f-Sensitivity Distance Oracles and Routing Schemes
An f-sensitivity distance oracle for a weighted undirected graph G ( V , E ) is a data structure capable of answering restricted distance queries between vertex pairs, i.e., calculating distances on a subgraph avoiding some forbidden edges. This paper presents an efficiently constructible f -sensiti...
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| Published in: | Algorithmica Vol. 63; no. 4; pp. 861 - 882 |
|---|---|
| Main Authors: | , , , |
| Format: | Journal Article Conference Proceeding |
| Language: | English |
| Published: |
New York
Springer-Verlag
01.08.2012
Springer |
| Subjects: | |
| ISSN: | 0178-4617, 1432-0541 |
| Online Access: | Get full text |
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| Summary: | An
f-sensitivity distance oracle
for a weighted undirected graph
G
(
V
,
E
) is a data structure capable of answering restricted distance queries between vertex pairs, i.e., calculating distances on a subgraph avoiding some forbidden edges. This paper presents an efficiently constructible
f
-sensitivity distance oracle that given a triplet (
s
,
t
,
F
), where
s
and
t
are vertices and
F
is a set of forbidden edges such that |
F
|≤
f
, returns an estimate of the distance between
s
and
t
in
G
(
V
,
E
∖
F
). For an integer parameter
k
≥1, the size of the data structure is
O
(
fkn
1+1/
k
log (
nW
)), where
W
is the heaviest edge in
G
, the stretch (approximation ratio) of the returned distance is (8
k
−2)(
f
+1), and the query time is
O
(|
F
|⋅log
2
n
⋅log log
n
⋅log log
d
), where
d
is the distance between
s
and
t
in
G
(
V
,
E
∖
F
).
Our result differs from previous ones in two major respects: (1) it is the first to consider
approximate
oracles for general graphs (and thus obtain a succinct data structure); (2) our result holds for an arbitrary number of forbidden edges. In contrast, previous papers concern
f
-sensitive
exact
distance oracles, which consequently have size Ω(
n
2
). Moreover, those oracles support forbidden sets
F
of size |
F
|≤2.
The paper also considers
f
-sensitive compact routing schemes, namely, routing schemes that avoid a given set of forbidden (or
failed
) edges. It presents a scheme capable of withstanding up to two edge failures. Given a message
M
destined to
t
at a source vertex
s
, in the presence of a forbidden edge set
F
of size |
F
|≤2 (unknown to
s
), our scheme routes
M
from
s
to
t
in a distributed manner, over a path of length at most
O
(
k
) times the length of the optimal path (avoiding
F
). The total amount of information stored in vertices of
G
is
O
(
kn
1+1/
k
log (
nW
)log
n
). To the best of our knowledge, this is the first result obtaining an
f
-sensitive compact routing scheme for general graphs. |
|---|---|
| ISSN: | 0178-4617 1432-0541 |
| DOI: | 10.1007/s00453-011-9543-0 |