Here is a segment of an algorithm I came up with:
for (int i = 0; i < n - 1; i++)
for (int j = i; j < n; j++)
(...)
I am using this "double loop" to test all possible 2-element sums in a an array of size n.
Apparently (and I have to agree with it), this "double loop" is O(n²):
n + (n-1) + (n-2) + ... + 1 = sum from 1 to n = (n (n - 1))/2
Here is where I am confused:
for (int i = 0; i < n; i++)
for (int j = 0; j < n; j++)
(...)
This second "double loop" also has a complexity of O(n²), when it is clearly (at worst) much (?) better than the first.
What am I missing? Is the information accurate? Can someone explain this "phenomenon"?
(n (n - 1))/2 simplifies to n²/2 - n/2. If you use really large numbers for n, the growth rate of n/2 will be dwarfed in comparison to n², so for the sake of calculating Big-O complexity, you effectively ignore it. Likewise, the "constant" value of 1/2 doesn't grow at all as n increases, so you ignore that too. That just leaves you with n².
Just remember that complexity calculations are not the same as "speed". One algorithm can be five thousand times slower than another and still have a smaller Big-O complexity. But as you increase n to really large numbers, general patterns emerge that can typically be classified using simple formulae: 1, log n, n, n log n, n², etc.
It sometimes helps to create a graph and see what kind of line appears:
Even though the zoom factors of these two graphs are very different, you can see that the type of curve it produces is almost exactly the same.
Constant factors.
Big-O notation ignores constant factors, so even though the second loop is slower by a constant factor, they end up with the same time complexity.
Right there in the definition it tells you that you can pick any old constant factor:
... if and only if there is a positive constant M ...
This is because we want to analyse the growth rate of an algorithm - constant factors just complicates things and are often system-dependent (operations may vary in duration on different machines).
You could just count certain types of operations, but then the question becomes which operation to pick, and what if that operation isn't predominant in some algorithm. Then you'll need to relate operations to each other (in a system-independent way, which is probably impossible), or you could just assign the same weight to each, but that would be fairly inaccurate as some operations would take significantly longer than others.
And how useful would saying O(15n² + 568n + 8 log n + 23 sqrt(n) + 17) (for example) really be? As opposed to just O(n²).
(For the purpose of the below, assume n >= 2)
Note that we actually have asymptotically smaller (i.e. smaller as we approach infinity) terms here, but we can always simplify that to a matter of constant factors. (It's n(n+1)/2, not n(n-1)/2)
n(n+1)/2 = n²/2 + n/2
and
n²/2 <= n²/2 + n/2 <= n²
Given that we've just shown that n(n+1)/2 lies between C.n² and D.n², for two constants C and D, we've also just shown that it's O(n²).
Note - big-O notation is actually strictly an upper bound (so we only care that it's smaller than a function, not between two), but it's often used to mean Θ (big-Theta), which cares about both bounds.
From The Big O page on Wikipedia
In typical usage, the formal definition of O notation is not used
directly; rather, the O notation for a function f is derived by the
following simplification rules:
If f(x) is a sum of several terms, the
one with the largest growth rate is kept, and all others omitted
Big-O is used only to give the asymptotic behaviour - that one is a bit faster than the other doesn't come into it - they're both O(N^2)
You could also say that the first loop is O(n(n-1)/2). The fancy mathematical definition of big-O is something like:
function "f" is big-O of function "g" if there exists constants c, n such that f(x) < c*g(x) for some c and all x > n.
It's a fancy way of saying g is an upper bound past some point with some constant applied. It then follows that O(n(n-1)/2) = O((n^2-n)/2) is big-O of O(n^2), which is neater for quick analysis.
AFAIK, your second code snippet
for(int i = 0; i < n; i++) <-- this loop goes for n times
for(int j = 0; j < n; j++) <-- loop also goes for n times
(...)
So essentially, it's getting a O(n*n) = O(n^2) time complexity.
Per BIG-O theory, constant factor is neglected and only higher order is considered. that's to say, if complexity is O(n^2+k) then actual complexity will be O(n^2) constant k will be ignored.
(OR) if complexity is O(n^2+n) then actual complexity will be O(n^2) lower order n will be ignored.
So in your first case where complexity is O(n(n - 1)/2) will/can be simplified to
O(n^2/2 - n/2) = O(n^2/2) (Ignoring the lower order n/2)
= O(1/2 * n^2)
= O(n^2) (Ignoring the constant factor 1/2)
I have the following algorithm:
for(int i = 1; i < n; i++)
for(int j = 0; j < i; j++)
if(j % i == 0) System.out.println(i + " " + j);
This will run max(n,0) times.
