Big O how do you calculate approximate it

Question

Most people with a degree in CS will certainly know what Big O stands for  It helps us to measure how well an algorithm scales    But I m curious  how do you calculate or approximate the complexity of your algorithms

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I ll do my best to explain it here on simple terms  but be warned that this topic takes my students a couple of months to finally grasp  You can find more information on the Chapter 2 of the Data Structures and Algorithms in Java book     There is no mechanical procedure that can be used to get the BigOh   As a  cookbook   to obtain the BigOh from a piece of code you first need to realize that you are creating a math formula to count how many steps of computations get executed given an input of some size   The purpose is simple  to compare algorithms from a theoretical point of view  without the need to execute the code  The lesser the number of steps  the faster the algorithm   For example  let s say you have this piece of code   int sum int  data  int N        int result   0                   1      for  int i   0  i  lt  N  i         2         result    data i             3            return result                    4     This function returns the sum of all the elements of the array  and we want to create a formula to count the computational complexity of that function   Number Of Steps   f N    So we have f N   a function to count the number of computational steps  The input of the function is the size of the structure to process  It means that this function is called such as   Number Of Steps   f data length    The parameter N takes the data length value  Now we need the actual definition of the function f    This is done from the source code  in which each interesting line is numbered from 1 to 4   There are many ways to calculate the BigOh  From this point forward we are going to assume that every sentence that doesn t depend on the size of the input data takes a constant C number computational steps   We are going to add the individual number of steps of the function  and neither the local variable declaration nor the return statement depends on the size of the data array   That means that lines 1 and 4 takes C amount of steps each  and the function is somewhat like this   f N    C         C   The next part is to define the value of the for statement  Remember that we are counting the number of computational steps  meaning that the body of the for statement gets executed N times  That s the same as adding C  N times   f N    C    C   C         C    C   C   N   C   C   There is no mechanical rule to count how many times the body of the for gets executed  you need to count it by looking at what does the code do  To simplify the calculations  we are ignoring the variable initialization  condition and increment parts of the for statement   To get the actual BigOh we need the Asymptotic analysis of the function  This is roughly done like this    Take away all the constants C  From f   get the polynomium in its standard form  Divide the terms of the polynomium and sort them by the rate of growth  Keep the one that grows bigger when N approaches infinity    Our f   has two terms   f N    2   C   N   0   1   C   N   1   Taking away all the C constants and redundant parts   f N    1   N   1   Since the last term is the one which grows bigger when f   approaches infinity  think on limits  this is the BigOh argument  and the sum   function has a BigOh of   O N      There are a few tricks to solve some tricky ones  use summations whenever you can   As an example  this code can be easily solved using summations   for  i   0  i  lt  2 n  i    2        1     for  j n  j  gt  i  j--           2         foo                        3           The first thing you needed to be asked is the order of execution of foo    While the usual is to be O 1   you need to ask your professors about it  O 1  means  almost  mostly  constant C  independent of the size N   The for statement on the sentence number one is tricky  While the index ends at 2   N  the increment is done by two  That means that the first for gets executed only N steps  and we need to divide the count by two   f N    Summation i from 1 to 2   N   2                   Summation i from 1 to N           The sentence number two is even trickier since it depends on the value of i  Take a look  the index i takes the values  0  2  4  6  8       2   N  and the second for get executed  N times the first one  N - 2 the second  N - 4 the third    up to the N   2 stage  on which the second for never gets executed   On formula  that means   f N    Summation i from 1 to N   Summation j                Again  we are counting the number of steps  And by definition  every summation should always start at one  and end at a number bigger-or-equal than one   f N    Summation i from 1 to N   Summation j   1 to  N -  i - 1    2   C        We are assuming that foo   is O 1  and takes C steps    We have a problem here  when i takes the value N   2   1 upwards  the inner Summation ends at a negative number  That s impossible and wrong  We need to split the summation in two  being the pivotal point the moment i takes N   2   1   f N    Summation i from 1 to N   2   Summation j   1 to  N -  i - 1    2       C       Summation i from 1 to N   2      C     Since the pivotal moment i  gt  N   2  the inner for won t get executed  and we are assuming a constant C execution complexity on its body   Now the summations can be simplified using some identity rules    Summation w from 1 to N   C     N   C Summation w from 1 to N   A    -  B     Summation w from 1 to N   A      -  Summation w from 1 to N   B   Summation w from 1 to N   w   C     C   Summation w from 1 to N   w    C is a constant  independent of w  Summation w from 1 to N   w      N    N   1     2   Applying some algebra   f N    Summation i from 1 to N   2    N -  i - 1    2      C        N   2   C    f N    C   Summation i from 1 to N   2    N -  i - 1    2      N   2   C    f N    C    Summation i from 1 to N   2   N   - Summation i from 1 to N   2    i - 1    2      N   2   C    f N    C      N   2   2   - 2   Summation i from 1 to N   2   i - 1       N   2   C      gt  Summation i from 1 to N   2   i - 1     Summation i from 1 to N   2 - 1   i    f N    C      N   2   2   - 2   Summation i from 1 to N   2 - 1   i       N   2   C    f N    C      N   2   2   - 2      N   2 - 1     N   2 - 1   1    2       N   2   C      gt   N   2 - 1     N   2 - 1   1    2         N   2 - 1     N   2    2          N   2   4  -  N   2     2         N   2   8  -  N   4   f N    C      N   2   2   - 2      N   2   8  -  N   4        N   2   C    f N    C      N   2   2   -    N   2   4  -  N   2        N   2   C    f N    C      N   2   2   -  N   2   4     N   2      N   2   C    f N    C     N   2   4     C    N   2    C    N   2   f N    C     N   2   4     2   C    N   2   f N    C     N   2   4     C   N  f N    C   1 4   N   2   C   N   And the BigOh is   O N

