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Leveraging Linialโs Locality limit
Christoph Lenzen and Roger Wattenhofer
Presented by Yam Chemel
Linial (1992)
3-coloring and MIS
on a ring (๐ ๐)
takes at least logโ(๐) rounds
Reminder
Prove logโ(๐) Lower bound for MaxIS algorithms in rings
Our Goal:
How?
MaxIS ๐(logโ ๐ )
3-coloring ๐ logโ ๐
Contradicts Linial!
MaxIS vs. MIS - Example
Independent Set MIS MaxIS
(general graphs)
MIS โธ MaxIS
MaxIS in rings
1 2
3 4
1 2
3
4
5
Q: What if each even node returns 1, and each odd node returns 0?
1 2
3 4
๐ 4 MaxIS
1 2
3
4
5 ๐ 5
MaxIS
1 3
2 4
MaxIS
Identifiers are not necessarily in order!
1 3
2 4
MaxIS vs. MIS โ in rings
MIS โธ MaxIS
Algorithm ๐ด MaxIS
๐(logโ ๐ )
Algorithm ๐ด MIS
๐ logโ ๐
Contradicts Linial!
f-approximation algorithm for MaxIS
๐ -MaxIS solution ๐ผ โ algorithm A solution
๐ผ = ๐
If ๐ด is ๐(๐)-approximation: ๐ผ โค ๐
and also:
๐ผ โ ๐(๐) โฅ |๐|
๐ด ๐ ๐ = ๐ ๐
A is 2-approximation for MaxIS
๐ผ =๐
4
๐
4โ 2 โฅ ๐ =
๐
2
Leveraging Linialโs Locality limit
MaxIS approximation ๐(๐๐๐โ ๐ )
3-coloring ๐ ๐๐๐โ ๐
Leveraging Linialโs Locality limit
MaxIS approximation ๐ ๐
3-coloring ๐(๐)
Leveraging Linialโs Locality limit
๐(๐)-alternating ๐(๐)
MaxIS approximation ๐ ๐
3-coloring ๐(๐)
๐(๐)-alternating algorithm - Definition:
Suppose A is an algorithm operating on ๐ ๐ which assigns each node ๐ฃ๐ โ ๐๐ a value ๐(๐ฃ๐) โ {0, 1}. We call A ๐(n)-alternating, if the length of any monochromatic sequence
๐ ๐ฃ๐ = ๐ ๐ฃ๐+1 = โฏ = ๐ ๐ฃ๐+๐ is smaller than ๐(n).
< ๐(๐)
๐ด ๐ ๐ = ๐ ๐
๐(๐)-alternating algorithm - examples
n is even
Longest monochromatic sequence = 1
n is odd
Longest monochromatic sequence = 2
A is 3-alternating (also 4-alternating, 5-alternatingโฆ)
(Not necessarily MaxIS algorithm)
๐ด ๐ ๐ = ๐ ๐
A is (n+1)-alternating
(Not necessarily MaxIS algorithm)
๐(๐)-alternating algorithm - examples
Lemma - Modified MaxIS approximation Suppose an f-approximation algorithm A for the MaxIS problem on the ring ๐ ๐ running in at most g(n) โฅ 1 rounds is given, where we have ๐(๐)๐(๐) โ ๐(logโ(๐)). Then an ๐(logโ(๐))-alternating algorithm Aโฒ requiring ๐ logโ ๐ communication rounds exists.
A
MaxIS, ๐(๐)-approximation,
๐(๐) rounds, ๐ ๐ ๐ ๐ โ ๐ ๐๐๐โ ๐
Aโ ๐ ๐๐๐โ ๐ -alternating,
๐ ๐๐๐โ ๐ rounds
Modified MaxIS approximation - Proof
General idea
๐๐
๐ ๐
๐ ๐โฒ ๐
๐๐โฒ
Observation: For each node ๐ฃ๐, ๐(๐ฃ๐) is a function of:
๐ฃ๐
๐(๐) ๐(๐)
๐ ๐ = #๐๐๐ข๐๐๐
โข The IDs of its ๐(๐) neighbors on each side
โข ๐ โข Its ID
๐(๐) ๐(๐)
Modified MaxIS approximation - Proof
โข ๐๐๐ก ๐ ๐ โ 10๐(๐)๐(๐)
๐๐ =
๐ฃ1, โฆ , ๐ฃ๐ ๐ , ๐ฃ๐ ๐ +1, โฆ, ๐ฃ๐ ๐ +๐ ๐ , ๐ฃ๐ ๐ +1+๐ ๐ , โฆ , ๐ฃ2๐ ๐ +๐ ๐ฃ
| โ๐ โ ๐ฃ๐ ๐ +1, โฆ, ๐ฃ๐ ๐ +๐ ๐ ๐ ๐ฃ๐ = 0
0 0
?