Would the Big-O notation be O(n)? If not, what is it and why?
Thank you.
You haven't stated what you are trying to measure with the Big-O notation. Let's assume it's time complexity. Next we have to define what the dependent variable is against which you want to measure the complexity. A reasonable choice here is the absolute value of n (as opposed to the bit-length), since you are dealing with fixed-length ints and not arbitrary-length integers.
You are right that the println is executed O(n) times, but that's counting how often a certain line is hit, it's not measuring time complexity.
It's easy to see that the if statement is hit O(n^2) times, so we have already established that the time complexity is bounded from below by Omega(n^2). As a commenter has already noted, the if-condition is only true for j=0, so I suspect that you actually meant to write i % j instead of j % i? This matters because the time complexity of the println(i + " " + j)-statement is certainly not O(1), but O(log n) (you can't possibly print x characters in less than x steps), so at first sight there is a possibility that the overall complexity is strictly worse than O(n^2).
Assuming that you meant to write i % j we could make the simplifying assumption that the condition is always true, in which case we would obtain the upper bound O(n^2 log n), which is strictly worse than O(n^2)!
However, noting that the number of divisors of n is bounded by O(Sqrt(n)), we actually have O(n^2 + n*Sqrt(n)*log(n)). But since O(Sqrt(n) * log(n)) < O(n), this amounts to O(n^2).
You can dig deeper into number theory to find tighter bounds on the number of divisors, but that doesn't make a difference since the n^2 stays the dominating factor.
So the tightest upper bound is indeed O(n^2), but it's not as obvious as it seems at first sight.
max(n,0) would indeed be O(n). However, your algorithm is in O(n**2). Your first loop goes n times, and the second loop goes i times which is on average n/2. That makes O(n**2 / 2) = O(n**2). However, unlike the runtime of the algorithm, the amount of times println is reached is in O(n), as this happens exactly n times.
So, the answer depends on what exactly you want to measure.
I don't know what the procedure of this would be. How do I think of this, how do I determine what the big-O will be? What is the process to solving?
Example1:
for ( i = 1; i <= n; i++)
for (j = 1; j <= n*3; j++)
System.out.println("Apple");
Example2:
for (i = 1; i < n*n*n; i *=n)
System.out.println("Banana");
Thank you
The short answer is that you count the loops. If there is no loop, it is O constant, if there is one it is O(N) if there are two nested loops it is O(N squared) and if there are three it is O(N cubed).
However that's only the short answer. You can also have loops which reduce an input by half on each iteration, so thats a log N term. And you can have pathological brute force functions which try every possibility, these are non-polynomial. Usually they are written to make heavy use of recursion and the problem is hardly chipped away at on each recursive step.
Be aware that library functions are often not O constant, and that has to be factored in.
Big-O measures efficiency. So say you were to loop through an array of size n and say n is 2,000. O(n) would signify that your algorithm for solving this is doing WORST CASE 2,000 total calculations. O is always the worst case scenario for your algorithm. There are other notation used for best case. You also have Ω(n) and Θ(n).
Check this out to kind of get an idea of the difference in efficiency:
http://bigocheatsheet.com/
Informally:
"T(n)T(n)T(n) is O(f(n))O(f(n))O(f(n))" basically means that f(n)f(n)f(n) describes the upper bound for T(n)T(n)T(n)
"T(n)T(n)T(n) is Ω(f(n))\Omega(f(n))Ω(f(n))" basically means that f(n)f(n)f(n) describes the lower bound for T(n)T(n)T(n)
"T(n)T(n)T(n) is Θ(f(n))\Theta(f(n))Θ(f(n))" basically means that f(n)f(n)f(n) describes the exact bound for T(n)T(n)T(n)
A good way to approach this for simple situations is to plug a couple of easy numbers in for n and see what happens. So say n is size 10:
in example 1:
for ( i = 1; i <= n; i++) //loop through this n times
for (j = 1; j <= n*3; j++) for each of those n times, loop through 3*n times
System.out.println("Apple"); //negligible time (O(1))
If it were just the outside loop, it would be O(n). However, since you add the inside loop, you get O(N^2) because although your input is (say) 10, you're doing 300 (30 prints for each of the 10; 30*10) operations. 3* O(N^2) but we generally leave the 3 out so O(n^2). Most nested for loops where you aren't modifying by n are O(n^2).
