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 running time 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 for efficient algorithms.
In theoretical analysis of algorithms it is common to estimate their complexity in the asymptotic sense, i.e., to estimate the complexity function for arbitrarily large input. Big O notation, omega notation and theta notation are used to this end. For instance, binary search is said to run in a number of steps proportional to the logarithm of the length of the list being searched, 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 a 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 list to which we apply binary search has n elements, and we can guarantee that each lookup of an element in the list can be done in unit time, then at most log_{2} n + 1 time units are needed to return an answer.
Contents
Cost models
Time efficiency estimates depend on what we define to be a step. For the analysis to correspond usefully to the actual execution time, 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 one step. This assumption may not be warranted in certain contexts. For example, if the numbers involved in a computation may be arbitrarily large, the time required by a single addition can no longer be assumed to be constant.
Two cost models are generally used:^{[1]}^{[2]}^{[3]}^{[4]}^{[5]}
 the uniform cost model, also called uniformcost measurement (and similar variations), assigns a constant cost to every machine operation, regardless of the size of the numbers involved
 the logarithmic cost model, also called logarithmiccost measurement (and variations thereof), assigns a cost to every machine operation proportional to the number of bits involved
The latter is more cumbersome to use, so it's only employed when necessary, for example in the analysis of arbitraryprecision arithmetic algorithms, like those used in cryptography.
A key point which is often overlooked is that published lower bounds for problems are often given for a model of computation that is more restricted than the set of operations that you could use in practice and therefore there are algorithms that are faster than what would naively be thought possible.^{[6]}
Runtime analysis
Runtime analysis is a theoretical classification that estimates and anticipates the increase in running time (or runtime) of an algorithm as its input size (usually denoted as n) increases. Runtime 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 (see also performance analysis, which is the analysis of an algorithm's runtime in practice).
Shortcomings of empirical metrics
Since algorithms are platformindependent (i.e. a given algorithm can be implemented in an arbitrary programming language on an arbitrary computer running an arbitrary 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 stateoftheart 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 runtime
(in nanoseconds)Computer B runtime
(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 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 that of Computer B. However, if the size of the inputlist is increased to a sufficient number, that conclusion is dramatically demonstrated to be in error:
n (list size) Computer A runtime
(in nanoseconds)Computer B runtime
(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 × 10^{12} 31,536 × 10^{12} ns,
or 1 year1,375,000 ns,
or 1.375 millisecondsComputer A, running the linear search program, exhibits a linear growth rate. The program's runtime is directly proportional to its input size. Doubling the input size doubles the run time, quadrupling the input size quadruples the runtime, and so forth. On the other hand, 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 runtime because it's running an algorithm with a much slower growth rate.
Orders of growth
Main article: Big O notationInformally, 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 runtime of that algorithm. In other words, for a given input size n greater than some n_{0} and a constant c, the running time of that algorithm will never be larger than c × f(n). This concept is frequently expressed using Big O notation. For example, since the runtime 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 worstcase scenario for a given algorithm, although it can also be used to express the averagecase — for example, the worstcase scenario for quicksort is O(n²), but the averagecase runtime is O(n log n).^{[7]}
Evaluating runtime complexity
The runtime complexity for the worstcase 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.^{[8]} Say that the actions carried out in step 1 are considered to consume time T_{1}, step 2 uses time T_{2}, and so forth.
In the algorithm above, steps 1, 2 and 7 will only be run once. For a worstcase evaluation, it should be assumed that step 3 will be run as well. Thus the total amount of time to run steps 13 and step 7 is:
The loops in steps 4, 5 and 6 are trickier to evaluate. The outer loop test in step 4 will execute ( n + 1 ) times (note that an extra step is required to terminate the for loop, hence n + 1 and not n executions), which will consume T_{4}( 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 T_{6} time, and the inner loop test (step 5) consumes 2T_{5} 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 2T_{6} time, and the inner loop test (step 5) consumes 3T_{5} time.
