Algorithms & Their Complexity - Blueprint, Key Concepts, Analysis

Quiz

Algorithm

An algorithm is a set of instructions following a specific sequence to solve problems. It is an easier way to solve programming problems using a set of sequential steps. This method is easy to learn and solves programming problems.

Complexity

The complexity of an algorithm refers to the time and space required to compute a processthe complexity of an algorithm is measured using time and space complexity.

Overall, an algorithm is a finite set of instructions executed in a precise order to complete a specific goal. It does not include the whole program or code; it is a simple logic to a problem expressed as an informal description in the form of a flowchart or pseudocode.

Blueprint of an Algorithm

The best way to understand an algorithm is to understand its blueprint. A blueprint is a structure of an algorithm. The following structure shows the algorithms blueprint.

Step 1: Start
Step 2: Variable declaration    //optional
Step 3: Input	
Step 4: Processing
Step 5: Get Results / Outcomes
Step 6: Show results
Step 7: End

Example of an Algorithm

Step 1: Start
Step 2: Take three integer variables like n1, n2 and n3
Step 3: Input two integer numbers and store them in n1 and n2
Step 4: n3 = n1 + n2
Step 5: Print n3
Step 6: End

Algorithm Analysis

Algorithm analysis is a process of analyzing an algorithm's efficiency and performance, with reference to time complexity (speed) and space complexity (memory usage). It determines the best, average, and worst-case scenarios for executing the algorithm and making informed decisions about its applicability to specific applications.

The efficiency and behavior of algorithms with respect to time and space complexity can be described using asymptotic notations. Asymptotic notations allow you to define how an algorithm's performance scales with the size of its input, and its growth rate rather than execution times. The three main asymptotic notations are Big-O, Big-, and Big-.

Asymptotic Notations (Big-O, Big-, Big-)

i. Big-O Notation (O) − Big-O Notation (O) describes how much time an algorithm takes to run (time complexity) or how much memory (space complexity) an algorithm will take to run a program or an application). It is a mathematical notation used to describe the upper bound or worst-case scenario of an algorithm's runtime complexity in terms of input size. Hence, it ensures that the method doesn't exceed a predetermined time or space complexity as the input size grows. It provides a clear and consistent approach to defining how an algorithm's performance scales as input size increases. Big-O notations ensure that the maximum resources an algorithm will require.

For example

Linear Search − let’s assume that we have an unsorted list of n numbers and want to search for a specific number (key) from the list.

Algorithm

Start searching from the first element of the list.
Compare each element with the element you are searching for.
Stop when the key is found or continue the search till the end of the list.

Time Complexity Analysis

Best Case	The key element is the first element in the list. The algorithm will stop searching after getting it.
Time Complexity	O(1) (constant time)
Worst Case	The key element is not in the list, or at the last position in the list. The algorithm makes n comparisons. Search time will increase.
Time Complexity	O(n) (linear time)
Average Case	The key element is in the middle of the list. In this case, n/2 comparisons will be made.
Time Complexity	O(n) (drop constants in Big-O)

From above mentioned example it is clarified that The Big-O notation simplifies performance descriptions by focusing on how the algorithm scales with input size.

ii. Big- Notation () − This provides a tight bound on an algorithm's performance, encompassing both upper and lower limits for a function f(n) and provides the average time complexity of an algorithm. If an algorithm is (n), its performance grows linearly with the input size with respect to best and worst cases. It provides the most exact description of an algorithm's evolution, probable its uses are less because finding tight constraints can be difficult.

Big-Ω Notation (Ω) − This describes the best-case scenario for an algorithm's performance at its lowest limit. If an algorithm is Ω(n), it will perform in at least linear time, independent of optimizations or input arrangements. Big-Ω is useful for theoretical study, but less practical for evaluating real-world performance because worst-case situations are more important to examine.

Factors of an Algorithm

A good algorithm always works with associated key factors. Some of the key factors of an Algorithm are as follows −

Correctness − The algorithm must produce the correct and desired output for all valid inputs.
Modularity − Modularity refers to breaking down a problem into smaller modules and then solving it easily.
Efficiency − The efficiency of an algorithm can be measured using time and space complexity where the time complexity can be measured using the time taken by the algorithm to execute a function of inputted size like O(n), O(log n) and Space complexity can be computed as the amount of memory used by the method as a function of input size.
Functionality − It considers multiple logical steps while solving a real-world situation.
Robustness − The algorithm should be able to handle edge cases and unexpected inputs without crashing.
Portability − The algorithm should be executable on multiple systems or platforms without requiring significant changes.
Resource Utilization − It refers to optimal uses of computational resources like CPU, memory, etc.
User-friendly − It should be user-friendly; if any problem occurs then it should guide users to understand the issue and its feasible solution.
Simplicity − The algorithm should be as simple as feasible to comprehend and apply.
Extensibility − Your algorithm should be extendable in case another algorithm creator or programmer wishes to utilize it.

Importance of an Algorithm

An algorithm is a step-by-step solution to a problem or completing a task. As it is the simplest and easiest method to solve problems; it has its significance in problem-solving in the area of computer programming, data structure, analysis and design of algorithms data and in different engineering applications. It gives a methodical approach to achieving efficiency, accuracy, and consistency. Well-designed algorithms maximize resource utilization, improve decision-making, and enable automation.

Some of the key significance of an algorithm is as follows −

It has efficiency in problem-solving − It is a systematic technique to efficiently solve complex problems and gives time and resource utilisation.
Optimal solutions − An algorithm provides optimal solutions by effective resource utilization, such as time, memory, or energy, hence improving process efficiency.
Automation − It is one of the most essential approaches to automate tasks, allowing computers to manage large amounts of data or operations with minimal manual intervention.
Serves as a foundation of technology − They support a wide range of technical developments, from basic applications to large systems like artificial intelligence, robots, and data analytics.
Consistency and Accuracy − Well-designed algorithms produce reliable and consistent outcomes and minimise the possibility of human error.
Adaptability across Domains − Algorithms can be effectively applicable in the sectors of healthcare, finance, engineering, and logistics, making them adaptable and broadly useful.
Decision-Making − Algorithms help to improve real-time or strategic decision-making by evaluating data and discovering patterns.
Supports development in modern systems − Algorithms drive innovation by addressing emerging challenges and allowing for the creation of advanced systems such as predictive models and machine learning frameworks.

Key Characteristics of an Algorithm

An algorithm is a well-defined sequence of steps to solve a specific problem. Some of the key characteristics of an algorithm are described below −

Finiteness − An algorithm must properly terminate after some finite number of steps. It cannot continue indefinitely.
Definiteness − An algorithm must be clearly and precisely defined.
Input − An algorithm may take null or more inputs to produce desired results.
Output − It must produce at least one output after the successful execution of defined steps.
Effectiveness − Each step of the algorithm should be simple and relevant to perform within the time limit.
Generalization − An algorithm should apply to a wide range of inputs, not specific.
Correctness − The algorithm must return the proper result for all valid inputs.
Efficiency − It should make the best use of resources like time (time complexity) and space (space complexity).
Deterministic or Non-Deterministic − A deterministic algorithm always produces the same result when given the same input. A non-deterministic algorithm may generate different outputs for the same input due to parallel execution.
Scalability − The algorithm should work well even if the input size increases significantly.

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