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Mathematical References

Sequences Finite sequence where n = 1, 2, 3, 4, 5

Infinite sequence generates thesequence

Observe that the general rule for the nth termi.e. 2n is inserted as an element in the middle.

nan 2

10;8;6;4;2 54321 aaaaa

nan 2

,2,,10;8;6;4;2 54321 naaaaa


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Logarithms

xb

a x

x

x y x

y xog y x

y x xy

x

b

ba

b y

b

bbb

bbb

a

log

loglog

log

loglog

log1log

loglog)(log

k k n 22 1

k nk log1

Theorem : If k and n are positiveintegers satisfying

Then

Theorem:


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Mathematical Induction

Using mathematical induction we can provethat a collection of statements are true. The statements are usually indexed by an integer

n. For example, the assertion that

Is true for all n, is an assertion about all n. i.e. aninfinitely many statements.

2)1(

1

nni

n

i


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Mathematical Induction

In order to prove that a sequence ofstatements s(1), s(2) , s(3) . are trueusing mathematical statement we mustfollow two steps. Basic step : prove that statement s(1) is true Inductive step : Assume that the nth statement

s(n) is true and use this assumption to provethat statement s(n+1) is true.


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Example Proof by induction

Prove that

Basic step Show that the statement is true

for n =1. This would be the first of the infinitelymany statements. Plug in n=1 into the question and see if it is true.

2)1(

1

nnin

i

12 )11(111

1ii Yes, it is true!


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Example proof by induction

Induction step: Suppose the equation is true for n, thismeans that we assume that

Is true for n. And we will use it to prove that the nextconsecutive statement i.e. (n+1)st statement is true aswell. State with the (n+1)st statement

2

)1(

1

nni

n

i

)1(1

1

1

niin

i

n

i

2)2)(1(

)1(2

)1( nnn

nn

Start withthe sum 2

)2)(1(1

1

nn

i

n

i

Need toshow


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Analysis of Algorithms Analysis of algorithms is the process of estimating the

running time and space (memory) utilization of analgorithm as a function of the input size.

These estimates are also referred to as time and space

complexity of an algorithm. The estimated time complexity is independent of themachine the algorithm will run on.

This allows us to compare time complexity ofdifferent algorithms.


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Best case, worst case and average case timecomplexity

Among all inputs of size n the one that takes the leastamount of time to run produces the best case timecomplexity.

Similarly the input of size n that takes the longest to runwill produce the worst case time complexity for analgorithm. We will mostly be interested in worst caserunning time of algorithms.

Average case time complexity is obtained by finding thetime complexity of all input classes of size n and findingtheir average. Often very complicated to find

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