CORE
πΊπ¦Β
Β make metadata, not war
Services
Services overview
Explore all CORE services
Access to raw data
API
Dataset
FastSync
Content discovery
Recommender
Discovery
OAI identifiers
OAI Resolver
Managing content
Dashboard
Bespoke contracts
Consultancy services
Support us
Support us
Membership
Sponsorship
Community governance
Advisory Board
Board of supporters
Research network
About
About us
Our mission
Team
Blog
FAQs
Contact us
Connectivity Labeling for Multiple Vertex Failures
Authors
Merav Parter
Asaf Petruschka
Seth Pettie
Publication date
13 July 2023
Publisher
View
on
arXiv
Abstract
We present an efficient labeling scheme for answering connectivity queries in graphs subject to a specified number of vertex failures. Our first result is a randomized construction of a labeling function that assigns vertices
O
(
f
3
log
β‘
5
n
)
O(f^3\log^5 n)
O
(
f
3
lo
g
5
n
)
-bit labels, such that given the labels of
F
βͺ
{
s
,
t
}
F\cup \{s,t\}
F
βͺ
{
s
,
t
}
where
β£
F
β£
β€
f
|F|\leq f
β£
F
β£
β€
f
, we can correctly report, with probability
1
β
1
/
p
o
l
y
(
n
)
1-1/\mathrm{poly}(n)
1
β
1/
poly
(
n
)
, whether
s
s
s
and
t
t
t
are connected in
G
β
F
G-F
G
β
F
. However, it is possible that over all
n
O
(
f
)
n^{O(f)}
n
O
(
f
)
distinct queries, some are answered incorrectly. Our second result is a deterministic labeling function that produces
O
(
f
7
log
β‘
13
n
)
O(f^7 \log^{13} n)
O
(
f
7
lo
g
13
n
)
-bit labels such that all connectivity queries are answered correctly. Both upper bounds are polynomially off from an
Ξ©
(
f
)
\Omega(f)
Ξ©
(
f
)
-bit lower bound. Our labeling schemes are based on a new low degree decomposition that improves the Duan-Pettie decomposition, and facilitates its distributed representation. We make heavy use of randomization to construct hitting sets, fault-tolerant graph sparsifiers, and in constructing linear sketches. Our derandomized labeling scheme combines a variety of techniques: the method of conditional expectations, hit-miss hash families, and
Ο΅
\epsilon
Ο΅
-nets for axis-aligned rectangles. The prior labeling scheme of Parter and Petruschka shows that
f
=
1
f=1
f
=
1
and
f
=
2
f=2
f
=
2
vertex faults can be handled with
O
(
log
β‘
n
)
O(\log n)
O
(
lo
g
n
)
- and
O
(
log
β‘
3
n
)
O(\log^3 n)
O
(
lo
g
3
n
)
-bit labels, respectively, and for
f
>
2
f>2
f
>
2
vertex faults,
O
~
(
n
1
β
1
/
2
f
β
2
)
\tilde{O}(n^{1-1/2^{f-2}})
O
~
(
n
1
β
1/
2
f
β
2
)
-bit labels suffice
Similar works
Full text
Available Versions
arXiv.org e-Print Archive
See this paper in CORE
Go to the repository landing page
Download from data provider
oai:arXiv.org:2307.06276
Last time updated on 14/07/2023