Generate All Combinations Sum K in DSA Rust - Deep Dive

Overview - Generate All Combinations Sum K

What is it?

Generate All Combinations Sum K means finding all possible groups of numbers from a list that add up exactly to a target number K. Each number can be used once or multiple times depending on the problem rules. The goal is to list every unique combination without missing any or repeating the same group. This helps solve problems where you want to explore all ways to reach a total using given parts.

Why it matters

Without this concept, solving problems like finding all ways to pay a bill with coins or all ingredient mixes for a recipe would be guesswork or incomplete. It helps computers explore all options systematically and efficiently. This is important in planning, resource allocation, and decision-making where every possible solution matters.

Where it fits

Before this, you should understand basic recursion and arrays or lists. After mastering this, you can learn more complex backtracking problems, dynamic programming, and optimization techniques.

Mental Model

Core Idea

Find all groups of numbers that add up to K by trying each number and exploring further only if it doesn't exceed K.

Think of it like...

It's like trying different combinations of ingredients to bake a cake that weighs exactly K grams, testing each mix and backtracking if it gets too heavy.

Start
  ↓
Choose a number -> Add to current sum
  ↓               ↓
If sum == K -> Save combination
If sum > K -> Backtrack
Else -> Try next number
  ↓
Repeat until all numbers tried

Build-Up - 7 Steps

1

Foundation

Understanding the Problem Setup

Concept: Introduce the problem of finding combinations that sum to K using a list of numbers.

Imagine you have a list of numbers, for example [2, 3, 6, 7], and a target sum K = 7. Your task is to find all unique groups of these numbers that add up exactly to 7. For instance, [7] and [2, 2, 3] are valid groups if repetition is allowed.

Result

You understand the goal: find all groups of numbers from the list that sum to K.

Understanding the problem clearly is the first step to solving it; knowing what counts as a valid combination guides the solution approach.

2

Foundation

Basics of Recursion for Exploration

Concept: Use recursion to explore choices by picking numbers and checking sums.

Recursion means a function calls itself to solve smaller parts of the problem. Here, you pick a number, add it to a running total, and call the function again to pick the next number. If the total equals K, you save the combination. If it exceeds K, you stop exploring that path.

Result

You can explore all possible combinations by trying each number recursively.

Recursion naturally fits problems where you explore many possible choices step-by-step.

3

Intermediate

Avoiding Duplicate Combinations

🤔Before reading on: do you think [2,3] and [3,2] are considered different combinations? Commit to your answer.

Concept: Prevent counting the same combination in different orders by controlling the order of picking numbers.

To avoid duplicates like [2,3] and [3,2], always pick numbers from the current position or forward in the list. This means you never go back to earlier numbers, ensuring combinations are built in a sorted order.

Result

All combinations are unique and no repeats occur due to order differences.

Controlling the order of choices prevents redundant work and keeps results clean.

4

Intermediate

Handling Number Repetition Rules

🤔Before reading on: do you think numbers can be used unlimited times or only once? Commit to your answer based on problem rules.

Concept: Adjust recursion to allow numbers to be reused or used once depending on the problem variant.

If repetition is allowed, after picking a number, you call recursion with the same index to pick it again. If repetition is not allowed, you move to the next index. This small change controls how many times a number can appear in combinations.

Result

You can solve both variants: with and without repetition.

A simple change in recursion parameters adapts the solution to different problem constraints.

5

Intermediate

Pruning to Improve Efficiency

🤔Before reading on: do you think checking sums early helps speed up the search? Commit to yes or no.

Concept: Stop exploring paths as soon as the sum exceeds K to save time.

During recursion, if the running sum goes beyond K, you stop exploring that path immediately. This pruning avoids unnecessary work and speeds up the search.

Result

The algorithm runs faster by ignoring impossible combinations early.

Pruning reduces the search space drastically, making the solution practical for larger inputs.

6

Advanced

Rust Implementation with Backtracking

🤔Before reading on: do you think Rust's ownership rules affect how you pass the current combination in recursion? Commit to your answer.

Concept: Implement the combination sum using Rust's recursion and backtracking with careful handling of mutable state.

