Generate All Combinations Sum K in DSA Javascript - Deep Dive

Overview - Generate All Combinations Sum K

What is it?

Generating all combinations that sum to a target number K means finding every possible group of numbers from a list that add up exactly to K. Each combination is unique and order does not matter. This helps solve problems where you want to explore all ways to reach a total using given numbers.

Why it matters

Without this concept, we would struggle to find all possible solutions to problems like making change with coins, selecting items to fit a budget, or splitting tasks evenly. It helps computers explore many options quickly and completely, which is important in planning, optimization, and decision-making.

Where it fits

Before this, you should understand basic recursion and arrays. After this, you can learn about backtracking optimization, dynamic programming for sum problems, and subset sum variations.

Mental Model

Core Idea

Find all unique groups of numbers that add up exactly to K by exploring choices step-by-step and backtracking when a path doesn't work.

Think of it like...

Imagine you have a box of different colored beads and want to make all bracelets that use some beads adding up to a certain length. You try adding beads one by one, and if the bracelet gets too long, you remove the last bead and try a different one.

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

Build-Up - 7 Steps

1

Foundation

Understanding the Problem Setup

Concept: Learn what it means to find combinations that sum to K and how to represent the problem with arrays and a target sum.

We have an array of numbers, for example [2, 3, 6, 7], and a target sum K, say 7. Our goal is to find all groups of numbers from the array that add up to 7. Each number can be used multiple times or once depending on the problem variant. We represent combinations as arrays like [7] or [2, 2, 3].

Result

Clear understanding of input (array and target) and output (list of combinations).

Understanding the problem clearly helps avoid confusion about what counts as a valid combination and what the output should look like.

2

Foundation

Basic Recursion for Exploring Choices

Concept: Use recursion to try adding each number to the current combination and explore further until the sum reaches or exceeds K.

We write a function that takes the current combination and sum so far. For each number, we add it to the combination and call the function again with the new sum. If sum equals K, we save the combination. If sum exceeds K, we stop exploring that path.

Result

A recursive function that can find one or more combinations summing to K.

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

3

Intermediate

Backtracking to Avoid Wrong Paths

🤔Before reading on: Do you think we should keep numbers in the combination even if the sum exceeds K? Commit to yes or no.

Concept: Backtracking means undoing the last choice when it leads to a sum greater than K, so we don't waste time exploring impossible combinations.

When the sum goes over K, we remove the last added number from the combination and try the next number. This way, we explore all possible combinations without repeating or going down wrong paths.

Result

Efficient exploration that only keeps valid partial combinations.

Knowing when and how to backtrack prevents unnecessary work and keeps the algorithm efficient.

4

Intermediate

Avoiding Duplicate Combinations

🤔Before reading on: Do you think [2,3] and [3,2] should be counted as different combinations? Commit to yes or no.

Concept: To avoid duplicates, we ensure combinations are built in a sorted or fixed order, so [2,3] and [3,2] are treated as the same and only one is included.

We do this by always choosing numbers from the current index or later in the array during recursion. This way, numbers are added in non-decreasing order, preventing duplicates.

Result

Unique combinations without repeats in different orders.

Controlling the order of choices is key to eliminating duplicates in combination problems.

5

Intermediate

Implementing Combination Sum in JavaScript

Concept: Putting all ideas together into a working JavaScript function using recursion and backtracking.

function combinationSum(candidates, target) { const result = []; function backtrack(start, path, sum) { if (sum === target) { result.push([...path]); return; } if (sum > target) return; for (let i = start; i < candidates.length; i++) { path.push(candidates[i]); backtrack(i, path, sum + candidates[i]); path.pop(); } } backtrack(0, [], 0); return result; } // Example usage: // combinationSum([2,3,6,7], 7) returns [[2,2,3],[7]]

Result

A function that returns all unique combinations summing to target.

Seeing the full code helps connect theory to practice and understand how recursion and backtracking work together.

6

Advanced

Handling Large Inputs and Optimization

🤔Before reading on: Do you think sorting the input array helps optimize the search? Commit to yes or no.

Concept: Sorting the input array allows early stopping in loops when the current number exceeds the remaining target, improving efficiency.

By sorting candidates, during recursion, if candidates[i] > remaining target, we break the loop early since no further numbers can fit. This reduces unnecessary recursive calls.

Result

Faster execution on large inputs by pruning impossible paths early.

Sorting combined with pruning is a simple but powerful optimization in backtracking.

7

Expert

Understanding Time Complexity and Limits

🤔Before reading on: Do you think the number of combinations can grow exponentially with input size? Commit to yes or no.

Concept: The number of combinations can be very large, causing exponential time complexity. Understanding this helps set realistic expectations and guides when to use approximate methods.

Because each number can be chosen multiple times, the search tree can grow exponentially. For example, with small numbers and large target, combinations explode. This means the algorithm is not always fast and may need limits or heuristics in practice.

Result

Awareness of performance limits and when to consider alternative approaches.

