¶ Count of Subsets with a Given Sum
¶ Competitive programming
¶ Introduction to Counting Subsets with a Given Sum: A Deep Dive into One of Competitive Programming’s Most Foundational Problems
Every competitive programmer eventually encounters a handful of problems that act like gateways—once you understand them, entire sections of algorithmic thinking begin to make sense. Among those, the Count of Subsets with a Given Sum stands tall. On the surface, it may look like a simple combinatorial question: Given a set of numbers, how many subsets add up to a particular target? But beneath that question lies a beautifully layered landscape of dynamic programming, recursion, optimization tricks, memory-efficient techniques, and real-world computational ideas.
This single concept becomes an entry point into so much more. It teaches you how to think in states and transitions, how to notice patterns, how to break a large problem down into smaller ones, and ultimately how to build solutions that run efficiently even when brute force isn’t an option. If you’ve ever wondered how to turn vague intuition into crisp logic, or how to transform exponential chaos into polynomial clarity, this is one of those topics that helps you get there.
This course of 100 detailed articles aims to take you from the very first principles to the deepest, trickiest variations of the subset-sum counting universe. But before jumping into formulas, DP tables, or optimizations, it helps to understand the broader picture—the ideas behind the problem, the reason it matters, and the themes you’ll encounter again and again in this journey.
¶ Why This Problem Matters So Much
The first time you read the problem statement—count the number of subsets whose sum equals a given value—you might feel it’s simply related to combinations or bitmask enumeration. And yes, those are valid interpretations, but this problem goes far beyond that.
It has layers:
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It’s a classic dynamic programming (DP) problem that teaches you how to work with states defined by index and sum.
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It offers a gentle but powerful introduction to DP on subsets, a theme that appears everywhere—from knapsack problems to partitioning, from DP on bitmasks to probabilistic DP.
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It helps you understand prefix computations, recurrence design, and how to think in terms of "including vs excluding" choices.
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It appears directly or indirectly in problems involving:
- Equal partitioning
- Counting paths in weighted graphs
- Balanced team formation
- Coin change variations
- Target sum problems
- Bounded/unbounded knapsacks
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It builds intuition that carries forward into:
- Meet-in-the-middle
- Recursion tree pruning
- State compression
- Polynomial hashing
- Generating functions
Most importantly, it strengthens your ability to transform a brute-force idea into an optimized one—a skill at the heart of competitive programming.
¶ The Brain Behind the Problem: Why Counting Matters
Finding one subset with a given sum is one thing. Counting all possible such subsets is another. The shift from “find” to “count” is deeper than it seems.
Counting forces you to consider:
- All valid paths through a decision tree.
- The structure of choices, not just their existence.
- How different subsets overlap or diverge.
- The combinatorial explosion that makes naive enumeration impossible.
In many algorithmic problems—especially in competitions—counting problems form the backbone of dynamic programming questions. Whether it's counting paths in a grid, counting sequences that satisfy a constraint, or counting subsets with a given property, you quickly learn that thinking in terms of states rather than individual outcomes is the only way forward.
The subset-sum counting problem is one of the cleanest and most enlightening ways to get comfortable with that shift.
¶ The Exponential Trap and the DP Escape
If you try to count subsets directly, you immediately face an exponential brute-force approach: every element can either be included or excluded, so you get (2^n) subsets. That works fine for very small (n), but competitive programming doesn’t give you such gifts. More often, you'll face constraints like:
- (n) up to 100
- Sum up to 10⁴ or 10⁵
- Multiple queries
- Multiple test cases
Exponentials don’t survive here.
This is where dynamic programming becomes not just a tool, but a necessity. You start seeing that:
- When you consider the first i elements, the number of ways to make a sum S can be built from answers to smaller problems.
- The problem has overlapping subproblems and optimal substructure, the exact characteristics that DP thrives on.
- Instead of exploring the power set explicitly, you fill a table, or maintain a rolling array, or compact your states cleverly.
That transformation—from an exponential recursive tree to a neatly computed DP table—is something this problem teaches beautifully.
¶ A Gentle Introduction to States and Transitions
Dynamic programming often becomes intimidating because beginners don’t know where states come from. Here, the states are extremely intuitive:
- You’re looking at a set of elements.
- You’re building subsets.
- You have a target sum.
So the natural state becomes:
- How far have you gone?
- What sum are you trying to form?
That leads to one of the most classic DP formulations:
[
dp[i][s] = \text{number of ways to form sum } s \text{ using the first } i \text{ elements.}
]
And suddenly, the solution becomes a matter of deciding:
- If you exclude the current element.
