Amortized Analysis — Aggregate, Accounting, and Potential Methods Explained | Chapter 16 of Intro to Algorithms

Amortized analysis provides a rigorous way to analyze the average performance of data structure operations over time—even when some individual operations are costly. Chapter 16 of introduction to algorithms introduces the three main techniques of amortized analysis: aggregate analysis, the accounting method, and the potential method. With clear stack and binary counter examples, plus practical guidance on dynamic table resizing, this chapter equips you with the tools to guarantee efficient performance in real-world algorithms.

For a full walkthrough of amortized analysis techniques, watch the chapter summary video below. Be sure to subscribe to Last Minute Lecture for more algorithm tutorials.

Aggregate Analysis

Aggregate analysis determines the total cost T(n) for n operations, then divides by n to find the amortized cost per operation. For example, a stack supporting PUSH, POP, and MULTIPOP has total cost O(n), since each object can be popped at most once per push. Similarly, incrementing a binary counter n times results in a total number of bit flips O(n), so the amortized cost per increment is O(1).

The Accounting Method

The accounting method assigns hypothetical costs (credits) to operations. Some operations are intentionally overcharged, storing “credit” that pays for later expensive steps. For example, charging $2 for PUSH (with $1 saved on each element) allows POP and MULTIPOP operations to use the stored credits, ensuring all pops are covered. The binary counter example charges $2 for setting a bit to 1, saving $1 to pay for future bit resets. The accounting method guarantees credits never go negative and gives a robust way to prove amortized costs.

The Potential Method

The potential method defines a potential function Φ(D) that represents prepaid work stored in the data structure’s state. Amortized cost is calculated as the actual cost plus the change in potential: amortized_cost = actual_cost + Φ(new) – Φ(old). For stacks, potential is the number of items; for binary counters, it’s the number of 1s. By ensuring potential never drops below zero, this method elegantly captures the idea of “stored energy” in the system, making it easy to prove amortized bounds.

Dynamic Tables — Expansion and Contraction

Dynamic tables (resizable arrays) demonstrate amortized analysis in practice. When a table is full, it doubles in size—this “expansion” can be expensive, but happens rarely. The accounting method charges extra for each insert, saving up credits to pay for future expansions. Contraction happens when the load factor drops below 1/4, with the potential method adjusting Φ based on the table’s size and number of elements. With proper analysis, both insert and delete operations are guaranteed to run in O(1) amortized time, ensuring efficient resizing without wasted space.

Glossary of Key Terms

• Amortized Analysis: Averages cost of operations over sequences
• Aggregate Analysis: Divides total cost by number of operations
• Accounting Method: Uses stored credits for costly steps
• Potential Method: Tracks prepaid work with a potential function
• Dynamic Table: Resizable array supporting efficient insert/delete
• Expansion/Contraction: Table grows or shrinks to match usage
• Credit: Overcharged units saved for future operations
• Potential Function (Φ): Quantifies stored work in data structure
• Load Factor: Ratio of items to table capacity
• Binary Counter: Bit array incremented sequentially
• MULTIPOP: Stack operation removing up to k items
• Aggregate Bound: Worst-case total cost for a sequence

Conclusion: Mastering Efficient Data Structures

Amortized analysis is crucial for designing data structures and algorithms that perform reliably in practice, not just in isolated cases. By learning aggregate, accounting, and potential methods, you can confidently guarantee efficiency over long-running processes. Chapter 16 of introduction to algorithms provides a blueprint for applying these tools, from stacks to dynamic tables and beyond.

Continue exploring algorithmic insights and deep-dives—subscribe to Last Minute Lecture and browse the complete introduction to algorithms chapter summary series.