Here is a curated list of 100 chapter titles focused on the "Count of Subsets with a Given Sum" problem, structured from basic theory to advanced competitive programming techniques. This guide explores variations, optimizations, real-world applications, and edge-case strategies.
¶ 🟢 Beginner Level (Chapters 1–30): Introduction and Foundational Understanding
- Introduction to Subset Sum Problems
- What Is “Count of Subsets with a Given Sum”?
- Understanding the Problem Statement
- Basic Brute Force Solution
- Exploring the Subset Space with Recursion
- Printing All Subsets Matching a Given Sum
- Why Counting Is Different from Existence
- Base Cases in Subset Sum Problems
- Role of the Target Sum in Subset Counting
- Time and Space Complexity of Brute Force
- Dry-Run Examples for Manual Understanding
- Recursion Tree of Subset Sum
- Optimizing Recursive Calls
- Introduction to Memoization
- Converting Recursion to Top-Down DP
- 2D DP Array: States and Transitions
- Initialization of the DP Table
- Filling the DP Table Correctly
- Edge Cases: Empty Array and Zero Sum
- Understanding Subset Count for Zero Target
- Visualizing the DP Table
- Building Intuition for DP Transitions
- Code Walkthrough: Top-Down DP Approach
- Code Walkthrough: Bottom-Up DP Approach
- Time Complexity Analysis of DP Approach
- Space Complexity and Optimization Techniques
- Handling Negative Numbers in Subset Sum
- Prefix Sum Approach: When It Fails
- When to Use Greedy vs DP in Subset Problems
- Role of Sorting in Subset Sum Problems
- Introduction to 1D DP for Subset Count
- Memory Optimization with Rolling Arrays
- Importance of Loop Ordering in 1D DP
- Counting Subsets with Duplicates
- Subsets with Minimum or Maximum Size
- Subsets with Exactly K Elements and Given Sum
- Subsets with Sum in a Given Range
- Multiple Target Sums in a Single Query
- Frequency Map Optimization
- Using Hash Maps in Subset Problems
- Bitmasking and Subset Enumeration
- Meet in the Middle Approach
- Binary Search for Half-Sum Matching
- Modular Arithmetic in Subset Counts
- Counting Subsets with Modulo Constraints
- Integer Overflow Handling in Large Subsets
- Sparse DP Tables for Large Input
- Transitioning from Counting to Constructing
- Reconstructing Subsets from DP Tables
- Subset Sum for Multi-Set Inputs
- Sum of Subset Counts Instead of Just Count
- Recurrence Relations for Subset Sum
- Optimizing for Many Test Cases
- Fast Input Techniques for CP
- Optimized Space 2D DP Table
- Combining Subset Count with Prefix DP
- DP with Sliding Window (When Applicable)
- Optimizing Subset Count in Tree Structures
- Inclusion-Exclusion Principle in Subsets
- Subset Sum with Element Reuse Allowed
- Number of Partitions into Two Equal Subsets
- Counting Subsets with Odd or Even Sums
- Fixed Element Subset Inclusion
- Handling Large Arrays with Small Target
- Bitwise DP Variants of Subset Count
- Floating Point Precision and Subsets
- Real-Life Applications of Subset Counting
- Practical Examples in Finance and Scheduling
- Reducing Time per Test Case in Contests
- When to Switch from DP to BFS/DFS Approaches
¶ 🔴 Advanced Level (Chapters 71–100): Competitive Programming Mastery and Edge Handling
- Subset Count with Negative and Positive Integers
- Balanced Partition Problems and Their Relation
- Total Ways to Partition an Array into Equal Sum Sets
- Dealing with Huge Constraints: N > 10⁵
- Using Multi-Source DP for Subset Count
- Subset Count Using Polynomial Multiplication
- Fast Fourier Transform (FFT) for Subset Count
- Probabilistic Approaches and Random Sampling
- Meet-in-the-Middle Optimization for Big Arrays
- Multi-Dimensional DP Optimization
- Using Sparse Hashing for Large Sum Values
- Counting All Subsets with Even Number of Elements
- Unique Subsets with Constraints on Position
- Counting Subsets in a Grid-Like Structure
- Finding Second Highest Count of Subsets
- Generalized Subset Sum Problems
- Dealing with Ties in Subset Sums
- Memory Pooling in Subset DP Implementations
- Profile-Guided Optimizations in CP
- Interleaving Subset Count with Other Algorithms
- Parallelizing Subset Computation
- Using Segment Trees to Track Subsets with Given Sum
- Subset Count from Streamed Input
- Distributed Programming and Subset Sum
- Combinatorial Identity Applications in Subset Count
- Ternary Search with Subset Sum Problems
- Counting with Backtracking + Pruning
- Custom Test Case Generation and Validation
- Pitfalls to Avoid in Subset Count Implementations
- Final Challenge: Subset Count in Real-Time CP Scenarios