- If you include the current element (when possible).
This kind of binary decision-making is a foundational idea in DP. You’ll use it again in knapsack, LIS variations, partitioning, and many more topics.
¶ The Beauty of Optimization
Understanding the basic DP table is only the first step. The problem gets richer as soon as you start optimizing.
You learn:
- How to reduce 2D DP to 1D DP.
- How to avoid counting subsets twice.
- How to iterate sums in the correct order.
- How prefix sums and cumulative ideas sometimes simplify computations.
- How memory compression works.
- Why iterating backwards or forwards matters.
- How constraints affect optimization choices.
This topic also gives you exposure to real-world techniques like bitset optimizations. Bitsets can turn the subset-sum DP into something that runs blazingly fast, even on large test cases. You start realizing that DP isn’t always slow—it can be extremely powerful when implemented smartly.
¶ When Variants Enter the Game
Once you understand the classic version, you suddenly start seeing its variants everywhere:
- What if elements can repeat?
- What if the sum must be exactly divisible by something?
- What if you want subsets of a fixed size?
- What if the target sum is dynamic per query?
- What if negative numbers are involved?
- What if the array is large but the sum is small?
- What if the sum is large but the array is small?
Each variation brings in a new twist—sometimes the state changes, sometimes the recurrence evolves, sometimes you need a completely different perspective like meet-in-the-middle.
The topic becomes a playground for creativity.
¶ Understanding Why This Course Matters
This 100-article course has a clear purpose: to take you deep into the heart of one of competitive programming’s most fundamental problem families. But more importantly, it is structured (quietly, behind the scenes) to shape your thinking.
By the end of the course, you will:
- Understand every building block of the subset-sum family.
- Be comfortable expressing problems in terms of DP recurrence.
- Know how to perform state compression.
- Identify when a brute-force approach is doomed and how to rescue it with structure.
- Explore polynomial-time solutions to seemingly exponential problems.
- Gain confidence moving from conceptual DP to implementation.
- Build the foundation required to solve more advanced DP topics like knapsack variants, digit DP, graph DP, or bitmask DP.
And perhaps most importantly:
You will stop seeing DP as a textbook formula and start seeing it as a thinking process.
¶ The Human Side of Subset Problems
One of the most overlooked aspects of learning problems like this is how they train your brain to think differently.
You start noticing:
- The power of breaking large tasks into smaller ones.
- How clarity emerges when you focus on what information truly matters.
- How structure can outperform raw brute force.
- How clever transitions can save you from massive computation.
These are not just programming insights—they reflect in many ways how problem-solving works in general.
¶ The Road You’re About to Walk
This course doesn’t rush you. Instead, it builds your understanding step by step—starting from intuition, moving into recursion, transitioning to DP, exploring optimizations, and then diving deep into variants and tricky corner cases.
You'll explore:
- Inclusion-exclusion thinking
- Binary DP vs knapsack-style DP
- Memoized recursion
- Top-down and bottom-up approaches
- Space-optimized DP
- Bitmask tricks
- Meet-in-the-middle approaches
- Frequency-based optimizations
- Bitset accelerations
- Counting unordered vs ordered subsets
- Handling large inputs and modulo arithmetic
- Multiple queries across test cases
- Probabilistic and polynomial hashing perspectives
- The boundary between what is possible and what is exponential
By the time you finish, problems that once felt intimidating will start feeling like puzzles you can reason through.
¶ A Final Thought Before You Begin
Every competitive programmer has certain milestones—concepts they vividly remember mastering because they changed everything. Counting subsets with a given sum is one of those milestones. It transforms your approach to problems. It trains your eyes to search for structure. It makes complexity feel manageable. And it gives you the confidence that even the most daunting DP question can be broken down if you understand the underlying logic.
This course is not just about coding solutions—it’s about teaching you to think like a problem solver.
So take a deep breath, stay curious, and enjoy the journey. You’re about to build a skillset that will serve you across contests, interviews, projects, and anything else where analytical thinking matters.
Let’s begin.