?
0
? ?
? ?
๐(๐) ๐(๐) ๐(๐)
Modified MaxIS approximation - Proof
โข Define exactly the set of sequences preventing that A is ๐(๐)-alternating
๐(๐) ๐(๐)
? ? ? 0 0 0 ? ? ?
Id=2 Id=11 Id=8 Id=9 Id=14 Id=21 Id=7 Id=6 Id=3
๐(๐)
๐ = 2, 11, 8, 9, 14, 21, 7, 6, 3 โ ๐๐
Take 2๐ ๐ + ๐(๐) consecutive nodes in ๐ ๐
Assign them identifiers
Run A on the sequence / on ๐ ๐ containing the sequence.
If the ๐(๐) center nodes compute 0, add s to ๐๐
Id=2 Id=11 Id=8 Id=9 Id=14 Id=21 Id=7 Id=6 Id=3
Building ๐๐
2๐ ๐ + ๐(๐)
Construct a sequence of identifiers for ๐ ๐:
๐ 2
๐ 1 ๐ โ ๐
๐ ๐ ๐ ๐
Modified MaxIS approximation - Proof
1. Choose an arbitrary sequence s from ๐๐ and assign the identifiers to ๐ฃ1, โฆ , ๐ฃ ๐ .
2. Assuming we already assigned labels to the nodes ๐ฃ1, โฆ , ๐ฃ๐:
While there exists a sequence ๐ โ ๐๐ that can be appended to ๐ฃ1, โฆ , ๐ฃ๐ without duplicating an identifier, we do so.
3. If no further sequence fits, we add ๐ โ ๐ arbitrary identifiers not yet present in ๐ฃ1, โฆ , ๐ฃ๐ to
complete the labeling (๐ฃ1, โฆ , ๐ฃ๐) of ๐ ๐.
Assume for contradiction, that for arbitrarily large n it is possible to
label ๐ ๐ as described, with at least ๐ โ๐
5๐ ๐ identifiers
stemming from sequences out of ๐๐.
# ๐๐๐๐๐ ๐กโ๐๐ก ๐๐๐๐๐ข๐ก๐ 1 ๐๐ ๐ ๐ โค ๐ โ๐ ๐ โ ๐
๐ ๐ + 2๐ ๐โ
๐
5๐ ๐
โฅ ๐๐๐๐๐ ๐กโ๐๐ก ๐๐๐๐๐ข๐ก๐ 0 ๐๐ ๐ โ ๐๐
๐๐๐๐๐ ๐๐ ๐ โ ๐๐โ ๐ โ #(๐๐๐๐๐ ๐๐๐ก ๐๐ ๐ โ ๐๐)
Modified MaxIS approximation - Proof
# ๐๐๐๐๐ ๐กโ๐๐ก ๐๐๐๐๐ข๐ก๐ 0 ๐๐ ๐ ๐ โฅ
=๐ ๐
๐ ๐ + 2๐(๐)โ ๐ โ
๐
5๐ ๐
# ๐๐๐๐๐ ๐กโ๐๐ก ๐๐๐๐๐ข๐ก๐ 1 ๐๐ ๐ ๐ โค ๐ โ๐ ๐ โ ๐
๐ ๐ + 2๐ ๐โ
๐
5๐ ๐
= ๐ 1 โ๐ ๐
๐ ๐ + 2๐ ๐+
1
5๐ ๐
๐ ๐ = 10๐ ๐ ๐(๐)
Modified MaxIS approximation - Proof
= ๐ 1 โ10๐ ๐ ๐ ๐
10๐ ๐ ๐ ๐ + 2๐ ๐+
1
5๐ ๐
= ๐ 1 โ5๐ ๐
5๐ ๐ + 1+
1
5๐ ๐
= ๐ 1 โ 1 โ1
5๐ ๐ + 1+
1
5๐ ๐
= ๐1
5๐ ๐ + 1+
1
5๐ ๐
โค ๐1
5๐ ๐ + 0+
1
5๐ ๐
=2๐
5๐ ๐
๐ผ = # ๐๐๐๐๐ ๐กโ๐๐ก ๐๐๐๐๐ข๐ก๐ 1 ๐๐ ๐ ๐ โค2๐
5๐ ๐
However, according to f-approximation definition: ๐ ๐ ๐ผ โฅ ๐
(M -an arbitrary MaxIS of G)
๐ 6
Contradicts the assumption! (โFor arbitrarily large n it is possible to label ๐ ๐ as described, with at least
๐ โ๐
5๐ ๐ identifiers stemming from sequences out of ๐๐.โ)
Modified MaxIS approximation - Proof
๐(๐) ๐ผ โค2
5๐
Choosing every other node in ๐ ๐ is a MaxIS solution:
๐ =๐
2
Combining the equations:
๐ ๐ ๐ผ โฅ๐
2
At least ๐
๐๐ ๐ identifiers remain which cannot form a further
sequence from ๐๐.
