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Outline

Balls and Bins (Advanced)

K. Subramani1

1 Lane Department of Computer Science and Electrical Engineering

West Virginia University

6 March, 2012

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Outline

Outline

1 Recap

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Outline

Outline

1 Recap

2 The Poisson Approximation

Some theorems and lemmas

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Outline

Outline

1 Recap

2 The Poisson Approximation

Some theorems and lemmas

3 Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom FiltersBreaking Symmetry

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Recap

The Poisson Approximation

Applications to Hashing

Recap

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Recap

The Poisson Approximation

Applications to Hashing

Recap

Main issues

The experiment of throwing m balls into nbins,

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Recap

The Poisson Approximation

Applications to Hashing

Recap

Main issues

The experiment of throwing m balls into nbins, each bin being chosen independently anduniformly at random.

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Recap

The Poisson Approximation

Applications to Hashing

Recap

Main issues

The experiment of throwing m balls into nbins, each bin being chosen independently anduniformly at random.Several questions regarding the above random process were examined,

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Recap

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Recap

The Poisson Approximation

Applications to Hashing

Recap

Main issues

The experiment of throwing m balls into nbins, each bin being chosen independently anduniformly at random.Several questions regarding the above random process were examined,

such as expected maximum load, expected number of balls in a bin,

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Recap

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Recap

The Poisson Approximation

Applications to Hashing

Recap

Main issues

The experiment of throwing m balls into nbins, each bin being chosen independently anduniformly at random.Several questions regarding the above random process were examined,

such as expected maximum load, expected number of balls in a bin, expected number of empty

bins, and expected number of bins with r balls.

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Recap

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Recap

The Poisson Approximation

Applications to Hashing

Recap

Main issues

The experiment of throwing m balls into nbins, each bin being chosen independently anduniformly at random.Several questions regarding the above random process were examined,

such as expected maximum load, expected number of balls in a bin, expected number of empty

bins, and expected number of bins with r balls. We also examined the Poisson random variable

and its applications to Balls and Bins questions.

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Recap

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p

The Poisson Approximation

Applications to Hashing

Some theorems and lemmas

Recap

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The Poisson Approximation

Applications to Hashing

Some theorems and lemmas

The Poisson Approximation

Main Issues

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Recap

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The Poisson Approximation

Applications to Hashing

Some theorems and lemmas

The Poisson Approximation

Main Issues

Is bin emptiness events independent?

Subramani Balls and Bins

Recap

Th P i A i i S h d l

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The Poisson Approximation

Applications to Hashing

Some theorems and lemmas

The Poisson Approximation

Main Issues

Is bin emptiness events independent?

We know that if m balls are thrown uniformly and independently into n bins, the distribution

is approximately Poisson with mean mn

.

Subramani Balls and Bins

Recap

The Poisson Approximation Some theorems and lemmas

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The Poisson Approximation

Applications to Hashing

Some theorems and lemmas

The Poisson Approximation

Main Issues

Is bin emptiness events independent?

We know that if m balls are thrown uniformly and independently into n bins, the distribution

is approximately Poisson with mean mn

.

We wish to approximate the load at each bin with independent Poisson random variables.

Subramani Balls and Bins

Recap

The Poisson Approximation Some theorems and lemmas

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The Poisson Approximation

Applications to Hashing

Some theorems and lemmas

The Poisson Approximation

Main Issues

Is bin emptiness events independent?

We know that if m balls are thrown uniformly and independently into n bins, the distribution

is approximately Poisson with mean mn

.

We wish to approximate the load at each bin with independent Poisson random variables.

We will show that this can be achieved by provable bounds.

Subramani Balls and Bins

Recap

The Poisson Approximation Some theorems and lemmas

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The Poisson Approximation

Applications to Hashing

Some theorems and lemmas

The Poisson Approximation

Main Issues

Is bin emptiness events independent?

We know that if m balls are thrown uniformly and independently into n bins, the distribution

is approximately Poisson with mean mn

.

We wish to approximate the load at each bin with independent Poisson random variables.

We will show that this can be achieved by provable bounds.

Note

There is a difference between throwing m balls randomly and assigning each bin a number of

balls that is Poisson distributed with mean mn .

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Recap

The Poisson Approximation Some theorems and lemmas

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The Poisson Approximation

Applications to Hashing

Some theorems and lemmas

The Poisson Approximation

Main Issues

Is bin emptiness events independent?

We know that if m balls are thrown uniformly and independently into n bins, the distribution

is approximately Poisson with mean mn

.

We wish to approximate the load at each bin with independent Poisson random variables.

We will show that this can be achieved by provable bounds.

Note

There is a difference between throwing m balls randomly and assigning each bin a number of

balls that is Poisson distributed with mean mn . However, if you use Poisson distribution and endwith m balls, the distributions are identical!

