Time and Space Complexity in Machine Learning Explained

Understanding the performance of various algorithms is necessary to select the most appropriate one to solve a particular problem in computer science. 

The chosen method must be independent of the machine’s configuration on which it is running. It should also directly relate to the number of inputs and differentiate between two algorithms without ambiguity. The two available methods include time complexity and space complexity

Many factors, such as operating systems, hardware, processors etc, affect time and space complexity. However, these factors are not considered when analysing an algorithm.

Time Complexity

Time complexity can be defined as the amount of time required for an algorithm to run as a function of the input length. The time complexity of any algorithm is the time needed for executing a statement of code in the algorithm. 

It depends on the operating system, programming language, processor, etc. Time complexity can only measure an algorithm’s execution time entirely dependent on the algorithm and its inputs. 

This computational complexity denotes the time needed to execute an algorithm. It showcases the variation – increase or decrease – in execution time when the number of operations in an algorithm alters.  

Space Complexity

Space complexity can be defined as the amount of memory an algorithm or problem takes during execution. Space complexity considers both the space used by the input values and the variables in the algorithm. 

Space complexity combines auxiliary space and the space needed for input variables. Auxiliary space is the space needed during the execution of an algorithm. Space complexity is calculated to analyse any algorithm and check its efficiency. It is directly proportional to the number of inputs an algorithm takes. 

There is often confusion, but it is necessary to understand that time complexity and space complexity are unrelated. 

Developing a Good Algorithm

The critical step to solving a problem is the development of an algorithm. Once the algorithm is developed, any computer programming language can execute it. Follow the steps given below to develop a good algorithm:

Understand the problem properly 

Understanding the problem is key to developing the algorithm. A developer’s job is to get the description of the problem and develop an algorithm to solve it. It is necessary to identify the starting and ending points of the problem. 

Create a high-level algorithm 

An algorithm is like a plan for solving a particular problem., A developer can develop a high-level algorithm with the gathered details. Creating a high-level algorithm will solve most problems; the details can be added later. 

Improve the algorithm by adding details

Once the high-level algorithm has been developed, it has the significant steps necessary to solve the problem. However, now is the time when the developer has to refine the algorithm. The algorithm is developed step-by-step while adding the necessary details. 

Review the algorithm 

Once the algorithm has been fully developed, the final part is to review the algorithm. The developer must identify if the algorithm solves the particular problem, if it can be further simplified and whether it solves any other problems etc. 

Why Are Time and Space Complexity Significant?

Significance of time complexity

The time complexity of an algorithm is the time needed for an algorithm to complete the execution. Every piece of code requires time to execute. The execution time increases with a decrease in time complexity. 

The time required for a program to run depends both on the code’s efficiency and the computer’s processing power. In a professional setting, where programmers have to write hundreds of lines of code, time complexity plays a significant role. Hence, choosing the correct algorithm is also necessary to reduce the time complexity. 

Significance of space complexity 

The space complexity of an algorithm plays an important role when analysing its efficiency. In cases where memory is limited, the necessary space or memory for a program to run is essential. Hence, reducing the space complexity will eventually minimise the time complexity. 

Asymptotic Notations Explained

Asymptotic notations can be described as mathematical notations used to describe an algorithm’s run time as the input tends towards a limiting or particular value. It cannot compare two given algorithms head-to-head. Instead, asymptotic analysis compares an algorithm’s space and time complexity. 

As the input size increases or decreases, these notations compare two algorithms based on their performance change. 

There are three main asymptotic notations:

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1. Big-Oh (O) notation

This notation stands for the upper bound of an algorithm’s run time. Hence it states the worst complexity case of the given algorithm. The Big-Oh (O) notation specifies the maximum time needed for the execution of an algorithm. 

If f(n) stands for the run time of an algorithm, f(n) is O(g(n)), if a positive constant n0 and C exists such that 0 ≤ f(n) ≤ cg(n) for every n ≥ n0.

2. Big-Omega (Ω) notation

This notation represents the lower bound of an algorithm’s run time. Hence, it states the best complexity case of the algorithm. This condition allows the algorithm to complete the execution of the statements in the least amount of time. 

Let f and g be the function set of natural numbers to itself. In case there exists a natural number n0 and a constant c>0 such that c*g(n) ≤ f(n) for every n ≥ n0.

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3. Big-Theta (Θ) notation

This notation encloses the function from below and above. It is used to analyse the average-case complexity of an algorithm as this notation stands for the lower and upper bound of an algorithm’s run time. 

Let f and g be the function from the set of natural numbers to itself. The function f is Θ(g) if there is a natural number n0 and constants c1 and c2 such that c1* g(n) ≤ f(n) ≤ c2 * g(n) for every n ≥ n0. 

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Best, Worst, and Average Asymptotic Analysis Cases

The concept of handling and analysing an algorithm is known as asymptotic analysis. An algorithm’s performance is measured using the input size. The mathematical boundaries of an algorithm’s run-time are defined with asymptotic analysis. It is used to compute an operation’ run time using mathematical computational units. 

