Generate All Combinations Sum K in DSA Go - Deep Dive

Overview - Generate All Combinations Sum K

What is it?

Generate All Combinations Sum K means finding all unique 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 possible combination that meets this sum. This helps solve problems where you want to explore all ways to reach a total using given parts.

Why it matters

This concept helps in many real-life problems like budgeting, resource allocation, or puzzle solving where you want to find all possible ways to combine items to reach a goal. Without it, you might miss important options or solutions. It also teaches how to explore choices systematically and avoid missing or repeating combinations.

Where it fits

Before this, you should understand basic recursion and arrays or lists. After this, you can learn optimization techniques like pruning or dynamic programming to handle bigger inputs efficiently.

Mental Model

Core Idea

Finding all combinations that sum to K is like exploring every path in a decision tree where each step chooses a number to add until the total reaches K or goes over.

Think of it like...

Imagine you have a box of different colored beads and want to make all possible bracelets that use beads adding up to a certain length. You try each bead one by one, adding it to your bracelet, and backtrack if the length is too long or you reach the exact length.

Combination Sum Tree:

Start (sum=0)
  ├─ Add 2 (sum=2)
  │    ├─ Add 2 (sum=4)
  │    └─ Add 3 (sum=5)
  ├─ Add 3 (sum=3)
  │    ├─ Add 2 (sum=5)
  │    └─ Add 3 (sum=6)
  └─ Add 6 (sum=6)

Each branch adds a number, exploring all paths until sum == K or sum > K.

Build-Up - 7 Steps

1

Foundation

Understanding the Problem Setup

Concept: Learn what it means to find combinations that sum to a target K.

Given a list of positive integers and a target sum K, we want to find all unique sets of numbers from the list that add up exactly to K. Each number can be chosen multiple times or once depending on the problem variant. The output is a list of these sets. Example: Input: candidates = [2,3,6,7], target = 7 Output: [[2,2,3],[7]]

Result

Clear understanding of the problem goal and input-output format.

Knowing exactly what the problem asks prevents confusion and guides the solution approach.

2

Foundation

Basic Recursion for Exploring Choices

Concept: Use recursion to try each number and explore all possible sums.

Pseudocode: func backtrack(start, currentSum): if currentSum == target: add current combination to result return if currentSum > target: return for i from start to end: add nums[i] to current backtrack(i or i+1, currentSum + nums[i]) remove nums[i]

Result

A recursive method that explores all possible combinations, though not optimized yet.

Recursion naturally models the decision process of including or excluding numbers.

3

Intermediate

Backtracking to Build Combinations

🤔Before reading on: Do you think we should keep numbers after sum > K or stop immediately? Commit to your answer.

Concept: Backtracking means undoing choices to explore new paths without extra memory.

In Go, we pass a pointer to the current combination slice to modify it in place: ```go comb = append(*comb, nums[i]) backtrack(...) *comb = (*comb)[:len(*comb)-1] ``` This adds, recurses, then removes (backtracks). Stop if sum > K.

Result

Efficient exploration of valid combinations only, reducing unnecessary work.

Backtracking helps prune the search space and keeps memory usage low by reusing the same combination list.

4

Intermediate

Avoiding Duplicate Combinations

🤔Before reading on: Do you think sorting the input helps avoid duplicates? Commit to your answer.

