Road coloring problem Trahtman A.N. BarIlan university

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Road coloring problemTrahtman A.N. Bar-Ilan
university

• Complete deterministic directed finite automaton
  with transition graph G

q ? q?
Complete for any vertex outgoing edges of all
colors from given alphabet
Deterministic a a
?
For edge q ? p suppose p q ?. For a set of
states Q and mapping ? consider a map Q? and Qs
for s?1?2 ?i . Gs presents a map of G.
If for some word s Gs1 then s is
synchronizing word of automaton with transition
graph G and the automaton is called synchronizing.
Synchronizing coloring of directed graph G turns
the graph into synchronizing automaton
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Adler, Goodwyn, Weiss, 1970 , 1977, Road coloring
problem

• AGW graph
• 1.directed finite strongly connected graph
• 2.constant outdegree of all its vertices
• 3. the greatest common divisor (gcd) of lengths
  of all its cycles is one.
• Has AGW graph synchronizing coloring?

the problem depends only on sink (minimal)
strongly connected component constant outdegree
- for to be complete and deterministic the
condition on gcd is necessary
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Known
A pair of distinct states p, q is called
synchronizing if ps qs for some s S. In
opposite case, if for any s ps ? qs we call the
pair deadlock. Synchronizing pair p, q is
called stable if for any s the pair (ps, qs) is
also synchronizing. (Culik, Karhumaki, Kari)
T 1. (Kari, 2001, Culik, Karhumaki, Kari, 2000)
Let us consider a coloring of AGW graph G. Let ?
be binary relation on the set of states. For any
stable pair of states p, q suppose p ? q. Then ?
is a congruence relation, G /? presents an AGW
graph and synchronizing coloring of G /? implies
synchronizing recoloring of G.
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The subset of states of maximal size such that
every pair of its states is deadlock will be
called an F-clique.
Whiller and Beal, Perrin simplification

• L.2 Let F be F-clique via some coloring of AGW
  graph G. For any word s the set Fs is also an
  F-clique and any state p belongs to some
  F-clique.

L.3 Let A and B (Agt1) be F-cliques via some
coloring of the AGW graph G and A -A n B 1.
Then the coloring has a stable couple.
1
A B
We call the set of all outgoing edges of a vertex
a bunch if all these edges are incoming edges of
only one vertex. L.4 Let some vertex of AGW
graph G have two incoming bunches. Then any
coloring of G has a stable couple.
stable couple (p,q)
p
q
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Spanning subgraph

• Let us call a subgraph S of the AGW graph G a
  spanning subgraph if to S belong all vertices of
  G and exactly one outgoing edge of any vertex.
• A maximal tree of the spanning subgraph S with
  root on cycle of S having no common edges with
  cycles from S is called a tree of S.
• The length of path from vertex p through edges of
  the spanning subgraph S to the root of its tree
  is called a level of p in S.

Any spanning subgraph S consists of disjoint
cycles and trees with roots on cycles any tree
and cycle of S is defined identically, the level
of the cycle vertex is zero, the vertices of
trees except root have positive level, the vertex
of maximal positive level has no ingoing edge
from S.
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F-cliques and maximal level
L. 5 Let N be a set of vertices of level n from
some tree of the spanning set S of AGW graph G.
Then via a coloring such that all edges of S have
the same color, for any F-clique F holds F n N
1.
L. 6 Let spanning subgraph of an AGW graph G have
no trees. Then G has a spanning subgraph with
one vertex of maximal positive level.
L. 7 Let any vertex of an AGW graph G have no two
incoming bunches. Then G has a spanning subgraph
such that all its vertices of maximal positive
level belong to one non-trivial tree.
tree b
p
c
root
a
p of maximal level (b,c) stable?
cycle
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Road coloring problem for AGW graph
T. 3 Any AGW graph has a coloring with stable
pairs. 1.The ancestors of a root on tree and
cycle of a tree with all vertices of
maximal level 2. The beginnings of two
bunches having common end
form stable pair.
T. 4 Any AGW graph has a synchronizing
coloring.

• T.5 Let every vertex of strongly connected
  directed finite graph G have the same number of
  outgoing edges. Then G has synchronizing coloring
  if and only if the greatest common divisor of
  lengths of all its cycles is one.

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Algorithms for road coloring

• L.7a Any spanning subgraph of AGW graph G can
  be transformed in a spanning subgraph having only
  one tree with vertices of maximal level by
  changing edges to root and to vertex of maximal
  level.

p tree b
p of maximal level (b,c) stable by coloring?
c
root
. cycle
L.8 Let AGW graph G have two cycles with one
common vertex. Then G has a spanning subgraph
with vertices of maximal level in one tree of the
subgraph.