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CSci 2011

Discrete Mathematics

Lecture 12

3.1 Algorithms

Algorithm 1: Maximum element

Algorithm for finding the maximum element

in a list:

procedure max ( a 1 , a 2 , …, an : integers)

max := a 1

for i := 2 to n

if max < ai then max := ai

{ max is the largest element}

How long does this take?

If the list has n elements, worst case

scenario is that it takes n “steps”

Algorithm 2: Linear search

Given a list, find a specific element in the list

 List does NOT have to be sorted!

procedure linear_search ( x : integer; a 1 , a 2 , …, an : integers) i := 1 while ( i ≤ n and x ≠ ai ) i := i + 1 if i ≤ n then location := i else location := 0

{ location is the subscript of the term that equals x, or it is 0 if x is not found}

Binary search running time

How long does this take (worst case)?

If the list has 8 elements

 It takes 3 steps

If the list has 16 elements

 It takes 4 steps

If the list has 64 elements

 It takes 6 steps

If the list has n elements

 It takes log 2 n steps

ch3.

Growth of Functions

Binary Search running time

 The binary search takes log 2 n “steps”

 Let’s say the binary search takes the following number of steps on specific CPUs:  Intel Pentium IV CPU: 58* log 2 n /  Motorola CPU: 84.4(log 2 n + 1)/  Intel Pentium V CPU: 44(log 2 n)/

Notice that each has an log 2 n term

 As n increases, the other terms will drop out

As processors change, the constants will always

change

 The exponent on n will not

Big-Oh notation

If f(x) and g(x) are two functions of a single

variable, the statement f(x)∈O(g(x)) means that

∃k∈ R , ∃c∈ R , ∀x∈ R , x>k ⇒ 0 ≤|f(x)|≤c|g(x)|.

Note: O(g(x)): a set of functions

Informally

 c g(x) is greater than f(x) for sufficiently large x.  f(x) grows no faster than g(x), as x gets large.

How proof goes

 Need to find k and c to show f(x)∈O(g(x)).

Conventionally people use f(x)=O(g(x)).

Big-Oh proofs

A variant of the last question

Show that f ( x ) = 3 x +7 is O ( x^2 )

 In other words, show that 3 x +7 ≤ c* x^2

Function growth rates

 For input size n = 1000

 O(1) 1

 O(log n) ≈

 O(n) 103

 O(n log n) ≈10^4

 O(n 2 ) 106

 O(n 3 ) 109

 O(n 4 ) 1012

 O(n c^ ) 10 3*c^ c is a consant

 2 n^ ≈10^301

 n! ≈10^2568

 n n^103000 Many interesting

problems fall into

this category

More Examples

Show that x log x is O(x^2 ).

Proof) Take c = 1, k=1. Then, x 2 > x log x for all x > 1, since x log x = 0 when x = 1, while x 2 = 1.

Show that x^2 is not O(x log x).

Proof) Suppose x^2 is O(x log x). Then ∃ c and k such that c x log x ≥ x 2 for x > k. Note that c>2. Otherwise, cx log x < x 2 for x = 3. Consider x = c 3. Then lhs) cx log x = c 4 log c 3 = 2 c 4 log c rhs) x 2 = c 6. Since c>2, it is clear that c 2 > 2 log c. Therefore, cxlog x<x 2 for x=c 3. (Contradiction)

Little bit of Complexity

 Efficiency of your algorithm? Given input size,

 Time complexity: Time used by your computer to solve the problem  We use number of operations…  Memory (space) complexity: Memory used by your computer to solve the problem  We use number of variables allocated…

 Example: Find the maximal element in a set

max = a 1 , i = 2 While (i ≤ n) if (max < a (^) i ) max = a (^) i i++  Time complexity: 2 n - 2 comparison, n-1 addition, n+ assignment (worst case) = 4n - 2 operation = θ(n)  Memory complexity: 2 = θ(1)

 Other example: Hash Chain