Concept: Sorting the input and controlling recursion order prevents repeated combinations in different orders.

```go sort.Ints(candidates) // In backtrack, for i := start; i < len(candidates); i++ ``` Start from 'start' index and increment i, ensuring non-decreasing order in combinations.

Result

Unique combinations without repeats, making output clean and correct.

Ordering choices and controlling recursion direction is key to uniqueness in combination problems.

5

Intermediate

Handling Single Use vs Multiple Use of Numbers

🤔Before reading on: Should the recursion index move forward after choosing a number if numbers can be reused? Commit to your answer.

Concept: Adjust recursion index movement to allow or disallow reuse of numbers.

Unlimited reuse (LeetCode 39): `backtrack(..., i, ...)` // same i Used once (LeetCode 40): `backtrack(..., i+1, ...)` // next i Example for unlimited: ```go backtrack(cands, remain - cands[i], i /* reuse */, comb, res) ```

Result

Correct combinations respecting the problem's reuse rules.

Small changes in recursion index control drastically change the solution space.

6

Advanced

Implementing Combination Sum in Go

🤔Before reading on: Do you think passing the current combination by value or reference affects performance? Commit to your answer.

Concept: Write a complete Go function using backtracking to generate all combinations summing to K.

Full implementation for unlimited reuse (candidates unique, positive): ```go package main import ( "fmt" "sort" ) func combinationSum(candidates []int, target int) [][]int { sort.Ints(candidates) var res [][]int var comb []int backtrack(candidates, target, 0, &comb, &res) return res } func backtrack(cands []int, remain int, start int, comb *[]int, res *[][]int) { if remain == 0 { tmp := make([]int, len(*comb)) copy(tmp, *comb) *res = append(*res, tmp) return } for i := start; i < len(cands); i++ { if cands[i] > remain { break } *comb = append(*comb, cands[i]) backtrack(cands, remain - cands[i], i, comb, res) *comb = (*comb)[:len(*comb)-1] } } func main() { fmt.Println(combinationSum([]int{2,3,6,7}, 7)) // Output: [[2 2 3] [7]] } ``` Run it to see results.

Result

A runnable Go function that prints all unique combinations summing to K.

Implementing the algorithm solidifies understanding and reveals practical details like slice handling and recursion control.

7

Expert

Optimizing with Pruning and Early Stopping

🤔Before reading on: Do you think checking sum > K early saves time? Commit to your answer.

Concept: Add pruning to stop recursion early when sum exceeds K, improving efficiency.

Since sorted, `if cands[i] > remain { break }` skips larger numbers. Also, base case `if remain < 0 { return }` though not needed with prune. This reduces calls from O(2^n) towards better in practice.

Result

Faster execution on large inputs by cutting off impossible paths early.

Pruning leverages problem structure to optimize brute-force search without changing correctness.

Under the Hood

The algorithm builds a recursion tree where each node represents a choice to include a number or move on. Backtracking means the function explores one path, then returns to try others, reusing the same memory for the current combination. Sorting and index control ensure unique paths. Pruning stops branches that cannot reach the target sum, saving time.

Why designed this way?

Backtracking was chosen because it naturally fits problems requiring exploration of all combinations. Sorting and pruning were added to reduce duplicates and improve performance. Alternatives like dynamic programming exist but are less intuitive for generating all combinations explicitly.

Recursion Tree Example:

[Start sum=0]
  ├─ Include 2 (sum=2)
  │    ├─ Include 2 (sum=4)
  │    │    └─ Include 3 (sum=7) valid
  │    └─ Include 3 (sum=5) valid
  ├─ Include 3 (sum=3)
  │    └─ Include 3 (sum=6) valid
  └─ Include 6 (sum=6) valid

Backtrack after each leaf to explore other branches.

Myth Busters - 4 Common Misconceptions

Quick: Does sorting the input change the final combinations found? Commit to yes or no.

Common Belief: Sorting the input changes the combinations you get.

Reality: Sorting only changes the order in which combinations are found, not the combinations themselves.

Why it matters: Believing sorting changes results may cause confusion or unnecessary code complexity.

Quick: Can you reuse numbers if the problem says each number can be used only once? Commit to yes or no.

Common Belief: You can always reuse numbers regardless of problem rules.

Reality: You must follow problem rules; if reuse is not allowed, you must move forward in the list to avoid repeats.

Why it matters: Ignoring reuse rules leads to incorrect solutions with duplicates or invalid combinations.

Quick: Does backtracking always mean copying the current combination list? Commit to yes or no.

Common Belief: Backtracking requires copying the current combination every time.

Reality: Backtracking can reuse the same list by adding/removing elements, avoiding costly copies.

Why it matters: Misunderstanding this leads to inefficient code and higher memory use.

Quick: Is pruning optional and does not affect performance? Commit to yes or no.

Common Belief: Pruning is just a nice-to-have and does not impact speed much.

Reality: Pruning can drastically reduce the number of recursive calls, improving performance exponentially.

Why it matters: Skipping pruning causes slow solutions on large inputs, making the algorithm impractical.

Expert Zone

1

The order of recursion calls affects the shape of the recursion tree and can impact performance subtly.

2

Using slices in Go carefully to avoid unnecessary allocations during backtracking improves memory efficiency.

3

Pruning based on sorted input is a subtle optimization that requires understanding input properties.

When NOT to use

This approach is not suitable when the input size is very large and performance is critical; in such cases, dynamic programming or approximation algorithms are better. Also, if only one valid combination is needed, simpler greedy methods may suffice.

Production Patterns

Used in recommendation systems to find all item bundles matching a budget, in finance for portfolio combinations, and in puzzle games to generate all valid moves or solutions.

Connections

Subset Sum Problem

Builds-on

Understanding combination sum helps grasp subset sum, which asks if any subset sums to K, focusing on existence rather than listing all.

Backtracking Algorithms

Same pattern

Combination sum is a classic example of backtracking, a general method for exploring all solutions by building candidates incrementally.

Combinatorial Chemistry

Analogous process

Generating all chemical compound combinations to test properties is similar to generating all number combinations summing to a target, showing cross-domain pattern of exhaustive search.

Common Pitfalls

#1 Including numbers without checking if sum exceeds target.

Wrong approach:

```go
func backtrack(nums []int, target int, start int, comb []int, res *[][]int) {
if target == 0 {
*res = append(*res, append([]int{}, comb...))
return
}
for i := start; i < len(nums); i++ {
comb = append(comb, nums[i])
backtrack(nums, target-nums[i], i, comb, res) // no check for target < 0
comb = comb[:len(comb)-1]
}
}
```

Correct approach:

```go
func backtrack(nums []int, target int, start int, comb []int, res *[][]int) {
if target == 0 {
*res = append(*res, append([]int{}, comb...))
return
}
if target < 0 { return }
for i := start; i < len(nums); i++ {
if nums[i] > target { // prune here
break
}
comb = append(comb, nums[i])
backtrack(nums, target-nums[i], i, comb, res)
comb = comb[:len(comb)-1]
}
}
```

Root cause:

Not pruning leads to exploring invalid paths, wasting time and possibly causing stack overflow.

#2 Not sorting input and allowing duplicates in output.

Wrong approach:

// no sorting, no index control
func combinationSum(nums []int, target int) [][]int {
var res [][]int
backtrack(nums, target, 0, []int{}, &res)
return res
}

Correct approach:

```go
sort.Ints(nums)
func combinationSum(nums []int, target int) [][]int {
var res [][]int
backtrack(nums, target, 0, []int{}, &res)
return res
}
```

Root cause:

Without sorting and controlled recursion, duplicate combinations appear in different orders.

#3 Passing combination slice by value causing high memory use.

Wrong approach:

```go
func backtrack(nums []int, target int, start int, comb []int, res *[][]int) {
if target == 0 {
*res = append(*res, comb) // shares underlying array
return
}
for i := start; i < len(nums); i++ {
newComb := append(comb, nums[i]) // copy each time
backtrack(nums, target-nums[i], i, newComb, res)
}
}
```

Correct approach:

```go
func backtrack(nums []int, target int, start int, comb *[]int, res *[][]int) {
if target == 0 {
temp := make([]int, len(*comb))
copy(temp, *comb)
*res = append(*res, temp)
return
}
for i := start; i < len(nums); i++ {
*comb = append(*comb, nums[i])
backtrack(nums, target-nums[i], i, comb, res)
*comb = (*comb)[:len(*comb)-1]
}
}
```

Root cause:

Slices share underlying arrays; without proper copy on add and backtrack, results can be overwritten leading to incorrect output.

Key Takeaways

Generating all combinations that sum to K is a classic backtracking problem exploring all possible choices systematically.

Sorting input and controlling recursion order are essential to avoid duplicate combinations.

Backtracking uses recursion with add/remove steps to efficiently explore and prune the search space.

Pruning early when the sum exceeds K drastically improves performance by cutting impossible paths.

Understanding how to handle reuse of numbers by adjusting recursion index is key to solving different variants.

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 where order does not matter, 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 adding to K [OK]

Hint: Focus on sums that exactly equal K, ignoring order [OK]

Common Mistakes:

• Confusing combinations with permutations
• Looking for a single sum instead of all combinations
• Ignoring the target sum K

2. Which of the following Go function signatures is correct for a function that generates all combinations summing to K from a slice of integers?

easy

A. func combinationSum(nums []int, target int) int

B. func combinationSum(nums int, target []int) []int

C. func combinationSum(nums []int, target int) []int

D. func combinationSum(nums []int, target int) [][]int

Solution

1. Step 1: Identify input parameters

   The function should take a slice of integers and a target integer sum.

2. Step 2: Identify return type

   It should return a slice of slices of integers, representing all combinations.

3. Final Answer:

   func combinationSum(nums []int, target int) [][]int -> Option D

4. Quick Check:

   Input slice and target int, output slice of int slices [OK]

Hint: Return type must hold multiple combinations ([][]int) [OK]

Common Mistakes:

• Swapping parameter types
• Returning a single slice instead of slice of slices
• Returning an int instead of combinations

3. What is the output of the following Go code snippet?

func main() {
    nums := []int{2, 3, 6, 7}
    target := 7
    result := combinationSum(nums, target)
    fmt.Println(result)
}

func combinationSum(nums []int, target int) [][]int {
    var res [][]int
    var backtrack func(start, target int, path []int)
    backtrack = func(start, target int, path []int) {
        if target == 0 {
            temp := make([]int, len(path))
            copy(temp, path)
            res = append(res, temp)
            return
        }
        for i := start; i < len(nums); i++ {
            if nums[i] > target {
                continue
            }
            backtrack(i, target-nums[i], append(path, nums[i]))
        }
    }
    backtrack(0, target, []int{})
    return res
}

medium

A. [[2 2 3] [7]]

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

C. [[3 3 3] [7]]

D. [[2 2 2 3] [7]]

Solution

1. Step 1: Understand backtracking logic

   The function tries combinations starting from each index, allowing reuse of numbers, and collects paths summing to target.

2. Step 2: Trace combinations for target 7

   Possible sums are [2,2,2,3] and [7] because 2+2+2+3=9 is incorrect, so check carefully: 2+2+2+3=9 (too big), so that is invalid. The actual combinations are [2,2,3] (2+2+3=7) and [7].

3. Step 3: Verify options

   [[2 2 2 3] [7]] shows [2 2 2 3] which sums to 9, invalid. [[2 2 3] [7]] shows [2 2 3] and [7], which is correct.

4. Final Answer:

   [[2 2 3] [7]] -> Option A

5. Quick Check:

   Backtracking finds [2 2 3] and [7] sums [OK]

Hint: Look for combinations that sum exactly to 7 using given nums [OK]

Common Mistakes:

• Including invalid sums like [2 3 6]
• Ignoring reuse of numbers
• Confusing order of elements

4. Identify the bug in this Go function for generating combinations summing to K:

func combinationSum(nums []int, target int) [][]int {
    var res [][]int
    var backtrack func(start, target int, path []int)
    backtrack = func(start, target int, path []int) {
        if target == 0 {
            res = append(res, path)
            return
        }
        for i := start; i < len(nums); i++ {
            if nums[i] > target {
                continue
            }
            backtrack(i, target-nums[i], append(path, nums[i]))
        }
    }
    backtrack(0, target, []int{})
    return res
}

medium

A. The target check should be target < 0 instead of target == 0

B. The loop should start from 0 every time, not from start

C. Appending path directly causes shared slice reference issues

D. The function should return a single slice, not slice of slices

Solution

1. Step 1: Check how path is appended to result

   Appending path directly stores the reference, causing all results to point to the same underlying array.

2. Step 2: Fix by copying path before appending

   Make a copy of path slice before appending to res to avoid shared references.

3. Final Answer:

   Appending path directly causes shared slice reference issues -> Option C

4. Quick Check:

   Copy path before append to avoid shared slice bugs [OK]

Hint: Always copy slices before appending to result [OK]

Common Mistakes:

• Appending slice references directly
• Incorrect loop start index
• Wrong base case condition

5. Given a list of integers with duplicates, how would you modify the combination sum algorithm to generate all unique combinations summing to K without duplicates? Choose the best approach.

hard

A. Use a map to store sums and ignore duplicates after generation

B. Sort the list and skip duplicate numbers during recursion

C. Allow duplicates freely and filter results at the end

D. Increase target by 1 to avoid duplicates

Solution

1. Step 1: Sort the input list

   Sorting helps group duplicates together so they can be skipped during recursion.

2. Step 2: Skip duplicates in recursion

   When iterating, skip numbers equal to the previous one at the same recursion level to avoid duplicate combinations.

3. Final Answer:

   Sort the list and skip duplicate numbers during recursion -> Option B

4. Quick Check:

   Sorting + skipping duplicates = unique combinations [OK]

Hint: Sort input and skip duplicates while recursing [OK]

Common Mistakes:

• Filtering duplicates after generating all combinations
• Not sorting input before recursion
• Changing target incorrectly