johannes schneider –1 a log-star distributed maximal independent set algorithm for growth-bounded...
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Johannes Schneider – 1
A Log-Star Distributed Maximal Independent Set Algorithm
for Growth-Bounded Graphs
Johannes SchneiderRoger Wattenhofer
Johannes Schneider – 2
Motivation
• Maximal Independent Set (MIS) algorithms allow to get Connected Dominating Sets (CDS) and Minimum Dominating Sets (MDS) for wireless multi-hop networks
• MDS and CDS are useful for – Routing
– Media access control
– Coverage
– …
• Compute CDS/MDS with little communication to save valuable time and energy
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Model and Definitions
• Maximal Independent Set (MIS) – Node v in MIS or ≥1 neighbor in MIS
– Nodes u,v in MIS cannot be adjacent
• Unit Disk Graph (UDG) – Geometrical graph
– Edge between nodes u,v if dist(u,v) < 1
– Growth bounded– Maximum size of an independent set
in the neighborhood of a node is at most 5
• Every node has an ID in [1,n]
• A node communicates with neighbors in
synchronized rounds without interference
• Definition log*– How often one has to take the logarithm to get 1
– Example: log* 16 = 3 since log 16 = 4; loglog 16 = 2; logloglog 16 = 1
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• Every node performs competitions (with breaks) until it (or a neighbor) is in the MIS
• Competition – First one based on ID to obtain result r– Node v picks neighbor u with smallest ID– If ID_v ≤ ID_u
– result r_v is 0
– If ID_v > ID_u – result r_v is the maximum position where
ID_v has a 1 and ID_u has a 0. – Example: Position 4 3 2 1
ID_v 1 1 0 1
ID_u 1 0 1 0
r_v = 11 (binary)
Algorithm
ID_a 10 r_a 0
ID_u 1010r_u 100
ID_v 1101r_v 11 ID_d 1100
r_d 11
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What to do with the result of a competition?
0
101
10 10
100
111
110
110
110111
• Node v changes its state depending on its result and those of neighbors.
• Dominator– If result r_v < r_u for all neighbors u
– Joins the MIS
– Neighbors are dominated and stay quiet
• Ruler– if result r_v ≤ r_u for all neighbors u
and at least one has same result
– All neighbors become ruled (if not dominated or rulers themselves)
– Ruled nodes stay quiet until all neighbors become ruled or dominated.
– Rulers immediately become competitors again and compete again based on IDs
• Competitor– None of above conditions applies
– Compete again based on the result of the last competition
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How many competitions?
• How often must a competitor compete before changing its state?– at most log* n times
• The result of log* n consecutive competitions must be 1.• Proof
– The result of the 1st competition is in [0,log n]– The result gives an index of a bit of the ID
– An ID in [1,n] => needs log n bits
– … 2nd … in [0,loglog n]– Since the previous result has up to loglog n bits
– a.s.o.
• Once a node has result 1, it must change its state.– Either its own result is a minimum or a neighbor has
smallest result possible, i.e. 0.
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How often can a node be before changing to ?
• Let S be the set of connected competitors with v in S
• A node not in S cannot join before v is
ruled or dominated
v
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How often can a node be before changing to ?
• S shrinks with every transition– When v becomes a ruler, one 2-hop
neighbor w in S is not reachable
by a path of rulers!– Node w (and all its
neighbors) cannot be
in S any more.
vw
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How many of such 2-hop neighbors W exist?
• For the UDG there exist only 13 such 2 hop neighbors W for a node v.
vw
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• After a competitor has become a ruler 13 times (without becoming ruled), no 2 hop neighbor can be reached by a path of rulers.
• Thus all neighbors of ruler v, that are still rulers form a clique.
• In the next competition based on the ID, the ruler of the clique with the smallest ID becomes a dominator!101 10
1 100
101 10
1 100
How often can a node be before changing to ?
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• After log* n competitions a competitor changes its state.– If dominated or dominator it is done
• A competitor can become a ruler at most 13 times in a row.
• After 13·log* n competitions every node gets a dominator within distance 13.
• Within distance 13 there are at most 132 nodes in an independent set, thus the maximum comptetions the algorithm needs are 133 ·log* n.
How many competitions for an arbitrary node?
… … …
…
…
Distance <= 13
|W| 13 13 13 13 12 12 12 11 11 11 10 10
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Related work
• How many rounds of communication to get a MIS?– Lower bounds
– on ring (log* n) [Lineal92]
– on general graphs (log n/loglog n) [Kuhn05]
– Upper bounds– On general graphs O(log n) [Luby86]
• … a CDS?– Lower bounds
– on UDG (log* n) [Lenzen08]
– Upper bounds – on UDG O(loglog n log*n) [VicariGfeller07]
– on UDG with distance information O(log* n) [Kuhn05]
• Here: MIS, CDS, MDS and Coloring on UDG in O(log* n)