Distributed tree comparison
with nodes of limited memory
Emanuele G. Fusco
1
Andrzej Pelc
2
1 Computer Science Department
Sapienza, University of Rome, via Salaria 113 – 00198 Rome, Italy
[email protected]
2
Département d’informatique
Université du Québec en Outaouais, Gatineau, Québec J8X 3X7, Canada.
[email protected]
SIROCCO 2010 — June 7-11 2010,
Nesin Mathematics Village — Şirince, TURKEY
E.Fusco and A.Pelc
Distributed tree comparison with nodes of limited memory
Outline
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Introduction
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The synchronous scenario
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The asynchronous scenario
E.Fusco and A.Pelc
Distributed tree comparison with nodes of limited memory
Comparing trees
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We are given a (rooted) tree
E.Fusco and A.Pelc
Distributed tree comparison with nodes of limited memory
Comparing trees
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And another one to compare to
E.Fusco and A.Pelc
Distributed tree comparison with nodes of limited memory
Comparing trees
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Roots are connected by an edge
E.Fusco and A.Pelc
Distributed tree comparison with nodes of limited memory
Comparing trees
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And we look for differences (in a distributed setting)
E.Fusco and A.Pelc
Distributed tree comparison with nodes of limited memory
Related work
Symmetry-breaking task, closely related to leader election.
Anonymous networks (or networks where node labels are not
unique).
Tree canonization – give a tree with port labels a unique
isomorphism invariant name.
E.Fusco and A.Pelc
Distributed tree comparison with nodes of limited memory
Automata and memory usage
Each node is a copy of a finite state automaton A.
All nodes start in the same state S0 .
If the trees are isomorphic, all nodes must enter a state YES.
If the trees are not isomorphic, nodes in one tree must enter a
state NO0 and nodes in the other tree must enter a state NO1 ,
thus breaking the simmetry.
E.Fusco and A.Pelc
Distributed tree comparison with nodes of limited memory
Automata and memory usage
Each node is a copy of a finite state automaton A.
All nodes start in the same state S0 .
If the trees are isomorphic, all nodes must enter a state YES.
If the trees are not isomorphic, nodes in one tree must enter a
state NO0 and nodes in the other tree must enter a state NO1 ,
thus breaking the simmetry.
The number of states of the automaton A performing the
comparison gives us the measure on the memory usage.
An automaton with k states requires Θ log k memory bits.
E.Fusco and A.Pelc
Distributed tree comparison with nodes of limited memory
Synchronous and asynchronous communication
We consider both synchronous and asynchronous communication.
E.Fusco and A.Pelc
Distributed tree comparison with nodes of limited memory
Synchronous and asynchronous communication
We consider both synchronous and asynchronous communication.
For synchronous communication we consider memory vs time
tradeoff.
E.Fusco and A.Pelc
Distributed tree comparison with nodes of limited memory
Synchronous and asynchronous communication
We consider both synchronous and asynchronous communication.
For synchronous communication we consider memory vs time
tradeoff.
For asynchronous communication we consider memory vs #
messages tradeoffs.
