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CS 173
Spring 2021

Algorithms 1

Comparing the growth of two functions

When designing computer algorithms, we need to figure out how long they will take to run. The actual wallclock time isn't very helpful. It changes with the size of the input. It depends on how expensive a machine you're running it on. And it gets smaller over time, as hardware gets faster. We'd like to provide an analysis that stays useful in a wide range of circumstances.

Our main focus will be on how the algorithm's running time changes as the problem size grows. Computer Science is full of stories of programs that seemed to work ok on small test examples, but failed to handle larger ones. That's supposedly one reason our own registration system had so many problems when it was first introduced to U. Illinois: its previous tests had been at much smaller sites.

So we'll be looking at differences in the shape of the running time function, its "asymptotic complexity," ignoring several factors:

Running time on small inputs is often dominated by constant set-up costs, so doesn't accurately show how running time will scale up.

Ignoring multiplicative constants keeps the analysis stable across changes in hardware. This is a useful working approximation most of the time. However, be aware that there are times when you need to step away from it. For example, the standard quicksort library function uses a method that doesn't have the best asymptotic complexity but has extremely good constants.

Finally, the actual running-time function may wiggle. There may be good and bad input values. Discretization effects may mean that it has sudden jumps at certain input sizes. We will handle this by comparing our real wiggly function to a set of simple benchmark functions.

Asymptotically smaller and similar

A function f is "asymptotically smaller" than a function g (shorthand \(f(n) \ll g(n) \)) if and only if \[\lim_{n\rightarrow \infty} \frac{f(n)}{g(n)} = 0\]

This means that f(n) grows more slowly than g(n), as n gets large. This relationship is important for determining which benchmark functions grow faster than others, and for deciding which parts of a complex running-time expression will dominate for larger inputs. (See below.)

Notice that I'm using "f(n)" and "g(n)" to name the functions. We have previously been using just f and g. However, when talking about running times, it's convenient to include the variable because we'll be comparing named functions like f(n) to nameless functions like \(n^2\).

We can define a corresponding equality relationship "asymptotically similar" (shorthand \(f(n) \approx g(n) \)) by \[\lim_{n\rightarrow \infty} \frac{f(n)}{g(n)} = c\]

This forces the two functions to have similar shape as they approach infinity. Notice that the limit is required to be a constant c rather than 1. This makes the relationship ignore constant multipliers. So this satisfies two of our three goals. However, it doesn't handle functions that wiggle, because the wrong type of wiggling may cause the limit not to be defined.


Because of the problem with wiggling, comparisons involving actual running times of algorithms typically use "big O".

f(n) is O(g(n)) if and only if there are positive reals c and k such that \(0 \le f(n) \le cg(n)\) for every \(n \ge k\).

Looking at the key parts of this definition:

Big-O is a \(\le\) type relationship. So f(n) is O(g(n)) might mean that f(n) grows more slowly than g(n) or it might mean that they grow at the same rate (for large inputs). If we want to say that f(n) definitely grows more slowly, like <, we would use \(f(n) \ll g(n)\). If we want to say that f(n) and g(n) grow at the same rate, i.e. big-O holds in both directions, we say that f(n) is \(\Theta(g(n))\).

< \( f \ll g\)
\(\le\) f is O(g)
\(= \) f is \(\Theta (g)\)

To apply the definition of big-O, you need to select suitable values for c and k. You can often choose k=1. To find c in simple cases involving positive coefficients, we can figure out c by adding up the coefficients on the terms of f(n).

\(f(n) = 2n^2 + 3n + 7\)
\(g(n) = 2n^2 + n\)

The coefficients for f(n) sum to 12 and the coefficient on the leading term of g(n) is 2. So set c=6. Since we're considering only \(n \ge k\), we know that \(n^2 \ge n \ge 1\). Then:

\(f(n) = 2n^2 + 3n + 7 \le 2n^2 + 3n^2 + 7n^2 = 12n^2\)
    \(= 6(2n^2) = c(2n^2) \le c(2n^2 + n) = cg(n)\)

There are exceptions where k=1 does not work. For example, consider comparing \(\log n + 10\) to \(n\log n\). \(n\log n\) grows faster than \(\log n + 10\). However, when n=1, \(\log n = 0\). So \(n\log n = 0\) but \(\log n + 10 = 10\). So you'd need to set k larger in this case.

The dominant term method

In practice, most manipulation of big-O relationships uses the "dominant term" method. We strip any constant multipliers, then identify the fastest growing term for each of the two functions, and compare these two dominant terms.

To apply this method, we first need to understand the ordering of common primitive functions. For powers of n, we have this sequence:

\(1 \ll n \ll n^2 \ll n^3 \ll n^4 \ll \ldots \)

1 represents all constant functions.

Two common fast running times are \(\log n\) and \(n \log n\). We can insert these into the sequence:

\(1 \ll \log n \ll n \ll n\log n \ll n^2 \ll \ldots \)

Notice that the base of the log does not matter. The change of base formula means that \(\log_a n\) and \(\log_b n\) differ by a factor that depends on a and b, but not n.

On the upper end of the sequence, we can add exponential running times and n!:

\(\ldots n^3 \ll n^4 \ll \ldots 2^n \ll 3^n \ll 4^n \ll \ldots \ll n!\)

Now, suppose that we are comparing these two functions:

\(f(n) = n^4 + 3n! + 17\)
\(g(n) = 27 n^5 + n\log n + 4000\)

First, we remove the constant multipliers:

f(n): \( n^4 + n! + 1\)
g(n): \( n^5 + n\log n + 1\)

The dominant terms are

f(n): \( n!\)
g(n): \( n^5 \)

So f(n) grows faster than g(n). So g(n) is O(f(n)) but not vice versa.