Algebraic Algorithms

Definition

Combinatorial methods: Tries to construct the object explicitly piece-by-piece.

Algebraic methods: Implicitly sieves for the object by evaluating a sum.

The 4cycle Detection Problem

4cycle detection

1. Init a boolean nxn table T(:,:)=false.2. For each vertex v3. For each pair of neighbors u,w of v4. If T(u,w) then return true and

stop5. T(u,w)=true6. End7. End8. Return false

The Triangle Detection Problem

Square Matrix MultiplicationWhat is the computational

complexity in terms of number of scalar operations (*,+) of computing the product of two nxn-matrices A and B?

From definition Cij=Sk AikBkj we get O(n3) time.

There is a (highly theoretical) O(n2.373) time algorithm!

Strassen’s algorithm

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A1,1 A1,2

A2,1 A2,2

⎡ ⎣ ⎢

⎤ ⎦ ⎥B1,1 B1,2

B2,1 B2,2

⎡ ⎣ ⎢

⎤ ⎦ ⎥=C1,1 = A1,1B1,1 + A1,2B2,1 C1,2 = A1,1B1,2 + A1,2B2,2

C2,1 = A2,1B1,1 + A2,2B2,1 C2,2 = A2,1B1,2 + A2,2B2,2

⎡ ⎣ ⎢

⎤ ⎦ ⎥

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M1 = (A1,1 + A2,2)(B1,1 + B2,2)

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M2 = (A2,1 + A2,2)B1,1

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M3 = A1,1(B1,2 − B2,2)

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M4 = A2,2(B2,1 − B1,1)

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M5 = (A1,1 + A1,2)B2,2

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M6 = (A2,1 − A1,1)(B1,1 + B1,2)

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M7 = (A1,2 − A2,2)(B2,1 + B2,2)

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C1,1 = M1 +M4 −M5 +M7

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C1,2 = M3 +M5

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C2,1 = M2 +M4

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C2,2 = M1 −M2 +M3 +M6

Strassen’s algorithm

T(n)=cn2+7T(n/2)=cn2+7c(n/2)2+72T(n/4)=

cn2+7c(n/2)2+72c(n/4)2+73T(n/8)=…=

cn2(1+7/4+(7/4)2+…+(7/4)log2(n))+7log2(n)=

O(nlog2(7))=O(n2.81).

Verifying Matrix ProductsGiven three nxn matrices A,B, and C

you want to verify that AB=C.Can you do this faster than

recomputing the matrix product?

Verifying Matrix Products Idea: Project the difference matrix on

a randomly chosen 0/1-vector r. If (AB-C)r=A(Br)-Cr=0, we conclude

that AB=C, otherwise we deem them different.

No false negatives.False positives with probability

<=1/2.

Note that this is an O(n2) time algorithm!

Verifying Matrix Products

(AB-C)= r=

e1e2 e3 e4r1r2

r3

r4

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Pr( eiri = 0) ≤12i=1

4

∑

Polynomial Identity TestingGiven ”black box” multivariate

polynomial P, is P identically zero or not?

Think of polynomial P as given by arithmetic circuit:

x

+

x xx

+ + + + + +

x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14

+ +

The Schwartz Zippel LemmaSchwartz-Zippel Lemma: Let P(x1,x2,

…,xn) be a multivariate non-zero polynomial of total degree d over a finite field F. For uniformly and independently sampled values r1,r2,…,rn in F:

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Pr[P(r1,r2,...,rn ) = 0] ≤dF

Bipartite Matching

Pick as many end-point disjoint edges as possible.

Bipartite Matching: Augmentation paths

Classic matching algorithm:Use dfs to find augmentation paths.

Bipartite Matching: Biadjacency matrix

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1 0 1 0 00 1 1 0 01 0 0 0 10 1 0 1 00 0 0 1 1

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

1

2

3

4

5

A

B

C

D

E

1

2

3

4

5

A B C D E

Determinants

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1 0 1 0 00 1 1 0 01 0 0 0 10 1 0 1 00 0 0 1 1

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

1

2

3

4

5

A B C D E

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det(A) = sgn(σ ) Ai,σ (i)i=1

n

∏σ :[n ]→[n ]one−to−one

∑

In fields of characteristic two:

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det(A) = Ai,σ ( i)i=1

n

∏σ :[n ]→[n ]one−to−one

∑

Fields of characteristic two

+ 0 1 A B0 0 1 A B1 1 0 B AA A B 0 1B B A 1 0

* 0 1 A B0 0 0 0 01 0 1 A BA 0 A B 1B 0 B 1 A

F4

Determinants with indeterminates

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x1A 0 x1C 0 00 x2B x2C 0 0x3A 0 0 0 x3E

0 x4B 0 x4D 00 0 0 x5D x5E

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

1

2

3

4

5

A B C D E

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det(A) = Ai,σ ( i)i=1

n

∏σ :[n ]→[n ]one−to−one

∑

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det(A) = x i,σ (i)i=1

n

∏Matchingsσ :[1..5]→[A ..E ]

∑

Algorithm

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r1A 0 r1C 0 00 r2B r2C 0 0r3A 0 0 0 r3E0 r4B 0 r4D 00 0 0 r5D r5E

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥

1

2

3

4

5

A B C D E1. Replace indeterminates by random values xij=rij.2. Compute the determinant.3. If it is non-zero return Yes otherwise No.

General Matchings

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0 x12 0 0 x15 0x12 0 x23 x24 0 x26

0 x23 0 x34 x35 x36

0 x24 x34 0 0 x46

x15 0 x35 0 0 00 x26 x36 x46 0 0

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

1

2

3

4

5

1 2 3 4 5 6

6

12

3 4

5 6