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Recap

The Poisson Approximation Some theorems and lemmas

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pp

Applications to Hashing

Outline

1 Recap

2 The Poisson Approximation

Some theorems and lemmas

3 Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom FiltersBreaking Symmetry

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Recap

The Poisson Approximation Some theorems and lemmas

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Applications to Hashing

Theorem I

Theorem

Let X(m)i , 1 i n be the number of balls in the ith bin.

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The Poisson Approximation Some theorems and lemmas

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Applications to Hashing

Theorem I

Theorem

Let X(m)i , 1 i n be the number of balls in the ith bin. Let Y(m)i , 1 i n denote independentPoisson random variables with mean m

n.

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Recap

The Poisson Approximation

A li ti t H hi

Some theorems and lemmas

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Applications to Hashing

Theorem I

Theorem

Let X(m)i , 1 i n be the number of balls in the ith bin. Let Y(m)i , 1 i n denote independentPoisson random variables with mean m

n.

The distribution of(Y(m)

1 , . . . , Y(m)n ) conditioned oni Y

(m)i = k is the same as(X

(k)1 , . . . , X

(k)n ).

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Recap

The Poisson Approximation

Applications to Hashing

Some theorems and lemmas

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Applications to Hashing

Theorem II

Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Some theorems and lemmas

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Applications to Hashing

Theorem II

Theorem

Let f(x1, x2, . . . , xn) denote a nonnegative function.

Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Some theorems and lemmas

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Applications to Hashing

Theorem II

Theorem

Let f(x1, x2, . . . , xn) denote a nonnegative function. Then,

Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Some theorems and lemmas

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pp cat o s to as g

Theorem II

Theorem

Let f(x1, x2, . . . , xn) denote a nonnegative function. Then,

E[f(X(m)1 , . . . , X(m)n )] emE[f(Y(m)1 , . . . , Y(m)n )]

Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Some theorems and lemmas

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pp g

Theorem II

Theorem

Let f(x1, x2, . . . , xn) denote a nonnegative function. Then,

E[f(X(m)1 , . . . , X(m)n )] emE[f(Y(m)1 , . . . , Y(m)n )]

Corollary

Any event that takes place with probability p in the Poisson case, takes place with probability at

most

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Recap

The Poisson Approximation

Applications to Hashing

Some theorems and lemmas

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Theorem II

Theorem

Let f(x1, x2, . . . , xn) denote a nonnegative function. Then,

E[f(X(m)1 , . . . , X(m)n )] emE[f(Y(m)1 , . . . , Y(m)n )]

Corollary

Any event that takes place with probability p in the Poisson case, takes place with probability at

most pem in the exact case.

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Recap

The Poisson Approximation

Applications to Hashing

Some theorems and lemmas

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Theorem III

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RecapThe Poisson Approximation

Applications to Hashing

Some theorems and lemmas

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Theorem III

Theorem

Let f(x1, x2, . . . , xn) denote a nonnegative function,

Subramani Balls and Bins

RecapThe Poisson Approximation

Applications to Hashing

Some theorems and lemmas

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Theorem III

Theorem

Let f(x1, x2, . . . , xn) denote a nonnegative function, such that E[f(X(m)1 , . . . , X

(m)n )] is either

monotonically increasing or monotonically decreasing in m.

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RecapThe Poisson Approximation

Applications to Hashing

Some theorems and lemmas

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Theorem III

Theorem

Let f(x1, x2, . . . , xn) denote a nonnegative function, such that E[f(X(m)1 , . . . , X

(m)n )] is either

monotonically increasing or monotonically decreasing in m. Then,

E[f(X(m)1 , . . . , X

(m)n )] 2 E[f(Y(m)1 , . . . , Y(m)n )]

Corollary

Let be an event whose probability is either monotonically increasing or decreasing in thenumber of balls.

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RecapThe Poisson Approximation

Applications to Hashing

Some theorems and lemmas

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Theorem III

Theorem

Let f(x1, x2, . . . , xn) denote a nonnegative function, such that E[f(X(m)1 , . . . , X

(m)n )] is either

monotonically increasing or monotonically decreasing in m. Then,

E[f(X(m)1 , . . . , X

(m)n )] 2 E[f(Y(m)1 , . . . , Y(m)n )]

Corollary

Let be an event whose probability is either monotonically increasing or decreasing in thenumber of balls. If has probability p in the Poisson case, then it has probability at most 2 p inthe exact case.

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RecapThe Poisson Approximation

Applications to Hashing

Some theorems and lemmas

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Theorem III

Theorem

Let f(x1, x2, . . . , xn) denote a nonnegative function, such that E[f(X(m)1 , . . . , X

(m)n )] is either

monotonically increasing or monotonically decreasing in m. Then,

E[f(X(m)1 , . . . , X

(m)n )] 2 E[f(Y(m)1 , . . . , Y(m)n )]

Corollary

Let be an event whose probability is either monotonically increasing or decreasing in thenumber of balls. If has probability p in the Poisson case, then it has probability at most 2 p inthe exact case.

