The order of algorithms – A Level Further Mathematics OCR Revision

Time complexity: Analyses the amount of time an algorithm takes to run. Classical notations include O, Ω, and Θ, which are used to describe an algorithm’s upper bound, lower bound, and tight bound, respectively.
Space complexity: Evaluates the quantity of memory an algorithm needs to execute. Similar to time complexity, it is expressed by Big O notation.
P vs. NP problem: Though not typically directly tested, understanding of this topic can enhance comprehension of algorithm complexity. This talks about the distinction and relation between problems an algorithm can solve in polynomial time (P) and those for which solutions can be verified in polynomial time (NP).

The order of algorithms is fundamental in comparing the efficiency of different algorithms. Even if two algorithms can solve the same problem, the quicker and less memory-consuming algorithm is often preferred.
But it doesn’t stop there: understanding the order can also help optimise an algorithm’s performance and modify it for specific needs.
Hence, the order of algorithms is not just a theoretical concept, but a practical tool for improving computational efficiency.

Constant time (O(1)): The processing time of the algorithm remains constant irrespective of the size of input data. A simple example of a constant time operation is accessing an element in an array by index.
Logarithmic time(O(logn)): The execution time of the algorithm increases logarithmically with the size of input data. An example would be Binary search in a sorted array.
Linear time (O(n)): The running time of the algorithm increases proportionally with the input data size. For example, finding a specific value in an unsorted list or array.
Quadric time (O(n^2)): The processing time of the algorithm increases quadratically with the size of input data. Examples includes bubble or insertion sort.

There’s often a trade-off between time and space complexity - an algorithm that runs quicker may require more memory, and vice versa.
Deciding which priority - time efficiency or saving on space - can influence the choice of algorithm based on the specific requirements of the task at hand.
This involves considering factors such as available hardware, expected input size, and the importance of speed versus memory usage in the given scenario. The ideal is to find a balance between time and space efficiency that best suit the needs.