Freivalds' algorithm - Alchetron, The Free Social Encyclopedia

Freivalds' algorithm (named after Rūsiņš Mārtiņš Freivalds) is a probabilistic randomized algorithm used to verify matrix multiplication. Given three n × n matrices A , B , and C , a general problem is to verify whether A × B = C . A naïve algorithm would compute the product A × B explicitly and compare term by term whether this product equals C . However, the best known matrix multiplication algorithm runs in O ( n 2.3729 ) time. Freivalds' algorithm utilizes randomization in order to reduce this time bound to O ( n 2 ) with high probability. In O ( k n 2 ) time the algorithm can verify a matrix product with probability of failure less than 2 − k .

Input

Three n × n matrices A , B , and C .

Output

Yes, if A × B = C ; No, otherwise.

Procedure

Generate an n × 1 random 0/1 vector r → .
Compute P → = A × ( B r → ) − C r → .
Output "Yes" if P → = ( 0 , 0 , … , 0 ) T ; "No," otherwise.

Error

If A × B = C , then the algorithm always returns "Yes". If A × B ≠ C , then the probability that the algorithm returns "Yes" is less than or equal to one half. This is called one-sided error.

By iterating the algorithm k times and returning "Yes" only if all iterations yield "Yes", a runtime of O ( k n 2 ) and error probability of ≤ 1 / 2 k is achieved.

Example

Suppose one wished to determine whether:

A B = [ 2 3 3 4 ] [ 1 0 1 2 ] = ? [ 6 5 8 7 ] = C .

A random two-element vector with entries equal to 0 or 1 is selected — say r → = [ 1 1 ] — and used to compute:

A × ( B r → ) − C r → = [ 2 3 3 4 ] ( [ 1 0 1 2 ] [ 1 1 ] ) − [ 6 5 8 7 ] [ 1 1 ] = [ 2 3 3 4 ] [ 1 3 ] − [ 11 15 ] = [ 11 15 ] − [ 11 15 ] = [ 0 0 ] .

This yields the zero vector, suggesting the possibility that AB = C. However, if in a second trial the vector r → = [ 1 0 ] is selected, the result becomes:

A × ( B r → ) − C r → = [ 2 3 3 4 ] ( [ 1 0 1 2 ] [ 1 0 ] ) − [ 6 5 8 7 ] [ 1 0 ] = [ − 1 − 1 ] .

The result is nonzero, proving that in fact AB ≠ C.

There are four two-element 0/1 vectors, and half of them give the zero vector in this case ( r → = [ 0 0 ] and r → = [ 1 1 ] ), so the chance of randomly selecting these in two trials (and falsely concluding that AB=C) is 1/2² or 1/4. In the general case, the proportion of r yielding the zero vector may be less than 1/2, and a larger number of trials (such as 20) would be used, rendering the probability of error very small.

Error analysis

Let p equal the probability of error. We claim that if A × B = C, then p = 0, and if A × B ≠ C, then p ≤ 1/2.

Case A × B = C

P → = A × ( B r → ) − C r → = ( A × B ) r → − C r → = ( A × B − C ) r → = 0 →

This is regardless of the value of r → , since it uses only that A × B − C = 0 . Hence the probability for error in this case is:

Pr [ P → ≠ 0 ] = 0

Case A × B ≠ C

Let

P → = D × r → = ( p 1 , p 2 , … , p n ) T

Where

D = A × B − C = ( d i j ) .

Since A × B ≠ C , we have that some element of D is nonzero. Suppose that the element d i j ≠ 0 . By the definition of matrix multiplication, we have:

p i = ∑ k = 1 n d i k r k = d i 1 r 1 + ⋯ + d i j r j + ⋯ + d i n r n = d i j r j + y .

For some constant y . Using Bayes' Theorem, we can partition over y :

We use that:

Pr [ p i = 0 | y = 0 ] = Pr [ r j = 0 ] = 1 2 Pr [ p i = 0 | y ≠ 0 ] = Pr [ r j = 1 ∧ d i j = − y ] ≤ Pr [ r j = 1 ] = 1 2

Plugging these in the equation (1), we get:

Pr [ p i = 0 ] ≤ 1 2 ⋅ Pr [ y = 0 ] + 1 2 ⋅ Pr [ y ≠ 0 ] = 1 2 ⋅ Pr [ y = 0 ] + 1 2 ⋅ ( 1 − Pr [ y = 0 ] ) = 1 2

Therefore,

Pr [ P → = 0 ] = Pr [ p 1 = 0 ∧ ⋯ ∧ p i = 0 ∧ ⋯ ∧ p n = 0 ] ≤ Pr [ p i = 0 ] ≤ 1 2 .

This completes the proof.

Ramifications

Simple algorithmic analysis shows that the running time of this algorithm is O(n²), beating the classical deterministic algorithm's bound of O(n³). The error analysis also shows that if we run our algorithm k times, we can achieve an error bound of less than 1 2 k , an exponentially small quantity. The algorithm is also fast in practice due to wide availability of fast implementations for matrix-vector products. Therefore, utilization of randomized algorithms can speed up a very slow deterministic algorithm. In fact, the best known deterministic matrix multiplication verification algorithm known at the current time is a variant of the Coppersmith–Winograd algorithm with an asymptotic running time of O(n^2.3729).

Freivalds' algorithm frequently arises in introductions to probabilistic algorithms due to its simplicity and how it illustrates the superiority of probabilistic algorithms in practice for some problems.

References

Freivalds' algorithm Wikipedia

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