Lemma

When n balls are thrown independently into n bins, the maximum load is at least ln nlnln n

with

probability at least(1 1n

), for sufficiently large n.

Subramani Balls and Bins

RecapThe Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

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Outline

1 Recap

2 The Poisson Approximation

Some theorems and lemmas

3 Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Subramani Balls and Bins

RecapThe Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Ch i H hi

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Chain Hashing

Main Issues

Subramani Balls and Bins

RecapThe Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Ch i H hi

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Chain Hashing

Main Issues

The password checking problem.

Subramani Balls and Bins

RecapThe Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Ch i H hi

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Chain Hashing

Main Issues

The password checking problem.

The Hashing Approach and Hash functions.

Subramani Balls and Bins

RecapThe Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Chain Hashing

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Chain Hashing

Main Issues

The password checking problem.

The Hashing Approach and Hash functions. f : U [0, n1].

Subramani Balls and Bins

RecapThe Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Chain Hashing

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Chain Hashing

Main Issues

The password checking problem.

The Hashing Approach and Hash functions. f : U [0, n1].Chain Hashing.

Subramani Balls and Bins

RecapThe Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Chain Hashing

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Chain Hashing

Main Issues

The password checking problem.

The Hashing Approach and Hash functions. f : U [0, n1].Chain Hashing.

Assumption: Hash function maps words into bins in random fashion.

Subramani Balls and Bins

RecapThe Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Chain Hashing

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Chain Hashing

Main Issues

The password checking problem.

The Hashing Approach and Hash functions. f : U [0, n1].Chain Hashing.

Assumption: Hash function maps words into bins in random fashion.

For each x U, the probability that f(x) = j is 1n

,

Subramani Balls and Bins

RecapThe Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Chain Hashing

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Chain Hashing

Main Issues

The password checking problem.

The Hashing Approach and Hash functions. f : U [0, n1].Chain Hashing.

Assumption: Hash function maps words into bins in random fashion.

For each x U, the probability that f(x) = j is 1n

, and

Subramani Balls and Bins

RecapThe Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Chain Hashing

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Chain Hashing

Main Issues

The password checking problem.

The Hashing Approach and Hash functions. f : U [0, n1].Chain Hashing.

Assumption: Hash function maps words into bins in random fashion.

For each x U, the probability that f(x) = j is 1n

, and the values of f(x) for each x areindependent of each other.

Subramani Balls and Bins

RecapThe Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String HashingBloom Filters

Breaking Symmetry

Chain Hashing

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Chain Hashing

Main Issues

The password checking problem.

The Hashing Approach and Hash functions. f : U [0, n1].Chain Hashing.

Assumption: Hash function maps words into bins in random fashion.

For each x U, the probability that f(x) = j is 1n

, and the values of f(x) for each x areindependent of each other.

Search approach.

Subramani Balls and Bins

RecapThe Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String HashingBloom Filters

Breaking Symmetry

Chain Hashing

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Chain Hashing

Main Issues

The password checking problem.

The Hashing Approach and Hash functions. f : U [0, n1].Chain Hashing.

Assumption: Hash function maps words into bins in random fashion.

For each x U, the probability that f(x) = j is 1n

, and the values of f(x) for each x areindependent of each other.

Search approach.

If the word is not in dictionary, expected time is mn

,

Subramani Balls and Bins

RecapThe Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String HashingBloom Filters

Breaking Symmetry

Chain Hashing

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g

Main Issues

The password checking problem.

The Hashing Approach and Hash functions. f : U [0, n1].Chain Hashing.

Assumption: Hash function maps words into bins in random fashion.

For each x U, the probability that f(x) = j is 1n

, and the values of f(x) for each x areindependent of each other.

Search approach.

If the word is not in dictionary, expected time is mn

, otherwise, expected time is 1 + m1n

.

Subramani Balls and Bins

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RecapThe Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String HashingBloom Filters

Breaking Symmetry

Chain Hashing

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g

Main Issues

The password checking problem.

The Hashing Approach and Hash functions. f : U [0, n1].Chain Hashing.

Assumption: Hash function maps words into bins in random fashion.

For each x U, the probability that f(x) = j is 1n

, and the values of f(x) for each x areindependent of each other.

Search approach.

If the word is not in dictionary, expected time is mn

, otherwise, expected time is 1 + m1n

.

Choosing n= m, gives constant expected search time.

When n= m, the maximum load is ( ln nlnln n), w.h.p.;

Subramani Balls and Bins

RecapThe Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String HashingBloom Filters

Breaking Symmetry

Chain Hashing

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Main Issues

The password checking problem.

The Hashing Approach and Hash functions. f : U [0, n1].Chain Hashing.

Assumption: Hash function maps words into bins in random fashion.

For each x U, the probability that f(x) = j is 1n

, and the values of f(x) for each x areindependent of each other.

Search approach.

If the word is not in dictionary, expected time is mn

, otherwise, expected time is 1 + m1n

.

Choosing n= m, gives constant expected search time.

When n= m, the maximum load is ( ln nlnln n), w.h.p.; faster than binary search.

