Self-Stabilizing Labeling and Ranking in Ordered Trees Ajoy K. Datta - - PowerPoint PPT Presentation
Introduction Labeling with Guide Pairs Application: Ranking Conclusion Self-Stabilizing Labeling and Ranking in Ordered Trees Ajoy K. Datta 1 ephane Devismes 2 St Lawrence L. Larmore 1 Yvan Rivierre 2 1 School of Computer Science, University
Introduction Labeling with Guide Pairs Application: Ranking Conclusion
Ajoy K. Datta1 St´ ephane Devismes2 Lawrence L. Larmore1 Yvan Rivierre2
1School of Computer Science, University of Nevada Las Vegas, USA 2VERIMAG Laboratory, Universit´
e Joseph Fourier, Grenoble, France
Introduction Labeling with Guide Pairs Application: Ranking Conclusion
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Two self-stabilizing algorithms in ordered directed tree networks:
◮ Labeling processes, so to ease routing. ◮ Application: Ranking processes by weight.
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Model of Computation:
◮ Shared memory model ◮ Read/write atomicity ◮ Asynchronous setting ◮ Weakly fair scheduler
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Model of Computation:
◮ Shared memory model ◮ Read/write atomicity ◮ Asynchronous setting ◮ Weakly fair scheduler
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Model of Computation:
◮ Shared memory model ◮ Read/write atomicity ◮ Asynchronous setting ◮ Weakly fair scheduler
Topology:
◮ Ordered directed tree network
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Ordered directed tree network:
◮ tree ◮ rooted at some process ◮ directed toward the root ◮ left-to-right order on children
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Ordered directed tree network:
◮ tree ◮ rooted at some process ◮ directed toward the root ◮ left-to-right order on children
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Ordered directed tree network:
◮ tree ◮ rooted at some process ◮ directed toward the root ◮ left-to-right order on children
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Ordered directed tree network:
◮ tree ◮ rooted at some process ◮ directed toward the root ◮ left-to-right order on children 1 2 1
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Ordered directed tree network:
◮ tree ◮ rooted at some process ◮ directed toward the root ◮ left-to-right order on children
Computable in arbitrary rooted
1 2 1
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The guide pair of any process P is defined as gp(P) = (i, j), where:
◮ i is the rank of P in the preorder traversal
(left-to-right, top-down)
◮ j is the rank of P in the reverse postorder traversal
(right-to-left, top-down) Introduced by Flocchini & al., 2006.
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gp(P) = (i, j), where i is the preorder rank of P. 1, 2, 3, 4, 5, 6, 7, 8, 9,
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gp(P) = (i, j), where j is the reverse postorder rank of P.
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gp(P) = (i = preorder rank, j = reverse preorder rank) 1, 1 2, 6 3, 9 4, 7 5, 8 6, 5 7, 2 8, 4 9, 3
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We define a partial order
gp(P) ≤ gp(Q) ≡ i(P) ≤ i(Q) ∧ j(P) ≤ j(Q) 1, 1 2, 6 3, 9 4, 7 5, 8 6, 5 7, 2 8, 4 9, 3
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We define a partial order
gp(P) ≤ gp(Q) ≡ i(P) ≤ i(Q) ∧ j(P) ≤ j(Q) 1, 1 2, 6 3, 9 4, 7 5, 8 6, 5 7, 2 8, 4 9, 3
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We define a partial order
gp(P) ≤ gp(Q) ≡ i(P) ≤ i(Q) ∧ j(P) ≤ j(Q) Note that: gp(P) ≤ gp(Q) ≡ Q is a descendant of P 1, 1 2, 6 3, 9 4, 7 5, 8 6, 5 7, 2 8, 4 9, 3
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parents
Introduction Labeling with Guide Pairs Application: Ranking Conclusion
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parents ???
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1, 1 2, 6 3, 9 4, 7 5, 8 6, 5 7, 2 8, 4 9, 3 parents 1, 1
w, 5, 8
(5, 8)
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1, 1 2, 6 3, 9 4, 7 5, 8 6, 5 7, 2 8, 4 9, 3 parents 1, 1 2, 6 6, 5 7, 2
w, 5, 8
(2, 6) <= (5, 8)
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1, 1 2, 6 3, 9 4, 7 5, 8 6, 5 7, 2 8, 4 9, 3 parents 1, 1 2, 6 3, 9 4, 7 6, 5 7, 2
w, 5, 8
(4, 7) <= (5, 8)
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1, 1 2, 6 3, 9 4, 7 5, 8 6, 5 7, 2 8, 4 9, 3 parents 1, 1 2, 6 3, 9 4, 7 5, 8 6, 5 7, 2
w, 5, 8
(5, 8) <= (5, 8)
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1, 1 2, 6 3, 9 4, 7 5, 8 6, 5 7, 2 8, 4 9, 3 parents 1, 1 2, 6 3, 9 4, 7 5, 8 6, 5 7, 2
w, 5, 8
guide pairs Bidirectional communication between root and non-root.
