Online Graph Exploration with Advice

online graph exploration
lower bound
advice
optimality
Online Graph Exploration with
Advice
Stefan Dobrev1 and Rastislav Královič2 and
Euripides Markou3
Slovak Academy of Sciences, Bratislava.
Stefan.Dobrev@savba.sk
Comenius University, Bratislava.
kralovic@dcs.fmph.uniba.sk
University of Central Greece, Lamia.
emarkou@ucg.gr
ACAC 2012, Athens
algorithm
online graph exploration
Graph exploration
lower bound
advice
optimality
algorithm
online graph exploration
lower bound
advice
Graph exploration
general statement
Given a ... graph,
find a ... closed walk that visits all ... .
optimality
algorithm
online graph exploration
lower bound
advice
Graph exploration
our case
Given a weighted undirected graph,
find a shortest closed walk that visits all vertices.
optimality
algorithm
online graph exploration
lower bound
advice
problem statement
Given a weighted undirected graph,
find a shortest closed walk that visits all vertices.
optimality
algorithm
online graph exploration
lower bound
advice
problem statement
Given a weighted undirected graph,
find a shortest closed walk that visits all vertices.
optimality
algorithm
online graph exploration
lower bound
advice
optimality
algorithm
problem statement
Given a weighted undirected graph,
find a shortest closed walk that visits all vertices.
equivalent to TSP in the metric closure (vertices may repeat)
online graph exploration
lower bound
advice
optimality
algorithm
problem statement
Given a weighted undirected graph,
find a shortest closed walk that visits all vertices.
equivalent to TSP in the metric closure (vertices may repeat)
2·MST is 2-approximation
online graph exploration
online setting: agent
lower bound
advice
optimality
algorithm
online graph exploration
lower bound
online setting: agent
can move
advice
optimality
algorithm
online graph exploration
lower bound
online setting: agent
can not write
can move
advice
optimality
algorithm
online graph exploration
lower bound
online setting: agent
has (polynomial) memory
can not write
can move
advice
optimality
algorithm
online graph exploration
lower bound
advice
optimality
online setting: agent
has (polynomial) memory
can see
can not write
can move
agent in vertex v sees
algorithm
online graph exploration
lower bound
advice
optimality
online setting: agent
has (polynomial) memory
can see
can not write
can move
agent in vertex v sees
unique ID
algorithm
online graph exploration
lower bound
advice
optimality
online setting: agent
has (polynomial) memory
can see
can not write
can move
agent in vertex v sees
unique ID
weights
algorithm
online graph exploration
lower bound
advice
optimality
online setting: agent
has (polynomial) memory
can see
can not write
can move
agent in vertex v sees
unique ID
weights
neighbors’ IDs
algorithm
online graph exploration
lower bound
advice
optimality
algorithm
online graph exploration
traversed 10
lower bound
advice
optimality
algorithm
online graph exploration
traversed 20
lower bound
advice
optimality
algorithm
online graph exploration
traversed 45
lower bound
advice
optimality
algorithm
online graph exploration
traversed 55
lower bound
advice
optimality
algorithm
online graph exploration
traversed 60
lower bound
advice
optimality
algorithm
online graph exploration
traversed 61
lower bound
advice
optimality
algorithm
online graph exploration
traversed 62
lower bound
advice
optimality
algorithm
online graph exploration
traversed 67
lower bound
advice
optimality
algorithm
online graph exploration
traversed 97
lower bound
advice
optimality
algorithm
online graph exploration
traversed 107
lower bound
advice
optimality
algorithm
online graph exploration
lower bound
advice
optimality
main question
What is the (worst case) length of the agent’s traversal
compared to (offline) optimum?
Is there a constant competitive algorithm?
algorithm
online graph exploration
lower bound
advice
optimality
algorithm
main question
What is the (worst case) length of the agent’s traversal
compared to (offline) optimum?