Knowing complexity helps avoid surprises in real-world use and guides optimization or approximation choices.

Under the Hood

The algorithm uses recursion to explore all possible combinations by adding numbers step-by-step. It keeps track of the current sum and combination. When the sum matches the target, it saves the combination. If the sum exceeds the target, it backtracks by removing the last number and tries other options. This depth-first search explores a tree of choices, pruning branches early when sums get too large.

Why designed this way?

Backtracking was chosen because it naturally fits problems where all possible combinations must be explored without repetition. Alternatives like brute force would try all subsets blindly, wasting time. Dynamic programming can solve related problems but does not list all combinations easily. This method balances completeness and clarity.

Start
  │
  ├─ Choose number[i]
  │     │
  │     ├─ sum + number[i] == target -> Save combination
  │     ├─ sum + number[i] < target -> Recurse with number[i] and onwards
  │     └─ sum + number[i] > target -> Backtrack
  └─ Try next number[i+1]
Repeat until all numbers tried

Myth Busters - 4 Common Misconceptions

Quick: Can combinations like [2,3] and [3,2] both appear in the result? Commit yes or no.

Common Belief: People often think that order matters and both [2,3] and [3,2] are different combinations.

Reality: Order does not matter in combinations, so only one of these should appear to avoid duplicates.

Why it matters: Including both leads to duplicate results and confusion about the number of unique solutions.

Quick: Is it okay to stop exploring as soon as you find one combination? Commit yes or no.

Common Belief: Some believe finding one valid combination is enough and stop the search early.

Reality: The goal is to find all combinations, so stopping early misses other valid solutions.

Why it matters: Missing combinations can cause incomplete answers in applications like budgeting or planning.

Quick: Can you use each number unlimited times without any rule? Commit yes or no.

Common Belief: Many think numbers can always be reused unlimited times in combinations.

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

Why it matters: Using wrong reuse rules leads to incorrect combinations and wrong results.

Quick: Does sorting the input array affect the correctness of the result? Commit yes or no.

Common Belief: Some think sorting changes the result or is unnecessary.

Reality: Sorting does not change correctness but helps optimize and avoid duplicates.

Why it matters: Not sorting can cause slower performance and harder duplicate elimination.

Expert Zone

1

Backtracking can be combined with memoization to cache partial results and avoid repeated work in some variants.

2

Choosing to allow or disallow reuse of numbers changes the recursion logic subtly but significantly.

3

Pruning early by sorting and breaking loops is a simple optimization that drastically reduces runtime in practice.

When NOT to use

This approach is not suitable when the input size is very large or the target sum is huge, as the number of combinations grows exponentially. In such cases, dynamic programming can be used to count combinations efficiently without listing them, or approximation algorithms can be applied.

Production Patterns

In real-world systems, this pattern is used in financial software for coin change problems, in inventory management to find packing options, and in game development for resource allocation. Often combined with caching and pruning to handle large datasets.

Connections

Subset Sum Problem

Builds-on

Understanding combination sum helps grasp subset sum, which asks if any subset sums to a target, 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 and abandoning invalid ones.

Knapsack Problem (Optimization)

Related problem

Both involve selecting items to meet constraints, but knapsack maximizes value under weight limits, while combination sum lists all sums exactly matching a target.

Common Pitfalls

#1 Including duplicate combinations due to ignoring order.

Wrong approach:

function backtrack(start, path, sum) {
if (sum === target) {
result.push([...path]);
return;
}
if (sum > target) return;
for (let i = 0; i < candidates.length; i++) {
path.push(candidates[i]);
backtrack(0, path, sum + candidates[i]);
path.pop();
}
}

Correct approach:

function backtrack(start, path, sum) {
if (sum === target) {
result.push([...path]);
return;
}
if (sum > target) return;
for (let i = start; i < candidates.length; i++) {
path.push(candidates[i]);
backtrack(i, path, sum + candidates[i]);
path.pop();
}
}

Root cause:

Not restricting the start index causes permutations of the same combination to be counted multiple times.

#2 Not backtracking properly, causing incorrect combinations.

Wrong approach:

path.push(candidates[i]);
backtrack(i, path, sum + candidates[i]);
// Missing path.pop() here

Correct approach:

path.push(candidates[i]);
backtrack(i, path, sum + candidates[i]);
path.pop();

Root cause:

Forgetting to remove the last added number after recursion leads to growing combinations and wrong results.

#3 Not stopping recursion when sum exceeds target, wasting time.

Wrong approach:

if (sum === target) {
result.push([...path]);
}
// No check for sum > target, so recursion continues

Correct approach:

if (sum === target) {
result.push([...path]);
return;
}
if (sum > target) return;

Root cause:

Failing to prune branches early causes exponential blowup and slow performance.

Key Takeaways

Generating all combinations that sum to K requires exploring choices step-by-step using recursion and backtracking.

Backtracking helps avoid exploring impossible paths by undoing choices when the sum exceeds the target.

Controlling the order of number selection prevents duplicate combinations and keeps results unique.