Set ๐โฒ โ max ๐0, 5๐ ๐ โ ๐
In ๐ ๐โฒ
#๐๐๐๐๐๐๐๐๐ ๐๐๐๐๐ก๐๐๐๐๐๐
โฅ๐โฒ
5๐ ๐โฒ
For a large n (๐ > ๐0) it is possible to label ๐ ๐ as described, with
at most ๐ โ๐
5๐ ๐ identifiers stemming from sequences out of ๐บ๐
Modified MaxIS approximation - Proof
๐ 2
๐ 1 ๐ โ ๐
๐ ๐ ๐ ๐
โฅ5๐ ๐ โ ๐
5๐ ๐= ๐
๐ ๐ ๐ ๐โฒ
๐๐
Algorithm Aโ : ๐๐ข๐ โถ ๐ด(๐โฒ, ๐ ๐๐๐๐๐ก๐๐๐๐๐๐ ๐๐๐๐ ๐ ๐โฒ )
Modified MaxIS approximation - Proof
๐ 4 ๐ 11
1
2
3
4
3
8
1
4
๐4
Algorithm Aโ : ๐ ๐ข๐ ๐ด(11, 3, 8, 1, 4 )
Example:
Modified MaxIS approximation - Proof
Aโ is ๐ ๐โฒ -alternating
No ๐(๐โฒ) consecutive 0โs No ๐(๐โฒ) consecutive 1โs
๐ด computes an independent setโ no 2 neighbors in independent set โ
no 2 consecutive 1โs
No ๐ โ ๐๐โฒ in remaining n nodes of ๐ ๐โฒ โ No ๐ ๐โฒ consecutive nodes compute 0 in
Aโ run
Maximum monochromatic sequence : ๐ ๐โฒ โ 1 โ ๐ ๐ ๐โฒ ๐ ๐โฒ โ ๐ logโ ๐โฒ = ๐ logโ ๐
Aโ running time -๐(๐๐๐โ(๐)) rounds ๐ ๐ โค ๐ ๐ ๐ ๐ โ ๐(๐๐๐โ ๐)
๐ ๐โฒ โ ๐(๐๐๐โ ๐)
Aโ is ๐ ๐๐๐โ ๐ -alternating
Leveraging Linialโs Locality limit
๐(๐)-alternating ๐(๐)
MaxIS approximation ๐ ๐
3-coloring ๐(๐)
Leveraging Linialโs Locality limit
๐(๐)-alternating ๐(๐)
MaxIS approximation ๐ ๐
3-coloring ๐(๐)
2. Proof : Lemma โ Given a ๐(๐)-alternating algorithm A running in ๐(๐(๐)) rounds, a 3-coloring of the ring can be computed in ๐(๐(๐)) rounds.
1. Run A. Let ๐(๐ฃ) โ {0, 1} denote the result of this run.
2. Find a pair of neighboring nodes {๐ค1, ๐ค2} with ๐ ๐ค1 โ ๐ ๐ค2 which is closest to v.
0 1 ๐ ๐ค2
1 0 ๐ ๐ค2
If ๐ฃ โ {๐ค1, ๐ค2}: if ๐(๐ฃ) = 0: set ๐ ๐ฃ โ ๐ otherwise: set ๐(๐ฃ) โ ๐
0 1 ๐ ๐ค2
1 0 ๐ ๐ค2
1 0 1 1
๐ค2 ๐ค1 ๐ ๐ค
1 0 1 1
Else: denote by ๐ฟ the distance to the closer node in {๐ค1, ๐ค2}, w.l.o.g. ๐ค1. if ๐ฟ โ 2โ: set ๐ ๐ฃ โ ๐(๐ค1) else: set ๐ ๐ฃ โ ๐(๐ค2)
1 0 1 1
๐ค2 ๐ค1 ๐ ๐ค (๐ฟ โ 2โ)
๐ฟ
3 8 ๐ ๐ค
8 3 ๐ ๐ค
3. If v has a neighbor w with ๐(๐ฃ) = ๐(๐ค) and v > w, set ๐(๐ฃ) โ ๐.
3 8 ๐ ๐ค
8 3 ๐ ๐ค
3 8 ๐ ๐ค
1 2 3 8 ๐ ๐ค
1 2 (3)
4. If v has a neighbor w with ๐(๐ฃ) = ๐(๐ค) = ๐ and v > w, set c(v) to the color none of the neighbors of v has.
3 8 ๐ ๐ค
1 2 (4)
2. Proof : Lemma โ Given a ๐(๐)-alternating algorithm A running in ๐(๐(๐)) rounds, a 3-coloring of the ring can be computed in ๐(๐(๐)) rounds.
