How to find time complexity of an algorithm

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The Question  How to find time complexity of an algorithm   What have I done before posting a question on SO    I have gone through this  this and many other links  But no where I was able to find a clear and straight forward explanation for how to calculate time complexity   What do I know    Say for a code as simple as the one below   char h    y      This will be executed 1 time int abc   0     This will be executed 1 time   Say for a loop like the one below    for  int i   0  i  lt  N  i                  Console Write  Hello World          int i 0  This will be executed only once  The time is actually calculated to i 0 and not the declaration   i  lt  N   This will be executed N 1 times  i       This will be executed N times  So the number of operations required by this loop are   1  N 1  N    2N 2  Note  This still may be wrong  as I am not confident about my understanding on calculating time complexity  What I want to know    Ok  so these small basic calculations I think I know  but in most cases I have seen the time complexity as   O N   O n2   O log n   O n       and many other    Can anyone help me understand how does one calculate time complexity of an algorithm  I am sure there are plenty of newbies like me wanting to know this

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O n  is big O notation used for writing time complexity of an algorithm  When you add up the number of executions in an algoritm you ll get an expression in result like 2N 2  in this expression N is the dominating term the term having largest effect on expression if its value increases or decreases   Now O N  is the time comlexity while N is dominating term  Example  For i  1 to n    j  0  while j lt  n     j j 1    here total number of executions for inner loop are n 1 and total number of executions for outer loop are n n 1  2  so total number of executions for whole algorithm are n 1 n n 1 2     n 2 3n  2  here n 2 is the dominating term so the time complexity for this algorithm is O n 2

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How to find time complexity of an algorithm   You add up how many machine instructions it will execute as a function of the size of its input  and then simplify the expression to the largest  when N is very large  term  and can include any simplifying constant factor   For example  lets see how we simplify 2N   2 machine instructions to describe this as just O N    Why do we remove the two 2s    We are interested in the performance of the algorithm as N becomes large   Consider the two terms 2N and 2    What is the relative influence of these two terms as N becomes large  Suppose N is a million   Then the first term is 2 million and the second term is only 2   For this reason  we drop all but the largest terms for large N   So  now we have gone from 2N   2 to 2N   Traditionally  we are only interested in performance up to constant factors    This means that we don t really care if there is some constant multiple of difference in performance when N is large   The unit of 2N is not well-defined in the first place anyway   So we can multiply or divide by a constant factor to get to the simplest expression   So 2N becomes just N

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When you re analyzing  code  you have to analyse it line by line  counting every operation recognizing time complexity  in the end  you have to sum it to get whole picture   For example  you can have one simple loop with linear complexity  but later in that same program you can have a triple loop that has cubic complexity  so your program will have cubic complexity  Function order of growth comes into play right here    Let s look at what are possibilities for time complexity of an algorithm  you can see order of growth I mentioned above    Constant time has an order of growth 1  for example  a   b   c  Logarithmic time has an order of growth LogN  it usually occurs when you re dividing something in half  binary search  trees  even loops   or multiplying something in same way  Linear  order of growth is N  for example  int p   0  for  int i   1  i  lt  N  i      p   p   2   Linearithmic  order of growth is n logN  usually occurs in divide and conquer algorithms  Cubic  order of growth N 3  classic example is a triple loop where you check all triplets    int x   0  for  int i   0  i  lt  N  i       for  int j   0  j  lt  N  j          for  int k   0  k  lt  N  k              x   x   2  Exponential  order of growth 2 N  usually occurs when you do exhaustive search  for example check subsets of some set

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This is an excellent article   http   www daniweb com software-development computer-science threads 13488 time-complexity-of-algorithm  The below answer is copied from above  in case the excellent link goes bust   The most common metric for calculating time complexity is Big O notation  This removes all constant factors so that the running time can be estimated in relation to N as N approaches infinity  In general you can think of it like this   statement    Is constant  The running time of the statement will not change in relation to N   for   i   0  i  lt  N  i          statement    Is linear  The running time of the loop is directly proportional to N  When N doubles  so does the running time   for   i   0  i  lt  N  i         for   j   0  j  lt  N  j         statement      Is quadratic  The running time of the two loops is proportional to the square of N  When N doubles  the running time increases by N   N   while   low  lt   high       mid     low   high     2    if   target  lt  list mid        high   mid - 1    else if   target  gt  list mid        low   mid   1    else break      Is logarithmic  The running time of the algorithm is proportional to the number of times N can be divided by 2  This is because the algorithm divides the working area in half with each iteration   void quicksort   int list    int left  int right       int pivot   partition   list  left  right      quicksort   list  left  pivot - 1      quicksort   list  pivot   1  right        Is N   log   N    The running time consists of N loops  iterative or recursive  that are logarithmic  thus the algorithm is a combination of linear and logarithmic   In general  doing something with every item in one dimension is linear  doing something with every item in two dimensions is quadratic  and dividing the working area in half is logarithmic  There are other Big O measures such as cubic  exponential  and square root  but they re not nearly as common  Big O notation is described as O    lt type gt    where  lt type gt  is the measure  The quicksort algorithm would be described as O   N   log   N       Note that none of this has taken into account best  average  and worst case measures  Each would have its own Big O notation  Also note that this is a VERY simplistic explanation  Big O is the most common  but it s also more complex that I ve shown  There are also other notations such as big omega  little o  and big theta  You probably won t encounter them outside of an algorithm analysis course

