problem, and activity selection with deadlines and profits.
In Greedy (Intro), you saw simple greedy strategies that needed only sorting. Heaps & Priority Queue taught you O(log n) access to the minimum or maximum, which is exactly what these advanced greedy patterns rely on for their efficiency.
Next, Branch and Bound extends backtracking with the same kind of bounding-function pruning, applied to optimization search trees.
Topics:
What You'll Learn
By the end of this lesson, you will be able to:
Implement the monotonic stack pattern to find next greater/smaller elements in O(n) and apply it to largest rectangle in histogram.
Use a monotonic deque to compute sliding window maximum in O(n) and explain why each element is processed at most twice.
Apply the sweep line technique to solve interval overlap problems like meeting rooms and maximum concurrent events.
Solve advanced greedy problems including Huffman encoding with a priority queue and the gas station circular tour.
Analyze the amortized O(n) complexity of monotonic stack and queue operations.
Why This Matters
01
The monotonic stack pattern transforms O(n^2) brute-force solutions for next greater/smaller element into O(n), and solves classic problems like largest rectangle in histogram and trapping rain water.
02
Monotonic queues compute sliding window maximum/minimum in O(n) total, enabling efficient solutions to streaming data problems and DP optimization.
03
Sweep line algorithms process events in sorted order to solve interval problems (meeting rooms, rectangle area union) in O(n log n), a technique used in computational geometry and scheduling.
04
Advanced greedy problems like Huffman encoding and task scheduling combine local optimization with data structures (heaps, stacks) to achieve globally optimal solutions.
Key Terms
6 terms you'll encounter in this lesson
Monotonic Stack
A stack that maintains elements in strictly increasing or decreasing order. When pushing a new element, all elements that violate the monotonic property are popped first. Used for next greater/smaller element problems.
Next Greater Element
For each element in an array, the next greater element is the first element to its right that is larger. A monotonic decreasing stack solves this in O(n) by popping elements that are smaller than the current element.
Monotonic Queue (Deque)
A deque that maintains elements in monotonic order, supporting O(1) access to the maximum (or minimum) of a sliding window. Elements are removed from both ends as the window slides.
Sweep Line
An algorithmic technique that processes events (typically interval starts and ends) in sorted order, maintaining a running state. Used for interval scheduling, overlap counting, and computational geometry.
Largest Rectangle in Histogram
Given bar heights, find the area of the largest rectangle that fits under the histogram. Solved in O(n) using a monotonic increasing stack that processes each bar at most twice.
Huffman Encoding
A greedy algorithm that builds an optimal prefix-free binary encoding by repeatedly merging the two lowest-frequency symbols. Uses a min-heap (priority queue) and produces a Huffman tree.
Heads Up
Common misconceptions to watch out for
"Monotonic stack is O(n^2) because of the inner while loop"
Each element is pushed once and popped at most once, so total operations across all iterations is O(n). The while loop does not run n times for each element — it is amortized O(1) per element.
"Sliding window maximum requires a heap for O(n log n)"
A monotonic deque achieves O(n) total — each element enters and leaves the deque at most once. A heap approach gives O(n log n) because removal of arbitrary elements is O(log n), which is suboptimal.
"Sweep line only works for intervals on a number line"
Sweep line extends to 2D (rectangle union area, segment intersection), 3D, and non-geometric problems (event processing, scheduling). The core idea — process events in sorted order — applies broadly.