Deterministic Rendezvous

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Deterministic Rendezvous, Treasure Hunts
andStrongly Universal Exploration Sequences
Amnon Ta-Shma Uri Zwick
Tel Aviv University
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The Rendezvous problem
Two robots are activated, at different times and
in different locations, in an unknown environment
Should both follow deterministic sequences of
instructions
The instructions should guarantee that the two
robots meet in a polynomial number of steps,
after the activation of the second robot
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The unknown environment
An undirected (cubic) graph. Graph and its size
uknown to robots
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Vertices are undistinguishable
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Exits are numbers. No consistently assumed
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No pebbles allowed
Robots meet when they are in the same vertex at
the same time
Robots move synchronously
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Instructions memoryless model
Each robot gets a sequence s s1 s2 s3 ?
0,1,2,3. In the i-th step, if si0, stay in
place, otherwise, take exit no. si.
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Instructions backtracking model
When a robot enters a vertex, it learns the
number of the edge used to enter the vertex.
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Action taken may now depend on entrance labels
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In particular, backtracking is possible
s 3202
? 12?2
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Parameters
n size of the environment
l length (in bits) of smaller label
t difference in activation time
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Results
Dessmark, Fraigniaud, Pelc (2003)Dessmark,
Fraigniaud, Kowalski, Pelc (2006)
Rendezvous after O(n5(tl)1/2n10l) steps
Kowalski, Malinowski (2006)
Rendezvous after O(n15l3) stepswhen
backtracking allowed
Our result
Rendezvous after O(n5l) steps
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Additional results
The previous results guarantee a rendezvous after
a polynomial number of steps.
But, we do not know how to compute these steps
in polynomial time
We give the first polynomial stepsand polynomial
time solution in the backtracking model
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Symmetry breaking
If the two robots are identical, no
deterministic solutionis possible
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To break the symmetry, the robots are assigned
distinct labels L1 and L2.
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In this talk we assume that the labels are 0 and
1.
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Randomized rendezvous
If randomization is allowed, then achieving a
rendezvous is easy
Each robot simply performs a random walk.
Expected number of steps before the two robots
meet is O(n3) Coppersmith, Tetali, Winkler (1993)
Alternatively, one of the robots performs a
random walk while the other stays put.
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Universal Traversal Sequences (UTS)
A sequence s ? 1,2,3 is a UTS for (cubic)
graphs of size n if for every connected (cubic)
graph of size at most n, every labeling and every
starting point, the walk defined by s covers the
graph.
Aleliunas, Karp, Lipton, Lovasz, Rackoff (1979)A
random sequence of length O(d2n3log n) is, with
high probability, a UTS for d-regular graphs of
size n.
No efficient construction known!
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A natural solution -
That doesnt quite work
Robot 0 stays put
Robot 1 executes U1U2U4U2kwhere Un is a UTS
for graphs of size n
Fails as robot 0 may be activated when robot 1
is executing Uk, where k?gtgt n.
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Strongly Universal Traversal Sequences (SUTS)
An infinite sequence s ? 1,2,3? is a SUTS for
(cubic) graphs with cover time p(n), if for any
n1, any contiguous subsequence of s of length
p(n) is a UTS for (cubic) graphs of size n.
Main open problem Do SUTS exist?
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Treasure hunt
The version of the rendezvous problem in which
one of the robots is static.
Treasure may be activated after the seeking robot.
In the memoryless model, equivalent to the
existence of SUTS.
We obtain an efficient solution when
backtrackings are allowed.
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Solution of Rendezvous problem
A robot with label uses the
sequence
where Un is a UTS for graphs of size n
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Solution of rendezvous problem
A robot with label uses the
sequence
The k-th block of robot 1Run U2k twice, then
stay put for 2u2k steps.Stay put for u2k steps
between any two steps above.
Theorem The two robots meet after O(Un4)
steps, where n is the size of the environment.
A more complicated solution guarantees a meeting
after O(Un) steps, where Un are the best UTS
currently known
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Universal Exploration Sequences (UES)
Instructions are now interpreted as offsets
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UES are analogous to UTS
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Theorem (Reingold 05) UESs (of polynomial
length) can be constructed in polynomial time.
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Strongly Universal ExplorationSequences (SUES)
Suppose that Unu1u2un is a UES for graphs of
size na, for 0ltalt1.
Suppose that U2i is a prefix of U2j for iltj.
Theorem ??Sn is a SUES
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Open problems
Strongly Universal Traversal Sequences ???
Efficient construction of Universal Traversal
Sequences ???
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Solution of rendezvous problem
A robot with label uses the
sequence
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