Altogether, the total time required to run the inner loop body can be expressed as an arithmetic progression:
which can be factored^{[9]} as
The total time required to run the inner loop test can be evaluated similarly:
which can be factored as
Therefore the total running time for this algorithm is:
which reduces to
As a ruleofthumb, one can assume that the highestorder term in any given function dominates its rate of growth and thus defines its runtime order. In this example, n² is the highestorder term, so one can conclude that f(n) = O(n²). Formally this can be proven as follows:
Prove that
(for n ≥ 0)
Let k be a constant greater than or equal to [T_{1}..T_{7}]
(for n ≥ 1) = 12kn^{2}
Therefore for c = 12k,n_{0} = 1A more elegant approach to analyzing this algorithm would be to declare that [T_{1}..T_{7}] are all equal to one unit of time greater than or equal to [T_{1}..T_{7}].^{[clarification needed]} This would mean that the algorithm's running time breaks down as follows:^{[10]}
(for n ≥ 1) = O(n^{2}).
Growth rate analysis of other resources
The methodology of runtime 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:
while (file still open) let n = size of file for every 100,000 kilobytes of increase in file size 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(2^{n}).^{[11]}
Relevance
Algorithm analysis is important in practice because the accidental or unintentional use of an inefficient algorithm can significantly impact system performance. In timesensitive applications, an algorithm taking too long to run can render its results outdated or useless. An inefficient algorithm can also end up requiring an uneconomical amount of computing power or storage in order to run, again rendering it practically useless.
See also
 Amortized analysis
 Asymptotic analysis
 Best, worst and average case
 Big O notation
 Computational complexity theory
 NPComplete
 Numerical analysis
 Polynomial time
 Program optimization
 Profiling (computer programming)
 Smoothed analysis
 Time complexity — includes table of orders for common algorithms
Notes
 ^ Alfred V. Aho; John E. Hopcroft; Jeffrey D. Ullman (1974). The design and analysis of computer algorithms. AddisonWesley Pub. Co.., section 1.3
 ^ Juraj Hromkovič (2004). Theoretical computer science: introduction to Automata, computability, complexity, algorithmics, randomization, communication, and cryptography. Springer. pp. 177–178. ISBN 9783540140153. http://books.google.com/books?id=KpNetn262QC&pg=PA177.
 ^ Giorgio Ausiello (1999). Complexity and approximation: combinatorial optimization problems and their approximability properties. Springer. pp. 3–8. ISBN 9783540654315. http://books.google.com/books?id=Yxxw90d9AuMC&pg=PA3.
 ^ Wegener, Ingo (2005), Complexity theory: exploring the limits of efficient algorithms, Berlin, New York: SpringerVerlag, p. 20, ISBN 9783540210450, http://books.google.com/books?id=u7DZSDSUYlQC&pg=PA20
 ^ Robert Endre Tarjan (1983). Data structures and network algorithms. SIAM. pp. 3–7. ISBN 9780898711875. http://books.google.com/books?id=JiC7mIqgX4C&pg=PA3.
 ^ Examples of the price of abstraction?, cstheory.stackexchange.com
 ^ The term lg is often used as shorthand for log_{2}
 ^ However, this is not the case with a quantum computer
 ^ It can be proven by induction that
 ^ This approach, unlike the above approach, neglects the constant time consumed by the loop tests which terminate their respective loops, but it is trivial to prove that such omission does not affect the final result
 ^ Note that this is an extremely rapid and most likely unmanageable growth rate for consumption of memory resources
References
 Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L. & Stein, Clifford (2001). Introduction to Algorithms. Chapter 1: Foundations (Second ed.). Cambridge, MA: MIT Press and McGrawHill. pp. 3–122. ISBN 0262032937.
 Sedgewick, Robert (1998). Algorithms in C, Parts 14: Fundamentals, Data Structures, Sorting, Searching (3rd ed.). Reading, MA: AddisonWesley Professional. ISBN 9780201314526.
 Knuth, Donald. The Art of Computer Programming. AddisonWesley.
 Greene, Daniel A.; Knuth, Donald E. (1982). Mathematics for the Analysis of Algorithms (Second ed.). Birkhäuser. ISBN 374633102X.
 Goldreich, Oded (2008). Computational Complexity: A Conceptual Perspective. Cambridge University Press. ISBN 9780521884730.
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