Discrrete mathematics for computer science asymptotic warmup

Order Notation Warmup... The exact running time depends on details of the code initializations, for example, the quality of the compiler, the speed of the computer, etc... Let Tn be the

Order Notation Warmup

Analyzing Bubble Sort

Sort (a1,…,an):

for i := 1 to n-1

for j := 1 to n-i

if aj>aj+1 then interchange

aj↔aj+1

How many comparison steps as a function of n?

• Inner loop is executed

n-1 times when i=1

n-2 times when i=2

…

1 time when i=n-1

For a total of 1+2+3+…+(n-2)+(n-1)

=((n-1)∙n)/2.

The exact running time depends on

details of the code (initializations, for example), the quality of the compiler, the speed of the computer, etc.

Let T(n) be the maximum running time of this program for any array of length n (Small variations due to whether swaps actually occur)

• T(n) = a2n2 + a1n + a0 for some

constants a0, a1, a2

• But the last two terms become

increasing inconsequential as n

increases since a1n + a0 = o(a2n2)

• So the most useful thing to say is that T(n)=�(n2) (why both upper and lower bound?)