Analysis of Algorithms[ Section 4.1 ]

Examples of functions important in CS:

• the constant function:f(n) =

Analysis of Algorithms[ Section 4.1 ]

Examples of functions important in CS:

• the constant function:f(n) = c

• the logarithm function: f(n) = logb n

Analysis of Algorithms[ Section 4.1 ]

Examples of functions important in CS:

• the constant function:f(n) = c

• the logarithm function: f(n) = logb n

• the linear function: f(n) =

Analysis of Algorithms[ Section 4.1 ]

Examples of functions important in CS:

• the constant function:f(n) = c

• the logarithm function: f(n) = logb n

• the linear function: f(n) = n

• the n-log-n function: f(n) = n log n

Analysis of Algorithms[ Section 4.1 ]

Examples of functions important in CS:

• the constant function:f(n) = c

• the logarithm function: f(n) = logb n

• the linear function: f(n) = n

• the n-log-n function: f(n) = n log n

• the quadratic function: f(n) = n2

Analysis of Algorithms[ Section 4.1 ]

Examples of functions important in CS:

• the constant function:f(n) = c

• the logarithm function: f(n) = logb n

• the linear function: f(n) = n

• the n-log-n function: f(n) = n log n

• the quadratic function: f(n) = n2

• the cubic function and other polynomials

f(n) = n3

f(n) =

Analysis of Algorithms[ Section 4.1 ]

Examples of functions important in CS:

• the constant function:f(n) = c

• the logarithm function: f(n) = logb n

• the linear function: f(n) = n

• the n-log-n function: f(n) = n log n

• the quadratic function: f(n) = n2

• the cubic function and other polynomials

f(n) = n3

• the exponential function:

f(n) =

Analysis of Algorithms[ Section 4.1 ]

Comparing growth rates

• f(n) = c

• f(n) = logb n

• f(n) = n

• f(n) = n log n

• f(n) = n2

• f(n) = n3

• f(n) = bn

Analysis of Algorithms[ Section 4.2 ]

How to analyze algorithms

- Experimental studies

Analysis of Algorithms[ Section 4.2 ]

How to analyze algorithms

- Counting the number of primitive operations

Note: possible difference between the worst-case running time and the average-case running time

Analysis of Algorithms[ Section 4.2.3 ]

Asymptotic notation

Big-Oh notation:

Let f(n) and g(n) be functions from integers to reals. We say that f(n) = O(g(n)) if there are constants c>0 and n0>0 such that

f(n) · c g(n) for every n ¸ n0

We also say that f(n) is order of g(n).

Analysis of Algorithms[ Section 4.2.3 ]

Examples:

f(n) = 5n-3 g(n) = n

Analysis of Algorithms[ Section 4.2.3 ]

Examples:

f(n) = 7n2+(n3)/5 g(n) = n4

Analysis of Algorithms[ Section 4.2.3 ]

Examples:

Insert-sort algorithm

// input: array A, output: array A is sortedint i,j;int n = A.length;for (i=0; i<n-1; i++) { j = i; while ((j>=0) && (A[j+1]<A[j])) {

int tmp = A[j+1];A[j+1] = A[j];A[j] = tmp;j--;

}}

Analysis of Algorithms[ Section 4.2.3 ]

Asymptotic notation continued

Big-Omega notation:

Let f(n) and g(n) be functions from integers to reals. We say that f(n) = (g(n)) if there are constants c>0 and n0>0 such that

f(n) ¸ c g(n) for every n ¸ n0

Analysis of Algorithms[ Section 4.2.3 ]

Asymptotic notation continued

Big-Theta notation:

Let f(n) and g(n) be functions from integers to reals. We say that f(n) = £(g(n)) if f(n) = O(g(n)) and f(n) = (g(n)).

Analysis of Algorithms[ Section 4.2.5 ]

Words of caution:

• what is 10100n ?

• exponential algorithms are a big NO (unless the input is really small)