Distributed Vertex Coloring

1
Distributed Vertex Coloring

• Part II

2
A Randomized -Coloring Algorithm

• Distributed Algorithm

• Randomized Algorithm

Running time
with high probability
(similar with the -coloring algorithm, but
now the color palette size is )
3
Each node has a palette with
colors
Palette of node
Initially all colors in palette are available
(Recall is the nodes degree)
4
At the beginning of a phase
uncolored neighbors of
uncolored degree of
Example
5
In conflicts, the node with the highest
uncolored degree wins
Example 1
Beginning of phase
6
In conflicts, the node with the highest
uncolored degree wins
Random color choices
7
In conflicts, the node with the highest
uncolored degree wins
End of phase
8
In conflicts, the node with the highest
uncolored degree wins
Example 2
Beginning of phase
9
In conflicts, the node with the highest
uncolored degree wins
Random color choices
10
In conflicts, the node with the highest
uncolored degree wins
End of phase
11
If both nodes have same degree, both reject their
color
Example 3
Beginning of phase
12
If both nodes have same degree, both reject their
color
Random color choices
13
If both nodes have same degree, both reject their
color
End of phase
14
Algorithm for node
Repeat
(iteration phase)
Pick a color uniformly at random from
available palette colors
Send color to neighbors
If (some neighbor with chose same
color )
Then Reject color
Else Accept color
Inform neighbors about color
(so that they mark color as unavailable)
Until color is accepted
15
Example execution
16
Phase 1 (iteration 1)
Nodes pick random colors
17
Conflicts
For this phase, uncolored degree degree
The nodes of higher uncolored degree win
18
Successful colors
19
Phase 2
(iteration 2)
Nodes pick random colors
20
Conflicts
The nodes of higher uncolored degree win
21
Successful colors
22
Phase 3
(iteration 3)
Nodes pick random colors
23
Successful colors
End of execution
24
Consider phase
(iteration )
Analysis
Palette of uncolored node
available color
unavailable color
Set of available colors
Example
25
Number of available colors
Palette size
Maximum unavailable colors
26
If then every color choice is a
success
So suppose that
27
We want to compute the probability of
Event node successfully
accepts a color
in current phase
We will prove that is at least
constant
28
Event node successfully
picks and then accepts
color
(available color)
29
Note that the events and
are mutually exclusive for any pair of colors
Any two such events cannot occur simultaneously
30
since, success for node in current phase is
to successfully accept some color in
31
since are mutually exclusive
32
nodes in with same or higher
uncolored degree than
Example
33
Note that for any
34
Event that node picks randomly color
Event that no node in picks randomly color
35
Probability that node picks color
Since node picks randomly and uniformly a
color from the available colors
36
Event that no uncolored neighbor of picks
randomly color
Event that uncolored neighbor does not pick color
37
This holds because the events are
independent
38
Consider some
39
(No Transcript)
40
Fundamental inequalities
41
(No Transcript)
42
(No Transcript)
43
This holds since and are independent
events
(the nodes pick randomly their colors
independent from one another)
44
(No Transcript)
45
Probability of success
46
Probability that node succeeds in a phase
at least
Probability that node fails in a phase
at most
Note that is a constant
47
is the number of nodes
Probability that node fails for
phases
at most
48
Probability that node fails for
phases
at most
Probability that some node fails for
phases
at most
Probability that every node succeeds in the first
phases
at least
49
Duration of each phase
time steps
The algorithm terminates in phases
with probability at least
Total time steps
(with high probability)
END OF ANALYSIS
50
Alternative Randomized -Coloring
Algorithm

• Distributed Algorithm

• Randomized Algorithm

Running time
with high probability
51
Same with previous algorithm
(palette for node is )
in conflicts, every conflicting node rejects the
color
But
(regardless of the uncolored degrees)
52
Algorithm for node
Repeat
Pick a color uniformly at random from
available palette colors
Send color to neighbors
If (some neighbor chose same color )
Then Reject color
Else Accept color
inform neighbors about color
(so that they mark color as unavailable)
Until color is accepted
53
Example execution
54
Phase 1
55
Successful Colors
56
Phase 2
57
Successful Colors
58
Phase 3
59
End of execution
60
Consider phase
(iteration )
Analysis
Palette of node
available color
unavailable color
Set of available colors
Example
61
At the beginning of a phase
uncolored neighbors of
Example
62
Number of available colors
Palette size
Maximum unavailable colors
63
If then every color choice is a
success
So suppose that
and
64
We want to compute the probability of
Event node successfully
accepts a color
in current phase
We will prove that is at least
constant
65
Event node successfully
picks and then accepts
color
(available color)
66
Note that the events and
are mutually exclusive for any pair of colors
Any two such events cannot occur simultaneously
67
Success for node in current phase is to
successfully accept some color in
(available color)
68
since are mutually exclusive
69
Event that node picks randomly color
Event that no uncolored neighbor of picks
randomly color
70
This holds since and are independent
events
(the nodes pick randomly their colors independent
from one another)
71
Probability that node picks color
Since node picks randomly and uniformly a
color from the available colors
72
Event that no uncolored neighbor of picks
randomly color
Event that uncolored neighbor does not pick color
73
This holds because the events are
independent
74
(No Transcript)
75
(No Transcript)
76
Take any
77
Out of probability space of
78
Thus, we can safely assume that the available
palette of is with
79
where
80
We can show that
where
and
Suppose that
81
We first show
Since is minimum, we have
82
since
83
Since
84
As needed
85
Therefore, if then
86
Therefore, if then
Thus, we can safely assume that the Available
palette of is
with
The available colors of Have been reduced by one
87
Thus, node can repeatedly reduce the
available colors in its palette until it
consists of 2 available colors
A similar observation holds for any node
88
Similarly, any node will reduce
the available colors in its palette until it
consists of 2 colors
89
Therefore,
But assume that for any
Available colors
90
Palette of node
Available colors
91
Palette of node
Color overlaps with palettes of neighbors
92
Palette of node
1
3
1
1
3
number of incoming arrows for a color
93
Palette of node
1
3
1
1
3
and
Since,
Average
94
(No Transcript)
95
Let be the number of colors with
We have that
Colors exceeding average
Which implies
96
Since chooses its colors randomly and
uniformly in ,
97
Suppose that a color has
there are at most two neighbors that may choose
color
say
98
Probability of success for node in
current phase
constant
99
Since a node succeeds with constant probability
in a phase
we need phases for success of
all nodes
Since time duration of a phase is constant, total
time needed is