Use a recursive function that takes the current index, the current sum, and a mutable vector for the current combination. Add numbers, recurse, then remove them (backtrack) to try other options. Rust's ownership and borrowing require passing mutable references carefully. Example code: fn combination_sum(candidates: &[i32], target: i32) -> Vec> { fn backtrack(start: usize, target: i32, candidates: &[i32], current: &mut Vec, result: &mut Vec>) { if target == 0 { result.push(current.clone()); return; } for i in start..candidates.len() { if candidates[i] > target { break; } current.push(candidates[i]); backtrack(i, target - candidates[i], candidates, current, result); current.pop(); } } let mut result = Vec::new(); let mut current = Vec::new(); let mut sorted_candidates = candidates.to_vec(); sorted_candidates.sort_unstable(); backtrack(0, target, &sorted_candidates, &mut current, &mut result); result } fn main() { let candidates = vec![2, 3, 6, 7]; let target = 7; let combos = combination_sum(&candidates, target); for combo in combos { println!("{:?}", combo); } }

Result

[[2, 2, 3], [7]] printed as output

Understanding Rust's ownership and mutable references is key to implementing backtracking efficiently and safely.

7

Expert

Optimizing with Early Sorting and Deduplication

🤔Before reading on: do you think sorting candidates before recursion affects correctness or performance? Commit to your answer.

Concept: Sort input to enable pruning and handle duplicates efficiently in recursion.

Sorting candidates allows early stopping when the current number exceeds the remaining target. It also helps skip duplicates by checking if the current number equals the previous one at the same recursion level. This reduces redundant work and ensures unique combinations.

Result

Algorithm runs faster and produces unique combinations without extra data structures.

Sorting before recursion is a simple but powerful optimization that improves both speed and correctness.

Under the Hood

The algorithm uses recursion to explore a tree of choices. Each node represents picking a number or skipping it. Backtracking means after exploring a choice, the algorithm reverses it to try others. The recursion stack keeps track of the current combination and sum. Sorting and pruning reduce the number of nodes explored.

Why designed this way?

Backtracking was chosen because it naturally fits exploring all combinations without repetition. Sorting and pruning were added to handle duplicates and improve efficiency. Alternatives like dynamic programming solve related problems but don't list all combinations explicitly.

Root
├─ Pick candidate[0]
│  ├─ Pick candidate[0] again (if repetition allowed)
│  └─ Skip to candidate[1]
├─ Pick candidate[1]
│  ├─ Pick candidate[1] again
│  └─ Skip to candidate[2]
└─ ...

At each node:
- Check sum
- Save if sum == K
- Backtrack if sum > K

Myth Busters - 4 Common Misconceptions

Quick: Do you think [2,3] and [3,2] are different combinations? Commit yes or no.

Common Belief: People often think different orders count as different combinations.

Reality: Combinations are sets without order, so [2,3] and [3,2] are the same and should appear only once.

Why it matters: Counting duplicates inflates results and wastes computation, leading to incorrect answers.

Quick: Can you use each number unlimited times by default? Commit yes or no.

Common Belief: Many believe numbers can always be reused unlimited times.

Reality: Whether numbers can be reused depends on problem rules; some variants allow reuse, others don't.

Why it matters: Wrong assumptions cause incorrect solutions or missed combinations.

Quick: Does sorting candidates change the final combinations? Commit yes or no.

Common Belief: Some think sorting changes the solution set or order matters in output.

Reality: Sorting only helps efficiency and uniqueness; it does not change which combinations exist.

Why it matters: Misunderstanding sorting leads to avoiding useful optimizations.

Quick: Is pruning optional and only for speed? Commit yes or no.

Common Belief: Pruning is just a speed-up and not necessary for correctness.

Reality: Pruning is essential to avoid exploring impossible paths and keeps recursion manageable.

Why it matters: Without pruning, the algorithm can become too slow or crash on large inputs.

Expert Zone

1

Backtracking state must be carefully managed to avoid side effects due to mutable references in Rust.

2

Skipping duplicates requires checking the previous candidate at the same recursion depth, not globally.

3

Sorting candidates upfront enables both pruning and duplicate skipping without extra memory overhead.

When NOT to use

This approach is inefficient for very large inputs or when only the count of combinations is needed; dynamic programming or memoization is better then.

