The Four Core Patterns of Divide and Conquer
Understanding Divide and Conquer Patterns
Divide and conquer algorithms can be applied to a wide variety of problems, but most of them follow one of these four core patterns.
Each pattern represents a different way of thinking about and applying divide and conquer to solve problems.
Closest Pair of Points Pattern
Strassen's Matrix Multiplication
Learning Objective
Understand the core patterns of divide and conquer and how they can be applied to solve different types of problems.
The Four Core Patterns of Divide and Conquer
While divide and conquer can be applied to many different problems, most solutions follow one of these four core patterns. Understanding these patterns will help you recognize when and how to apply divide and conquer to new problems.
Merge-based Pattern
Divide the problem into smaller subproblems, solve them independently, and then merge the results.
Examples: Merge Sort, Counting Inversions
Partition-based Pattern
Partition the problem around a pivot, solve the partitions independently, and combine the results.
Examples: Quick Sort, Quickselect
Geometric Pattern
Divide the geometric space into regions, solve each region, and combine the results while checking boundaries.
Examples: Closest Pair of Points, Convex Hull
Matrix/Numerical Pattern
Break down matrices or numbers into smaller components, perform operations, and combine results.
Examples: Strassen's Matrix Multiplication, Karatsuba Algorithm
Closest Pair of Points Pattern
Choosing the Right Pattern
When faced with a problem that might be solved using divide and conquer, ask yourself these questions to identify the most appropriate pattern:
For Merge-based Pattern
"Can I divide the problem into independent subproblems and then merge their solutions efficiently?"
For Partition-based Pattern
"Can I partition the problem around a pivot element and solve the partitions independently?"
For Geometric Pattern
"Is this a geometric problem where I can divide the space and solve each region independently?"
For Matrix/Numerical Pattern
"Am I working with matrices or large numbers that can be broken down into smaller components?"