Discrrete mathematics for computer science series warmup

Series Warmup


A really bad way to maintain order

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Given a list L in order

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Unbeknownst to Joe, Sort does a bubble sort

Suppose elements get added dynamically
Problem is to keep it in order
Joe has a sort function available Sort(L)
He learned in CS50 not to reinvent any wheels
So when a new element x arrives he appends it to the end of L: L
<- append(L,x) and then calls Sort(L)
How many item comparisons, starting from an empty list and
adding n elements?


Recall Bubble Sort Analysis

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((n-1)∙n)/2 comparisons to sort a list of length n
So

(i − 1)ι 1 ν 2 1 ν
∑ 2 = 2 ∑ι − 2 ∑ι
ι=1
ι=1
ι=1
ν

But what is the sum of the first n squares?


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n=4, sum of first 4
squares
Area = n across and
tall
ν

∑

How big isi the white
space? i=1
Let

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ν

S(n) = ∑ ι2
ι=1
ν

Ι (ν) = ∑ ι =
ι=1

ν(ν + 1)
2


ν

S(n) = ∑ ι2
ι=1

ν(ν + 1)
Ι (ν) = ∑ ι =
2
ι=1
ν

ν

S(n) + ∑ Ι ( ϕ) =
ι=1

(ν + 1)(ν)(ν + 1)
2

ι2 ν ι (ν + 1)(ν)(ν + 1)
Σ(ν) + ∑ + ∑ =
2
ι=1 2
ι=1 2
ν

3
ν3
ν ν2 ν
2
Σ(ν) = + ν + − −
2
2
2 4 4
ν 3 ν2 ν
Σ(ν) = + +
3 2 6
1 ν 2 1 ν
ν 3 ν2 ν ν2 ν
∑ i − 2 ∑ ι = 6 + 4 + 12 − 4 − 4
2 i=1
ι=1

ν 3 3ν2 ν
= +
−
6
4
6

Of course this is Θ(n3)!


FINIS