Discrrete mathematics for computer science asymptotic warmup

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?)