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

Time Complexity: Generate All Combinations Sum K

O(n choose k * k)

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.

Specifically, how does the number of operations increase when we try to generate all possible combinations?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

void backtrack(int start, int k, int target, int* candidates, int candidatesSize, int* combination, int combSize) {
    if (target == 0 && combSize == k) {
        // store combination
        return;
    }
    if (target < 0 || combSize > k) return;
    for (int i = start; i < candidatesSize; i++) {
        combination[combSize] = candidates[i];
        backtrack(i + 1, k, target - candidates[i], candidates, candidatesSize, combination, combSize + 1);
    }
}

This code tries all combinations of numbers from candidates to find those with length k that sum to target.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Recursive calls exploring combinations by looping through candidates starting from current index.
• How many times: Potentially explores all subsets of size k, which can be many depending on input size.

How Execution Grows With Input

As input size n grows, the number of combinations of size k grows quickly.

Input Size (n)	Approx. Operations
10	About 252 combinations (if k=5)
20	About 15,504 combinations (if k=5)
30	About 142,506 combinations (if k=5)

Pattern observation: The number of combinations grows very fast, roughly like n choose k, which increases sharply as n increases.

Final Time Complexity

Time Complexity: O(n choose k * k)

This means the time grows roughly with the number of k-sized combinations from n elements, and each combination takes up to k steps to build.

Common Mistake

[X] Wrong: "The time complexity is just O(n) because we loop through the array once."

[OK] Correct: The code uses recursion to explore many combinations, not just a single pass, so the time grows much faster than linear.

Interview Connect

Understanding how recursive combination generation scales helps you explain your approach clearly and shows you grasp the cost of exploring many possibilities.

Self-Check

"What if we allowed repeated elements in combinations? How would the time complexity change?"

Practice

1. What is the main goal of the Generate All Combinations Sum K problem?

easy

A. Count the total number of elements in an array

B. Sort an array of numbers in ascending order

C. Find the maximum number in an array

D. Find all unique combinations of numbers that add up to a target sum K

Solution

1. Step 1: Understand the problem statement

   The problem asks to find all unique sets of numbers that add up exactly to a given sum K.

2. Step 2: Identify the goal

   The goal is to generate all possible combinations, not just one or count elements.

3. Final Answer:

   Find all unique combinations of numbers that add up to a target sum K -> Option D

4. Quick Check:

   Combination sum = find all sets adding to K [OK]

Hint: Focus on combinations that sum exactly to K [OK]

Common Mistakes:

• Confusing combination sum with sorting
• Looking for maximum instead of combinations
• Counting elements instead of sums

2. Which of the following is the correct way to start a recursive function header in C for generating combinations that sum to K from an array arr of size n?

easy

A. void generate(int arr[], int n, int target, int start, int currentSum, int combination[], int combSize)

B. int generate(int arr, int n, int target, int start, int currentSum, int combination[], int combSize)

C. void generate(int arr, int n, int target, int start, int currentSum, int combination[], int combSize)

D. int generate(int arr[], int n, int target, int start, int currentSum, int combination[], int combSize)

Solution

1. Step 1: Check parameter types for array and function return

   In C, arrays are passed as int arr[] and recursive functions for generating combinations usually return void.

2. Step 2: Verify function signature correctness

   void generate(int arr[], int n, int target, int start, int currentSum, int combination[], int combSize) correctly uses void return type and int arr[] for the array parameter.

3. Final Answer:

   void generate(int arr[], int n, int target, int start, int currentSum, int combination[], int combSize) -> Option A

4. Quick Check:

   Correct C syntax for recursive combination function [OK]

Hint: Use void return and array as int arr[] in C [OK]

Common Mistakes:

• Using int instead of int[] for array parameter
• Returning int instead of void for recursive function
• Missing array brackets in parameter

3. What will be the output combinations for the array [2, 3, 6, 7] and target sum 7 using combination sum logic?

medium

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

B. [[7], [3, 4]]

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

D. [[2, 2, 3], [6, 1]]

Solution

1. Step 1: Identify valid combinations summing to 7

   From the array, 7 alone is valid. Also, 2+2+3 = 7 is valid. 3+4 is invalid as 4 is not in array. 6+1 invalid as 1 not in array.

2. Step 2: Confirm combinations

   Valid combinations are [7] and [2, 2, 3].

3. Final Answer:

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

4. Quick Check:

   Sum 7 from array = [7], [2,2,3] [OK]

Hint: Check sums only from given array elements [OK]

Common Mistakes:

• Including numbers not in the array
• Ignoring repeated elements allowed
• Mixing invalid sums

4. Given the following snippet, what is the bug that prevents correct generation of combinations summing to K?

void generate(int arr[], int n, int target, int start, int currentSum, int combination[], int combSize) {
    if (currentSum == target) {
        // print combination
        return;
    }
    for (int i = start; i < n; i++) {
        if (currentSum + arr[i] <= target) {
            combination[combSize] = arr[i];
            generate(arr, n, target, i, currentSum + arr[i], combination, combSize + 1);
        }
    }
}

medium

A. Incorrect base case condition for currentSum == target

B. Loop should start from 0 instead of start

C. Missing backtracking step to remove last element after recursion

D. Function should return int instead of void

Solution

1. Step 1: Analyze recursion and combination building

   The function adds elements to combination but never removes them after recursion, causing incorrect combinations.

2. Step 2: Identify missing backtracking

   Backtracking requires removing last added element after recursive call to explore other paths.

3. Final Answer:

   Missing backtracking step to remove last element after recursion -> Option C

4. Quick Check:

   Backtracking needs element removal after recursion [OK]

Hint: Always remove last element after recursive call [OK]

Common Mistakes:

• Not removing last element after recursion
• Wrong base case check
• Loop starting index error

5. You want to generate all unique combinations from array [1, 2, 3] that sum to 4. Which of the following modifications ensures no duplicate combinations appear when numbers can be reused multiple times?

hard

A. Sort the array and skip duplicates by checking if current element equals previous

B. In recursive calls, pass i as start index to allow reuse and avoid duplicates

C. In recursive calls, always pass 0 as start index to restart from beginning

D. Use a global visited array to mark elements used

Solution

1. Step 1: Understand reuse and duplicates in combinations

   Allowing reuse means next recursion starts from current index i, not 0, to avoid duplicate sets in different orders.

2. Step 2: Identify correct start index usage

   Passing i as start index allows reuse and prevents duplicates by not revisiting earlier elements.

3. Final Answer:

   In recursive calls, pass i as start index to allow reuse and avoid duplicates -> Option B

4. Quick Check:

   Start index = i to reuse and avoid duplicates [OK]

Hint: Pass current index as start to reuse elements [OK]

Common Mistakes:

• Restarting from 0 causing duplicates
• Using visited array unnecessarily
• Not sorting or skipping duplicates properly