SLIDE 1
Shortest-path oracle
What is the shortest path from A to B?
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Hardness of exact distance queries in sparse graphs through hub - - PowerPoint PPT Presentation
Hardness of exact distance queries in sparse graphs through hub - - PowerPoint PPT Presentation
Hardness of exact distance queries in sparse graphs through hub labeling Adrian Kosowski, Przemysaw Uznaski and Laurent Viennot Inria University of Wrocaw Irif (Paris Univ.) Shortest-path oracle What is the shortest path from A to
SLIDE 2
SLIDE 3
Distance oracle
What is the distance from A to B? Trade-off data-structure size vs query time. Fastest oracles in road networks use hub labeling
[Abraham, Delling, Fiat, Goldberg, Werneck 2016]
Huge gap between lower and upper bounds for sparse graphs. This talk : better understand why.
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SLIDE 4
Distance oracle
What is the distance from A to B? Trade-off data-structure size vs query time. Fastest oracles in road networks use hub labeling
[Abraham, Delling, Fiat, Goldberg, Werneck 2016]
Huge gap between lower and upper bounds for sparse graphs. This talk : better understand why.
⇐ ? ⇒
2 / 5 3 / 14
SLIDE 5
Distance oracle
What is the distance from A to B? Trade-off data-structure size vs query time. Fastest oracles in road networks use hub labeling
[Abraham, Delling, Fiat, Goldberg, Werneck 2016]
Huge gap between lower and upper bounds for sparse graphs. This talk : better understand why.
⇐ ? ⇒
3 / 5 3 / 14
SLIDE 6
Distance oracle
What is the distance from A to B? Trade-off data-structure size vs query time. Fastest oracles in road networks use hub labeling
[Abraham, Delling, Fiat, Goldberg, Werneck 2016]
Huge gap between lower and upper bounds for sparse graphs. This talk : better understand why.
⇐ ? ⇒
4 / 5 3 / 14
SLIDE 7
Distance oracle
What is the distance from A to B? Trade-off data-structure size vs query time. Fastest oracles in road networks use hub labeling
[Abraham, Delling, Fiat, Goldberg, Werneck 2016]
Huge gap between lower and upper bounds for sparse graphs. This talk : better understand why.
⇐ ? ⇒
5 / 5 3 / 14
SLIDE 8
Distance oracles
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SLIDE 9
Distance labelings
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SLIDE 10
Motivation : understand gaps for sparse graphs
Sparse : m = O(n) or ∆ = O(1) 1/ Ω(√n) ≤ DistLab(n) ≤ O(n log log n
log n )
2/ Ω(
√n log n) ≤ HubLab(n) ≤ O( n log n)
[Gavoille, Peleg, Pérennes, Raz 2004] [Alstrup, Dahlgaard, Bæk, Knudsen 2016] [Gawrychowski, Kosovski, Uznański 2016]
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SLIDE 11
Our results (∆ = O(1))
1/
1 2O(√
log n) SumIndex(n) ≤ DistLab(n)
Ω( √ n) ≤ SumIndex(n) ≤ O( n 2 √
log n )
[Nisan, Wigderson 1993] [Babai, Gal, Kimmel, Lokan 1995, 2003] [Pudlak, Rodl, Sgall 1997]
2/
n 2O(√
log n) ≤ HubLab(n) ≤ O(
n RS(n)1/7 )
2Ωlog∗ n ≤ RS(n) ≤ 2O(√
log n)
[Ruzsa, Szemerédi 1978] [Behrand 1946] [Elkin 2010] [Fox 2011]
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SLIDE 12
Our results (∆ = O(1))
1/
1 2O(√
log n) SumIndex(n) ≤ DistLab(n)
Ω( √ n) ≤ SumIndex(n) ≤ O( n 2 √
log n )
[Nisan, Wigderson 1993] [Babai, Gal, Kimmel, Lokan 1995, 2003] [Pudlak, Rodl, Sgall 1997]
2/
n 2O(√
log n) ≤ HubLab(n) ≤ O(
n RS(n)1/7 )
2Ωlog∗ n ≤ RS(n) ≤ 2O(√
log n)
[Ruzsa, Szemerédi 1978] [Behrand 1946] [Elkin 2010] [Fox 2011]
⇐ ? ⇒
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SLIDE 13
Our results (∆ = O(1))
1/
1 2O(√
log n) SumIndex(n) ≤ DistLab(n)
Ω( √ n) ≤ SumIndex(n) ≤ O( n 2 √
log n )
[Nisan, Wigderson 1993] [Babai, Gal, Kimmel, Lokan 1995, 2003] [Pudlak, Rodl, Sgall 1997]
2/
n 2O(√
log n) ≤ HubLab(n) ≤ O(
n RS(n)1/7 )
2Ωlog∗ n ≤ RS(n) ≤ 2O(√
log n)
[Ruzsa, Szemerédi 1978] [Behrand 1946] [Elkin 2010] [Fox 2011]
⇐ ? ⇒
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SLIDE 14
A hard instance : 2ℓ + 1 grids of dim. ℓ = √ log n
(1,0) (3,2) (2,1)
V V V
l 2l
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SLIDE 15
Connection with Ruzsa-Szemerédi
RS-graph : can be decomposed into n induced matchings.
n2 RS(n) is the maximum number of edges in an RS-graph.
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SLIDE 16
(1,0) (3,2) (2,1)
V V V
l 2l
GD
y =
{ x0z2ℓ | y = x+z
2
and dG(x, z) = D } ∃Ds.t.| ∪y GD
y | ≥ n2 2O(√
log n) ⇐ ? ⇒
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SLIDE 17
Converse
Any cst. deg. graph G has hub sets of av. size O(
n RS(n)1/7 ).
Idea : use a vertex cover of each GD
y (VC ≤ 2MM).
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SLIDE 18
Connection with SumIndex
SumIndex(n) = minEncoder maxX |MA| + |MB|
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SLIDE 19
(1,0) (3,2) (2,1)
V V V
l 2l
GX = G \ { yℓ | Xy = 0 } , send x = 2a, Lx0, z = 2b, Lz2ℓ, check d(x0, z2ℓ).
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SLIDE 20
Thanks
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