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Don t forget to also allow for space complexities that can also be a cause for concern if one has limited memory resources   So for example you may hear someone wanting a constant space algorithm which is basically a way of saying that the amount of space taken by the algorithm doesn t depend on any factors inside the code   Sometimes the complexity can come from how many times is something called  how often is a loop executed  how often is memory allocated  and so on is another part to answer this question   Lastly  big O can be used for worst case  best case  and amortization cases where generally it is the worst case that is used for describing how bad an algorithm may be

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For code A  the outer loop will execute for n 1 times  the  1  time means the process which checks the whether i still meets the requirement  And inner loop runs n times  n-2 times     Thus 0 2     n-2  n   0 n  n 1  2  O n      For code B  though inner loop wouldn t step in and execute the foo    the inner loop will be executed for n times depend on outer loop execution time  which is O n

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Familiarity with the algorithms data structures I use and or quick glance analysis of iteration nesting   The difficulty is when you call a library function  possibly multiple times - you can often be unsure of whether you are calling the function unnecessarily at times or what implementation they are using   Maybe library functions should have a complexity efficiency measure  whether that be Big O or some other metric  that is available in documentation or even IntelliSense

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What often gets overlooked is the expected behavior of your algorithms  It doesn t change the Big-O of your algorithm  but it does relate to the statement  premature optimization         Expected behavior of your algorithm is -- very dumbed down -- how fast you can expect your algorithm to work on data you re most likely to see    For instance  if you re searching for a value in a list  it s O n   but if you know that most lists you see have your value up front  typical behavior of your algorithm is faster    To really nail it down  you need to be able to describe the probability distribution of your  input space   if you need to sort a list  how often is that list already going to be sorted  how often is it totally reversed  how often is it mostly sorted   It s not always feasible that you know that  but sometimes you do

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As to  how do you calculate  Big O  this is part of Computational complexity theory  For some  many  special cases you may be able to come with some simple heuristics  like multiplying loop counts for nested loops   esp  when all you want is any upper bound estimation  and you do not mind if it is too pessimistic - which I guess is probably what your question is about   If you really want to answer your question for any algorithm the best you can do is to apply the theory  Besides of simplistic  worst case  analysis I have found Amortized analysis very useful in practice

