Hashing Out Random Graphs Nick Jones Sean Porter Erik Weyers Andy Schieber Jon Kroening

Hashing Out Random Graphs

Nick Jones

Sean Porter

Erik Weyers

Andy Schieber

Jon Kroening

IntroductionWe will be looking at some applications of probability in computer science, hash functions, and also applications of probability with random graphs.

Hash Functions

We are going to map at set of n records, denoted , r1, r2, … rn, in m, m > n, locations with only one record in each location in m.

A hashing function is a function that maps the record values into the m locations.

We use a sequence of hash functions, denoted h1, h2, h3, …, to map the ri records in the m locations.

The records are placed sequentially as indicated below: h1(r1) = m1.

h1(r2), h2(r2), h3(r3), …

Every time we are unsuccessful in placing a record (because it is already full), a collision occurs.

We will let the random variable X denote the number of collisions that occur when placing n records.

We would like to find E[X] and Var(X).

These values are very hard to figure out but we can come up with a formula for each of these two problems.

In order to do this we need to define some other random variables.

Yk = #of collisions in placing rk

Therefore,

n

n

k

k YYYYX

...21

1

1 kk YZ

nZZZX n ...21

(geometric with p = (m-k+1)/m)

We can then find E[Zk].

1

1

km

m

p]E[Zk

nZEZEZEXE n ][...][][][ 21

n

k

kZEn1

][

n

k km

mn

1 1

1

1...

1

11

nmmmmn

m

nm

xdxmn1

/

1log

nm

mmn

nnm

mmXE

1log

We would also like to find Var(X).

1

1122

km

km

p

pZVar k

n

k

n

k

k

kmk

mZVarXVar1

21 1

1

1

1...

2

2

1

1222

nm

n

mmm

dxxm

xm

n

1

1

2

We now know the formula for E[X] and the Var(X).

dxxm

xmXVar

n

1

1

2

nnm

mmXE

1log

Alfred Renyi

March 30, 1921– Feb. 1, 1970

49 years old

The Hungarian mathematician spent six months in hiding after being forced into a Fascist Labor Camp in 1944

During that time he rescued his parents from a Budapest prison by dressing up in a soldiers uniform

He got his Ph.D. at the University of Szeged in Hungary

Renyi worked with Erdös on Random Graphs, they published joint work

He worked on number theory and graph theory, which led him to results about the measures of the dependency of random variables

Paul Erdös

“A Mathematician is a machine for turning coffee into theorems”

Born: March 26, 1913

May have been the most prolific mathematician of all time

Written and Co-Authored over

1475 Papers

Erdös was born to two high school

math teachers

His mother kept him out of school until his teen years because she

feared its influence

At home he did mental arithmetic and at three he could multiply numbers

in his head

Fortified by espresso Erdös did math for 19 hours a day, 7 days a week

He devoted his life to a single narrow mission: uncovering mathematical

truth

He traveled around for six decades with a suit case looking for

mathematicians to pick his brain

His motto was:“Another roof, another proof”

“Property is a nuisance”

“Erdös posed and solved thorny problems in number theory and other areas and founded the field of discrete mathematics which is a foundation of computer science”

Awarded his doctorate in 1934 at the University of Pazmany Peter in Budapest

Graphs

A graph consists of a set of elements V called vertices and a set E of pairs of vertices called edges

A path is a set of vertices i,i1,i2,..,ik,j for which (i,i1),(i1,i2),..,(ik,j) Є E is called a path from i to j

Connected Graphs

A graph is said to be connected if there is a path between each pair of vertices

If a graph is not connected it is called disconnected

Random Graphs

In a random graph, we start with a set of vertices and put in edges at random, thus creating paths

So an interesting question is to find P(graph is connected) such that there is a path to every vertex in the set

James Stirling

Who is James Stirling?Lived 1692 – 1770.

Family is Roman Catholic in a Protestant England.Family supported Jacobite Cause.Matriculated at Balliol College Oxford Believed to have studied and matriculated at two other universities but this is not certain.Did not graduate because h refused to take an oath because of his Jacobite beliefs.Spent years studying, traveling, and making friends with people such as Sir Isaac Newton and Nicolaus(I) Bernoulli.

Methodus DifferentialisStirling became a teacher in London.

There he wrote the book Methodus Differentialis in 1730.

The book’s purpose is to speed up the convergence of a series.

Stirling’s Formula is recorded in this book in Example 2 of Proposition 28.

nnennn 2!

Stirling’s Formula

Used to approximate n!Is an Asymptotic Expansion.Does not converge.Can be used to approximate a lower bound in a series.Percentile error is extremely low.The bigger the number inserted, the lower the percentile error.

nnennn 2!