Subramani Balls and Bins

RecapThe Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String HashingBloom Filters

Breaking Symmetry

Chain Hashing

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Main Issues

The password checking problem.

The Hashing Approach and Hash functions. f : U [0, n1].Chain Hashing.

Assumption: Hash function maps words into bins in random fashion.

For each x U, the probability that f(x) = j is 1n

, and the values of f(x) for each x areindependent of each other.

Search approach.

If the word is not in dictionary, expected time is mn

, otherwise, expected time is 1 + m1n

.

Choosing n= m, gives constant expected search time.

When n= m, the maximum load is ( ln nlnln n), w.h.p.; faster than binary search.

Wasted space.

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Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String HashingBloom Filters

Breaking Symmetry

Bit String Hashing

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Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String HashingBloom Filters

Breaking Symmetry

Bit String Hashing

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Main Issues

Subramani Balls and Bins

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Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String HashingBloom Filters

Breaking Symmetry

Bit String Hashing

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Main Issues

Goal is to save space instead of time.

Assume that passwords are restricted to 64 bits and that a hash function maps words into

32 bits.

Fingerprinting and false positives.

Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String HashingBloom Filters

Breaking Symmetry

Bit String Hashing

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Main Issues

Goal is to save space instead of time.

Assume that passwords are restricted to 64 bits and that a hash function maps words into

32 bits.

Fingerprinting and false positives.

The approximate set membership problem:

Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String HashingBloom Filters

Breaking Symmetry

Bit String Hashing

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Main Issues

Goal is to save space instead of time.

Assume that passwords are restricted to 64 bits and that a hash function maps words into

32 bits.

Fingerprinting and false positives.

The approximate set membership problem:

Given a set S= {s1, s2, . . . , sm} of m elements from a large universe U,

Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String HashingBloom Filters

Breaking Symmetry

Bit String Hashing

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Main Issues

Goal is to save space instead of time.

Assume that passwords are restricted to 64 bits and that a hash function maps words into

32 bits.

Fingerprinting and false positives.

The approximate set membership problem:

Given a set S= {s1, s2, . . . , sm} of m elements from a large universe U, we would like torepresent elements in such a way,

Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String HashingBloom Filters

Breaking Symmetry

Bit String Hashing

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Main Issues

Goal is to save space instead of time.

Assume that passwords are restricted to 64 bits and that a hash function maps words into

32 bits.

Fingerprinting and false positives.

The approximate set membership problem:

Given a set S= {s1, s2, . . . , sm} of m elements from a large universe U, we would like torepresent elements in such a way, as to efficiently answer queries of the form Is x S?

Subramani Balls and Bins

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Recap

The Poisson ApproximationApplications to Hashing

Chain Hashing

Bit String HashingBloom Filters

Breaking Symmetry

Bit String Hashing

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71/131

Main Issues

Goal is to save space instead of time.

Assume that passwords are restricted to 64 bits and that a hash function maps words into

32 bits.

Fingerprinting and false positives.

The approximate set membership problem:

Given a set S= {s1, s2, . . . , sm} of m elements from a large universe U, we would like torepresent elements in such a way, as to efficiently answer queries of the form Is x S?The disallowed passwords correspond to S. We also want to be space efficient and are

willing to tolerate some error.

Subramani Balls and Bins

Recap

The Poisson ApproximationApplications to Hashing

Chain Hashing

Bit String HashingBloom Filters

Breaking Symmetry

Bit String Hashing

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72/131

Main Issues

Goal is to save space instead of time.

Assume that passwords are restricted to 64 bits and that a hash function maps words into

32 bits.

Fingerprinting and false positives.

The approximate set membership problem:

Given a set S= {s1, s2, . . . , sm} of m elements from a large universe U, we would like torepresent elements in such a way, as to efficiently answer queries of the form Is x S?The disallowed passwords correspond to S. We also want to be space efficient and are

willing to tolerate some error.

Assume that we use b bits to create a fingerprint.

Subramani Balls and Bins

Recap

The Poisson ApproximationApplications to Hashing

Chain Hashing

Bit String HashingBloom Filters

Breaking Symmetry

Bit String Hashing

7/27/2019 Balls Bins Adv

73/131

Main Issues

Goal is to save space instead of time.

Assume that passwords are restricted to 64 bits and that a hash function maps words into

32 bits.

Fingerprinting and false positives.

The approximate set membership problem:

Given a set S= {s1, s2, . . . , sm} of m elements from a large universe U, we would like torepresent elements in such a way, as to efficiently answer queries of the form Is x S?The disallowed passwords correspond to S. We also want to be space efficient and are

willing to tolerate some error.

Assume that we use b bits to create a fingerprint. The probability that an acceptable

password has a fingerprint that is different from any specific password is

Subramani Balls and Bins

Recap

The Poisson ApproximationApplications to Hashing

Chain Hashing

Bit String HashingBloom Filters

Breaking Symmetry

Bit String Hashing

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Main Issues

Goal is to save space instead of time.