Is there a constant competitive algorithm?
what has been known about competitive ratio
[Rosenkranz et al., 1977] NN: Θ(log n) even on unweighted planar
[Myazaki et al., 2009] cycles:
√
1+ 3
2
≈ 1.366, unweighted graphs: 2
[Kalyanasundaram et al., 1994] planar: 16-competitive (general?)
[Megow et al., 2011] K+ algorithm is not constant competitive
genus g : 16(1 + 2g )-competitive
k distinct weights: 2k-competitive
online graph exploration
lower bound
advice
optimality
first task
Improve the 2 − ε lower bound from [Myazaki et al., 2009]
algorithm
online graph exploration
lower bound
advice
optimality
first task
Improve the 2 − ε lower bound from [Myazaki et al., 2009]
we got
Any algorithm A is at least
5
2
− ε competitive on some graph.
algorithm
online graph exploration
lower bound
advice
optimality
we got
Any algorithm A is at least
5
2
− ε competitive on some graph.
rough idea
algorithm
online graph exploration
lower bound
advice
optimality
algorithm
we got
Any algorithm A is at least
5
2
− ε competitive on some graph.
rough idea
fixed movement
online graph exploration
lower bound
advice
optimality
algorithm
we got
Any algorithm A is at least
5
2
− ε competitive on some graph.
rough idea
fixed movement
block of size x
online graph exploration
lower bound
advice
optimality
algorithm
we got
Any algorithm A is at least
5
2
− ε competitive on some graph.
rough idea
fixed movement
block of size x
orientation
online graph exploration
lower bound
advice
optimality
algorithm
we got
Any algorithm A is at least
5
2
− ε competitive on some graph.
rough idea
fixed movement
block of size x
orientation
additional edges
online graph exploration
lower bound
advice
optimality
algorithm
we got
Any algorithm A is at least
5
2
− ε competitive on some graph.
rough idea
fixed movement
block of size x
orientation
additional edges
must traverse
5
2 (x − 1)
online graph exploration
lower bound
advice
optimality
algorithm
we got
Any algorithm A is at least
5
2
− ε competitive on some graph.
rough idea
fixed movement
block of size x
orientation
additional edges
must traverse
5
2 (x − 1)
ratio with x blocks:
online graph exploration
lower bound
advice
optimality
algorithm
we got
Any algorithm A is at least
5
2
− ε competitive on some graph.
rough idea
fixed movement
block of size x
orientation
additional edges
must traverse
5
2 (x − 1)
ratio with x blocks:
5
(x−1)
2
online graph exploration
lower bound
advice
optimality
algorithm
we got
Any algorithm A is at least
5
2
− ε competitive on some graph.
rough idea
fixed movement
block of size x
orientation
additional edges
must traverse
5
2 (x − 1)
ratio with x blocks:
(x−2) 52 (x−1)
online graph exploration
lower bound
advice
optimality
algorithm
we got
Any algorithm A is at least
5
2
− ε competitive on some graph.
rough idea
fixed movement
block of size x
orientation
additional edges
must traverse
5
2 (x − 1)
ratio with x blocks:
(x−2) 52 (x−1)+2(x−1)
online graph exploration
lower bound
advice
optimality
algorithm
we got
Any algorithm A is at least
5
2
− ε competitive on some graph.
rough idea
fixed movement
block of size x
orientation
additional edges
must traverse
5
2 (x − 1)
ratio with x blocks:
(x−2) 52 (x−1)+2(x−1)+x(x−2)
online graph exploration
lower bound
advice
optimality
algorithm
we got
Any algorithm A is at least
5
2
− ε competitive on some graph.
rough idea
fixed movement
block of size x
orientation
additional edges
must traverse
5
2 (x − 1)
ratio with x blocks:
(x−2) 52 (x−1)+2(x−1)+x(x−2)
x(x−1)+x(x−2)
online graph exploration
lower bound
advice
optimality
algorithm
we got
Any algorithm A is at least
5
2
− ε competitive on some graph.