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Seeing the answers here I think we can conclude that most of us do indeed approximate the order of the algorithm by looking at it and use common sense instead of calculating it with  for example  the master method as we were thought at university  With that said I must add that even the professor encouraged us  later on  to actually think about it instead of just calculating it   Also I would like to add how it is done for recursive functions   suppose we have a function like  scheme code     define  fac n       if    n 0          1                n  fac  - n 1        which recursively calculates the factorial of the given number   The first step is to try and determine the performance characteristic for the body of the function only in this case  nothing special is done in the body  just a multiplication  or the return of the value 1    So the performance for the body is  O 1   constant    Next try and determine this for the number of recursive calls  In this case we have n-1 recursive calls   So the performance for the recursive calls is  O n-1   order is n  as we throw away the insignificant parts    Then put those two together and you then have the performance for the whole recursive function     1    n-1    O n     Peter  to answer your raised issues  the method I describe here actually handles this quite well  But keep in mind that this is still an approximation and not a full mathematically correct answer  The method described here is also one of the methods we were taught at university  and if I remember correctly was used for far more advanced algorithms than the factorial I used in this example  Of course it all depends on how well you can estimate the running time of the body of the function and the number of recursive calls  but that is just as true for the other methods

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I don t know how to programmatically solve this  but the first thing people do is that we sample the algorithm for certain patterns in the number of operations done  say 4n 2   2n   1 we have 2 rules    If we have a sum of terms  the term with the largest growth rate is kept  with other terms omitted   If we have a product of several factors constant factors are omitted     If we simplify f x   where f x  is the formula for number of operations done   4n 2   2n   1 explained above   we obtain the big-O value  O n 2  in this case   But this would have to account for Lagrange interpolation in the program  which may be hard to implement  And what if the real big-O value was O 2 n   and we might have something like O x n   so this algorithm probably wouldn t be programmable  But if someone proves me wrong  give me the code

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For the 1st case  the inner loop is executed n-i times  so the total number of executions is the sum for i going from 0 to n-1  because lower than  not lower than or equal  of the n-i  You get finally n  n   1    2  so O n   2    O n      For the 2nd loop  i is between 0 and n included for the outer loop  then the inner loop is executed when j is strictly greater than n  which is then impossible

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Big O gives the upper bound for time complexity of an algorithm   It is usually used in conjunction with processing data sets  lists  but can be used elsewhere     A few examples of how it s used in C code   Say we have an array of n elements  int array n     If we wanted to access the first element of the array this would be O 1  since it doesn t matter how big the array is  it always takes the same constant time to get the first item   x   array 0     If we wanted to find a number in the list   for int i   0  i  lt  n  i         if array i     numToFind   return i        This would be O n  since at most we would have to look through the entire list to find our number   The Big-O is still O n  even though we might find our number the first try and run through the loop once because Big-O describes the upper bound for an algorithm  omega is for lower bound and theta is for tight bound    When we get to nested loops   for int i   0  i  lt  n  i         for int j   i  j  lt  n  j             array j     2            This is O n 2  since for each pass of the outer loop   O n    we have to go through the entire list again so the n s multiply leaving us with n squared   This is barely scratching the surface but when you get to analyzing more complex algorithms complex math involving proofs comes into play   Hope this familiarizes you with the basics at least though

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Basically the thing that crops up 90  of the time is just analyzing loops   Do you have single  double  triple nested loops   The you have O n   O n 2   O n 3  running time   Very rarely  unless you are writing a platform with an extensive base library  like for instance  the  NET BCL  or C   s STL  you will encounter anything that is more difficult than just looking at your loops  for statements  while  goto  etc