If it's easier you can visualize it as the polynomial 3n * n = 3n^2 worst case.
I'll let you try the next one... hint in the bold statement above.
I have gone through Google and Stack Overflow search, but nowhere I was able to find a clear and straightforward explanation for how to calculate time complexity.
What do I know already?
Say for 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 < 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 < 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. (But this still may be wrong, as I am not confident about my understanding.)
OK, so these small basic calculations I think I know, but in most cases I have seen the time complexity as O(N), O(n^2), O(log n), O(n!), and many others.
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.
This is an excellent article: 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 < 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 < N; i++ ) {
for ( j = 0; j < 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 <= high ) {
mid = ( low + high ) / 2;
if ( target < list[mid] )
high = mid - 1;
else if ( target > 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 ( <type> ) where <type> 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. ;)
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.
A 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, and 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 < 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 "twenty questions" 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
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 <= n; i += c) {
// some O(1) expressions
}
for (int i = n; i > 0; i -= c) {
// some O(1) expressions
}
O(nc) 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(n2) time complexity
for (int i = 1; i <=n; i += c) {
for (int j = 1; j <=n; j += c) {
// some O(1) expressions
}
}
for (int i = n; i > 0; i += c) {
for (int j = i+1; j <=n; j += c) {
// some O(1) expressions
}
For example, selection sort and insertion sort have O(n2) time complexity.
O(log n) time complexity of a loop is considered as O(log n) if the loop variables is divided / multiplied by a constant amount.
for (int i = 1; i <=n; i *= c) {
// some O(1) expressions
}
for (int i = n; i > 0; i /= c) {
// some O(1) expressions
}
For example, [binary search][3] has _O(log n)_ time complexity.
O(log log n) time complexity of a loop is considered as O(log log n) 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 <=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 > 0; i = fun(i)) {
// some O(1) expressions
}
One example of time complexity analysis
int fun(int n)
{
for (int i = 1; i <= n; i++)
{
for (int j = 1; j < 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 around to O(log n). So the time complexity of the above code is O(n·log n).
References:
1
2
3
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 < 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 loops 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 & 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 = 1,000,000, T(n) would be 21,000,000. Brute Force algorithm has this running time.
O(n!) – Factorial time
The slowest!!! Example: Travelling salesman problem (TSP)
It is taken from this article. It is very well explained and you should give it a read.
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 log N. It usually occurs when you're dividing something in half (binary search, trees, and even loops), or multiplying something in same way.
Linear. The order of growth is N, for example
int p = 0;
for (int i = 1; i < N; i++)
p = p + 2;
Linearithmic. The order of growth is n·log N. It usually occurs in divide-and-conquer algorithms.
Cubic. The order of growth is N3. A classic example is a triple loop where you check all triplets:
int x = 0;
for (int i = 0; i < N; i++)
for (int j = 0; j < N; j++)
for (int k = 0; k < N; k++)
x = x + 2
Exponential. The order of growth is 2N. It usually occurs when you do exhaustive search, for example, check subsets of some set.
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 & 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.
For the mathematically-minded people: The master theorem is another useful thing to know when studying complexity.
O(n) is big O notation used for writing time complexity of an algorithm. When you add up the number of executions in an algorithm, 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 complexity while N is dominating term.
Example
For i = 1 to n;
j = 0;
while(j <= n);
j = j + 1;
Here the total number of executions for the inner loop are n+1 and the total number of executions for the outer loop are n(n+1)/2, so the total number of executions for the whole algorithm are n + 1 + n(n+1/2) = (n2 + 3n)/2.
Here n^2 is the dominating term so the time complexity for this algorithm is O(n2).
Other answers concentrate on the big-O-notation and practical examples. I want to answer the question by emphasizing the theoretical view. The explanation below is necessarily lacking in details; an excellent source to learn computational complexity theory is Introduction to the Theory of Computation by Michael Sipser.
Turing Machines
The most widespread model to investigate any question about computation is a Turing machine. A Turing machine has a one dimensional tape consisting of symbols which is used as a memory device. It has a tapehead which is used to write and read from the tape. It has a transition table determining the machine's behaviour, which is a fixed hardware component that is decided when the machine is created. A Turing machine works at discrete time steps doing the following:
It reads the symbol under the tapehead.
