Roger Wattenhofer On Local Fixing ETH Zurich Distributed Computing wwwdiscoethzch Motivation Motivation Motivation by Example Motivation by Example Motivation by Example Motivation by Example ID: 284610
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Slide1
Michael KönigRoger Wattenhofer
On Local Fixing
ETH Zurich – Distributed Computing – www.disco.ethz.chSlide2
MotivationSlide3
MotivationSlide4
Motivation by ExampleSlide5
Motivation by ExampleSlide6
Motivation by ExampleSlide7
Motivation by ExampleSlide8
Motivation by ExampleSlide9
Motivation by ExampleSlide10
Motivation by ExampleSlide11
Motivation by ExampleSlide12
Model
Neighborhood model
Unbounded message sizes
Synchronous rounds
t = 0
t = 1
t = 2Slide13
2
1
Model (cont.)
We say a problem can be
fixed locally
if any solution can be fixed within O(1) rounds after a graph change.
(+e)
Edge Insertion
(-e)
Edge Deletion
(w → w')
Weight Change
(+v
1
)
(+v
*
)
Node Insertion
(-v
1
)
(-v
*
)
Node DeletionSlide14
Computation vs. Fixing
Computing solutions is very well-studied
Different “complexity classes” have been defined:
Γ
1
-Count
“local”
3
3
3
4
4
5
5
Maximal Matching
“polylog”
Spanning Tree
“global”
Previous work
on local fixing
1995: Maximal Independent Sets (MIS), by Kutten and Peleg
2007: O(1)-Maximum Weighted Matchings, by Lotker, Patt-Shamir and RosénSlide15
LBound UBound
+e -e w
→
w
'
+v
1
-v
1
+v
*
-v
*
Γ
1
-Count
Ω
(1) O(1)
o(n)-MDS
Ω
(1) O(1)
MIS
Ω
(√log(n)) O(log(n))
O(1)-MWM
Ω
(√log(n)) O(log(n))
MM
Ω
(√log(n)) O(log(n))
2-MVC
Ω
(√log(n)) O(log(n))
Γ
log(n)
-Count
Ω
(log(n)) O(log(n))
ST
Ω
(D) O(D)
MST
Ω
(D) O(D)
SPT
Ω
(D) O(D)
Flow
Ω
(D) O(D)
Leader Ω
(D) O(D)Count Ω
(D) O(D)
ProblemsSlide16
LBound UBound
+e -e w
→
w
'
+v
1
-v
1
+v
*
-v
*
Γ
1
-Count
Ω
(1) O(1)
o(n)-MDS
Ω
(1) O(1)
MIS
Ω
(√log(n)) O(log(n))
O(1)-MWM
Ω
(√log(n)) O(log(n))
MM
Ω
(√log(n)) O(log(n))
2-MVC
Ω
(√log(n)) O(log(n))
Γ
log(n)
-Count
Ω
(log(n)) O(log(n))
ST
Ω
(D) O(D)
MST
Ω
(D) O(D)
SPT
Ω
(D) O(D)
Flow
Ω
(D) O(D)
Leader Ω
(D) O(D)Count Ω
(D) O(D)
û û û û
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ProblemsSlide17
ü
û ü
û
ü
û
û
ü
û û
ü
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ü
ü û ü
û
û ü û û
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û
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û üProblems
LBound UBound
+e -e w
→ w'
+v1 -v1
+v* -v*
Γ1
-Count Ω(1) O(1)
o(n)-MDS Ω(1) O(1)
MIS
Ω(√log(n)) O(log(n))
O(1)-MWM Ω
(√log(n)) O(log(n))MM
Ω
(√log(n)) O(log(n))
2-MVC
Ω
(√log(n)) O(log(n))
Γ
log(n)
-Count
Ω
(log(n)) O(log(n))
ST
Ω
(D) O(D)
MST
Ω
(D) O(D)
SPT
Ω
(D) O(D)
Flow
Ω
(D) O(D)
Leader
Ω
(D) O(D)
Count
Ω
(D) O(D)Slide18
ü
û ü
ü
ü
û
û û û
ü
ü
û û
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ü
ü
û ûû û û ü
ü û û
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ü ü
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û û û û û û
ü
ü
ü
ü
ü üû û û
û û ûResults
LBound UBound
+e -e w
→ w'
+v1 -v1
+v* -v*
Γ1
-Count Ω(1) O(1)
o(n)-MDS Ω(1) O(1)
MIS
Ω(√log(n)) O(log(n))
O(1)-MWM Ω
(√log(n)) O(log(n))
MM
Ω
(√log(n)) O(log(n))
2-MVC
Ω
(√log(n)) O(log(n))
Γ
log(n)
-Count
Ω
(log(n)) O(log(n))
ST
Ω
(D) O(D)
MST
Ω
(D) O(D)
SPT
Ω
(D) O(D)
Flow
Ω
(D) O(D)
Leader
Ω
(D) O(D)
Count
Ω
(D) O(D)Slide19
Maximal Independent Set (MIS)
?
?
?
?
?Slide20
“Last Will”: during settlingSlide21
“Last Will”: in action
!
!Slide22
o(n)-Minimum Dominating Set
We know algorithms which compute a o(n)-MDS in constant time
[Kuhn et al., 2005]
But we can construct o(n)-MDS solutions which cannot be fixed locally!
Recipe for a
k
-MDS which cannot be fixed within
c
steps:
x = ⌊(k - 1)(c + 1)⌋, y = 3c + 1
V = (a
1
, a
2
, …, a
x
, b
1
, b
2
, …, b
y
)
E = {(a
i
, b
1
) | 1 ≤ i ≤ x} ∪ {(b
i
, b
i+1
) | 1 ≤ i ≤ y}
a
3
b
1
b
4
b
2
b
3
b
5
b
6
b
7
a
2
a
1
a
4
a
5
For k = 2.8, c = 2, x = 5 and y = 7:
a
3
b
1
b
4
b
2
b
3
b
5
b
6
b
7
a
2
a
1
a
4
a
5Slide23
ConclusionTraditional distributed complexity classes don’t tell the whole story.A set of orthogonal “fixing complexity” classes may be interesting!Slide24
Thanks!Questions & Comments?Slide25
Motivation by Example