Assume that passwords are restricted to 64 bits and that a hash function maps words into

32 bits.

Fingerprinting and false positives.

The approximate set membership problem:

Given a set S= {s1, s2, . . . , sm} of m elements from a large universe U, we would like torepresent elements in such a way, as to efficiently answer queries of the form Is x S?The disallowed passwords correspond to S. We also want to be space efficient and are

willing to tolerate some error.

Assume that we use b bits to create a fingerprint. The probability that an acceptable

password has a fingerprint that is different from any specific password is (1 12b ).

Subramani Balls and Bins

Recap

The Poisson ApproximationApplications to Hashing

Chain Hashing

Bit String HashingBloom Filters

Breaking Symmetry

Bit String Hashing (contd.)

7/27/2019 Balls Bins Adv

75/131

Subramani Balls and Bins

Recap

The Poisson ApproximationApplications to Hashing

Chain Hashing

Bit String HashingBloom Filters

Breaking Symmetry

Bit String Hashing (contd.)

7/27/2019 Balls Bins Adv

76/131

Main Issues (contd.)

Subramani Balls and Bins

Recap

The Poisson ApproximationApplications to Hashing

Chain Hashing

Bit String HashingBloom Filters

Breaking Symmetry

Bit String Hashing (contd.)

7/27/2019 Balls Bins Adv

77/131

Main Issues (contd.)

Since S has size m, what is the probability that an acceptable password fingerprint will

match the fingerprints of one of the elements of S?

Subramani Balls and Bins

Recap

The Poisson ApproximationApplications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Bit String Hashing (contd.)

7/27/2019 Balls Bins Adv

78/131

Main Issues (contd.)

Since S has size m, what is the probability that an acceptable password fingerprint will

match the fingerprints of one of the elements of S? 1 (1 12b

)m

Subramani Balls and Bins

Recap

The Poisson ApproximationApplications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Bit String Hashing (contd.)

7/27/2019 Balls Bins Adv

79/131

Main Issues (contd.)

Since S has size m, what is the probability that an acceptable password fingerprint will

match the fingerprints of one of the elements of S? 1 (1 12b

)m 1em2b .

Subramani Balls and Bins

Recap

The Poisson ApproximationApplications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Bit String Hashing (contd.)

7/27/2019 Balls Bins Adv

80/131

Main Issues (contd.)

Since S has size m, what is the probability that an acceptable password fingerprint will

match the fingerprints of one of the elements of S? 1 (1 12b

)m 1em2b .Since we want the probability of a false positive to be less than c, we need,

Subramani Balls and Bins

Recap

The Poisson ApproximationApplications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Bit String Hashing (contd.)

7/27/2019 Balls Bins Adv

81/131

Main Issues (contd.)

Since S has size m, what is the probability that an acceptable password fingerprint will

match the fingerprints of one of the elements of S? 1 (1 12b

)m 1em2b .Since we want the probability of a false positive to be less than c, we need,

1

em2b

c

Subramani Balls and Bins

Recap

The Poisson ApproximationApplications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Bit String Hashing (contd.)

7/27/2019 Balls Bins Adv

82/131

Main Issues (contd.)

Since S has size m, what is the probability that an acceptable password fingerprint will

match the fingerprints of one of the elements of S? 1 (1 12b

)m 1em2b .Since we want the probability of a false positive to be less than c, we need,

1

em2b

c

em2b 1 c

Subramani Balls and Bins

Recap

The Poisson ApproximationApplications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Bit String Hashing (contd.)

7/27/2019 Balls Bins Adv

83/131

Main Issues (contd.)

Since S has size m, what is the probability that an acceptable password fingerprint will

match the fingerprints of one of the elements of S? 1 (1 12b

)m 1em2b .Since we want the probability of a false positive to be less than c, we need,

1

em2b

c

em2b 1 c b log2

m

ln 11c

Subramani Balls and Bins

Recap

The Poisson ApproximationApplications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Bit String Hashing (contd.)

7/27/2019 Balls Bins Adv

84/131

Main Issues (contd.)

Since S has size m, what is the probability that an acceptable password fingerprint will

match the fingerprints of one of the elements of S? 1 (1 12b

)m 1em2b .Since we want the probability of a false positive to be less than c, we need,

1

em2b

c

em2b 1 c b log2

m

ln 11c

If we choose b= 2 log2 m, the probability of a false positive is:

Subramani Balls and Bins

Recap

The Poisson ApproximationApplications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Bit String Hashing (contd.)

7/27/2019 Balls Bins Adv

85/131

Main Issues (contd.)

Since S has size m, what is the probability that an acceptable password fingerprint will

match the fingerprints of one of the elements of S? 1 (1 12b

)m 1em2b .Since we want the probability of a false positive to be less than c, we need,

1

em2b

c

em2b 1 c b log2

m

ln 11c

If we choose b= 2 log2 m, the probability of a false positive is: 1 (1 1m2 )m

Subramani Balls and Bins

Recap

The Poisson ApproximationApplications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Bit String Hashing (contd.)