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Composition of two modules:
◮ Counting processes by subtrees (bottom-up). ◮ Labeling based on those counts (top-down).
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Count processes in each subtree, in O(h) rounds.
? ? ? ? ? ? ? ? ?
h = height, δ = degree, n = number of processes
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Count processes in each subtree, in O(h) rounds.
? 1 ? 1 ? 1 1 1 ?
h = height, δ = degree, n = number of processes
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Count processes in each subtree, in O(h) rounds.
? 1 3 1 2 1 1 1 ?
h = height, δ = degree, n = number of processes
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Count processes in each subtree, in O(h) rounds.
4 1 3 1 2 1 1 1 ?
h = height, δ = degree, n = number of processes
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Count processes in each subtree, in O(h) rounds.
4 1 3 1 2 1 1 1 9
h = height, δ = degree, n = number of processes
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Preliminary: Copy counts from children.
4 4 1 1 3 3 1 1 2 2 1 1 1 1 1 1 9
h = height, δ = degree, n = number of processes
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Compute guide pairs from counts, in O(h) rounds.
4 1 3 1 2 1 1 1
h = height, δ = degree, n = number of processes
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Compute guide pairs from counts, in O(h) rounds. 1, 1
4 1 3 1 2 1 1 1
h = height, δ = degree, n = number of processes
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Compute guide pairs from counts, in O(h) rounds. 1, 1 2,
4 1 3 1 2 1 1 1
1 + 1 = 2
h = height, δ = degree, n = number of processes
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Compute guide pairs from counts, in O(h) rounds. 1, 1 2, 6,
4 1 3 1 2 1 1 1
1 + 4 + 1 = 6
h = height, δ = degree, n = number of processes
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Compute guide pairs from counts, in O(h) rounds. 1, 1 2, 6, 7,
4 1 3 1 2 1 1 1
1 + 4 + 1 + 1 = 7
h = height, δ = degree, n = number of processes
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Compute guide pairs from counts, in O(h) rounds. 1, 1 2, 6, 7, 2
4 1 3 1 2 1 1 1
1 + 1 = 2
h = height, δ = degree, n = number of processes
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Compute guide pairs from counts, in O(h) rounds. 1, 1 2, 6, 5 7, 2
4 1 3 1 2 1 1 1
1 + 3 + 1 = 5
h = height, δ = degree, n = number of processes
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Compute guide pairs from counts, in O(h) rounds. 1, 1 2, 6 6, 5 7, 2
4 1 3 1 2 1 1 1
1 + 3 + 1 + 1 = 6
h = height, δ = degree, n = number of processes
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Compute guide pairs from counts, in O(h) rounds. 1, 1 2, 6 6, 5 7, 2 3, 9 4, 7 8, 4 9, 3
4 1 3 1 2 1 1 1
h = height, δ = degree, n = number of processes
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Compute guide pairs from counts, in O(h) rounds. 1, 1 2, 6 6, 5 7, 2 3, 9 4, 7 8, 4 9, 3 5, 8
4 1 3 1 2 1 1 1
h = height, δ = degree, n = number of processes
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Algorithm that computes guide pairs,
◮ is self-stabilizing, ◮ is silent, ◮ stabilizes in O(h) rounds, ◮ uses O(δ log n) bits
and allows for bidirectional communication.
h = height, δ = degree, n = number of processes
Introduction Labeling with Guide Pairs Application: Ranking Conclusion
Introduction Labeling with Guide Pairs Application: Ranking Conclusion
19 / 28 ◮ Input: weight, of ordered
type.
◮ Output: rank, by order of
ascending weight.
6 8 3 2 9 1 7
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19 / 28 ◮ Input: weight, of ordered
type.
◮ Output: rank, by order of
ascending weight.
6 8 3 2 9 1 7 0#1
#2 #3 #4 #5 #6 #7 #8
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Self-stabilizing, but not silent: Endlessly compute ranks.
◮ Compute ranks ◮ Repeat computation waves ◮ Handle errors
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21 / 28 ◮ Weight packages are
(bottom-up) routed by priority of weight.
