Dealing with Algorithm Complexity

By John Paul Mueller, Luca Massaron

Part of Algorithms For Dummies Cheat Sheet

You already know that algorithms are complex. However, you need to know how complex an algorithm is because the more complex one is, the longer it takes to run. The following table helps you understand the various levels of complexity presented in order of running time (from fastest to slowest).

Complexity Description
Constant complexity O(1) Provides an unvarying execution time, no matter how much input you provide. Each input requires a single unit of execution time.
Logarithmic complexity O(log n) The number of operations grows at a slower rate than the input, making the algorithm less efficient with small inputs and more efficient with larger ones. A typical algorithm of this class is the binary search.
Linear complexity O(n) Operations grow with the input in a 1:1 ratio. A typical algorithm is iteration, when you scan input once and apply an operation to each element of it.
Linearithmic complexity O(n log n) Complexity is a mix between logarithmic and linear complexity. It is typical of some smart algorithms used to order data, such as Mergesortsort, Heapsort, and Quicksort.
Quadratic complexity O(n2) Operations grow as a square of the number of inputs. When you have one iteration inside another iteration (called nested iterations in computer science), you have quadratic complexity. For instance, you have a list of names and, in order to find the most similar ones, you compare each name against all the other names. Some less efficient ordering algorithms present such complexity: bubble sort, selection sort, and insertion sort. This level of complexity means that your algorithms may run for hours or even days before reaching a solution.
Cubic complexity O(n3) Operations grow even faster than quadratic complexity because now you have multiple nested iterations. When an algorithm has this order of complexity and you need to process a modest amount of data (100,000 elements), your algorithm may run for years. When you have a number of operations that is a power of the input, it is common to refer to the algorithm as running in polynomial time.
Exponential complexity O(2n) The algorithm takes twice the number of previous operations for every new element added. When an algorithm has this complexity, even small problems may take forever. Many algorithms doing exhaustive searches have exponential complexity. However, the classic example for this level of complexity is the calculation of Fibonacci numbers.
Factorial complexity O(n!) This algorithm presents a real nightmare of complexity because of the large number of possible combinations between the elements. Just imagine: If your input is 100 objects, and an operation on your computer takes 10-6 seconds (a reasonable speed for every computer nowadays), you will need about 10140 years to complete the task successfully (an impossible amount of time because the age of the universe is estimated as being 1014 years). A famous factorial complexity problem is the traveling salesman problem, in which a salesman has to find the shortest route for visiting many cities and coming back to the starting city.