In this tutorial, you will explore computational complexity (space and time complexity), developed by Juris Hartmanis and Richard E. Stearns to assess the difficulty of an algorithm. As you all know, human nature strives to find the most efficient way to complete their daily tasks. The overarching thought process behind innovation and technology is to make people's lives easier by providing solutions to problems they may face.
In the world of computer science and digital products, the same thing occurs. To perform better, you need to write algorithms that are time efficient and use less memory.
An algorithm's complexity is a measure of the amount of data that it must process in order to be efficient. Domain and range of this function are generally expressed in natural units.
What Is Time Complexity?
Time complexity is defined in terms of how many times it takes to run a given algorithm, based on the length of the input. Time complexity is not a measurement of how much time it takes to execute a particular algorithm because such factors as programming language, operating system, and processing power are also considered.
Time complexity is a type of computational complexity that describes the time required to execute an algorithm. The time complexity of an algorithm is the amount of time it takes for each statement to complete. As a result, it is highly dependent on the size of the processed data. It also aids in defining an algorithm's effectiveness and evaluating its performance.
Also Read: What Is An Algorithm?
What Is Space Complexity?
When an algorithm is run on a computer, it necessitates a certain amount of memory space. The amount of memory used by a program to execute it is represented by its space complexity. Because a program requires memory to store input data and temporal values while running, the space complexity is auxiliary and input space.
What Does It Take To Develop a Good Algorithm?
A good algorithm executes quickly and saves space in the process. You should find a happy medium of space and time (space and time complexity), but you can do with the average. Now, take a look at a simple algorithm for calculating the "mul" of two numbers.
Step 1: Start.
Step 2: Create two variables (a & b).
Step 3: Store integer values in ‘a’ and ‘b.’ -> Input
Step 4: Create a variable named 'mul'
Step 5: Store the mul of 'a' and 'b' in a variable named 'mul" -> Output
Step 6: End.
You will now see how significant space and time complexity is after understanding what they are.
How Significant Are Space and Time Complexity?
Significant in Terms of Time Complexity
The input size has a strong relationship with time complexity. As the size of the input increases, so does the runtime, or the amount of time it takes the algorithm to run.
Here is an example.
Assume you have a set of numbers S= (10, 50, 20, 15, 30)
There are numerous algorithms for sorting the given numbers. However, not all of them are effective. To determine which is the most effective, you must perform computational analysis on each algorithm.
Here are some of the most critical findings from the graph:
- This test revealed the following sorting algorithms: Quicksort, Insertion sort, Bubble sort, and Heapsort.
- Python is the programming language used to complete the task, and the input size ranges from 50 to 500 characters.
- The results were as follows: "Heap Sort algorithms performed well despite the length of the lists; on the other hand, you discovered that Insertion sort and Bubble sort algorithms performed far worse, significantly increasing computing time." See the graph above for the results.
- Before you can run an analysis on any algorithm, you must first determine its stability. Understanding your data is the most important aspect of conducting a successful analysis.
What Are Asymptotic Notations?
Asymptotic Notations are programming languages that allow you to analyze an algorithm's running time by identifying its behavior as its input size grows. This is also referred to as an algorithm's growth rate. When the input size increases, does the algorithm become incredibly slow? Is it able to maintain its fast run time as the input size grows? You can answer these questions thanks to Asymptotic Notation.
You can't compare two algorithms head to head. It is heavily influenced by the tools and hardware you use for comparisons, such as the operating system, CPU model, processor generation, and so on. Even if you calculate time and space complexity for two algorithms running on the same system, the subtle changes in the system environment may affect their time and space complexity.
As a result, you compare space and time complexity using asymptotic analysis. It compares two algorithms based on changes in their performance as the input size is increased or decreased.
Asymptotic notations are classified into three types:
- Big-Oh (O) notation
- Big Omega ( Ω ) notation
- Big Theta ( Θ ) notation
Now, go over each of these notations one by one.
1. Big-Oh (O) Notation
Paul Bachmann invented the big-O notation in 1894. He inadvertently introduced this notation in his discussion of function approximation.
From the definition: O (g(n)) =
f(n) : there exist positive constant c and n0 such that 0 <= f(n) <= c*g(n)
For all n >= n0
'n' denotes the upper bound value. If a function is O(n), it is also O(n2) and O(n3).
It is the most widely used notation for Asymptotic analysis. It specifies the upper bound of a function, i.e., the maximum time required by an algorithm or the worst-case time complexity. In other words, it returns the highest possible output value (big-O) for a given input.
2. Big-Omega (Ω) notation
Big-Omega is an Asymptotic Notation for the best case or a floor growth rate for a given function. It gives you an asymptotic lower bound on the growth rate of an algorithm's runtime.
From the definition: The function f( n ) is Ω (g(n)) if there exists a positive number c and N, such that f(n) >= cg(n) for all n >= N.
3. Big-Theta (Θ) notation
Big theta defines a function's lower and upper bounds, i.e., it exists as both, most, and least boundaries for a given input value.
From the definition : f(n) is Θ(g(n)) if there exists positive numbers c1, c2 and N such that c1g(n) <= f(n) <= c2g(n) for all n >= N.
Best Case, Worst Case, and Average Case in Asymptotic Analysis
Best Case: It is defined as the condition that allows an algorithm to complete statement execution in the shortest amount of time. In this case, the execution time serves as a lower bound on the algorithm's time complexity.
