Definitions

# Analysis of algorithms

To analyze an algorithm is to determine the amount of resources (such as time and storage) necessary to execute it. Most algorithms are designed to work with inputs of arbitrary length. Usually the efficiency or complexity of an algorithm is stated as a function relating the input length to the number of steps (time complexity) or storage locations (space complexity).

Algorithm analysis is an important part of a broader computational complexity theory, which provides theoretical estimates for the resources needed by any algorithm which solves a given computational problem. These estimates provide an insight into reasonable directions of search of efficient algorithms.

In theoretical analysis of algorithms it is common to estimate their complexity in asymptotic sense, i.e., to estimate the complexity function for reasonably large length of input. Big O notation, omega notation and theta notation are used to this end. For instance, binary search is said to run an amount of steps proportional to a logarithm, or in O(log(n)), colloquially "in logarithmic time". Usually asymptotic estimates are used because different implementations of the same algorithm may differ in efficiency. However the efficiencies of any two "reasonable" implementations of a given algorithm are related by a constant multiplicative factor called hidden constant.

Exact (not asymptotic) measures of efficiency can sometimes be computed but they usually require certain assumptions concerning the particular implementation of the algorithm, called model of computation. A model of computation may be defined in terms of an abstract computer, e.g., Turing machine, and/or by postulating that certain operations are executed in unit time. For example, if the sorted set to which we apply binary search has N elements, and we can guarantee that a single binary lookup can be done in unit time, then at most log2 N + 1 time units are needed to return an answer.

Exact measures of efficiency are useful to the people who actually implement and use algorithms, because they are more precise and thus enable them to know how much time they can expect to spend in execution. To some people (e.g. game programmers), a hidden constant can make all the difference between success and failure.

Time efficiency estimates depend on what we define to be a step. For the analysis to make sense, the time required to perform a step must be guaranteed to be bounded above by a constant. One must be careful here; for instance, some analyses count an addition of two numbers as a step. This assumption may not be warranted in certain contexts. For example, if the numbers involved in a computation may be arbitrarily large, addition no longer can be assumed to require constant time (compare the time you need to add two 2-digit integers and two 1000-digit integers using a pen and paper).

## Run-time analysis

Run-time analysis is a theoretical classification that estimates and anticipates the increase in running time (or run-time) of an algorithm as its input size (usually denoted as "n") increases. Run-time efficiency is a topic of great interest in Computer Science: a program can take seconds, hours or even years to finish executing, depending on which algorithm it implements. (This should not be confused with the analysis of an algorithm's actual run-time which is covered in the article performance analysis).

### Shortcomings of empirical metrics

Since algorithms are platform-independent (i.e. a given algorithm can be implemented in any programming language on any computer running any operating system), there are significant drawbacks to using an empirical approach to gauge the comparative performance of a given set of algorithms.

Take as an example a program that looks up a specific entry in a sorted list of size n. Suppose this program were implemented on Computer A, a state-of-the-art machine, using a linear search algorithm, and on Computer B, a much slower machine, using a binary search algorithm. Benchmark testing on the two computers running their respective programs might look something like the following:

n (list size) Computer A run-time
(in nanoseconds)
Computer B run-time
(in nanoseconds)
15 7 ns 100,000 ns
65 32 ns 150,000 ns
250 128 ns 200,000 ns
1,000 500 ns 250,000 ns

Based on these metrics, it would be easy to jump to the conclusion that Computer A is running an algorithm that is far superior in efficiency to what Computer B is running. However, if the size of the input-list is increased to a sufficient number, that conclusion is dramatically demonstrated to be in error:

n (list size) Computer A run-time
(in nanoseconds)
Computer B run-time
(in nanoseconds)
15 7 ns 100,000 ns
65 32 ns 150,000 ns
250 125 ns 200,000 ns
1,000 500 ns 250,000 ns
... ... ...
1,000,000 500,000 ns 500,000 ns
4,000,000 2,000,000 ns 550,000 ns
16,000,000 8,000,000 ns 600,000 ns
... ... ...
63,072 * 1012 31,536 * 1012 ns,
or 1 year
1,375,000 ns,
or 1.375 milliseconds

