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Path-Reporting Distance Oracles with Near-Logarithmic Stretch and Linear Size
Authors
Michael Elkin
Idan Shabat
Publication date
11 April 2023
Publisher
View
on
arXiv
Abstract
Given an
n
n
n
-vertex undirected graph
G
=
(
V
,
E
,
w
)
G=(V,E,w)
G
=
(
V
,
E
,
w
)
, and a parameter
k
β₯
1
k\geq1
k
β₯
1
, a path-reporting distance oracle (or PRDO) is a data structure of size
S
(
n
,
k
)
S(n,k)
S
(
n
,
k
)
, that given a query
(
u
,
v
)
β
V
2
(u,v)\in V^2
(
u
,
v
)
β
V
2
, returns an
f
(
k
)
f(k)
f
(
k
)
-approximate shortest
u
β
v
u-v
u
β
v
path
P
P
P
in
G
G
G
within time
q
(
k
)
+
O
(
β£
P
β£
)
q(k)+O(|P|)
q
(
k
)
+
O
(
β£
P
β£
)
. Here
S
(
n
,
k
)
S(n,k)
S
(
n
,
k
)
,
f
(
k
)
f(k)
f
(
k
)
and
q
(
k
)
q(k)
q
(
k
)
are arbitrary functions. A landmark PRDO due to Thorup and Zwick, with an improvement of Wulff-Nilsen, has
S
(
n
,
k
)
=
O
(
k
β
n
1
+
1
k
)
S(n,k)=O(k\cdot n^{1+\frac{1}{k}})
S
(
n
,
k
)
=
O
(
k
β
n
1
+
k
1
β
)
,
f
(
k
)
=
2
k
β
1
f(k)=2k-1
f
(
k
)
=
2
k
β
1
and
q
(
k
)
=
O
(
log
β‘
k
)
q(k)=O(\log k)
q
(
k
)
=
O
(
lo
g
k
)
. The size of this oracle is
Ξ©
(
n
log
β‘
n
)
\Omega(n\log n)
Ξ©
(
n
lo
g
n
)
for all
k
k
k
. Elkin and Pettie and Neiman and Shabat devised much sparser PRDOs, but their stretch was polynomially larger than the optimal
2
k
β
1
2k-1
2
k
β
1
. On the other hand, for non-path-reporting distance oracles, Chechik devised a result with
S
(
n
,
k
)
=
O
(
n
1
+
1
k
)
S(n,k)=O(n^{1+\frac{1}{k}})
S
(
n
,
k
)
=
O
(
n
1
+
k
1
β
)
,
f
(
k
)
=
2
k
β
1
f(k)=2k-1
f
(
k
)
=
2
k
β
1
and
q
(
k
)
=
O
(
1
)
q(k)=O(1)
q
(
k
)
=
O
(
1
)
. In this paper we make a dramatic progress in bridging the gap between path-reporting and non-path-reporting distance oracles. We devise a PRDO with size
S
(
n
,
k
)
=
O
(
β
k
log
β‘
log
β‘
n
log
β‘
n
β
β
n
1
+
1
k
)
S(n,k)=O(\lceil\frac{k\log\log n}{\log n}\rceil\cdot n^{1+\frac{1}{k}})
S
(
n
,
k
)
=
O
(β
l
o
g
n
k
l
o
g
l
o
g
n
β
β
β
n
1
+
k
1
β
)
, stretch
f
(
k
)
=
O
(
k
)
f(k)=O(k)
f
(
k
)
=
O
(
k
)
and query time
q
(
k
)
=
O
(
log
β‘
β
k
log
β‘
log
β‘
n
log
β‘
n
β
)
q(k)=O(\log\lceil\frac{k\log\log n}{\log n}\rceil)
q
(
k
)
=
O
(
lo
g
β
l
o
g
n
k
l
o
g
l
o
g
n
β
β)
. We can also have size
O
(
n
1
+
1
k
)
O(n^{1+\frac{1}{k}})
O
(
n
1
+
k
1
β
)
, stretch
O
(
k
β
β
k
log
β‘
log
β‘
n
log
β‘
n
β
)
O(k\cdot\lceil\frac{k\log\log n}{\log n}\rceil)
O
(
k
β
β
l
o
g
n
k
l
o
g
l
o
g
n
β
β)
and query time
q
(
k
)
=
O
(
log
β‘
β
k
log
β‘
log
β‘
n
log
β‘
n
β
)
q(k)=O(\log\lceil\frac{k\log\log n}{\log n}\rceil)
q
(
k
)
=
O
(
lo
g
β
l
o
g
n
k
l
o
g
l
o
g
n
β
β)
. Our results on PRDOs are based on novel constructions of approximate distance preservers, that we devise in this paper. Specifically, we show that for any
Ο΅
>
0
\epsilon>0
Ο΅
>
0
, any
k
=
1
,
2
,
.
.
.
k=1,2,...
k
=
1
,
2
,
...
, and any graph
G
G
G
and a collection
P
\mathcal{P}
P
of
p
p
p
vertex pairs, there exists a
(
1
+
Ο΅
)
(1+\epsilon)
(
1
+
Ο΅
)
-approximate preserver with
O
(
Ξ³
(
Ο΅
,
k
)
β
p
+
n
log
β‘
k
+
n
1
+
1
k
)
O(\gamma(\epsilon,k)\cdot p+n\log k+n^{1+\frac{1}{k}})
O
(
Ξ³
(
Ο΅
,
k
)
β
p
+
n
lo
g
k
+
n
1
+
k
1
β
)
edges, where
Ξ³
(
Ο΅
,
k
)
=
(
log
β‘
k
Ο΅
)
O
(
log
β‘
k
)
\gamma(\epsilon,k)=(\frac{\log k}{\epsilon})^{O(\log k)}
Ξ³
(
Ο΅
,
k
)
=
(
Ο΅
l
o
g
k
β
)
O
(
l
o
g
k
)
. These new preservers are significantly sparser than the previous state-of-the-art approximate preservers due to Kogan and Parter.Comment: 61 pages, 3 figure
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Last time updated on 14/04/2023