What is the time complexity of the combination sum?

 




Do you know?


The time complexity varies for different coding problems! 


The amount of time an algorithm needs to run as the input size initiates is measured by its time complexity. 


The time complexity is denoted by Big ‘O’ notation, it describes the running time's upper bound. 


In this blog, you will get to know about a specific coding problem known as combination sum time complexity. 


Combination sums is where you have to achieve a target using various given values. 


Different approaches have different time complexities, finding the one that is the most efficient is important.


So, read till end you will get to know the one which can help you solve the sum combination problem with reduced time complexity. 


First, let’s take a brief about combination sum problem.



What is the Combination sum?

Combination sum is a problem where you are provided with a set of unique positive integers along with a target value. The goal is to identify every single combination in the collection where the selected values add up to the target integer. 


For instance, if your target is 9 and the range is {2, 3, 4}, then the available combinations are {3,3,3}, and {2, 3, 4}.


Now, let’s see some of the most popular ways to solve this problem.



How to solve the Combination Sum problem?

You can solve the problems related to the Sum combination using the following ways listed below:


  1. Backtracking

It involves recursively running the operation with the changed combinations but with the same target. Backtracking begins with a null combination and iteratively adds each value to it.


  1. Dynamic Programming

This method builds solutions from the bottom to upside location for smaller subproblems and then leverages the solutions to find the answer to the main problem.


  1. Depth-First Search (DFS) 

A DFS algorithm can be employed to examine all potential pairings by finding the values from the depth first to determine whether they will provide the same target.


  1. Breadth-First Search (BFS)

A BFS algorithm ensures that the shortest combination is found first, it can also be employed to explore all potential combinations of the values.


  1. Backtracking with memoization

To avoid recalculating the value, you can utilise a memoization table to keep a track of all the subsets that you have already tracked before.


  1. Iteration

To iterate through all possible subsets and determine whether they will reach the target, you can employ a combination of loops. You can use this approach to find various other problems such as the largest number in k swaps, string problems, etc.


The given values must sum up to the target number for each of the aforementioned approaches to be successful. 


Now, you know the ways but what will be the time complexity for solving these problems. In the next section, we are going to discuss that in detail.


Time complexity of the combination sum 

Depending on the algorithm employed to answer the Sum Combination problem, the time complexity varies  for different approaches. Get to know about them in detail here: 


  1. Backtracking

When there are n values and m targets, the worst-case time difficulty of backtracking is O(nm). There will be m steps of recursion, and in each phase, there are n possibilities from which to choose a value.


  1. Dynamic Programming: 

Dynamic programming has an O(nm) time complexity, where n is the number of values while m is its target. This is due to the fact that a 2D dp arrays of size n*m needs to be filled.


  1. Depth-First Search (DFS)

In this way, we have two possibilities for selecting a value in each recursive step, So, DFS has an O(2n) time complexity where n represents the number of values.


  1. Breadth-First Search (BFS)

BFS has a time complexity of O(2n), where n is the total number of values.


  1. Backtracking with memoization

This strategy has an O(nm) time complexity, where n is the total number of values while m is the target.


  1. Iteration

This method has an O(2n*n) time complexity, where n is the total number of values.


It is preferable to employ a Backtracking with Memoization approach or using Dynamic Programming because they are less time-consuming and quicker than alternative ways.


It was just an overview, let’s discuss Time Complexity with simple examples.



Understanding Time Complexity with Simple Examples

To better comprehend time complexity, consider you can use these examples:


  1. Linear time complexity (O(n)):

An algorithm is considered to be linear time complexity (O(n)) if the execution time grows linearly as the size of the input does. 


Example-Take an array of integers and assume you want to calculate the total of all the entries in the array. This algorithm's execution time is inversely correlated with the array's element count. Consequently, this algorithm's time complexity is (O(n)).


  1. Quadratic Time Complexity (O(n^2))

If the running time of an algorithm is equal to the square of the input size, that algorithm is quadratic time complexity. 


Example-Bubble sort is an illustration of O(n2) quadratic time complexity. When using the bubble sort sorting method, nearby elements are frequently changed if they aren't in the right order. Since the number of operations grows quadratically with size of the array, bubble sort has an O(n2) time complexity.


  1. Constant Time Complexity (O(1))

If an algorithm's running time is unaffected by the input size, then it has constant time complexity.


Example-You want to locate the first entry in an array of numbers, for instance. Regardless of the array's element count, this algorithm will run in a fixed amount of time. Consequently, this algorithm's time complexity is (O(1)).


  1. Logarithmic Time Complexity (O(log n))

If the running time of an algorithm grows logarithmically with the input size, then the algorithm is logarithmic time complexity. 


Example-Finding the largest number in k swaps  has a time complexity of O(n log n), where n represents the size of digits in the input number. By beginning each iteration with the lowest digit and gradually working up to the most significant digit, you can reduce this to O(n).



Conclusion 

Summing it up, we hope now you know the ways to solve the combination sum problem. Different methods have different time complexity, so based on your requirements you can utilise the method. 


Select the most appropriate method and reduce your time complexity.



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