Average Case: You add the running times for each possible input combination and take the average in the average case. Here, the execution time serves as both a lower and upper bound on the algorithm's time complexity.
Worst Case: It is defined as the condition that allows an algorithm to complete statement execution in the shortest amount of time possible. In this case, the execution time serves as an upper bound on the algorithm's time complexity.
You will now see how to calculate space and time complexity after grasping the significance of space and time complexity.
Significant in Terms of Space Complexity
Space complexity refers to the total amount of memory space used by an algorithm/program, including the space of input values for execution. Calculate the space occupied by variables in an algorithm/program to determine space complexity.
However, people frequently confuse Space-complexity with auxiliary space. Auxiliary space is simply extra or temporary space, and it is not the same as space complexity. To put it another way,
Auxiliary space + space use by input values = Space Complexity
The best algorithm/program should have a low level of space complexity. The less space required, the faster it executes.
Method for Calculating Space and Time Complexity
Methods for Calculating Time Complexity
To calculate time complexity, you must consider each line of code in the program. Consider the multiplication function as an example. Now, calculate the time complexity of the multiply function:
Let T(n) be a function of the algorithm's time complexity. Lines 1 and 2 have a time complexity of O. (1). Line 3 represents a loop. As a result, you must repeat lines 4 and 5 (n -1) times. As a result, the time complexity of lines 4 and 5 is O. (n).
Finally, adding the time complexity of all the lines yields the overall time complexity of the multiple function fT(n) = O(n).
The iterative method gets its name because it calculates an iterative algorithm's time complexity by parsing it line by line and adding the complexity.
Aside from the iterative method, several other concepts are used in various cases. The recursive process, for example, is an excellent way to calculate time complexity for recurrent solutions that use recursive trees or substitutions. The master's theorem is another popular method for calculating time complexity.
Methods for Calculating Space Complexity
With an example, you will go over how to calculate space complexity in this section. Here is an example of computing the multiplication of array elements:
Let S(n) denote the algorithm's space complexity. In most systems, an integer occupies 4 bytes of memory. As a result, the number of allocated bytes would be the space complexity.
Line 1 allocates memory space for two integers, resulting in S(n) = 4 bytes multiplied by 2 = 8 bytes. Line 2 represents a loop. Lines 3 and 4 assign a value to an already existing variable. As a result, there is no need to set aside any space. The return statement in line 6 will allocate one more memory case. As a result, S(n)= 4 times 2 + 4 = 12 bytes.
Because the array is used in the algorithm to allocate n cases of integers, the final space complexity will be fS(n) = n + 12 = O (n).
As you progress through this tutorial, you will see some differences between space and time complexity.
Time Complexity vs. Space Complexity
You now understand space and time complexity fundamentals and how to calculate it for an algorithm or program. In this section, you will summarise all previous discussions and list the key differences in a table.
Calculates the time required
Estimates the space memory required
Time is counted for all statements
Memory space is counted for all variables, inputs, and outputs.
The size of the input data is the primary determinant.
Primarily determined by the auxiliary variable size
More crucial in terms of solution optimization
More essential in terms of solution optimization
Now that you have reached the end of the tutorial on space and time complexity, sum up what you’ve learned thus far.
Algorithm analysis is usually performed at two stages - before implementation and after implementation.
A priori analysis is the explanation of an algorithm's theoretical analysis. Assuming that all other variables, such as processor speed, are fixed and have no impact on implementation, an algorithm's efficiency is determined.
An algorithm's empirical analysis is referred to as a posterior analysis. Programming language is used to carry out the selected algorithm. The chosen algorithm is then put to use on the intended computer system. Actual data, such as running time and space requirements, are gathered for this investigation.
If N is used to represent the size of the input data and X is assumed to represent an algorithm, then the efficiency of X is mostly influenced by how much time and space it takes to implement the method.
Time Factor - The number of crucial operations, such as comparisons in a sorting algorithm, is counted or quantified to determine how much time has passed.
Space Factor - The amount of space is determined or assessed by adding together how much memory the algorithm can use.
When N is used as the size of the input data, the complexity of an algorithm f(N) gives the amount of running time and/or storage space required by the method.
Factors Affecting Long-term Usage of an Algorithm
The following are the factors that are crucial to an algorithm's long-term effectiveness:
- Efficiency — Efficiency plays a key role in developing a successful algorithm. Efficiency is what minimizes computation durations and delivers swift results.
- Finiteness – The algorithm must conclude a predetermined number of steps. Otherwise, it will consume more RAM and is not recommended. If it continues through infinite loops or recursion, stack overflow and out-of-bounds problems may happen.
- Correctness – Regardless of the magnitude of the input given, a good algorithm should generate a proper result.
Importance of Time Complexity and Space Complexity in Data Structures
Real-world programmers are bound by the physical memory of the systems on which they intend to run. This is where space complexity enters the picture because we never want to run a function or process that takes up more space than the system has available at any one time. On the other side, we don't want our operations to take too long, clogging and slowing down our apps.
Some algorithms perform better than others. We always prefer to select an efficient algorithm, hence metrics for assessing algorithm efficiency would be useful.
The complexity of an algorithm is a function that describes the algorithm's efficiency in terms of the amount of data it must process. There are usually natural units for the domain and range of this function. There are two basic complexity metrics of the efficiency of an algorithm:
- Time complexity is a function that describes how long an algorithm takes in terms of the quantity of input it receives.
- Space complexity is a function that describes how much memory (space) an algorithm requires to the quantity of input to the method.
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