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Lets start from the beginning   First of all  accept the principle that certain simple operations on data can be done in O 1  time  that is  in time that is independent of the size of the input  These primitive operations in C consist of   Arithmetic operations  e g    or     Logical operations  e g    amp  amp    Comparison operations  e g    lt     Structure accessing operations  e g  array-indexing like A i   or pointer fol- lowing with the -  operator   Simple assignment such as copying a value into a variable  Calls to library functions  e g   scanf  printf     The justification for this principle requires a detailed study of the machine instructions  primitive steps  of a typical computer  Each of the described operations can be done with some small number of machine instructions  often only one or two instructions are needed  As a consequence  several kinds of statements in C can be executed in O 1  time  that is  in some constant amount of time independent of input  These simple include   Assignment statements that do not involve function calls in their expressions  Read statements  Write statements that do not require function calls to evaluate arguments  The jump statements break  continue  goto  and return expression  where expression does not contain a function call    In C  many for-loops are formed by initializing an index variable to some value and incrementing that variable by 1 each time around the loop  The for-loop ends when the index reaches some limit  For instance  the for-loop   for  i   0  i  lt  n-1  i           small   i      for  j   i 1  j  lt  n  j            if  A j   lt  A small               small   j      temp   A small       A small    A i       A i    temp      uses index variable i  It increments i by 1 each time around the loop  and the iterations stop when i reaches n - 1   However  for the moment  focus on the simple form of for-loop  where the difference between the final and initial values  divided by the amount by which the index variable is incremented tells us how many times we go around the loop  That count is exact  unless there are ways to exit the loop via a jump statement  it is an upper bound on the number of iterations in any case    For instance  the for-loop iterates   n - 1  - 0  1   n - 1 times  since 0 is the initial value of i  n - 1 is the highest value reached by i  i e   when i reaches n-1  the loop stops and no iteration occurs with i   n-1   and 1 is added to i at each iteration of the loop   In the simplest case  where the time spent in the loop body is the same for each iteration  we can multiply the big-oh upper bound for the body by the number of times around the loop  Strictly speaking  we must then add O 1  time to initialize the loop index and O 1  time for the first comparison of the loop index with the limit  because we test one more time than we go around the loop  However  unless it is possible to execute the loop zero times  the time to initialize the loop and test the limit once is a low-order term that can be dropped by the summation rule     Now consider this example    1  for  j   0  j  lt  n  j     2    A i  j    0    We know that line  1  takes O 1  time  Clearly  we go around the loop n times  as we can determine by subtracting the lower limit from the upper limit found on line  1  and then adding 1  Since the body  line  2   takes O 1  time  we can neglect the time to increment j and the time to compare j with n  both of which are also O 1   Thus  the running time of lines  1  and  2  is the product of n and O 1   which is O n    Similarly  we can bound the running time of the outer loop consisting of lines  2  through  4   which is   2  for  i   0  i  lt  n  i     3      for  j   0  j  lt  n  j     4          A i  j    0    We have already established that the loop of lines  3  and  4  takes O n  time  Thus  we can neglect the O 1  time to increment i and to test whether i  lt  n in each iteration  concluding that each iteration of the outer loop takes O n  time   The initialization i   0 of the outer loop and the  n   1 st test of the condition i  lt  n likewise take O 1  time and can be neglected  Finally  we observe that we go around the outer loop n times  taking O n  time for each iteration  giving a total O n 2  running time     A more practical example

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Big O notation is useful because it s easy to work with and hides unnecessary complications and details  for some definition of unnecessary   One nice way of working out the complexity of divide and conquer algorithms is the tree method  Let s say you have a version of quicksort with the median procedure  so you split the array into perfectly balanced subarrays every time   Now build a tree corresponding to all the arrays you work with  At the root you have the original array  the root has two children which are the subarrays  Repeat this until you have single element arrays at the bottom    Since we can find the median in O n  time and split the array in two parts in O n  time  the work done at each node is O k  where k is the size of the array  Each level of the tree contains  at most  the entire array so the work per level is O n   the sizes of the subarrays add up to n  and since we have O k  per level we can add this up   There are only log n  levels in the tree since each time we halve the input   Therefore we can upper bound the amount of work by O n log n      However  Big O hides some details which we sometimes can t ignore  Consider computing the Fibonacci sequence with  a 0  b 1  for  i   0  i  lt n  i          tmp   b      b   a   b      a   tmp      and lets just assume the a and b are BigIntegers in Java or something that can handle arbitrarily large numbers  Most people would say this is an O n  algorithm without flinching  The reasoning is that you have n iterations in the for loop and O 1  work in side the loop    But Fibonacci numbers are large  the n-th Fibonacci number is exponential in n so just storing it will take on the order of n bytes  Performing addition with big integers will take O n  amount of work  So the total amount of work done in this procedure is   1   2   3         n   n n-1  2   O n 2   So this algorithm runs in quadradic time

