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

Time Complexity: Generate All Combinations Sum K

O(2^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 size changes.

How does the number of possible combinations affect the work done?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

public void FindCombinations(int[] nums, int target, int start, List<int> current, List<List<int>> result) {
    if (target == 0) {
        result.Add(new List<int>(current));
        return;
    }
    for (int i = start; i < nums.Length; i++) {
        if (nums[i] > target) break;
        current.Add(nums[i]);
        FindCombinations(nums, target - nums[i], i + 1, current, result);
        current.RemoveAt(current.Count - 1);
    }
}
    

This code finds all unique combinations of numbers from the array that sum to the target.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Recursive calls exploring combinations.
• How many times: Potentially exploring many subsets, each recursive call tries next elements.

How Execution Grows With Input

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

Input Size (n)	Approx. Operations
10	Thousands of combinations
20	Millions of combinations
30	Billions of combinations

Pattern observation: The work grows exponentially because we explore many subsets.

Final Time Complexity

Time Complexity: O(2^n)

This means the time roughly doubles with each additional input element because we explore all possible subsets.

Common Mistake

[X] Wrong: "The time grows linearly because we just loop through the array once."

[OK] Correct: The code uses recursion to explore many combinations, not just a single loop, 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 to be efficient.

Self-Check

"What if we allowed repeated use of the same element 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. The largest number less than K

B. The sum of all numbers from 1 to K

C. All unique combinations of numbers that add up 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 K.

2. Step 2: Identify the correct description

   All unique combinations of numbers that add up to K correctly describes finding all unique combinations summing to K.

3. Final Answer:

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

4. Quick Check:

   Combinations sum K = all sets adding to K [OK]

Hint: Look for options describing sets adding exactly to K [OK]

Common Mistakes:

• Confusing sum with product
• Thinking about single numbers instead of combinations
• Choosing options about divisors or ranges

2. Which of the following is the correct C# syntax to start a recursive method named FindCombinations that takes a list of integers and a target sum?

easy

A. void FindCombinations(List nums, int target) { }

B. void FindCombinations(List<int> nums, int target) { }

C. void FindCombinations(int nums[], int target) { }

D. void FindCombinations(List<int> nums; int target) { }

Solution

1. Step 1: Check parameter types and syntax

   In C#, a list of integers is declared as List<int>. The method parameters must be separated by commas.

2. Step 2: Identify correct syntax

   void FindCombinations(List<int> nums, int target) { } uses List<int> and comma-separated parameters correctly. Others have syntax errors or wrong types.

3. Final Answer:

   void FindCombinations(List<int> nums, int target) { } -> Option B

4. Quick Check:

   Correct C# method syntax = void FindCombinations(List<int> nums, int target) { } [OK]

Hint: Remember C# uses List<int> and commas between parameters [OK]

Common Mistakes:

• Using array syntax int nums[] in method parameters
• Missing commas between parameters
• Using untyped List without <int>

3. What is the output of the following C# code snippet that generates all combinations of numbers from 1 to 3 that sum to 3?

void Generate(int start, int target, List<int> current, List result) {
  if (target == 0) {
    result.Add(new List<int>(current));
    return;
  }
  for (int i = start; i <= 3; i++) {
    if (i <= target) {
      current.Add(i);
      Generate(i + 1, target - i, current, result);
      current.RemoveAt(current.Count - 1);
    }
  }
}

var result = new List();
Generate(1, 3, new List<int>(), result);
foreach(var combo in result) {
  Console.WriteLine(string.Join(",", combo));
}

medium

A. 3
2,1

B. 1,1,1
3

C. 1,2
2,1
3

D. 1,2
3

Solution

1. Step 1: Trace recursive calls and combinations

   The code tries numbers from start to 3, adding to current if <= target. It avoids repeats by increasing start each call.

2. Step 2: Identify valid combinations summing to 3

   Possible combinations are [1,2] and [3]. The code prints these two lines: "1,2" and "3".

3. Final Answer:

   1,2 3 -> Option D

4. Quick Check:

   Unique combos summing 3 = 1,2 and 3 [OK]

Hint: Check recursion start index to avoid duplicates [OK]

Common Mistakes:

• Expecting repeated numbers like 1,1,1
• Confusing order of output lines
• Including reversed duplicates like 2,1

4. The following code is intended to generate all combinations of numbers from 1 to 4 that sum to 5. What is the main error causing it to miss some combinations?

void Generate(int start, int target, List<int> current, List result) {
  if (target == 0) {
    result.Add(new List<int>(current));
    return;
  }
  for (int i = start; i < 4; i++) {
    if (i <= target) {
      current.Add(i);
      Generate(i + 1, target - i, current, result);
      current.RemoveAt(current.Count - 1);
    }
  }
}

medium

A. The loop condition should be <= 4, not < 4

B. The recursive call should use i, not i + 1

C. The target check should be target < i, not i <= target

D. The current list should not be copied when adding to result

Solution

1. Step 1: Analyze the for loop range

   The loop runs from start to i < 4, so it stops at i=3, missing i=4.

2. Step 2: Identify impact on combinations

   Because 4 is excluded, combinations including 4 are missed. Changing to i <= 4 fixes this.

3. Final Answer:

   The loop condition should be <= 4, not < 4 -> Option A

4. Quick Check:

   Loop range excludes 4, missing combos [OK]

Hint: Check loop boundaries carefully for inclusive ranges [OK]

Common Mistakes:

• Using wrong loop end condition
• Changing recursion start incorrectly
• Not copying list before adding to result

5. You want to generate all combinations of numbers from 1 to 7 that sum to 7, but each number can be used multiple times. Which change to the recursive call in this C# code will allow repeated numbers?

void Generate(int start, int target, List<int> current, List result) {
  if (target == 0) {
    result.Add(new List<int>(current));
    return;
  }
  for (int i = start; i <= 7; i++) {
    if (i <= target) {
      current.Add(i);
      Generate(??? , target - i, current, result);
      current.RemoveAt(current.Count - 1);
    }
  }
}

hard

A. Use i to allow repeats

B. Use start to allow repeats

C. Use i + 1 to avoid repeats

D. Use 1 to restart from beginning

Solution

1. Step 1: Understand how recursion controls repeats

   Using i + 1 moves start forward, preventing repeats. Using i keeps start same, allowing repeats.

2. Step 2: Choose correct recursive call argument

   To allow repeated numbers, pass i as start in recursive call.

3. Final Answer:

   Use i to allow repeats -> Option A

4. Quick Check:

   Passing i keeps current number available [OK]

Hint: Pass current index to allow repeats, increment to avoid [OK]

Common Mistakes:

• Using i + 1 disables repeats
• Using start (unchanged) may cause infinite loops
• Restarting from 1 causes duplicates and inefficiency