Running time: ๐ถ ๐ ๐
Step 1: Running A- ๐ ๐ ๐
Step 2: Finding a pair of neighbors with different d - ๐ ๐ .
No more than ๐(๐ฃ) consecutive nodes take the same decision d(v)
since A is ๐(๐)-alternating.
Valid 3-coloring of ๐น๐
Step 2: Each node ๐ฃ chooses different from one of its neighbors,
1 1 0 1 1 1 1 0 1
so at most one of the neighbors of ๐ฃ may take the same choice.
Step 3: From each pair of neighbors with the same color one chooses g.
Step 4: If that same color was g, v chooses the color non of its neighbors has.
v v
v
1 1 0
v
1 1 1 1 0 1
v v
Leveraging Linialโs Locality limit
๐(๐)-alternating ๐(๐)
MaxIS approximation ๐ ๐
3-coloring ๐(๐)
Therefore, there isnโt a MaxIS approximation algorithms running on a ring
in less than log*(n)
Proof - Summary
Assume by contradiction that there exists a MaxIS approximation algorithm A running in less than ๐๐๐โ(๐).
Construct a ๐ ๐๐๐โ ๐ -alternating algorithm running in ๐(๐๐๐โ(๐)) using algorithm A.
By lemma 2, a 3-coloring of the ring can be computed in ๐(๐๐๐โ(๐)) rounds.
This contradicts Linialโs 3-coloring lower bound.
- Discuss MDS lower bound - Compare MDS and MaxIS difficulty
Our (new) Goal:
No! Weโll show a case where MIS is easier than MDS
MDS on rings O(1)
Is MDS always easier than MIS?
How?
MDS โ Minimum Dominating Set
DS MDS
MDS in rings
Taking every third node gives a minimum dominating set
๐ =๐
3
Taking every node gives a 3-approximation MDS in 1 round โ ๐ logโ ๐
๐ ๐
๐ ๐
There is no ๐๐๐โ(๐) bound MDS approximation in rings! MDS approximation in rings takes ๐ถ(๐) rounds
MDS approximation in rings
Can we compare MaxIS and MDS difficulty?
MDS
MaxIS
We saw that in ๐ ๐ MDS can be computed in 1 round, but MaxIS requires at least ๐๐๐โ(๐) round.
Is it always easier to compute MDS than MaxIS?
MaxIS graph family
๐บ ๐ฃ ๐ค
Any graph that can be constructed this way
Lemma (Local computation of a MaxIS on MaxIS Graphs): The set {๐ฃ โ ๐ | |๐1
+(๐ฃ)|๐๐๐ 2 = 1} is a MaxIS for any MaxIS Graph.
Proof โ part 1:
- For ๐ฃ๐ โ ๐๐ ๐ โ {3,4}, ๐ฃ๐ has 2|๐1+ ๐ฃ | neighbors.
- ๐1+ ๐ฃ๐ = 2 ๐1
+ ๐ฃ + 1 โ ๐1+ ๐ฃ๐ ๐๐ ๐๐๐
- For ๐ฃ๐ โ ๐๐ ๐ โ {1,2}, ๐1+ ๐ฃ๐ = 4 ๐1
+ ๐ฃ โ ๐1+ ๐ฃ๐ ๐๐ ๐๐ฃ๐๐
- {๐ฃ โ ๐ | |๐1+(๐ฃ)|๐๐๐ 2 = 1} = ๐3 โช ๐4
Proof โ part 2:
- ๐3 โช ๐4 is an Independent set โ according to the construction of
MaxIS graph
- (๐ฃ1, ๐ฃ3, ๐ฃ2, ๐ฃ4) forms a cycle, so for each 4 nodes as such, only 2 can be in the IS.
- MaxIS canโt be larger than ๐
2
- ๐3 โช ๐4 =๐
2
โ ๐3 โช ๐4 ๐๐ ๐ ๐๐๐ฅ๐ผ๐
Lemma (Local computation of a MaxIS on MaxIS Graphs): The set {๐ฃ โ ๐ | |๐1
+(๐ฃ)|๐๐๐ 2 = 1} is a MaxIS for any MaxIS Graph.
Conclusion โ MaxIS on a MaxIS graph can be determined locally, without communication (in ๐(1) rounds).
MDS on MaxIS graphs
We can prove that MDS on MaxIS graphs is as efficient as in general graphs, meaning:
ฮฉlog ๐
log log ๐
MaxIS
MDS
Ring graphs - ๐ ๐ MaxIS graphs
MaxIS
MDS ๐(1)
ฮฉlog ๐
log log ๐
MDS
MaxIS ๐(1)
ฮฉ logโ ๐
MaxIS and MDS are not comparable in general graphs!