7/27/2019 Balls Bins Adv

86/131

Main Issues (contd.)

Since S has size m, what is the probability that an acceptable password fingerprint will

match the fingerprints of one of the elements of S? 1 (1 12b

)m 1em2b .Since we want the probability of a false positive to be less than c, we need,

1

em2b

c

em2b 1 c b log2

m

ln 11c

If we choose b= 2 log2 m, the probability of a false positive is: 1 (1 1m2 )m < 1m.

If our dictionary has 216 words, using 32 bits when hashing, leads to an error probability of atmost 1

65,536.

Subramani Balls and Bins

Recap

The Poisson ApproximationApplications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Outline

7/27/2019 Balls Bins Adv

87/131

1 Recap

2 The Poisson Approximation

Some theorems and lemmas

3 Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Subramani Balls and Bins

Recap

The Poisson ApproximationApplications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Bloom Filters

7/27/2019 Balls Bins Adv

88/131

Subramani Balls and Bins

7/27/2019 Balls Bins Adv

89/131

Recap

The Poisson ApproximationApplications to Hashing

Chain Hashing

Bit String Hashing

Bloom FiltersBreaking Symmetry

Bloom Filters

7/27/2019 Balls Bins Adv

90/131

Main Issues

Chain hashing optimizes time,

Subramani Balls and Bins

Recap

The Poisson ApproximationApplications to Hashing

Chain Hashing

Bit String Hashing

Bloom FiltersBreaking Symmetry

Bloom Filters

7/27/2019 Balls Bins Adv

91/131

Main Issues

Chain hashing optimizes time, while bit string hashing optimizes space.

Subramani Balls and Bins

Recap

The Poisson ApproximationApplications to Hashing

Chain Hashing

Bit String Hashing

Bloom FiltersBreaking Symmetry

Bloom Filters

7/27/2019 Balls Bins Adv

92/131

Main Issues

Chain hashing optimizes time, while bit string hashing optimizes space. Can we get a

better trade-off?

Subramani Balls and Bins

Recap

The Poisson ApproximationApplications to Hashing

Chain Hashing

Bit String Hashing

Bloom FiltersBreaking Symmetry

Bloom Filters

7/27/2019 Balls Bins Adv

93/131

Main Issues

Chain hashing optimizes time, while bit string hashing optimizes space. Can we get a

better trade-off? Bloom filters!

Subramani Balls and Bins

Recap

The Poisson ApproximationApplications to Hashing

Chain Hashing

Bit String Hashing

Bloom FiltersBreaking Symmetry

Bloom Filters

7/27/2019 Balls Bins Adv

94/131

Main Issues

Chain hashing optimizes time, while bit string hashing optimizes space. Can we get a

better trade-off? Bloom filters!

A Bloom filter consists of an array of nbits, A[0] to A[n1],

Subramani Balls and Bins

Recap

The Poisson ApproximationApplications to Hashing

Chain Hashing

Bit String Hashing

Bloom FiltersBreaking Symmetry

Bloom Filters

7/27/2019 Balls Bins Adv

95/131

Main Issues

Chain hashing optimizes time, while bit string hashing optimizes space. Can we get a

better trade-off? Bloom filters!

A Bloom filter consists of an array of nbits, A[0] to A[n1], initially set to 0.

Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom FiltersBreaking Symmetry

Bloom Filters

7/27/2019 Balls Bins Adv

96/131

Main Issues

Chain hashing optimizes time, while bit string hashing optimizes space. Can we get a

better trade-off? Bloom filters!

A Bloom filter consists of an array of nbits, A[0] to A[n1], initially set to 0.k hash function h1, h2, . . . , hk with range {0, . . . , n1} are used.

Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom FiltersBreaking Symmetry

Bloom Filters

7/27/2019 Balls Bins Adv

97/131

Main Issues

Chain hashing optimizes time, while bit string hashing optimizes space. Can we get a

better trade-off? Bloom filters!

A Bloom filter consists of an array of nbits, A[0] to A[n1], initially set to 0.k hash function h1, h2, . . . , hk with range {0, . . . , n1} are used.We wish to represent a set S= {s1, s2, . . . , sm} of m elements.

Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom FiltersBreaking Symmetry

Bloom Filters

7/27/2019 Balls Bins Adv

98/131

Main Issues

Chain hashing optimizes time, while bit string hashing optimizes space. Can we get a

better trade-off? Bloom filters!

A Bloom filter consists of an array of nbits, A[0] to A[n1], initially set to 0.k hash function h1, h2, . . . , hk with range {0, . . . , n1} are used.We wish to represent a set S= {s1, s2, . . . , sm} of m elements.Set A[hi(s)] to 1, for each 1 i k and each s S.

Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom FiltersBreaking Symmetry

Bloom Filters

7/27/2019 Balls Bins Adv

99/131

Main Issues

Chain hashing optimizes time, while bit string hashing optimizes space. Can we get a

better trade-off? Bloom filters!

