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Algorithm
the theory behind it

Algorithm - the theory behind it


NAME

Kleene's Algorithm - the theory behind it

brief introduction


DESCRIPTION

Semi-Rings

A Semi-Ring (S, +, ., 0, 1) is characterized by the following properties:

  1. )
  2. a) (S, +, 0) is a Semi-Group with neutral element 0

    b) (S, ., 1) is a Semi-Group with neutral element 1

    c) 0 . a = a . 0 = 0 for all a in S

  3. )
  4. "+" is commutative and idempotent, i.e., a + a = a

  5. )
  6. Distributivity holds, i.e.,

    a) a . ( b + c ) = a . b + a . c for all a,b,c in S

    b) ( a + b ) . c = a . c + b . c for all a,b,c in S

  7. )
  8. SUM_{i=0}^{+infinity} ( a[i] )

    exists, is well-defined and unique

    for all a[i] in S

    and associativity, commutativity and idempotency hold

  9. )
  10. Distributivity for infinite series also holds, i.e.,
    
      ( SUM_{i=0}^{+infty} a[i] ) . ( SUM_{j=0}^{+infty} b[j] )
    
      = SUM_{i=0}^{+infty} ( SUM_{j=0}^{+infty} ( a[i] . b[j] ) )

EXAMPLES:

Operator '*'

(reflexive and transitive closure)

Define an operator called ``*'' as follows:


    a in S   ==>   a*  :=  SUM_{i=0}^{+infty} a^i

where


    a^0  =  1,   a^(i+1)  =  a . a^i

Then, also


    a*  =  1 + a . a*,   0*  =  1*  =  1

hold.

Kleene's Algorithm

In its general form, Kleene's algorithm goes as follows:


    for i := 1 to n do

        for j := 1 to n do

        begin

            C^0[i,j] := m(v[i],v[j]);

            if (i = j) then C^0[i,j] := C^0[i,j] + 1

        end

    for k := 1 to n do

        for i := 1 to n do

            for j := 1 to n do

                C^k[i,j] := C^k-1[i,j] +

                            C^k-1[i,k] . ( C^k-1[k,k] )* . C^k-1[k,j]

    for i := 1 to n do

        for j := 1 to n do

            c(v[i],v[j]) := C^n[i,j]

Kleene's Algorithm and Semi-Rings

Kleene's algorithm can be applied to any Semi-Ring having the properties listed previously (above). (!)

EXAMPLES:

  • S1 = ({0,1}, |, &, 0, 1)

    G(V,E) be a graph with set of vortices V and set of edges E:

    m(v[i],v[j]) := ( (v[i],v[j]) in E ) ? 1 : 0

    Kleene's algorithm then calculates

    c^{n}_{i,j} = ( path from v[i] to v[j] exists ) ? 1 : 0

    using

    C^k[i,j] = C^k-1[i,j] | C^k-1[i,k] & C^k-1[k,j]

    (remember 0* = 1* = 1 )

  • S2 = (pos. reals with 0 and +infty, min, +, +infty, 0)

    G(V,E) be a graph with set of vortices V and set of edges E, with costs m(v[i],v[j]) associated with each edge (v[i],v[j]) in E:

    m(v[i],v[j]) := costs of (v[i],v[j])

    for all (v[i],v[j]) in E

    Set m(v[i],v[j]) := +infinity if an edge (v[i],v[j]) is not in E.

    ==> a* = 0 for all a in S2

    ==> C^k[i,j] = min( C^k-1[i,j] ,

    C^k-1[i,k] + C^k-1[k,j] )

    Kleene's algorithm then calculates the costs of the ``shortest'' path from any v[i] to any other v[j]:

    C^n[i,j] = costs of "shortest" path from v[i] to v[j]

  • S3 = (Pot(Sigma*), union, concat, {}, {''})

    M in DFA(Sigma) be a Deterministic Finite Automaton with a set of states Q, a subset F of Q of accepting states and a transition function delta : Q x Sigma --> Q.

    Define

    m(v[i],v[j]) :=

    { a in Sigma | delta( q[i] , a ) = q[j] }

    and

    C^0[i,j] := m(v[i],v[j]);

    if (i = j) then C^0[i,j] := C^0[i,j] union {''}

    ({''} is the set containing the empty string, whereas {} is the empty set!)

    Then Kleene's algorithm calculates the language accepted by Deterministic Finite Automaton M using

    C^k[i,j] = C^k-1[i,j] union

    C^k-1[i,k] concat ( C^k-1[k,k] )* concat C^k-1[k,j]

    and

    L(M) = UNION_{ q[j] in F } C^n[1,j]

    (state q[1] is assumed to be the ``start'' state)

    finally being the language recognized by Deterministic Finite Automaton M.

Note that instead of using Kleene's algorithm, you can also use the ``*'' operator on the associated matrix:

Define A[i,j] := m(v[i],v[j])

==> A*[i,j] = c(v[i],v[j])

Proof:

A* = SUM_{i=0}^{+infty} A^i

where A^0 = E_{n}

(matrix with one's in its main diagonal and zero's elsewhere)

and A^(i+1) = A . A^i

Induction over k yields:

A^k[i,j] = c_{k}(v[i],v[j])

k = 0:
c_{0}(v[i],v[j]) = d_{i,j}

with d_{i,j} := (i = j) ? 1 : 0

and A^0 = E_{n} = [d_{i,j}]

k-1 -> k:
c_{k}(v[i],v[j])

= SUM_{l=1}^{n} m(v[i],v[l]) . c_{k-1}(v[l],v[j])

= SUM_{l=1}^{n} ( a[i,l] . a[l,j] )

= [a^{k}_{i,j}] = A^1 . A^(k-1) = A^k

qed

In other words, the complexity of calculating the closure and doing matrix multiplications is of the same order O( n^3 ) in Semi-Rings!


SEE ALSO

Math::MatrixBool(3), Math::MatrixReal(3), DFA::Kleene(3).

Dijkstra's algorithm for shortest paths.


AUTHOR

This document is based on lecture notes and has been put into POD format by Steffen Beyer <sb@sdm.de>.


COPYRIGHT

Copyright (c) 1997 by Steffen Beyer. All rights reserved.

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