Sorting the input array enables early stopping and improves efficiency by pruning search space.

The number of combinations can grow exponentially, so understanding complexity guides when to optimize or use alternative methods.

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 count of numbers greater than K

D. The product of all numbers up to K

Solution

1. Step 1: Understand the problem goal

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

2. Step 2: Clarify what combinations mean

   Combinations mean groups of numbers without regard to order, all adding to K.

3. Final Answer:

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

4. Quick Check:

   Combinations sum to K = All unique combinations of numbers that add up to K [OK]

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

Common Mistakes:

• Confusing combinations with permutations
• Looking for counts instead of actual combinations
• Trying to find products instead of sums

2. Which of the following is the correct way to start a recursive function in JavaScript to generate combinations summing to K?

easy

A. function findCombos() { return; }

B. function findCombos(nums, target) { return nums + target; }

C. function findCombos(nums, target, start, path, result) { ... }

D. function findCombos(nums) { console.log(target); }

Solution

1. Step 1: Identify correct recursive function signature

   The function needs parameters for numbers, target sum, start index, current path, and result storage.

2. Step 2: Check options for these parameters

   Only function findCombos(nums, target, start, path, result) { ... } includes all necessary parameters for recursion and backtracking.

3. Final Answer:

   function findCombos(nums, target, start, path, result) { ... } -> Option C

4. Quick Check:

   Recursive function with all params = function findCombos(nums, target, start, path, result) { ... } [OK]

Hint: Include start index and path for backtracking recursion [OK]

Common Mistakes:

• Missing parameters needed for recursion
• Returning wrong values instead of building combinations
• Not including start index to avoid duplicates

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

function combinationSum(nums, target) {
  const result = [];
  function backtrack(start, path, sum) {
    if (sum === target) {
      result.push([...path]);
      return;
    }
    if (sum > target) return;
    for (let i = start; i < nums.length; i++) {
      path.push(nums[i]);
      backtrack(i + 1, path, sum + nums[i]);
      path.pop();
    }
  }
  backtrack(0, [], 0);
  return result;
}

console.log(combinationSum([2,3,6,7], 7));

medium

A. [[7]]

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

C. [[2,3,7]]

D. [[2,2,3]]

Solution

1. Step 1: Understand the code logic

   The function finds combinations of numbers from the array that sum to 7 without reuse (i + 1).

2. Step 2: Trace the combinations

   Only 7 alone sums to 7; others like 2+3=5, 2+3+6=11 exceed or don't match target.

3. Final Answer:

   [[7]] -> Option A

4. Quick Check:

   Only [7] sums to 7 without repeats = [[7]] [OK]

Hint: Check sum and index increment to avoid reuse [OK]

Common Mistakes:

• Assuming numbers can be reused multiple times
• Including combinations that exceed target
• Not copying path array before pushing

4. Identify the bug in this code snippet for generating combinations summing to K:

function findCombos(nums, target) {
  const result = [];
  function backtrack(start, path, sum) {
    if (sum === target) {
      result.push(path);
      return;
    }
    if (sum > target) return;
    for (let i = start; i < nums.length; i++) {
      path.push(nums[i]);
      backtrack(i + 1, path, sum + nums[i]);
      path.pop();
    }
  }
  backtrack(0, [], 0);
  return result;
}

medium

A. The base case sum === target is missing

B. The path array is pushed directly without copying, causing reference issues

C. The loop should start from 0 instead of start

D. The function does not return any result

Solution

1. Step 1: Check how results are stored

   The code pushes 'path' directly, which is a reference and will change later.

2. Step 2: Identify correct way to store combinations

   Must push a copy of path (e.g., [...path]) to save current state.

3. Final Answer:

   The path array is pushed directly without copying, causing reference issues -> Option B

4. Quick Check:

   Push copy of path, not reference = The path array is pushed directly without copying, causing reference issues [OK]

Hint: Always push a copy of path array to result [OK]

Common Mistakes:

• Pushing path reference instead of copy
• Incorrect loop start index causing duplicates
• Missing base case for sum equal to target

5. Given the array [1, 2, 3, 4] and target sum 5, which of the following is the correct set of all unique combinations that sum to 5, assuming numbers can be used only once?

hard

A. [[1,2,2]]

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

C. [[5]]

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

Solution

1. Step 1: List all combinations without reuse that sum to 5

   Possible pairs: 1+4=5, 2+3=5; triples or singles don't sum to 5 without reuse.

2. Step 2: Verify options against these combinations

   [[1,4],[2,3]] matches these pairs exactly; others include duplicates or invalid sums.

3. Final Answer:

   [[1,4],[2,3]] -> Option D

4. Quick Check:

   Unique pairs summing to 5 = [[1,4],[2,3]] [OK]

Hint: Sum pairs without reuse to find valid combos [OK]

Common Mistakes:

• Including repeated numbers when only single use allowed
• Assuming single number 5 exists in array
• Counting invalid sums or duplicates