A Bloom filter consists of an array of nbits, A[0] to A[n1], initially set to 0.k hash function h1, h2, . . . , hk with range {0, . . . , n1} are used.We wish to represent a set S= {s1, s2, . . . , sm} of m elements.Set A[hi(s)] to 1, for each 1 i k and each s S.How to check if x S?

Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom FiltersBreaking Symmetry

Bloom Filters

7/27/2019 Balls Bins Adv

100/131

Main Issues

Chain hashing optimizes time, while bit string hashing optimizes space. Can we get a

better trade-off? Bloom filters!

A Bloom filter consists of an array of nbits, A[0] to A[n1], initially set to 0.k hash function h1, h2, . . . , hk with range {0, . . . , n1} are used.We wish to represent a set S= {s1, s2, . . . , sm} of m elements.Set A[hi(s)] to 1, for each 1 i k and each s S.How to check if x S? Check all locations A[hi(x)], 1 i k.

Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom FiltersBreaking Symmetry

Bloom Filters

7/27/2019 Balls Bins Adv

101/131

Main Issues

Chain hashing optimizes time, while bit string hashing optimizes space. Can we get a

better trade-off? Bloom filters!

A Bloom filter consists of an array of nbits, A[0] to A[n1], initially set to 0.k hash function h1, h2, . . . , hk with range {0, . . . , n1} are used.We wish to represent a set S= {s1, s2, . . . , sm} of m elements.Set A[hi(s)] to 1, for each 1 i k and each s S.How to check if x S? Check all locations A[hi(x)], 1 i k.How could we go wrong?

Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom FiltersBreaking Symmetry

Bloom Filters

7/27/2019 Balls Bins Adv

102/131

Main Issues

Chain hashing optimizes time, while bit string hashing optimizes space. Can we get a

better trade-off? Bloom filters!

A Bloom filter consists of an array of nbits, A[0] to A[n1], initially set to 0.k hash function h1, h2, . . . , hk with range {0, . . . , n1} are used.We wish to represent a set S= {s1, s2, . . . , sm} of m elements.Set A[hi(s)] to 1, for each 1 i k and each s S.How to check if x S? Check all locations A[hi(x)], 1 i k.How could we go wrong? False positives.

Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom FiltersBreaking Symmetry

Analyzing error probability

7/27/2019 Balls Bins Adv

103/131

Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom FiltersBreaking Symmetry

Analyzing error probability

7/27/2019 Balls Bins Adv

104/131

Analysis

Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom FiltersBreaking Symmetry

Analyzing error probability

7/27/2019 Balls Bins Adv

105/131

Analysis

Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom FiltersBreaking Symmetry

Analyzing error probability

7/27/2019 Balls Bins Adv

106/131

Analysis

What is the probability that a specific bit is 0, after the preprocessing?

Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom FiltersBreaking Symmetry

Analyzing error probability

7/27/2019 Balls Bins Adv

107/131

Analysis

What is the probability that a specific bit is 0, after the preprocessing? (1 1n

)km

Subramani Balls and Bins

7/27/2019 Balls Bins Adv

108/131

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom FiltersBreaking Symmetry

Analyzing error probability

7/27/2019 Balls Bins Adv

109/131

Analysis

What is the probability that a specific bit is 0, after the preprocessing? (1 1n

)km = ekm

n .

Assume that a fraction p= ekm

n of the entries are still 0, after S has been hashed into the

Bloom filter.

Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom FiltersBreaking Symmetry

Analyzing error probability

7/27/2019 Balls Bins Adv

110/131

Analysis

What is the probability that a specific bit is 0, after the preprocessing? (1 1n

)km = ekm

n .

Assume that a fraction p= ekm

n of the entries are still 0, after S has been hashed into the

Bloom filter.

The probability of a false positive is then precisely the probability that all the hash functions

map the input string x S, to 1,

Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom FiltersBreaking Symmetry

Analyzing error probability

7/27/2019 Balls Bins Adv

111/131

Analysis

What is the probability that a specific bit is 0, after the preprocessing? (1 1n

)km = ekm

n .

Assume that a fraction p= ekm

n of the entries are still 0, after S has been hashed into the

Bloom filter.

The probability of a false positive is then precisely the probability that all the hash functions

map the input string x S, to 1, which is,

Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom FiltersBreaking Symmetry

Analyzing error probability

7/27/2019 Balls Bins Adv

112/131

Analysis

What is the probability that a specific bit is 0, after the preprocessing? (1 1n

)km = ekm

n .

Assume that a fraction p= ekm

n of the entries are still 0, after S has been hashed into the

Bloom filter.

The probability of a false positive is then precisely the probability that all the hash functions

map the input string x S, to 1, which is,

(1 (1 1n

)km)k

Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom FiltersBreaking Symmetry

Analyzing error probability

7/27/2019 Balls Bins Adv

113/131

Analysis

What is the probability that a specific bit is 0, after the preprocessing? (1 1n

)km = ekm

n .