◮ Rank packages are
(top-down) routed thanks to guide pairs.
counter = 0 weight
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21 / 28 ◮ Weight packages are
(bottom-up) routed by priority of weight.
◮ Rank packages are
(top-down) routed thanks to guide pairs. #1
weight counter = 1 rank
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21 / 28 ◮ Weight packages are
(bottom-up) routed by priority of weight.
◮ Rank packages are
(top-down) routed thanks to guide pairs. #1 #2
counter = 2 weight rank
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21 / 28 ◮ Weight packages are
(bottom-up) routed by priority of weight.
◮ Rank packages are
(top-down) routed thanks to guide pairs. #1 #2 #3
counter = 3 weight rank
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21 / 28 ◮ Weight packages are
(bottom-up) routed by priority of weight.
◮ Rank packages are
(top-down) routed thanks to guide pairs. #1 #2 #3 #4
counter = 4 weight rank
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21 / 28 ◮ Weight packages are
(bottom-up) routed by priority of weight.
◮ Rank packages are
(top-down) routed thanks to guide pairs. #1 #2 #3 #4 #5
counter = 5 weight rank
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21 / 28 ◮ Weight packages are
(bottom-up) routed by priority of weight.
◮ Rank packages are
(top-down) routed thanks to guide pairs. #1 #2 #3 #4 #5 #6
counter = 6 weight rank
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21 / 28 ◮ Weight packages are
(bottom-up) routed by priority of weight.
◮ Rank packages are
(top-down) routed thanks to guide pairs. #1 #2 #3 #4 #5 #6 #7
counter = 7 weight rank
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21 / 28 ◮ Weight packages are
(bottom-up) routed by priority of weight.
◮ Rank packages are
(top-down) routed thanks to guide pairs. #1 #2 #3 #4 #5 #6 #7 #8
counter = 8 weight rank
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21 / 28 ◮ Weight packages are
(bottom-up) routed by priority of weight.
◮ Rank packages are
(top-down) routed thanks to guide pairs. #1 #2 #3 #4 #5 #6 #7 #8
counter = 8
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Status variable:
◮ 1: new epoch (reset) ◮ 2: new comp. wave ◮ 3: end of comp. wave ◮ 4: end of epoch
root
. . .
leaves 1 1 1 2 2 2 w w w d d r
. . . . . . . . .
w d r r r 3 3 3 4 4 4 1 1 1 . . . time
w = weight package, d = done package, r = rank package
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23 / 28 ◮ 0: error correction ◮ propagated up and down
the tree, except below status 1 (reset).
3 3 3 3 2 2 4 2 2 status 0 propagation
Introduction Labeling with Guide Pairs Application: Ranking Conclusion
23 / 28 ◮ 0: error correction ◮ propagated up and down
the tree, except below status 1 (reset).
3 3 3 3 2 2 4 2 2 status 0 propagation
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23 / 28 ◮ 0: error correction ◮ propagated up and down
the tree, except below status 1 (reset).
3 3 3 3 2 2 4 2 2 status 0 propagation
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23 / 28 ◮ 0: error correction ◮ propagated up and down
the tree, except below status 1 (reset).
3 3 3 3 2 2 4 2 2 status 0 propagation
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23 / 28 ◮ 0: error correction ◮ propagated up and down
the tree, except below status 1 (reset).
3 3 3 3 2 2 4 2 2 1 status 1 global reset
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23 / 28 ◮ 0: error correction ◮ propagated up and down
the tree, except below status 1 (reset).
3 3 3 3 2 2 4 2 2 1 1 1 1 status 1 global reset
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23 / 28 ◮ 0: error correction ◮ propagated up and down
the tree, except below status 1 (reset).
3 3 3 3 2 2 4 2 2 1 1 1 1 1 1 1 1 1 1 status 1 global reset
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Algorithm that ranks processes by weight,
◮ is self-stabilizing, ◮ is not silent, ◮ stabilizes in O(n) rounds and
(previously O(n2) by Alari & al., 1998)
◮ uses O(b + δ log n) bits.
b = weight’s bits, δ = degree, n = number of processes
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Introduction Labeling with Guide Pairs Application: Ranking Conclusion
26 / 28 ◮ Labeling algorithm that computes guide pairs,
thus allows for bidirectionnal communication.
(first self-stabilizing solution)
◮ Application: Distributed ranking of processes.
(better stab. time than Alari & al., 1998)
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27 / 28 ◮ What about an unfair scheduler?
Which step complexity? in O(n2)?
◮ Find other applications to guide pairs.
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Thanks for your attention