rough idea
fixed movement
block of size x
orientation
additional edges
must traverse
5
2 (x − 1)
ratio with x blocks:
(x−2) 52 (x−1)+2(x−1)+x(x−2)
x(x−1)+x(x−2)
≈
7
4
online graph exploration
lower bound
advice
refinement: hierarchical construction
r1
t1
o1
e1
ẽ1
=
=
=
≥
≥
1
(x − 1)
2
x −2
x −1
5
(x − 1)
2
x −1
ratio x blocks, k levels:
(x−5)ek +5ẽk +xtk
xok +xtk
optimality
algorithm
online graph exploration
lower bound
advice
optimality
algorithm
refinement: hierarchical construction
r1
t1
o1
e1
ẽ1
=
=
=
≥
≥
1
(x − 1)
2
x −2
x −1
5
(x − 1)
2
x −1
ratio x blocks, k levels:
ri+1
ti+1
oi+1
ei+1
ẽi+1
=
=
=
=
≥
1
(x + 1)ri + 12 (x − 1)ti
2
(x − 1)ti + xri
xoi + (x − 1)ti
(x − 4)ei + 4ẽi + 52 (x − 1)ti +
(x − 5)ei + 5ẽi + (x − 1)ti
(x−5)ek +5ẽk +xtk
xok +xtk
≈
5
2
√
− 2−O(
log n)
3x−1
ri
2
online graph exploration
lower bound
advice complexity
online setting: agent
has (polynomial) memory
can see
can not write
can move
advice
optimality
algorithm
online graph exploration
lower bound
advice
optimality
algorithm
advice complexity
online setting: agent
advice
given to the agent at start
function of input graph
advice
01100101
s-bit binary string
”relevant” topology information
has (polynomial) memory
can see
can not write
can move
online graph exploration
lower bound
advice
optimality
algorithm
advice complexity
online setting: agent
advice
given to the agent at start
function of input graph
advice
01100101
s-bit binary string
”relevant” topology information
has (polynomial) memory
can see
can not write
can move
advice size vs. solution quality
online graph exploration
lower bound
advice
optimality
algorithm
lower bound on advice for optimality
Any optimal algorithm requires Ω(n log n) bits in the worst case.
w (vi , vj ) = 4 − min{i, j}
online graph exploration
lower bound
advice
optimality
algorithm
lower bound on advice for optimality
Any optimal algorithm requires Ω(n log n) bits in the worst case.
w (vi , vj ) = 4 − min{i, j}
unique optimal solution
online graph exploration
lower bound
advice
optimality
algorithm
lower bound on advice for optimality
Any optimal algorithm requires Ω(n log n) bits in the worst case.
w (vi , vj ) = 4 − min{i, j}
unique optimal solution
agent needs log n advice
online graph exploration
lower bound
advice
optimality
algorithm
lower bound on advice for optimality
Any optimal algorithm requires Ω(n log n) bits in the worst case.
w (vi , vj ) = 4 − min{i, j}
unique optimal solution
agent needs log n advice
cannot distinguish next
problem: reversal
online graph exploration
lower bound
advice
optimality
algorithm
lower bound on advice for optimality
Any optimal algorithm requires Ω(n log n) bits in the worst case.
v5
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v7 1 v70
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online graph exploration
lower bound
advice
optimality
algorithm
There is a constant-competitive algorithm with linear advice.
rough idea
traverse some tree
algorithm
online graph exploration
lower bound
advice
optimality
algorithm
There is a constant-competitive algorithm with linear advice.
rough idea
traverse some tree
DFS has a problem
algorithm
online graph exploration
lower bound
advice
optimality
algorithm
There is a constant-competitive algorithm with linear advice.
rough idea
traverse some tree
DFS has a problem
MST of weight M
cheap edge ≤
M
n
it is always safe to explore a cheap edge
algorithm
online graph exploration
lower bound
advice
optimality
algorithm
algorithm
There is a constant-competitive algorithm with linear advice.