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If your cost is a polynomial  just keep the highest-order term  without its multiplier  E g       O  n 2   1   n 2     O n2 4   n 2    O n2 4    O n2    This doesn t work for infinite series  mind you  There is no single recipe for the general case  though for some common cases  the following inequalities apply      O log N   lt  O N   lt  O N log N   lt  O N2   lt  O Nk   lt  O en   lt  O n

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While knowing how to figure out the Big O time for your particular problem is useful  knowing some general cases can go a long way in helping you make decisions in your algorithm   Here are some of the most common cases  lifted from http   en wikipedia org wiki Big O notation Orders of common functions   O 1  - Determining if a number is even or odd  using a constant-size lookup table or hash table  O logn  - Finding an item in a sorted array with a binary search  O n  - Finding an item in an unsorted list  adding two n-digit numbers  O n2  - Multiplying two n-digit numbers by a simple algorithm  adding two n  n matrices  bubble sort or insertion sort  O n3  - Multiplying two n  n matrices by simple algorithm  O cn  - Finding the  exact  solution to the traveling salesman problem using dynamic programming  determining if two logical statements are equivalent using brute force  O n   - Solving the traveling salesman problem via brute-force search  O nn  - Often used instead of O n   to derive simpler formulas for asymptotic complexity

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If you want to estimate the order of your code empirically rather than by analyzing the code  you could stick in a series of increasing values of n and time your code   Plot your timings on a log scale   If the code is O x n   the values should fall on a line of slope n   This has several advantages over just studying the code   For one thing  you can see whether you re in the range where the run time approaches its asymptotic order  Also  you may find that some code that you thought was order O x  is really order O x 2   for example  because of time spent in library calls

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Less useful generally  I think  but for the sake of completeness there is also a Big Omega O  which defines a lower-bound on an algorithm s complexity  and a Big Theta T  which defines both an upper and lower bound

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Small reminder  the big O notation is used to denote asymptotic complexity  that is  when the size of the problem grows to infinity   and it hides a constant   This means that between an algorithm in O n  and one in O n2   the fastest is not always the first one  though there always exists a value of n such that for problems of size  n  the first algorithm is the fastest    Note that the hidden constant very much depends on the implementation   Also  in some cases  the runtime is not a deterministic function of the size n of the input  Take sorting using quick sort for example  the time needed to sort an array of n elements is not a constant but depends on the starting configuration of the array    There are different time complexities     Worst case  usually the simplest to figure out  though not always very meaningful  Average case  usually much harder to figure out           A good introduction is An Introduction to the Analysis of Algorithms by R  Sedgewick and P  Flajolet   As you say  premature optimisation is the root of all evil  and  if possible  profiling really should always be used when optimising code  It can even help you determine the complexity of your algorithms

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Break down the algorithm into pieces you know the big O notation for  and combine through big O operators  That s the only way I know of   For more information  check the Wikipedia page on the subject

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In addition to using the master method  or one of its specializations   I test my algorithms experimentally   This can t prove that any particular complexity class is achieved  but it can provide reassurance that the mathematical analysis is appropriate   To help with this reassurance  I use code coverage tools in conjunction with my experiments  to ensure that I m exercising all the cases   As a very simple example say you wanted to do a sanity check on the speed of the  NET framework s list sort   You could write something like the following  then analyze the results in Excel to make sure they did not exceed an n log n  curve   In this example I measure the number of comparisons  but it s also prudent to examine the actual time required for each sample size   However then you must be even more careful that you are just measuring the algorithm and not including artifacts from your test infrastructure   int nCmp   0  System Random rnd   new System Random        measure the time required to sort a list of n integers void DoTest int n       List lt int gt  lst   new List lt int gt  n      for  int i 0  i lt n  i           lst i    rnd Next 0 1000          as we sort  keep track of the number of comparisons performed     nCmp   0     lst Sort  delegate  int a  int b     nCmp    return  a lt b  -1   a gt b  1 0          System Console Writeline    0   1    n  nCmp           Perform measurement for a variety of sample sizes     It would be prudent to check multiple random samples of each size  but this is OK for a quick sanity check for  int n   0  n lt 1000  n        DoTest n