Computer A, running the linear search program, exhibits a linear growth rate. The program's run-time is directly proportional to its input size. Doubling the input size doubles the run time, quadrupling the input size quadruples the run-time, and so forth. Whereas Computer B, running the binary search program, exhibits a logarithmic growth rate. Doubling the input size only increases the run time by a constant amount (in this example, 25,000 ns). Even though Computer A is ostensibly a faster machine, Computer B will inevitably surpass Computer A in run-time because it's running an algorithm with a much slower growth rate.

### Orders of growth

Informally, an algorithm can be said to exhibit a growth rate on the order of a mathematical function if beyond a certain input size n, the function f(n) times a positive constant provides an upper bound or limit for the run-time of that algorithm. In other words, for a given input size n greater than some n0 and a constant c, an algorithm can run no slower than c * f(n). This concept is frequently expressed using Big O notation (see Big O notation for a more formal discussion). For example, since the run-time of insertion sort grows quadratically as its input size increases, insertion sort can be said to be of order O(n²).

Big O notation is a convenient way to express the worst-case scenario for a given algorithm, although it can also be used to express the average-case — for example, the worst-case scenario for quicksort is O(n²) but the average-case run-time is O(n lg n).

### Evaluating run-time complexity

The run-time complexity for the worst-case scenario of a given algorithm can sometimes be evaluated by examining the structure of the algorithm and making some simplifying assumptions. Consider the following pseudocode:

`1    get a positive integer from input`
`2    if n > 10`
`3        print "This might take a while..."`
`4    for i = 1 to n`
`5        for j = 1 to i`
`6            print i * j`
`7    print "Done!"`

A given computer will take a discrete amount of time to execute each of the instructions involved with carrying out this algorithm. The specific amount of time to carry out a given instruction will vary depending on which instruction is being executed and which computer is executing it, but on a conventional computer, this amount will be deterministic. Say that the actions carried out in step 1 are considered to consume time T1, step 2 uses time T2, and so forth.

In the algorithm above, steps 1, 2 and 7 will only be run once. For a worst-case evaluation, it should be assumed that step 3 will be run as well. Thus the total amount of time to run steps 1-3 and step 7 is:

$T_1 + T_2 + T_3 + T_7 ,$

The loops in steps 4, 5 and 6 are trickier to evaluate. The outer loop test in step 4 will execute (n + 1 ) times, which will consume T4(n + 1 ) time. The inner loop, on the other hand, is governed by the value of i, which iterates from 1 to n. On the first pass through the outer loop, j iterates from 1 to 1: the inner loop makes one pass, so running the inner loop body (step 6) consumes T6 time, and the inner loop test (step 5) consumes 2T5 time. During the next pass through the outer loop, j iterates from 1 to 2: the inner loop makes two passes, so running the inner loop body (step 6) consumes 2T6 time, and the inner loop test (step 5) consumes 3T5 time.

Altogether, the total time required to run the inner loop body can be expressed as an arithmetic progression:

$T_6 + 2T_6 + 3T_6 + cdots + \left(n-1\right) T_6 + n T_6$

which can be factored as

$T_6 left\left[1 + 2 + 3 + cdots + \left(n-1\right) + n right\right] = T_6 left\left[frac\left\{1\right\}\left\{2\right\} \left(n^2 + n\right) right\right]$

The total time required to run the inner loop test can be evaluated similarly:

$2T_5 + 3T_5 + 4T_5 + cdots + \left(n-1\right) T_5 + n T_5 + \left(n + 1\right) T_5$