Production Patterns

Used in constraint solvers, puzzle games, and resource allocation tools where all possible solutions must be enumerated and filtered.

Connections

Backtracking

Builds-on

Understanding backtracking is essential because combination sum is a classic example of exploring all possibilities with pruning.

Subset Sum Problem

Related problem

Combination sum generalizes subset sum by listing all subsets that meet the sum, not just deciding existence.

Combinatorial Chemistry

Application domain

Generating all combinations that sum to a target is similar to finding all chemical compound mixtures with exact properties, showing cross-domain relevance.

Common Pitfalls

#1 Allowing duplicates by not controlling recursion start index.

Wrong approach:

for i in 0..candidates.len() {
current.push(candidates[i]);
backtrack(0, target - candidates[i], candidates, current, result);
current.pop();
}

Correct approach:

for i in start..candidates.len() {
current.push(candidates[i]);
backtrack(i, target - candidates[i], candidates, current, result);
current.pop();
}

Root cause:

Not advancing the start index causes revisiting earlier candidates, producing duplicate combinations.

#2 Not pruning when sum exceeds target, causing slow execution.

Wrong approach:

if target < 0 {
// no return here
}
// continue recursion

Correct approach:

if target < 0 {
return;
}
// continue recursion

Root cause:

Failing to stop recursion early wastes time exploring impossible paths.

#3 Modifying the current combination without backtracking (not popping).

Wrong approach:

current.push(candidates[i]);
backtrack(i, target - candidates[i], candidates, current, result);
// missing current.pop() here

Correct approach:

current.push(candidates[i]);
backtrack(i, target - candidates[i], candidates, current, result);
current.pop();

Root cause:

Not removing last choice corrupts the state for other recursive calls.

Key Takeaways

Generating all combinations that sum to K requires exploring choices recursively and backtracking to try alternatives.

Controlling the order of picking numbers prevents duplicate combinations caused by different orders.

Pruning paths that exceed the target sum improves efficiency and keeps the algorithm practical.

Sorting candidates before recursion enables early stopping and duplicate skipping without extra memory.

Rust's ownership and mutable references require careful handling of state during recursion for safe and efficient backtracking.

Practice

1. What does the problem "Generate All Combinations Sum K" ask you to find?

easy

A. All unique combinations of numbers that add up to K

B. The largest number less than K

C. The sum of all numbers from 1 to K

D. The product of all numbers that divide K

Solution

1. Step 1: Understand the problem goal

   The problem asks to find all unique sets of numbers that add exactly to the target sum K.

2. Step 2: Clarify what combinations mean

   Combinations mean groups of numbers without regard to order, and each number can be used once or multiple times depending on problem rules.

3. Final Answer:

   All unique combinations of numbers that add up to K -> Option A

4. Quick Check:

   Combinations sum K = unique sets summing to K [OK]

Hint: Focus on sums equal to K, not products or counts [OK]

Common Mistakes:

• Confusing sum with product
• Looking for single numbers instead of combinations
• Ignoring uniqueness of combinations

2. Which Rust syntax correctly defines a function header to generate combinations summing to k with candidates as a slice of integers?

easy

A. fn combination_sum(candidates: Vec<i32>, k: i32) -> Vec<i32>

B. fn combination_sum(candidates: Vec<i32>, k: i32) -> i32

C. fn combination_sum(candidates: &[i32], k: i32) -> i32

D. fn combination_sum(candidates: &[i32], k: i32) -> Vec

Solution

1. Step 1: Identify input types

   The candidates are passed as a slice reference &[i32] to avoid ownership issues and allow read-only access.

2. Step 2: Identify return type

   The function returns a vector of vectors of i32, representing all combinations found.