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great question   Disclaimer  this answer contains false statements see the comments below   If you re using the Big O  you re talking about the worse case  more on what that means later   Additionally  there is capital theta for average case and a big omega for best case   Check out this site for a lovely formal definition of Big O  https   xlinux nist gov dads HTML bigOnotation html     f n    O g n   means there are positive constants c and k  such that 0   f n    cg n  for all n   k  The values of c and k must be fixed for the function f and must not depend on n       Ok  so now what do we mean by  best-case  and  worst-case  complexities   This is probably most clearly illustrated through examples  For example if we are using linear search to find a number in a sorted array then the worst case is when we decide to search for the last element of the array as this would take as many steps as there are items in the array  The best case would be when we search for the first element since we would be done after the first check   The point of all these adjective-case complexities is that we re looking for a way to graph the amount of time a hypothetical program runs to completion in terms of the size of particular variables  However for many algorithms you can argue that there is not a single time for a particular size of input  Notice that this contradicts with the fundamental requirement of a function  any input should have no more than one output  So we come up with multiple functions to describe an algorithm s complexity  Now  even though searching an array of size n may take varying amounts of time depending on what you re looking for in the array and depending proportionally to n  we can create an informative description of the algorithm using best-case  average-case  and worst-case classes   Sorry this is so poorly written and lacks much technical information  But hopefully it ll make time complexity classes easier to think about  Once you become comfortable with these it becomes a simple matter of parsing through your program and looking for things like for-loops that depend on array sizes and reasoning based on your data structures what kind of input would result in trivial cases and what input would result in worst-cases

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I think about it in terms of information  Any problem consists of learning a certain number of bits   Your basic tool is the concept of decision points and their entropy  The entropy of a decision point is the average information it will give you  For example  if a program contains a decision point with two branches  it s entropy is the sum of the probability of each branch times the log2 of the inverse probability of that branch  That s how much you learn by executing that decision   For example  an if statement having two branches  both equally likely  has an entropy of 1 2   log 2 1    1 2   log 2 1    1 2   1   1 2   1   1  So its entropy is 1 bit   Suppose you are searching a table of N items  like N 1024  That is a 10-bit problem because log 1024    10 bits  So if you can search it with IF statements that have equally likely outcomes  it should take 10 decisions   That s what you get with binary search   Suppose you are doing linear search  You look at the first element and ask if it s the one you want  The probabilities are 1 1024 that it is  and 1023 1024 that it isn t  The entropy of that decision is 1 1024 log 1024 1    1023 1024   log 1024 1023    1 1024   10   1023 1024   about 0   about  01 bit  You ve learned very little  The second decision isn t much better  That is why linear search is so slow  In fact it s exponential in the number of bits you need to learn   Suppose you are doing indexing  Suppose the table is pre-sorted into a lot of bins  and you use some of all of the bits in the key to index directly to the table entry  If there are 1024 bins  the entropy is 1 1024   log 1024    1 1024   log 1024        for all 1024 possible outcomes  This is 1 1024   10 times 1024 outcomes  or 10 bits of entropy for that one indexing operation  That is why indexing search is fast   Now think about sorting   You have N items  and you have a list  For each item  you have to search for where the item goes in the list  and then add it to the list  So sorting takes roughly N times the number of steps of the underlying search   So sorts based on binary decisions having roughly equally likely outcomes all take about O N log N  steps  An O N  sort algorithm is possible if it is based on indexing search   I ve found that nearly all algorithmic performance issues can be looked at in this way

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I would like to explain the Big-O in a little bit different aspect   Big-O is just to compare the complexity of the programs which means how fast are they growing when the inputs are increasing and not the exact time which is spend to do the action   IMHO in the big-O formulas you better not to use more complex equations  you might just stick to the ones in the following graph   However you still might use other more precise formula  like 3 n  n 3       but more than that can be sometimes misleading  So better to keep it as simple as possible     I would like to emphasize once again that here we don t want to get an exact formula for our algorithm  We only want to show how it grows when the inputs are growing and compare with the other algorithms in that sense  Otherwise you would better use different methods like bench-marking

[algorithm] Big O, how do you calculate/approximate it?

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