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Time complexity with examples  1 - Basic Operations  arithmetic  comparisons  accessing array   s elements  assignment    The running time is always constant O 1   Example    read x                                   O 1  a   10                                   O 1  a   1 000 000 000 000 000 000            O 1    2 - If then else statement  Only taking the maximum running time from two or more possible statements   Example   age   read x                                    1 1    2 if age  lt  17 then begin                         1       status    Not allowed                    1 end else begin       status    Welcome  Please come in        1       visitors   visitors   1                  1 1   2 end    So  the complexity of the above pseudo code is T n    2   1   max 1  1 2    6  Thus  its big oh is still constant T n    O 1    3 - Looping  for  while  repeat   Running time for this statement is the number of looping multiplied by the number of operations inside that looping   Example   total   0                                      1 for i   1 to n do begin                         1 1  n   2n       total   total   i                         1 1  n   2n end  writeln total                                  1   So  its complexity is T n    1 4n 1   4n   2  Thus  T n    O n    4 - Nested Loop  looping inside looping   Since there is at least one looping inside the main looping  running time of this statement used O n 2  or O n 3    Example   for i   1 to n do begin                         1 1  n    2n    for j   1 to n do begin                      1 1 n n   2n 2        x   x   1                                1 1 n n   2n 2        print x                                  n n       n 2    end  end    Common Running Time  There are some common running times when analyzing an algorithm    O 1      Constant Time Constant time means the running time is constant  it   s not affected by the input size  O n      Linear Time When an algorithm accepts n input size  it would perform n operations as well  O log n      Logarithmic Time Algorithm that has running time O log n  is slight faster than O n   Commonly  algorithm divides the problem into sub problems with the same size  Example  binary search algorithm  binary conversion algorithm  O n log n      Linearithmic Time This running time is often found in  divide  amp  conquer algorithms  which divide the problem into sub problems recursively and then merge them in n time  Example  Merge Sort algorithm  O n2      Quadratic Time Look Bubble Sort algorithm  O n3      Cubic Time It has the same principle with O n2   O 2n      Exponential Time It is very slow as input get larger  if n   1000 000  T n  would be 21000 000  Brute Force algorithm has this running time  O n       Factorial Time THE SLOWEST     Example   Travel Salesman Problem  TSP    Taken from this article  Very well explained should give a read

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Taken from here - Introduction to Time Complexity of an Algorithm 1  Introduction In computer science  the time complexity of an algorithm quantifies the amount of time taken by an algorithm to run as a function of the length of the string representing the input  2  Big O notation The time complexity of an algorithm is commonly expressed using big O notation  which excludes coefficients and lower order terms  When expressed this way  the time complexity is said to be described asymptotically  i e   as the input size goes to infinity  For example  if the time required by an algorithm on all inputs of size n is at most 5n3   3n  the asymptotic time complexity is O n3   More on that later  Few more Examples   1   O n  n   O n2  log n    O n  2 n   1   O n   3  O 1  Constant Time  An algorithm is said to run in constant time if it requires the same amount of time regardless of the input size  Examples   array  accessing any element fixed-size stack  push and pop methods fixed-size queue  enqueue and dequeue methods  4  O n  Linear Time An algorithm is said to run in linear time if its time execution is directly proportional to the input size  i e  time grows linearly as input size increases  Consider the following examples  below I am linearly searching for an element  this has a time complexity of O n   int find   66  var numbers   new int     33  435  36  37  43  45  66  656  2232    for  int i   0  i  lt  numbers Length - 1  i          if find    numbers i                 return           More Examples   Array  Linear Search  Traversing  Find minimum etc ArrayList  contains method Queue  contains method  5  O log n  Logarithmic Time  An algorithm is said to run in logarithmic time if its time execution is proportional to the logarithm of the input size  Example  Binary Search Recall the  quot twenty questions quot  game - the task is to guess the value of a hidden number in an interval  Each time you make a guess  you are told whether your guess is too high or too low  Twenty questions game implies a strategy that uses your guess number to halve the interval size  This is an example of the general problem-solving method known as binary search 6  O n2  Quadratic Time An algorithm is said to run in quadratic time if its time execution is proportional to the square of the input size  Examples   Bubble Sort Selection Sort Insertion Sort  7  Some Useful links  Big-O Misconceptions Determining The Complexity Of Algorithm Big O Cheat Sheet

