Growth of Functions

Asymptotic Analysis for Algorithms

T(n) = the maximum number of steps taken by an algorithm for any input of size n (worst-case runtime)

• Always talking about worst-case runtime– Will become more important later

• We are not concerned with how good things can be

• Don’t want to be surprised by a “bad” case after implementation– Especially if “bad” cases are not uncommon

Asymptotic Analysis for Algorithms

T(n) = the maximum number of steps taken by an algorithm for any input of size n (worst-case runtime)

• If T(n)<u(n) for all n>n0 then T(n) is in O(u(n))– Need to show that the number of steps taken is

less than u(n) for all inputs of size n>n0

Asymptotic Analysis for Algorithms

T(n) = the maximum number of steps taken by an algorithm for any input of size n (worst-case runtime)

• If T(n)>L(n) for all n>n0 then T(n) is in Ω(L(n))– Need to show that there exists at least one input

that takes more than L(n) steps for all n>n0

Input Size

• Asymptotic runtime is always measured with respect to the input size

• Input size represented by n– Later: n = Number of elements for the data

structure• Runtime is a function of n

Common Runtimes

• Constant: O(1)– No dependence on n– As n grows, the runtime is constant

• Logarithmic: O(log(n))– Very efficient

• Linear: O(n)• O(n*log(n))• Squared: O(n2)• Cubic: O(n3)

– Starting to be impractical except for small n• Exponential: O(2n)

– Impractical

Example

• How to find an element in a sorted array?– Linear search• Check each element 1-by-1 and compare• If the element is found, return true• Linear runtime: O(n)

– Binary search• Recursively compare with center element• If center element is smaller, check the larger half• If center element is larger, check the smaller half• Eliminate half the search space each comparison• Logarithmic runtime: O(log(n))