Discrrete mathematics for computer science series warmup

A really bad way to maintain order• Given a list L in order • Suppose elements get added dynamically • Problem is to keep it in order • Joe has a sort function available SortL • He learn

Series Warmup

A really bad way to maintain order

• Given a list L in order

• 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)

• Unbeknownst to Joe, Sort does a bubble sort

• How many item comparisons, starting from an empty list and

adding n elements?

Recall Bubble Sort Analysis

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

• So

• But what is the sum of the first n squares?

(i −1) ι

2

ι =1

ν

2 ι2

ι =1

ν

ι =1

ν

∑

• n=4, sum of first 4

squares

• Area = n across and tall

• How big is the white space?

• Let

•

i

i=1

ν

∑

S(n) = ι2

ι=1

ν

∑

Ι ( ν ) = ι =

ι=1 ν

∑ ν ( ν 2 +1)

Of course this is Θ(n3)!

S(n) = ι2

ι=1

ν

∑

Ι ( ν ) = ι =

ι=1

ν

∑ ν ( ν 2 +1)

S(n) + Ι ( ϕ )

ι=1

ν

∑ = ( ν +1)( ν )( ν +1)

2

Σ ( ν ) + ι2

2

ι=1

ν

∑ + ι

2

ι=1

ν

∑ = ( ν +1)( ν )( ν +1)

2 3

2 Σ ( ν ) = ν3

2 + ν2 + ν

2 −

ν2

4 −

ν

4

Σ ( ν ) = ν3

3 +

ν2

2 +

ν

6

1

2

i=1

ν

ι=1

ν

ν2

ν

ν2

ν

4

ν2

ν

6

FINIS