Generate All Combinations Sum K in DSA Go - Time & Space Complexity

Time Complexity: Generate All Combinations Sum K

O(2^n * n)

Understanding Time Complexity

We want to understand how the time needed to find all combinations that add up to a target grows as the input changes.

How does the number of combinations affect the work done?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

func combinationSumK(nums []int, k int) [][]int {
    var res [][]int
    var comb []int
    var backtrack func(start, target int)

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

    backtrack(0, k)
    return res
}
    

This code finds all unique combinations of numbers from the list that add up to k, using backtracking.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Recursive backtracking calls exploring combinations.
• How many times: Potentially explores all subsets of nums, which can be up to 2^n combinations.

How Execution Grows With Input

As the input list grows, the number of possible combinations grows very fast.

Input Size (n)	Approx. Operations
10	Up to 2^10 = 1,024 combinations explored
20	Up to 2^20 = 1,048,576 combinations explored
30	Up to 2^30 = 1,073,741,824 combinations explored

Pattern observation: The work doubles roughly with each additional element, growing exponentially.

Final Time Complexity

Time Complexity: O(2^n * n)

This means the time grows exponentially with input size because it explores many subsets, and copying combinations takes extra time.

Common Mistake

[X] Wrong: "The algorithm runs in linear time because it just loops through the list once."

[OK] Correct: The code uses recursion to explore many subsets, not just one pass, so the work grows much faster than linear.

Interview Connect

Understanding this exponential growth helps you explain why some problems need careful pruning or different approaches, a key skill in interviews.

Self-Check

"What if we allowed repeated use of the same number in combinations? How would the time complexity change?"

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