Depending on the symbol and its internal state, which can only take finitely many values, it reads three values s, σ, and X from its transition table, where s is an internal state, σ is a symbol, and X is either Right or Left.
It changes its internal state to s.
It changes the symbol it has read to σ.
It moves the tapehead one step according to the direction in X.
Turing machines are powerful models of computation. They can do everything that your digital computer can do. They were introduced before the advent of digital modern computers by the father of theoretical computer science and mathematician: Alan Turing.
Time Complexity
It is hard to define the time complexity of a single problem like "Does white have a winning strategy in chess?" because there is a machine which runs for a single step giving the correct answer: Either the machine which says directly 'No' or directly 'Yes'. To make it work we instead define the time complexity of a family of problems L each of which has a size, usually the length of the problem description. Then we take a Turing machine M which correctly solves every problem in that family. When M is given a problem of this family of size n, it solves it in finitely many steps. Let us call f(n) the longest possible time it takes M to solve problems of size n. Then we say that the time complexity of L is O(f(n)), which means that there is a Turing machine which will solve an instance of it of size n in at most C.f(n) time where C is a constant independent of n.
Isn't it dependent on the machines? Can digital computers do it faster?
Yes! Some problems can be solved faster by other models of computation, for example two tape Turing machines solve some problems faster than those with a single tape. This is why theoreticians prefer to use robust complexity classes such as NL, P, NP, PSPACE, EXPTIME, etc. For example, P is the class of decision problems whose time complexity is O(p(n)) where p is a polynomial. The class P do not change even if you add ten thousand tapes to your Turing machine, or use other types of theoretical models such as random access machines.
A Difference in Theory and Practice
It is usually assumed that the time complexity of integer addition is O(1). This assumption makes sense in practice because computers use a fixed number of bits to store numbers for many applications. There is no reason to assume such a thing in theory, so time complexity of addition is O(k) where k is the number of bits needed to express the integer.
Finding The Time Complexity of a Class of Problems
The straightforward way to show the time complexity of a problem is O(f(n)) is to construct a Turing machine which solves it in O(f(n)) time. Creating Turing machines for complex problems is not trivial; one needs some familiarity with them. A transition table for a Turing machine is rarely given, and it is described in high level. It becomes easier to see how long it will take a machine to halt as one gets themselves familiar with them.
Showing that a problem is not O(f(n)) time complexity is another story... Even though there are some results like the time hierarchy theorem, there are many open problems here. For example whether problems in NP are in P, i.e. solvable in polynomial time, is one of the seven millennium prize problems in mathematics, whose solver will be awarded 1 million dollars.
I'm studying the running time of programs and have come across the Big O notation. One is asked to prove that T(n) is O(f(n)) by proving that there exists integer x and constant c > 0 such that for all integers n >= x, T(n) <= cf(n).
The examples I've seen prove this by "picking" values for x and c. I understand that you can plug values into the equation and see if they are correct, but is there a way to actually calculate x or c? Or, at least, some rules of thumb on how to pick them so one isn't plugging in values endlessly?
The values come from an examination from the algorithm T. For example, when you have a simple loop:
for (i=0; i < n; ++i) {
sum += i;
}
then you perform the operations i<n, ++i and sum+=i n times, and i=0 once. So f(n)==n, c==4 (for the four operations, elevating the "once" to "n times" for correctness of values), x==1 (for n==0, you still perform i=0 and i<n, so the formula would not work). This gives you an O(n) performance (linear in the number of inputs).
For the nested loops:
for (i=0; i < n; ++i) {
for (j=0; j<n; ++j) {
sum += j;
}
}
The calculations are similar, with f(n)==n^2, giving you O(n^2).
So there is no cut-n-dry way of telling the exact values of c and x, but most of the time the hard part is coming up with f -- and the "smallest" of that too (an O(n^2) algorithm is also an O(n^3) algorithm according to the definition you provided, but you want to characterize that algorithm with O(n^2) instead of O(n^3)). The ordering of fs is based on their growth when n approaches infinity: f(n)=n^3 grows slower than f(n)=2^n, even if for small ns the former is larger than the latter.
Note that in theory the actual values of x and c become irrelevant as n approaches infinity, that is why they don't show up in the O(n) notation itself. This does not mean, however that for (relatively) small values if n, the number of instructions are not much larger than f(n) (e.g. you have 1000 instructions within the for loop).
Also, the O(n) notation gives you worst performance, which might be much higher than what you observe in real life (average-case cost) or in the overall usage of a data structure (amortized cost), for example.