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Algorithm complexity

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Algorithm complexity is a measure which evaluates the order of the count of operations, performed by a given or algorithm as a function of the size of the input data. To put this simpler, complexity is a rough approximation of the number of steps necessary to execute an algorithm. When we evaluate complexity we speak of order of operation count, not of their exact count. For example if we have an order of N2 operations to process N elements, then N2/2 and 3*N2 are of one and the same quadratic order.

Algorithm complexity is commonly represented with the O(f) notation, also known as asymptotic notation or “Big O notation”, where f is the function of the size of the input data. The asymptotic computational complexity O(f) measures the order of the consumed resources (CPU time, memory, etc.) by certain algorithm expressed as function of the input data size.

Complexity can be constant, logarithmic, linear, n*log(n), quadratic, cubic, exponential, etc. This is respectively the order of constant, logarithmic, linear and so on, number of steps, are executed to solve a given problem. For simplicity, sometime instead of “algorithms complexity” or just “complexity” we use the term “running time”.

Info
Algorithm complexity is a rough approximation of the number of steps, which will be executed depending on the size of the input data. Complexity gives the order of steps count, not their exact count.

Typical Algorithm Complexities

Complexity Running Time Description
constant O(1) It takes a constant number of steps for performing a given operation (for example 1, 5, 10 or other number) and this count does not depend on the size of the input data.
logarithmic O(log(N)) It takes the order of log(N) steps, where the base of the logarithm is most often 2, for performing a given operation on N elements. For example, if N = 1,000,000, an algorithm with a complexity O(log(N)) would do about 20 steps (with a constant precision). Since the base of the logarithm is not of a vital importance for the order of the operation count, it is usually omitted.
linear O(N) It takes nearly the same amount of steps as the number of elements for performing an operation on N elements. For example, if we have 1,000 elements, it takes about 1,000 steps. Linear complexity means that the number of elements and the number of steps are linearly dependent, for example the number of steps for N elements can be N/2 or 3*N.
O(n*log(n)) It takes N*log(N) steps for performing a given operation on N elements. For example, if you have 1,000 elements, it will take about 10,000 steps.
quadratic O(n2 ) It takes the order of N^2 number of steps, where the N is the size of the input data, for performing a given operation. For example if N = 100, it takes about 10,000 steps. Actually we have a quadratic complexity when the number of steps is in quadratic relation with the size of the input data. For example for N elements the steps can be of the order of 3*N^2 /2.
cubic O(n3) It takes the order of N^3 steps, where N is the size of the input data, for performing an operation on N elements. For example, if we have 100 elements, it takes about 1,000,000 steps.
exponential O(2n ), O(N!), O(nk ), … It takes a number of steps, which is with an exponential dependability with the size of the input data, to perform an operation on N elements. For example, if N = 10, the exponential function 2^N has a value of 1024, if N = 20, it has a value of 1 048 576, and if N = 100, it has a value of a number with about 30 digits. The exponential function N! grows even faster: for N = 5 it has a value of 120, for N = 10 it has a value of 3,628,800 and for N = 20 – 2,432,90,008,176,640,000 .

Complexity and Execution Time

The execution speed of a program depends on the complexity of the algorithm, which is executed. If this complexity is low, the program will execute fast even for a big number of elements. If the complexity is high, the program will execute slowly or will not even work (it will hang) for a big number of elements.

Algorithm 10 20 50 100 1,000 10,000 100,000
O(1) < 1 sec. < 1 sec. < 1 sec. < 1 sec. < 1 sec. < 1 sec. < 1 sec.
O(log(n)) < 1 sec. < 1 sec. < 1 sec. < 1 sec. < 1 sec. < 1 sec. < 1 sec.
O(n) < 1 sec. < 1 sec. < 1 sec. < 1 sec. < 1 sec. < 1 sec. < 1 sec.
O(n*log(n)) < 1 sec. < 1 sec. < 1 sec. < 1 sec. < 1 sec. < 1 sec. < 1 sec.
O(n^2 ) < 1 sec. < 1 sec. < 1 sec. < 1 sec. < 1 sec. 2 sec. 3-4 min.
O(n^3 ) < 1 sec. < 1 sec. < 1 sec. < 1 sec. 20 sec. 5.55 hours 231.5 days
O(2n ) < 1 sec. < 1 sec. 260 days hangs hangs hangs hangs
O(n!) < 1 sec. hangs hangs hangs hangs hangs hangs
O(n^n ) 3-4 min. hangs hangs hangs hangs hangs hangs

We can draw many conclusions from the above table:

  • Algorithms with a constant, logarithmic or linear complexity are so fast that we cannot feel any delay, even with a relatively big size of the input data.

  • Complexity O(nlog(n)) is similar to the linear and works nearly as fast as linear, so it will be very difficult to feel any delay.

  • Quadratic algorithms work very well up to several thousand elements.

  • Cubic algorithms work well if the elements are not more than 1,000.

  • Generally these so called polynomial algorithms (any, which are not exponential) are considered to be fast and working well for thousands of elements.

  • Generally the exponential algorithms do not work well and we should avoid them (when possible). If we have an exponential solution to a task, maybe we actually do not have a solution, because it will work only if the number of the elements is below 10-20. Modern cryptography is based exactly on this – there are not any fast (non-exponential) algorithms for finding the secret keys used for data encryption.

Best, Worst and Average Case

Complexity of algorithms is usually evaluated in the worst case (most unfavorable scenario). This means in the average case they can work faster, but in the worst case they work with the evaluated complexity and not slower.

Let’s take an example: searching in array. To find the searched key in the worst case, we have to check all the elements in the array. In the best case we will have luck and we will find the element at first position. In the average case we can expect to check half the elements in the array until we find the one we are looking for. Hence in the worst case the complexity is O(N) – linear. In the average case the complexity is O(N/2) = O(N) – linear, because when evaluating complexity one does not take into account the constants. In the best case we have a constant complexity O(1), because we make only one step and directly find the element.

Futher reading

Chapter 19. Data Structures and Algorithm Complexity