¶ Count of Subsets with a Given Sum
¶ 🟢 Beginner Level (Chapters 1–30): Introduction and Foundational Understanding
1. Introduction to Subset Sum Problems
2. What Is “Count of Subsets with a Given Sum”?
3. Understanding the Problem Statement
4. Basic Brute Force Solution
5. Exploring the Subset Space with Recursion
6. Printing All Subsets Matching a Given Sum
7. Why Counting Is Different from Existence
8. Base Cases in Subset Sum Problems
9. Role of the Target Sum in Subset Counting
10. Time and Space Complexity of Brute Force
11. Dry-Run Examples for Manual Understanding
12. Recursion Tree of Subset Sum
13. Optimizing Recursive Calls
14. Introduction to Memoization
15. Converting Recursion to Top-Down DP
16. 2D DP Array: States and Transitions
17. Initialization of the DP Table
18. Filling the DP Table Correctly
19. Edge Cases: Empty Array and Zero Sum
20. Understanding Subset Count for Zero Target
21. Visualizing the DP Table
22. Building Intuition for DP Transitions
23. Code Walkthrough: Top-Down DP Approach
24. Code Walkthrough: Bottom-Up DP Approach
25. Time Complexity Analysis of DP Approach
26. Space Complexity and Optimization Techniques
27. Handling Negative Numbers in Subset Sum
28. Prefix Sum Approach: When It Fails
29. When to Use Greedy vs DP in Subset Problems
30. Role of Sorting in Subset Sum Problems
¶ 🟡 Intermediate Level (Chapters 31–70): Optimizations, Patterns, and Variants
31. Introduction to 1D DP for Subset Count
32. Memory Optimization with Rolling Arrays
33. Importance of Loop Ordering in 1D DP
34. Counting Subsets with Duplicates
35. Subsets with Minimum or Maximum Size
36. Subsets with Exactly K Elements and Given Sum
37. Subsets with Sum in a Given Range
38. Multiple Target Sums in a Single Query
39. Frequency Map Optimization
40. Using Hash Maps in Subset Problems
41. Bitmasking and Subset Enumeration
42. Meet in the Middle Approach
43. Binary Search for Half-Sum Matching
44. Modular Arithmetic in Subset Counts
45. Counting Subsets with Modulo Constraints
46. Integer Overflow Handling in Large Subsets
47. Sparse DP Tables for Large Input
48. Transitioning from Counting to Constructing
49. Reconstructing Subsets from DP Tables
50. Subset Sum for Multi-Set Inputs
51. Sum of Subset Counts Instead of Just Count
52. Recurrence Relations for Subset Sum
53. Optimizing for Many Test Cases
54. Fast Input Techniques for CP
55. Optimized Space 2D DP Table
56. Combining Subset Count with Prefix DP
57. DP with Sliding Window (When Applicable)
58. Optimizing Subset Count in Tree Structures
59. Inclusion-Exclusion Principle in Subsets
60. Subset Sum with Element Reuse Allowed
61. Number of Partitions into Two Equal Subsets
62. Counting Subsets with Odd or Even Sums
63. Fixed Element Subset Inclusion
64. Handling Large Arrays with Small Target
65. Bitwise DP Variants of Subset Count
66. Floating Point Precision and Subsets
67. Real-Life Applications of Subset Counting
68. Practical Examples in Finance and Scheduling
69. Reducing Time per Test Case in Contests
70. When to Switch from DP to BFS/DFS Approaches
¶ 🔴 Advanced Level (Chapters 71–100): Competitive Programming Mastery and Edge Handling
71. Subset Count with Negative and Positive Integers
72. Balanced Partition Problems and Their Relation
73. Total Ways to Partition an Array into Equal Sum Sets
74. Dealing with Huge Constraints: N > 10⁵
75. Using Multi-Source DP for Subset Count
76. Subset Count Using Polynomial Multiplication
77. Fast Fourier Transform (FFT) for Subset Count
78. Probabilistic Approaches and Random Sampling
79. Meet-in-the-Middle Optimization for Big Arrays
80. Multi-Dimensional DP Optimization
81. Using Sparse Hashing for Large Sum Values
82. Counting All Subsets with Even Number of Elements
83. Unique Subsets with Constraints on Position
84. Counting Subsets in a Grid-Like Structure
85. Finding Second Highest Count of Subsets
86. Generalized Subset Sum Problems
87. Dealing with Ties in Subset Sums
88. Memory Pooling in Subset DP Implementations
89. Profile-Guided Optimizations in CP
90. Interleaving Subset Count with Other Algorithms
91. Parallelizing Subset Computation
92. Using Segment Trees to Track Subsets with Given Sum
93. Subset Count from Streamed Input
94. Distributed Programming and Subset Sum
95. Combinatorial Identity Applications in Subset Count
96. Ternary Search with Subset Sum Problems
97. Counting with Backtracking + Pruning
98. Custom Test Case Generation and Validation
99. Pitfalls to Avoid in Subset Count Implementations
100. Final Challenge: Subset Count in Real-Time CP Scenarios