Mathematical Backgroundand Linked Lists

2

Iterative Algorithm for Sum

Find the sum of the first n integers stored in an array v:

sum (v[], n)

temp_sum 0

for i 0 to n – 1 do

temp_sum temp_sum + v[i]

return temp_sum

3

Programming Using Recursion

Write a recursive function that solves the same problem:

sum (v[], n)

if (n = 0) then

return 0

else

return (v[n-1] + sum (v, n-1))

4

Review: Induction

Suppose S(c) is true for a fixed constant c

Often c = 0

S(k) implies S(k+1) for all k >= c

Then S(n) is true for all n >= c

5

Proof By Induction

Claim: S(n) is true for all n >= c

Basis:Show S(n) is true for n = c

Inductive hypothesis:Assume S(n) is true for n = k

Step:Show that S(n) is then true for n = k + 1

6

Induction Example:Geometric Closed Form

Prove:x0 + x1 + … + xn = (xn+1 – 1) / (x – 1)

for all x 1

Basis: show that x0 = (x0+1 - 1) / (x - 1):x0 = 1 = (x1 – 1) / (x – 1)

Inductive hypothesis: assume thatx0 + x1 + … + xk = (xk+1 – 1) / (x – 1)

7

Induction Example:Geometric Closed Form

Step (show true for n+1):x0 + x1 + … + xk+1 =

(x0 + x1 + … + xk) + xk+1 = (by hypothesis)

(xk+1 – 1) / (x – 1) + xk+1 =

(xk+1 – 1 + xk+1 (x – 1)) / (x – 1) =

(xk+1 – 1 + xk+2 – xk+1) / (x – 1) =

(xk+2 – 1) / (x – 1)

8

Proving Program Correctness Using Induction

Basis Step: sum (v, 0) = 0

Inductive Hypothesis (n = k):Assume that sum (v, k) correctly returns the sum of the first k elements of v, i.e. v[0] + v[1] + … + v[k-1]

Inductive Step (n = k + 1): sum (v, n) returns v[k] + sum (v, k) which is the sum of the first k+1 elements of v

9

Powers of 2

Many of the numbers we use will be powers of 2

Binary numbers (base 2) are easily represented in digital computers

Each "bit" is a 0 or a 1

20=1, 21=2, 22=4, 24=16, 28=256, …

An n-bit wide field can hold 2n positive integers: 0 k 2n-1

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Unsigned Binary NumbersEach bit represents a power of 2

For unsigned numbers:The minimum value is 0The maximum value is 2n-1, where n is the number of bits

So, for example5 bits => 32 possible values (25)10 bits => 1024 possible values (210)

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Binary and Decimal

20=1

21=2

22=4

23=8

24=16

25=32

26=64

27=128

28=256

Decimal10

1 1 3

1 0 0 1 9

1 0 1 0 10

1 1 1 1 15

0 0 0 0 161

1 1 1 1 311

1 1 1 1 127111

1 1 1 1 2551111

12

Logs and ExponentsDefinition: log2x = y means x = 2y

The log of x in base 2, is the value of y that gives x = 2y

8 = 23, so log28 = 3

65536= 216, so log265536 = 16

Notice that log2x tells you how many bits are needed to hold x values

8 bits hold 256 numbers: 0 to 255 (28-1)

log2256 = 8

13

2x and log2x – for Small x’s

14

2x and log2x – for Large x’s

15

Floor and Ceiling

X

X

– Floor function: the largest integer < X

– Ceiling function: the smallest integer > X

2232.722.7

2222.332.3

16

Floor and Ceiling

Properties:

More examples:

1XXX2.

XX1X1.

33.6

463 .

3333

4 5.4

17

Example:log2x and Tree Height

n items in an almost full binary tree,

the tree height (length of longest path) is log2n

4

2 6

51 3

18

Properties of logs

Usually, we work in log base 2

In base 2:a = 2log

2a

log2an = n·log2a

Similarly, in any base b:a = blog

ba

logban = n·logba

19

Properties of logs

Claim:log a·b = log a + log b

Proof:a = 2log

2a and b = 2log

2b

a·b = 2log2a · 2log

2b = 2log

2a+log

2b

Therefore: log2a·b = log2a + log2b

Note: log a·b log a·log b

20

Other log Properties

log a/b = log a – log bSpecial case: log 2 1/a = – log 2 a

Base change: logan = logbn / logba

log log X < log X < X for all X > 0 log log X = Y means

log X grows slower than X Called a “sub-linear” function

X2Y2

21

Log Base Change

Any base x log is equivalent to base 2 log within a constant factor

Example:log10n = log2n / log210

log210 = 10/3

Therefore:log10n = 0.3 log2n

22

Monotonic Functions

A function is called monotonically increasing if for all x, y such that x > y:f(x) > f(y)

Example: f(x) = x

A function is called monotonically non-decreasing if for all x, y such that x > y:f(x) ≥ f(y)

Example: f(x) = 5

23

Monotonic Functions

A function is called monotonically decreasing if for all x, y such that x > y:f(x) < f(y)

Example: f(x) = -x

A function is called monotonically non-increasing if for all x, y such that x > y:f(x) ≤ f(y)

24

Monotonic Functions

monotonically non-decreasing

monotonically non-increasing

neither

25

Monotonic Functions – Examples

f(x) = x2

Decreasing for x < 0, increasing for x > 0, therefore does not fit any definition

Monotonous non-decreasing

1xx

1x01

0xx

f(x)

26

Arithmetic Series

The sum isS(1) = 1

S(2) = 1 + 2 = 3

S(3) = 1 + 2 + 3 = 6

n

1i

in21S(n)

n

1i 2

1)n(ni

27

Algorithm Analysis

Consider the following program segment:x 0

for i 1 to n do

for j 1 to i do

x x + 1

What is the value of x at the end?

28

Analyzing the Loop

Total number of times x is incremented:

The running time of the program is proportional to n(n+1)/2 for all n

O(n2)

2

nnin21S(n)

n

1i

)1(

29

Geometric Series

General geometric series:

Common special case, for x = 2:

n+1n

k 2 n

k=0

x -1x =1+ x + x +...+ x =

x -1

n

k 2 n n+1

k=0

2 =1+2+2 +...+2 = 2 -1

30

Infinite Geometric Seriesfor |x| < 1

When |x| < 1, we can compute the sum of an infinite geometric series:

Example, x = ½:

k

k=0

1x =

1- x

k

k=0

1(1 2) =1+1 2+1 4+... = = 2

1-1 2