Project 11: Combinatorial Object Generator

Generate permutations, combinations, and Gray codes with minimal change.

Quick Reference

Attribute Value
Difficulty Advanced
Time Estimate 2 weeks
Language C
Prerequisites Recursion, iteration
Key Topics combinatorics, generation order, Gray codes

1. Learning Objectives

  1. Generate permutations in lexicographic order.
  2. Generate combinations and subsets.
  3. Implement Gray code and minimal-change generation.
  4. Validate counts against combinatorial identities.

2. Theoretical Foundation

2.1 Core Concepts

  • Lexicographic generation: next permutation algorithm.
  • Gray codes: minimal-change sequences.
  • Counting identities: n!, C(n,k), 2^n.

2.2 Why This Matters

TAOCP volume 4 emphasizes generating objects efficiently, not just counting them.

2.3 Historical Context / Background

Knuth formalized generation algorithms and minimal-change methods.

2.4 Common Misconceptions

  • “Recursion is the only way”: Iterative algorithms are often faster.
  • “Any order is fine”: Order affects algorithmic efficiency.

3. Project Specification

3.1 What You Will Build

A library that generates permutations, combinations, and Gray codes with configurable order.

3.2 Functional Requirements

  1. Next permutation algorithm.
  2. k-combination generator.
  3. Binary Gray code sequence.
  4. Count verification and output options.

3.3 Non-Functional Requirements

  • Correctness: All objects generated exactly once.
  • Performance: Minimal overhead per object.
  • Usability: Iterative and callback APIs.

3.4 Example Usage / Output

permute 1 2 3 -> 123, 132, 213, 231, 312, 321

3.5 Real World Outcome

You can generate combinatorial objects for testing and algorithm input generation.

4. Solution Architecture

4.1 High-Level Design

┌──────────────┐     ┌──────────────┐     ┌──────────────┐
│ generator    │────▶│ sequence     │────▶│ callback     │
└──────────────┘     └──────────────┘     └──────────────┘

4.2 Key Components

Component Responsibility Key Decisions
Permuter next permutation lexicographic order
Combiner k-combinations iterative algorithm
Gray code minimal change binary reflected

4.3 Data Structures

typedef struct {
    int *arr;
    size_t n;
} PermState;

4.4 Algorithm Overview

Key Algorithm: Next permutation

  1. Find pivot i where a[i] < a[i+1].
  2. Swap with rightmost larger element.
  3. Reverse suffix.

Complexity Analysis:

  • Time: O(n) per permutation step.
  • Space: O(1).

5. Implementation Guide

5.1 Development Environment Setup

gcc --version

5.2 Project Structure

project-root/
├── src/gen.c
├── include/gen.h
├── tests/
└── Makefile

5.3 The Core Question You’re Answering

“How do I enumerate objects without duplicates or omissions?”

5.4 Concepts You Must Understand First

  • Lexicographic ordering
  • Combinatorial counting
  • Gray code properties

5.5 Questions to Guide Your Design

  1. How will you ensure uniqueness of generated objects?
  2. What order is most useful for downstream algorithms?
  3. How will you verify counts efficiently?

5.6 Thinking Exercise

Prove that binary reflected Gray code changes only one bit at a time.

5.7 The Interview Questions They’ll Ask

  1. How do you generate the next permutation?
  2. Why use Gray codes?
  3. How do you ensure no duplicates?

5.8 Hints in Layers

Hint 1: Implement next permutation. Hint 2: Add combination generator. Hint 3: Add Gray code sequence.

5.9 Books That Will Help

Topic Book Chapter
Combinatorial generation TAOCP Vol 4A Ch. 7.2.1

5.10 Implementation Phases

Phase 1: Foundation (4-5 days)

  • Permutations with next-perm.

Phase 2: Core Functionality (4-5 days)

  • Combinations and subsets.

Phase 3: Polish & Edge Cases (3-4 days)

  • Gray codes and performance.

5.11 Key Implementation Decisions

Decision Options Recommendation Rationale
Generation order lexicographic vs minimal change both learning value

6. Testing Strategy

6.1 Test Categories

Category Purpose Examples
Counts completeness n! and C(n,k)
Uniqueness no duplicates hash set checks
Order lexicographic known sequences

6.2 Critical Test Cases

  1. n=3 permutation order matches known sequence.
  2. Combinations count matches C(n,k).
  3. Gray code changes one bit at each step.

6.3 Test Data

n=3,4; k=2

7. Common Pitfalls & Debugging

7.1 Frequent Mistakes

Pitfall Symptom Solution
Missing objects count mismatch validate counts
Duplicate generation repeated outputs use visited set
Wrong order mismatched sequence compare to references

7.2 Debugging Strategies

  • Log the first N outputs.
  • Validate counts at end of run.

7.3 Performance Traps

Storing all objects can be memory heavy; stream results via callback.

8. Extensions & Challenges

8.1 Beginner Extensions

  • Add integer partitions.

8.2 Intermediate Extensions

  • Generate permutations with constraints.

8.3 Advanced Extensions

  • Implement cool-lex order.

9. Real-World Connections

  • Test case generation, search algorithms, and combinatorial optimization.

10. Resources

  • TAOCP Vol 4A Chapter 7.2.1.

11. Self-Assessment Checklist

  • I can explain next-permutation.
  • My generators match correct counts.

12. Submission / Completion Criteria

Minimum: Permutations and combinations. Full: Gray codes and counts verification. Excellence: Advanced generation orders.

This guide was generated from LEARN_TAOCP_DEEP_DIVE.md. For the complete learning path, see the parent directory README.