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Loosely speaking  time complexity is a way of summarising how the number of operations or run-time of an algorithm grows as the input size increases   Like most things in life  a cocktail party can help us understand   O N   When you arrive at the party  you have to shake everyone s hand  do an operation on every item   As the number of attendees N increases  the time work it will take you to shake everyone s hand increases as O N    Why O N  and not cN   There s variation in the amount of time it takes to shake hands with people  You could average this out and capture it in a constant c  But the fundamental operation here --- shaking hands with everyone --- would always be proportional to O N   no matter what c was  When debating whether we should go to a cocktail party  we re often more interested in the fact that we ll have to meet everyone than in the minute details of what those meetings look like   O N 2   The host of the cocktail party wants you to play a silly game where everyone meets everyone else  Therefore  you must meet N-1 other people and  because the next person has already met you  they must meet N-2 people  and so on  The sum of this series is x 2 2 x 2  As the number of attendees grows  the x 2 term gets big fast  so we just drop everything else   O N 3   You have to meet everyone else and  during each meeting  you must talk about everyone else in the room   O 1   The host wants to announce something  They ding a wineglass and speak loudly  Everyone hears them  It turns out it doesn t matter how many attendees there are  this operation always takes the same amount of time   O log N   The host has laid everyone out at the table in alphabetical order  Where is Dan  You reason that he must be somewhere between Adam and Mandy  certainly not between Mandy and Zach    Given that  is he between George and Mandy  No  He must be between Adam and Fred  and between Cindy and Fred  And so on    we can efficiently locate Dan by looking at half the set and then half of that set  Ultimately  we look at O log 2 N  individuals   O N log N   You could find where to sit down at the table using the algorithm above  If a large number of people came to the table  one at a time  and all did this  that would take O N log N  time  This turns out to be how long it takes to sort any collection of items when they must be compared   Best Worst Case  You arrive at the party and need to find Inigo - how long will it take  It depends on when you arrive  If everyone is milling around you ve hit the worst-case  it will take O N  time  However  if everyone is sitting down at the table  it will take only O log N  time  Or maybe you can leverage the host s wineglass-shouting power and it will take only O 1  time   Assuming the host is unavailable  we can say that the Inigo-finding algorithm has a lower-bound of O log N  and an upper-bound of O N   depending on the state of the party when you arrive   Space  amp  Communication  The same ideas can be applied to understanding how algorithms use space or communication   Knuth has written a nice paper about the former entitled  The Complexity of Songs       Theorem 2  There exist arbitrarily long songs of complexity O 1        PROOF   due to Casey and the Sunshine Band   Consider the songs Sk defined by  15   but with   V k    That s the way   U  I like it    U U      uh huh    uh huh       for all k

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I know this question goes a way back and there are some excellent answers here  nonetheless I wanted to share another bit for the mathematically-minded people that will stumble in this post  The Master theorem is another usefull thing to know when studying complexity  I didn t see it mentioned in the other answers

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Although there are some good answers for this question  I would like to give another answer here with several examples of loop    O n   Time Complexity of a loop is considered as O n  if the loop variables is incremented   decremented by a constant amount  For example following functions have O n  time complexity      Here c is a positive integer constant    for  int i   1  i  lt   n  i    c             some O 1  expressions    for  int i   n  i  gt  0  i -  c           some O 1  expressions    O n c   Time complexity of nested loops is equal to the number of times the innermost statement is executed  For example the following sample loops have O n 2  time complexity  for  int i   1  i  lt  n  i    c       for  int j   1  j  lt  n  j    c             some O 1  expressions         for  int i   n  i  gt  0  i    c       for  int j   i 1  j  lt  n  j    c             some O 1  expressions     For example Selection sort and Insertion Sort have O n 2  time complexity  O Logn  Time Complexity of a loop is considered as O Logn  if the loop variables is divided   multiplied by a constant amount   for  int i   1  i  lt  n  i    c          some O 1  expressions   for  int i   n  i  gt  0  i    c          some O 1  expressions     For example Binary Search has O Logn  time complexity  O LogLogn  Time Complexity of a loop is considered as O LogLogn  if the loop variables is reduced   increased exponentially by a constant amount      Here c is a constant greater than 1    for  int i   2  i  lt  n  i   pow i  c            some O 1  expressions     Here fun is sqrt or cuberoot or any other constant root for  int i   n  i  gt  0  i   fun i            some O 1  expressions        One example of time complexity analysis  int fun int n            for  int i   1  i  lt   n  i                  for  int j   1  j  lt  n  j    i                           Some O 1  task                         Analysis    For i   1  the inner loop is executed n times  For i   2  the inner loop is executed approximately n 2 times  For i   3  the inner loop is executed approximately n 3 times  For i   4  the inner loop is executed approximately n 4 times                                                                For i   n  the inner loop is executed approximately n n times    So the total time complexity of the above algorithm is  n   n 2   n 3         n n   Which becomes n    1 1   1 2   1 3         1 n   The important thing about series  1 1   1 2   1 3         1 n  is equal to O Logn   So the time complexity of the above code is O nLogn      Ref  1 2 3

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