Finding All Possible unique K size combinations till N Using Python

In many programming scenarios, we often come across the need to find all possible combinations of a certain size from a given set of elements. These combinations can be useful in various applications such as generating permutations, solving combinatorial problems, or exploring different subsets of data. In this blog post, we will explore an efficient approach to find all unique combinations of size K until a given number N using the Python programming language.

Understanding the Problem

Before we dive into the solution, let's clearly define the problem we are trying to solve. Given a range of numbers from 1 to N and a desired combination size K, our goal is to generate all possible unique combinations of K numbers from the given range.

For example, let's say we have N = 5 and K = 3. The expected output would be −

[[1, 2, 3], [1, 2, 4], [1, 2, 5], [1, 3, 4], [1, 3, 5], [1, 4, 5], [2, 3, 4], [2, 3, 5], [2, 4, 5], [3, 4, 5]]

Approach and Algorithm

• Create an empty result list to store the combinations.

result = []

• Define a recursive function, backtrack, that takes the following parameters: start and current_combination.

def backtrack(start, current_combination):
   # ... code for the recursive function ...

• If the length of current_combination is equal to K, add it to the result list and return.

if len(current_combination) == k:
   result.append(current_combination)
   return

• Iterate from start to N −

• Append the current number to current_combination.

• Recursively call the backtrack function with start incremented by 1.

• Remove the last element from current_combination to backtrack and explore other possibilities.

for i in range(start, n + 1):
    backtrack(i + 1, current_combination + [i])

• Call the backtrack function initially with start set to 1 and an empty current_combination.

backtrack(1, [])

Handling Invalid Inputs

To ensure the robustness of our solution, we can add some input validation checks. For example, we can check if the given value of N is greater than or equal to K. If not, we can raise an exception or return an empty list, indicating that it is not possible to form combinations of size K from a range smaller than K.

def combinations(n, k):
   if n < k:
      raise ValueError("Invalid input: N must be greater than or equal to K.")

   # ... 

Optimizing the Algorithm

The current implementation generates all possible combinations by exploring all branches of the recursion tree. However, if the desired combination size K is relatively small compared to the range N, we can optimize the algorithm by pruning certain branches. For example, if the remaining numbers available for selection are not sufficient to form a combination of size K, we can stop exploring that branch.

def combinations(n, k):
   # ... existing code ...

   def backtrack(start, current_combination):
      if len(current_combination) == k:
         result.append(current_combination)
         return

      # Optimization: Check if remaining numbers are enough for a valid combination
      if k - len(current_combination) > n - start + 1:
         return

      for i in range(start, n + 1):
         backtrack(i + 1, current_combination + [i])

   # ... 

This optimization reduces unnecessary computations and can significantly improve the performance of the algorithm for larger values of N and K.

Example Output for Invalid Input

Let's consider an example with an invalid input to demonstrate the handling of such cases −

Input

combinations(2, 4)
try:
   print(combinations(2, 4))
except ValueError as e:
   print(e)

Output

Invalid input: N must be greater than or equal to K.

In this case, we raise a ValueError to indicate that the input is invalid because the range (2) is smaller than the desired combination size (4).

Implementing the Solution in Python

Here's the complete implementation of the solution −

def combinations(n, k):
   result = []

   def backtrack(start, current_combination):
      if len(current_combination) == k:
         result.append(current_combination)
         return

      for i in range(start, n + 1):
         backtrack(i + 1, current_combination + [i])

   backtrack(1, [])
   return result

Testing the Solution

Let's test the solution with a few example inputs −

Example

print(combinations(5, 3))

Output

[[1, 2, 3], [1, 2, 4], [1, 2, 5], [1, 3, 4], [1, 3, 5], [1, 4, 5], [2, 3, 4], [2, 3, 5], [2, 4, 5], [3, 4, 5]]

Example

print(combinations(4, 2))

Output

[[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]]

Explanation

Let's analyze the first example in detail −

Input: combinations(5, 3)

• Initially, result is an empty list.

• The backtrack function is called with start = 1 and current_combination = [].

• In the first iteration of the loop (i = 1), we append 1 to the current_combination and make a recursive call to backtrack(2, [1]).

• In the first iteration of the loop (i = 2), we append 2 to the current_combination and make a recursive call to backtrack(3, [1, 2]).

• Since the length of [1, 2] is equal to 3 (K), we add it to the result.

• Backtracking to the previous state, we remove the last element from the current_combination to explore other possibilities ([1]).

• In the second iteration of the loop (i = 3), we append 3 to the current_combination and make a recursive call to backtrack(4, [1, 3]).

• Since the length of [1, 3] is equal to 3 (K), we add it to the result.

• Backtracking to the previous state, we remove the last element from the current_combination to explore other possibilities ([1]).

• We continue this process until all possible combinations are generated.

• The final result is [[1, 2, 3], [1, 2, 4], [1, 2, 5], [1, 3, 4], [1, 3, 5], [1, 4, 5], [2, 3, 4], [2, 3, 5], [2, 4, 5], [3, 4, 5]], which represents all unique combinations of size 3 from the numbers 1 to 5.

Conclusion

In this approach, we utilize the backtracking technique to generate all possible unique combinations of a specific size from a given range of numbers. By incrementally building the combinations and backtracking when necessary, we explore all possible solutions systematically. The provided code snippet demonstrates the implementation, and the example outputs validate the correctness of the solution.

The approach we discussed involved defining a recursive function, backtrack, which incrementally built valid combinations. By iterating through the range of numbers and recursively calling the backtrack function, we explored all possible combinations, backtracking whenever an invalid state was encountered. The result was a comprehensive list of unique combinations satisfying the specified size constraint.

To validate the correctness of our solution, we tested it with example inputs. The outputs demonstrated that the code successfully generated the expected combinations, showcasing the reliability of the implemented algorithm.