  • Best case: In this case, the time needed for the code to run is minimum. 
  • Average case: Average time is needed for the execution of the program. 
  • Worst case: In this case, maximum time is needed for the code to execute. 

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How to Calculate Space and Time Complexity

Let’s see how you can calculate an algorithm’s time and space complexity.

1. Calculating time complexity

Let us consider the following pseudocode to find the sum of two numbers:

# Pseudocode : Sum(p , q) { return p + q }

p = 7

q = 8

def sum (p, q):

     return p+q

print (sum(p , q))

Output: 15

The given code is going to take two units of time (constant):

  • One for the sum (arithmetic operation)
  • One for return 

Hence the total cost for performing a sum operation (Tsum) is 2 (1+1)

Time complexity = O(2) = O(1), as 2 is a constant. 

2. Sorting algorithm’s time complexity

The time complexity of different sorting algorithms has been listed below:

Algorithm  Best case Average case Worst case
Bubble sort time complexity Ω(n) θ(n^2) O(n^2)
Selection sort time complexity Ω(n^2) θ(n^2) O(n^2)
Insertion sort time complexity Ω(n) θ(n^2) O(n^2)
Merge sort time complexity Ω(n log(n)) θ(n log(n)) O(n log(n))
Quick sort time complexity Ω(n log(n)) θ(n log(n)) O(n^2)
Heap sort time complexity Ω(n log(n)) θ(n log(n)) O(n log(n))

3. Searching algorithm’s time complexity

Algorithm  Best case Worst case
Linear search time complexity  O(1) O(n)
Binary search time complexity  O(1) O(log n)

4. Space complexity of different algorithms 

  • Space complexity of bubble sort is 1. 
  • Space complexity of selection sort is 1. 
  • Space complexity of insertion sort is 1. 
  • Space complexity of merge sort is n. 
  • Space complexity of quick sort is n. 
  • Space complexity of heap sort is 1. 

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5. Calculating space complexity

Let us consider the following code to understand space complexity:

int j, multi;

While j < = n do;

   multi <- multi * array[j]

   j <- j + 1

end while;

return multi

In the given code, the space complexity of the given algorithm is denoted by S(n). Two integers are given memory allocation, and the return statement allocates another memory. As integer values require 4-byte data, S(n) = 4 x 2 + 4 = 12 bytes. As n number of integers are allocated in the algorithm, the final space complexity is fS(n) = n + 12 = O (n).

Comparison Between Time and Space Complexity

See the table below to understand the time and space complexity in data structure.

Time complexity  Space complexity
Calculates the time needed for the execution of an algorithm.  Calculates the space needed for the execution of an algorithm 
Depends mainly on the input data size Depends mainly on the auxiliary variables’ size
Calculates the time of every statement Calculates the memory space of all input and output variables. 
Handles computational size as the input size changes  Handles the extra space necessary as the input size changes 

What Are Algorithm Analysis and Algorithm Complexity?

Algorithm Analysis 

Algorithm analysis theoretically estimates the required algorithm resources to solve a particular computational problem. 

It can predict an algorithm’s behaviour without implementing it on a machine. Even though it’s impossible to predict an algorithm’s behaviour precisely, this method can analyse different algorithms and choose the best one. 

Algorithm Complexity

For an input of size n, the algorithm’s complexity helps calculate the amount of time and space needed by the algorithm. Time and space are the two factors determining an algorithm’s efficiency. 

Crucial Factors for Long-Term Usage of an Algorithm

Listed below are factors affecting an algorithm’s effectiveness:

  • Finiteness – The algorithm must not continue through infinite recursions or loops. It must conclude within a predetermined number of steps to reduce RAM usage. 
  • Efficiency – Efficiency helps in delivering swift results and minimises computational durations. It plays a crucial role in creating a good algorithm. 
  • Accuracy – A well-built algorithm should give the correct result, irrespective of the input.

What Is Algorithm Efficiency?

An algorithm’s efficiency is the time needed to give desired output and the computational resources employed by the algorithm. For algorithms with only linear functions (no loops or recursions), algorithm efficiency is found by the number of instructions it has. 

On the other hand, for algorithms with loops, efficiency depends on the number and runtime of each loop.

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As a developer, it is essential to compare an algorithm’s time and space complexities to choose the best option. An algorithm’s efficiency algorithm can be estimated with time and space complexity

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What are some standard notations used to express time complexity, such as Big O notation?

Some standard time complexity notations are: Big O notation: worst-case time complexity Omega notation: best-case time complexity Theta notation: average-case time complexity

Can you explain the trade-off between time complexity and space complexity in machine learning?

The trade-off between time and space complexity is often inversely proportional - improving one will worsen the other.

How can high space complexity affect the practicality and deployment of machine learning models?

An ML model cannot execute successfully if the algorithm loads much data into the machine’s working memory.

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