E.Fusco and A.Pelc
Distributed tree comparison with nodes of limited memory
A simple code for (rooted) trees
The tree
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The code
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E.Fusco and A.Pelc
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Distributed tree comparison with nodes of limited memory
A simple code for (rooted) trees
The tree
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The code
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E.Fusco and A.Pelc
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Distributed tree comparison with nodes of limited memory
A simple code for (rooted) trees
The tree
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The code
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E.Fusco and A.Pelc
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Distributed tree comparison with nodes of limited memory
A simple code for (rooted) trees
The tree
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The code
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E.Fusco and A.Pelc
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Distributed tree comparison with nodes of limited memory
A simple code for (rooted) trees
The tree
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The code
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E.Fusco and A.Pelc
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Distributed tree comparison with nodes of limited memory
A simple code for (rooted) trees
The tree
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The code
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E.Fusco and A.Pelc
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Distributed tree comparison with nodes of limited memory
A simple code for (rooted) trees
The tree
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The code
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E.Fusco and A.Pelc
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Distributed tree comparison with nodes of limited memory
A simple code for (rooted) trees
The tree
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The code
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E.Fusco and A.Pelc
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Distributed tree comparison with nodes of limited memory
A simple code for (rooted) trees
The tree
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The code
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E.Fusco and A.Pelc
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Distributed tree comparison with nodes of limited memory
A simple code for (rooted) trees
The tree
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The code
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E.Fusco and A.Pelc
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Distributed tree comparison with nodes of limited memory
A simple code for (rooted) trees
The tree
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The code
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E.Fusco and A.Pelc
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Distributed tree comparison with nodes of limited memory
Augmenting the code
The code can be augmented to handle arbitrary port labelings by
inserting the value of the port number leading to the parent of each
node v immediately after the 1 corresponding to the first visit of v .
E.Fusco and A.Pelc
Distributed tree comparison with nodes of limited memory
Augmenting the code
The code can be augmented to handle arbitrary port labelings by
inserting the value of the port number leading to the parent of each
node v immediately after the 1 corresponding to the first visit of v .
Proposition
The length of the code B(T ) of a n-node tree T is O(n).
E.Fusco and A.Pelc
Distributed tree comparison with nodes of limited memory
The synchronous scenario
In the synchronous scenario we establish trade-offs between the
memory size and the time needed to accomplish the tree
comparison task.
E.Fusco and A.Pelc
Distributed tree comparison with nodes of limited memory
The synchronous scenario
In the synchronous scenario we establish trade-offs between the
memory size and the time needed to accomplish the tree
comparison task.
Theorem
Let x ≥ 9 dlog ne. There exists an automaton with x bits of memory
that solves the tree comparison problem in the class of all trees of
size at most n and of height at most h in time O(max(h, n/x)).
E.Fusco and A.Pelc
Distributed tree comparison with nodes of limited memory
Sketch of Algorithm Sync Compare
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Distributed tree comparison with nodes of limited memory
Sketch of Algorithm Sync Compare
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Distributed tree comparison with nodes of limited memory
Sketch of Algorithm Sync Compare
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E.Fusco and A.Pelc
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Distributed tree comparison with nodes of limited memory
Sketch of Algorithm Sync Compare
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Distributed tree comparison with nodes of limited memory
A matching lower bound
Theorem
If the automaton has x bits of memory, then time Ω(max(h, n/x))
is needed to solve the tree comparison problem in the class of all
trees of size at most n and of height at most h, where h > 1.
E.Fusco and A.Pelc
Distributed tree comparison with nodes of limited memory
The asynchronous scenario
In the asynchronous scenario we establish trade-offs between the
memory size of an automaton and the number of messages needed
to accomplish the tree comparison task.
E.Fusco and A.Pelc
Distributed tree comparison with nodes of limited memory
The asynchronous scenario
In the asynchronous scenario we establish trade-offs between the
memory size of an automaton and the number of messages needed
to accomplish the tree comparison task.
Algorithm idea
A token travels through the tree, by mean of message exchanges,
piggybacking a code segment.
When the segment is long enough (depending on the memory size
of the nodes), it is sent up to the root for comparison with the
corresponding segment from the other tree.
E.Fusco and A.Pelc
Distributed tree comparison with nodes of limited memory
Upper and lower bounds
Theorem
There exists an automaton with x bits of memory that solves the
tree comparison problem in the class of all trees of size at most n,
using O(n2 /x) messages, for 4 dlog ne ≤ x ≤ n.
Theorem
If the automaton has x bits of memory, then Ω(n2 /x) messages are
needed to solve the tree comparison problem in the class of all trees
of size at most n.
E.Fusco and A.Pelc
Distributed tree comparison with nodes of limited memory
Thanks!
Thank you for your attention!
Any questions?
E.Fusco and A.Pelc
Distributed tree comparison with nodes of limited memory