rough idea
traverse some tree
DFS has a problem
MST of weight M
cheap edge ≤
M
n
it is always safe to explore a cheap edge
advice tells the value dlog( M
n )e (M is unbounded, but can be done
using O(log n) bits and traversing O(M) total cost.)
online graph exploration
lower bound
advice
optimality
algorithm
Advice tells the value dlog( Mn )e by encoding: (n, n0 , p, l 0 )
Search for the first encountered edge e with w (e) ∈ [ nM2 , M]
keep traversing the cheapest outgoing edge until n0 −th vertex
is encountered,
consider the p−th incident edge e and let w (e) be its weight,
dlog(M)e = dlog(w (e))e + l 0
(n0 , p, l 0 ) are chosen in such way that e has the right property.
O(log n) bits are sufficient to encode n, n0 , p, l 0 ; Cost: O(M)
Such an edge e must exist (otherwise the MST weight < M).
l 0 ≤ dlog(M)e − dlog( nM2 )e ≤ 2 log n
at most n cheapest ( nM2 ) outgoing edges; the cost to reach
each one is at most O( M
n ).
online graph exploration
lower bound
advice
optimality
cheap edges always explored by DFS
cheap clusters connected with expensive edges
algorithm
online graph exploration
lower bound
advice
optimality
cheap edges always explored by DFS
cheap clusters connected with expensive edges
some expensive edges are tree edges (i.e. from MST)
advice must tell which ones in an efficient way
algorithm
online graph exploration
lower bound
cluster edges
level 0: cheap
level i: w (e) ≤ 2i M
n
≤
n
2i
level-i edges in MST
advice
optimality
algorithm
online graph exploration
lower bound
cluster edges
level 0: cheap
level i: w (e) ≤ 2i M
n
≤
n
2i
level-i edges in MST
advice
optimality
identify tree edges
levels in parallel
separate advice for levels
O(log i) bits per i-edge
algorithm
online graph exploration
lower bound
cluster edges
level 0: cheap
level i: w (e) ≤ 2i M
n
≤
n
2i
level-i edges in MST
all i-edges out: OUT
advice
optimality
identify tree edges
levels in parallel
separate advice for levels
O(log i) bits per i-edge
algorithm
online graph exploration
lower bound
cluster edges
level 0: cheap
level i: w (e) ≤ 2i M
n
≤
n
2i
level-i edges in MST
all i-edges out: OUT
in edges: WAIT
advice
optimality
identify tree edges
levels in parallel
separate advice for levels
O(log i) bits per i-edge
algorithm
online graph exploration
lower bound
cluster edges
level 0: cheap
level i: w (e) ≤ 2i M
n
≤
n
2i
level-i edges in MST
all i-edges out: OUT
in edges: WAIT
later TRIGGER
advice
optimality
identify tree edges
levels in parallel
separate advice for levels
O(log i) bits per i-edge
algorithm
online graph exploration
lower bound
cluster edges
level 0: cheap
level i: w (e) ≤ 2i M
n
≤
n
2i
level-i edges in MST
all i-edges out: OUT
in edges: WAIT
later TRIGGER
managing multiple triggers is
the core of the algorithm
advice
optimality
identify tree edges
levels in parallel
separate advice for levels
O(log i) bits per i-edge
algorithm
online graph exploration
lower bound
advice
optimality
open problems
so is there a constant competitive algorithm or not?
improve the lower bound
5
2
−ε
any general algorithm better than O(log n)?
what can be done with polylogarithmic advice? or o(n)?
lower bounds on advice / trade-off
algorithm
online graph exploration
lower bound
advice
optimality
open problems
so is there a constant competitive algorithm or not?
improve the lower bound
5
2
−ε
any general algorithm better than O(log n)?
what can be done with polylogarithmic advice? or o(n)?
lower bounds on advice / trade-off
The End
algorithm