Stirling’s Formula Error Probability

About 8.00% wrong for 1!About 0.80% wrong for 10!About 0.08% wrong for 100!Etc…

Percentile Error is close to so if the formula is multiplied by , it only gets better with errors only at .

n12

1

n12

11

2

1

n

Probability Background

Normal Distribution and Central Limit theorem

Poisson Distribution

Multinomial Distribution

The Normal Distribution

A continuous random variable x with pdf

e 2σμ)(x

σ2π

1f(x) 2

2

normal called is x

Normal Distribution

), shown that becan It X~N(μ

Normal Distribution

Note: When the mean = 0 and standard deviation = 1, we get the standard normal random variable

Z~N(0,1)

Central Limit Theorem

If X1, X2,… are independent identically distributed with common mean µ, and standard deviation σ, then

x

n

ii

n

dyy

xn

n

P ex

21

2

2

1lim

Central Limit Theorem

normalely approximat is S

then,large isn , S If

n

n

1iin x

N(0,1)~σ

μxZ

then,),N(~X If

Poisson Distribution

λ toequalboth varianceandMean

210

x

n p(x) p)(1plim

xnx

n

...,, , xx!

p(x) eλλx

Multinomial Distribution

n independent identical trials of events A1, A2,…,Ak with probabilities P1,P2,...Pk

Define Xi = number times Ai occurs j=1…k

(X1+X2+…+Xk = n) then,

Multinomial Distribution

Where n is sum of ni

PPPnnn

nxnxnxk21 n

k

n

2

n

1k21

kk2211

...!!...!

n!

,...,P

Connected Graphs

Recall: A random graph G consists of vertices, V={1,2,…,n}, random variables x(i) where i=1,..,n along with probabilities

P P P x i j Pj j j ( ) { ( ) } 1

Connected Graphs

The set of random edges is then

which is the edge emanating from vertex i

},..,1 : ))( ,{( niixiE

Connected GraphsThe probability that a random graph is connected P {graph is connected} = ?

A special case: suppose vertex 1 is ‘dead’ (doesn’t spawn an edge)

N = 2 P 1P 2

P P1 2 1 + =

P graph connected P{ } 1

Dead Vertex Lemma

Consider a random graph consisting of vertices 0,1,2,..,r and edges ,

i=1,2,…,r where are independent and , j=0,1,..,r

( , )i Yi

YiP Y j Qi j{ }

00

= }connectedgraph { then )1( if QPQn

jj

Dead Vertex Lemma

1

2

4

3 5

6

Maximal Non-Self Intersecting(MNSI)

Consider the maximal non-self intersecting path emanating from vertex 1:1 1 1 1 12 1, ( ), ( ),..., ( ) ( ( ))x x x x xk k

1 2

45

3k = 3

Maximal Non-Self Intersecting(MNSI)

Define

and set )})1(),..,1(,1{ )1(:min( 1 kk XXXkN

1

1)1(1

N

ixiPPW

Maximal Non-Self Intersecting(MNSI)

By using the MNSI path as the Dead Vertex Lemma,

P graph connected N X X WN{ | , , ( ),..., ( )} 1 1 11

12

3

45

6

7

k = 4

Conditional Probability

}{)(

averages.y probabilit lconditiona are

variablesrandom discrete of nsExpectatio

}{} | {}{

:yprobabilit lconditiona of idea The

xXPxXE

scenarioPscenarioeventPeventP

x

Conditional Probability

} {

scenario

)}1(),..,1(,1,{

)}1(),..,1(,1,| {)(

scenario event

:nsexpectatio Taking

1

1

connectedgraphP

XXNP

XXNconnectedgraphPWE

N

N

Conditional ProbabilitySpecial Case of Interest:

equiprobable vertices)

Pn

WN

nE W

nE N

E N P N i

j

i

n

1

1

0

1

(

[ ] [ ]

[ ] { }

Conditional Probability

E Wn

P N i

n

n n n i

n

n

n

n n i

i

n

i

n

i

[ ] { }

( )( )...( )

!

( )!

( )!

1

1 1 2

1 1

1

0

1

1

1

Conditional Probability

( )!

( )!

( )

( )!

[ ]( )!

!

n

n n n i

n

n

n

n i

j n i

E Wn

n

n

j

ii

n

n

n

n i

i

n

n

j

1 1

1

1

1

1

1

0

1

1

0

1

Let

Poisson Distribution

nk

k

ek

n

ek

kXP

nX

!

!

}{

mean th Poisson wi is Suppose

Poisson Distribution

1

0

1

0

1

0

!

!

}{}{

pick So

n

j

jn

n

k

nk

n

k

j

ne

ek

n

kXPnXP

Central Limit Theorem

2!2

1

!

)(2

1)(

),(

largefor Thm,Limit Central By the

nmean ofPoisson each ... :Recall 21

njjn

n

n

e

j

n

j

ne

asymptoticnXP

nnNS

n

XXXX

Conditional Probability

2

)1( )1(2][on substitutiby So

!

)!1(][ Recall

)1( )1(2)!1(

2!

Formula sStirling' :Recall

2

)1(1

)1(1

n

eennWE

j

n

n

nWE

ennn

ennn

nnn

j

n

nn

nn

Conditional Probability

xn

n

n

n

n

n

n

en

x

n

en

n

enn

enn

n

n

en

)1( lim 1

))1(1(22

1))1((

2

2

)1()1(

2

2

2

)1(2

n

21

2

21

Conditional Probability

nWEconnectedisgraphP

nnn

ee

2][} {

2

2

1=

n2

2=

12

2

1

122

Thank You “The first sign of senility is when a

man forgets his theorems. The second is when he forgets to zip up. The third is when he forgets to zip down.”

--Paul Erdös

References

http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Erdos.html

http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Renyi.html

http://www.lassp.cornell.edu/sethna/Cracks/Stirling.html

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Stirling.html