$= T_5 + 2T_5 + 3T_5 + 4T_5 + cdots + \left(n-1\right)T_5 + nT_5 + \left(n+1\right)T_5 - T_5$

which can be factored as

$T_5 left\left[1+2+3+cdots + \left(n-1\right) + n + \left(n + 1\right) right\right] - T_5 = T_5 left\left[frac\left\{1\right\}\left\{2\right\} \left(n^2 + 3n + 2\right) right\right] - T_5$

Therefore the total running time for this algorithm is:

$f\left(n\right) = T_1 + T_2 + T_3 + T_7 + \left(n + 1\right)T_4 + left\left[frac\left\{1\right\}\left\{2\right\} \left(n^2 + n\right) right\right] T_6 + left\left[frac\left\{1\right\}\left\{2\right\} \left(n^2+3n+2\right) right\right] T_5 - T_5$

which reduces to

$f\left(n\right) = left\left[frac\left\{1\right\}\left\{2\right\} \left(n^2 + n\right) right\right] T_6 + left\left[frac\left\{1\right\}\left\{2\right\} \left(n^2 + 3n\right) right\right] T_5 + \left(n + 1\right)T_4 + T_1 + T_2 + T_3 + T_7$

As a rule-of-thumb, one can assume that the highest-order term in any given function dominates its rate of growth and thus defines its run-time order. In this example, n² is the highest-order term, so one can conclude that f(n) = O(n²). Formally this can be proven as follows:

Prove that $left\left[frac\left\{1\right\}\left\{2\right\} \left(n^2 + n\right) right\right] T_6 + left\left[frac\left\{1\right\}\left\{2\right\} \left(n^2 + 3n\right) right\right] T_5 + \left(n + 1\right)T_4 + T_1 + T_2 + T_3 + T_7 le cn^2, n ge n_0$

$left\left[frac\left\{1\right\}\left\{2\right\} \left(n^2 + n\right) right\right] T_6 + left\left[frac\left\{1\right\}\left\{2\right\} \left(n^2 + 3n\right) right\right] T_5 + \left(n + 1\right)T_4 + T_1 + T_2 + T_3 + T_7$

$le \left(n^2 + n \right)T_6 + \left(n^2 + 3n \right)T_5 + \left(n + 1\right)T_4 + T_1 + T_2 + T_3 + T_7$ (for n ≥ 0)

Let k be a constant greater than or equal to [T1..T7]

$T_6\left(n^2 + n \right) + T_5\left(n^2 + 3n \right) + \left(n + 1\right)T_4 + T_1 + T_2 + T_3 + T_7 le k\left(n^2 + n \right) + k\left(n^2 + 3n \right) + kn + 5k$
$= 2kn^2 + 5kn + 5k le 2kn^2 + 5kn^2 + 5kn^2$ (for n ≥ 1) $= 12kn^2$

Therefore $left\left[frac\left\{1\right\}\left\{2\right\} \left(n^2 + n\right) right\right] T_6 + left\left[frac\left\{1\right\}\left\{2\right\} \left(n^2 + 3n\right) right\right] T_5 + \left(n + 1\right)T_4 + T_1 + T_2 + T_3 + T_7 le cn^2, n ge n_0$ for $c = 12k, n_0 = 1$

A more elegant approach to analyzing this algorithm would be to declare that [T1..T7] are all equal to one unit of time greater than or equal to [T1..T7]. This would mean that the algorithm's running time breaks down as follows:

$4+sum_\left\{i=1\right\}^n ileq 4+sum_\left\{i=1\right\}^n n=4+n^2leq5n^2$(for n ≥ 1)$=O\left(n^2\right)$

### Growth rate analysis of other resources

The methodology of run-time analysis can also be utilized for predicting other growth rates, such as consumption of memory space. As an example, consider the following pseudocode which manages and reallocates memory usage by a program based on the size of a file which that program manages:

`1    While file is still open`
`2        Let n = file size`
`3        For every 100,000 kilobytes of increase in file size`
`4            Double the amount of memory reserved`

In this instance, as the file size n increases, memory will be consumed at an exponential growth rate, which is order O(2n).

## References

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