Assume that a fraction p= ekm

n of the entries are still 0, after S has been hashed into the

Bloom filter.

The probability of a false positive is then precisely the probability that all the hash functions

map the input string x S, to 1, which is,

(1 (1 1n

)km)k = (1e kmn )k

Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Analyzing error probability

7/27/2019 Balls Bins Adv

114/131

Analysis

What is the probability that a specific bit is 0, after the preprocessing? (1 1n

)km = ekm

n .

Assume that a fraction p= ekm

n of the entries are still 0, after S has been hashed into the

Bloom filter.

The probability of a false positive is then precisely the probability that all the hash functions

map the input string x S, to 1, which is,(1 (1 1

n)km)k = (1e kmn )k

= f()

Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Analyzing error probability

7/27/2019 Balls Bins Adv

115/131

Analysis

What is the probability that a specific bit is 0, after the preprocessing? (1 1n

)km = ekm

n .

Assume that a fraction p= ekm

n of the entries are still 0, after S has been hashed into the

Bloom filter.

The probability of a false positive is then precisely the probability that all the hash functions

map the input string x S, to 1, which is,(1 (1 1

n)km)k = (1e kmn )k

= f()

Optimizing for k, we get k = ln2 and f (0.6185)mn .

Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Outline

7/27/2019 Balls Bins Adv

116/131

1 Recap

2 The Poisson Approximation

Some theorems and lemmas

3 Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Breaking Symmetry

7/27/2019 Balls Bins Adv

117/131

Main Issues

Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Breaking Symmetry

7/27/2019 Balls Bins Adv

118/131

Main IssuesResource contention.

Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Breaking Symmetry

7/27/2019 Balls Bins Adv

119/131

Main IssuesResource contention.

Sequential ordering.

Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Breaking Symmetry

7/27/2019 Balls Bins Adv

120/131

Main IssuesResource contention.

Sequential ordering.

The Hashing approach.

Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Breaking Symmetry

7/27/2019 Balls Bins Adv

121/131

Main IssuesResource contention.

Sequential ordering.

The Hashing approach. Hash each user identifier into b bits.

Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Breaking Symmetry

Main Issues

7/27/2019 Balls Bins Adv

122/131

Resource contention.

Sequential ordering.

The Hashing approach. Hash each user identifier into b bits.

How likely that two users will have the same hash value?

Subramani Balls and Bins

7/27/2019 Balls Bins Adv

123/131

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Breaking Symmetry

Main Issues

7/27/2019 Balls Bins Adv

124/131

Resource contention.

Sequential ordering.

The Hashing approach. Hash each user identifier into b bits.

How likely that two users will have the same hash value? Fix one user. What is the

probability that some other user obtains the same hash value?

Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Breaking Symmetry

Main Issues

7/27/2019 Balls Bins Adv

125/131

Resource contention.

Sequential ordering.

The Hashing approach. Hash each user identifier into b bits.

How likely that two users will have the same hash value? Fix one user. What is the

probability that some other user obtains the same hash value?

1 (1 12b

)n1

Subramani Balls and Bins

7/27/2019 Balls Bins Adv

126/131

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Breaking Symmetry

Main Issues

7/27/2019 Balls Bins Adv

127/131

Resource contention.

Sequential ordering.

The Hashing approach. Hash each user identifier into b bits.

How likely that two users will have the same hash value? Fix one user. What is the

probability that some other user obtains the same hash value?

1 (1 12b

)n1 n12b

By the union bound, the probability that any user has the same hash values as the fixed

user is

Subramani Balls and Bins

7/27/2019 Balls Bins Adv

128/131

7/27/2019 Balls Bins Adv

129/131

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Breaking Symmetry

Main Issues

7/27/2019 Balls Bins Adv

130/131

Resource contention.

Sequential ordering.

The Hashing approach. Hash each user identifier into b bits.

How likely that two users will have the same hash value? Fix one user. What is the

probability that some other user obtains the same hash value?

1 (1 12b

)n1 n12b

By the union bound, the probability that any user has the same hash values as the fixed

user isn(n1)

2b. In order to guarantee success with probability (1 1

n), choose

b= 3 log n2.

Subramani Balls and Bins

Recap

The Poisson Approximation

Applications to Hashing

Chain Hashing

Bit String Hashing

Bloom Filters

Breaking Symmetry

Breaking Symmetry

Main Issues

7/27/2019 Balls Bins Adv

131/131

Resource contention.

Sequential ordering.

The Hashing approach. Hash each user identifier into b bits.

How likely that two users will have the same hash value? Fix one user. What is the

probability that some other user obtains the same hash value?

1 (1 12b

)n1 n12b

By the union bound, the probability that any user has the same hash values as the fixed

user isn(n1)

2b. In order to guarantee success with probability (1 1

n), choose

b= 3 log n2.Can also be used for the leader election problem.

Subramani Balls and Bins