Algorithm Efficiency, Complexity, and Performance

Algorithm Efficiency

Algorithm efficiency refers to how much computational resources an algorithm needs to produce its results. The two major resources are time (i.e., how fast the algorithm runs) and space (i.e., how much memory the algorithm uses).
The efficiency of an algorithm can vary significantly based on the specifics of the data it operates on. For instance, an algorithm may perform well with small data sets but become prohibitively slow as the size of the data set increases.

Complexity of Algorithms

Complexity of an algorithm is a measure of the amount of resources needed by an algorithm to run. It is usually expressed in terms of time complexity (how many steps the algorithm takes to run) and space complexity (how much memory it uses).
The worst-case complexity measures the longest time the algorithm could possibly take to complete, or the most space it could possibly require. This is an upper bound on performance.
The average-case complexity measures the expected performance of the algorithm under average conditions. This is often more informative than worst-case complexity, but is more difficult to calculate.
The best-case complexity measures the least time the algorithm could possibly take to complete, or the least space it could possibly require. Although this may seem like valuable information, it tends not to be very helpful since best-case scenarios are often unrepresentative of typical inputs.
Certain algorithms demonstrate linear complexity where the running time increases linearly with the size of the input, noted as O(n).
Other algorithms show logarithmic complexity, where the running time increases logarithmically with the size of the input, denoted by O(log n).
Algorithms exhibiting constant complexity perform the same number of operations regardless of the size of the input, expressed as O(1). These are the most efficient algorithms.

Performance of Algorithms

The performance of an algorithm relates to how an algorithm’s time and space requirements grow as the size of the problem grows. It provides an estimate of how an algorithm will respond to changes in problem size.
Time performance is concerned with the number of operations an algorithm performs. This is usually the primary concern for algorithm performance.
Space performance looks at the maximum amount of memory an algorithm requires at any point during its execution. Depending on the problem and the environment, space performance can also be a vital consideration.
The choice between time efficiency and space efficiency often depends on specific limitations and project circumstances. In some cases, a faster algorithm is preferred even if it consumes more memory, while in other situations, a slower but less memory-intensive solution might be optimal.
Time-space trade-off is a term used when it has been intentionally decided to increase the time efficiency of an algorithm (but more memory will be consumed), or vice versa.

Big-O Notation in Algorithm Complexity

Big-O Notation is a way of expressing the worst-case scenario of an algorithm’s time or space complexity.
Big-O notation abstracts the performance of an algorithm into a single notation that allows algorithms to be compared to each other in terms of their respective growth rates as inputs grow larger.
It is concerned with the growth rate of an algorithm’s difficulty in terms of the size of the input. Thus O(n²) means that the algorithm’s complexity grows quadratically with the size of the input, while O(n log n) means the complexity grows logarithmically with the size of the input.
Comparing algorithms in this way makes it easier to choose the most efficient algorithm for a particular task, disregarding implementation-specific factors such as hardware or programming language.