3. Final Answer:

   fn combination_sum(candidates: &[i32], k: i32) -> Vec -> Option D

4. Quick Check:

   Function signature with slice input and Vec output [OK]

Hint: Use &[i32] for input slice and Vec for combinations [OK]

Common Mistakes:

• Returning i32 instead of combinations
• Using Vec<i32> input causing ownership issues
• Wrong return type for multiple combinations

3. What is the output of this Rust code snippet?

fn main() {
    let candidates = vec![2, 3, 6, 7];
    let target = 7;
    let result = combination_sum(&candidates, target);
    println!("{:?}", result);
}

fn combination_sum(candidates: &[i32], target: i32) -> Vec {
    fn backtrack(start: usize, target: i32, path: &mut Vec<i32>, res: &mut Vec, candidates: &[i32]) {
        if target == 0 {
            res.push(path.clone());
            return;
        }
        for i in start..candidates.len() {
            if candidates[i] > target {
                continue;
            }
            path.push(candidates[i]);
            backtrack(i, target - candidates[i], path, res, candidates);
            path.pop();
        }
    }
    let mut res = vec![];
    let mut path = vec![];
    backtrack(0, target, &mut path, &mut res, candidates);
    res
}

medium

A. [[3, 3, 3], [7]]

B. [[2, 3, 6], [7]]

C. [[2, 2, 3], [7]]

D. [[2, 2, 2, 3]]

Solution

1. Step 1: Understand backtracking logic

   The function tries combinations starting from each candidate, allowing repeats, and collects those summing to target 7.

2. Step 2: Trace valid combinations

   Valid combinations are [2, 2, 3] (2+2+3=7) and [7] itself.

3. Final Answer:

   [[2, 2, 3], [7]] -> Option C

4. Quick Check:

   Backtracking finds combinations summing to 7 [OK]

Hint: Check sums carefully; 2+2+3 and 7 match target [OK]

Common Mistakes:

• Including invalid sums like 2+3+6
• Ignoring repeated use of candidates
• Missing single element combination [7]

4. Identify the bug in this Rust function for generating combinations summing to k:

fn combination_sum(candidates: &[i32], target: i32) -> Vec {
    fn backtrack(start: usize, target: i32, path: &mut Vec<i32>, res: &mut Vec, candidates: &[i32]) {
        if target == 0 {
            res.push(path.clone());
            return;
        }
        for i in start..candidates.len() {
            if candidates[i] >= target {
                continue;
            }
            path.push(candidates[i]);
            backtrack(i, target - candidates[i], path, res, candidates);
            path.pop();
        }
    }
    let mut res = vec![];
    let mut path = vec![];
    backtrack(0, target, &mut path, &mut res, candidates);
    res
}

medium

A. The recursive call should use i + 1 instead of i

B. The condition should be candidates[i] > target, not >= target

C. The path vector should not be cloned when pushing to res

D. The function should return i32 instead of Vec

Solution

1. Step 1: Analyze the pruning condition

   The condition 'if candidates[i] >= target' skips candidates equal to target, missing valid combinations like [target] itself.

2. Step 2: Correct the condition

   Changing to 'candidates[i] > target' allows candidates equal to target to be included, fixing the bug.

3. Final Answer:

   The condition should be candidates[i] > target, not >= target -> Option B

4. Quick Check:

   Pruning must allow candidates equal to target [OK]

Hint: Use >, not >=, to include candidates equal to target [OK]

Common Mistakes:

• Skipping candidates equal to target
• Incorrect recursive index causing duplicates
• Not cloning path before pushing

5. Given candidates = [1, 2, 3] and target = 4, which of the following is the correct set of all unique combinations summing to 4, assuming candidates can be used unlimited times?

hard

A. [[1,1,1,1], [1,1,2], [1,3], [2,2]]

B. [[1,1,1,1], [1,1,2], [2,2], [3,1]]

C. [[1,1,1,1], [1,1,2], [1,3], [2,2], [3,1]]

D. [[1,1,1,1], [1,1,2], [1,3], [2,2], [4]]

Solution

1. Step 1: List all combinations summing to 4

   Possible combinations using unlimited candidates are [1,1,1,1], [1,1,2], [1,3], and [2,2].

2. Step 2: Check for duplicates and invalid entries

   [3,1] is a duplicate of [1,3] (order doesn't matter), and [4] is invalid as 4 is not in candidates.

3. Final Answer:

   [[1,1,1,1], [1,1,2], [1,3], [2,2]] -> Option A

4. Quick Check:

   Unique combinations without duplicates or invalid numbers [OK]

Hint: Exclude duplicates and numbers not in candidates [OK]

Common Mistakes:

• Including duplicates with different order
• Adding numbers not in candidates
• Missing valid combinations