Multiple-Edge-Fault-Tolerant Approximate Shortest-Path Trees
Let G be an n -node and m -edge positively real-weighted undirected graph. For any given integer f ≥ 1 , we study the problem of designing a sparse f -edge-fault-tolerant ( f -EFT) σ -approximate single-source shortest-path tree ( σ -ASPT), namely a subgraph of G having as few edges as possible and...
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| Vydáno v: | Algorithmica Ročník 84; číslo 1; s. 37 - 59 |
|---|---|
| Hlavní autoři: | , , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
New York
Springer US
01.01.2022
Springer Nature B.V |
| Témata: | |
| ISSN: | 0178-4617, 1432-0541 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | Let
G
be an
n
-node and
m
-edge positively real-weighted undirected graph. For any given integer
f
≥
1
, we study the problem of designing a sparse
f
-edge-fault-tolerant (
f
-EFT)
σ
-approximate single-source shortest-path tree (
σ
-ASPT), namely a subgraph of
G
having as few edges as possible and which, following the failure of a set
F
of at most
f
edges in
G
, contains paths from a fixed source that are stretched by a factor of at most
σ
. To this respect, we provide an algorithm that efficiently computes an
f
-EFT
(
2
|
F
|
+
1
)
-ASPT of size
O
(
fn
). Our structure improves on a previous related construction designed for
unweighted
graphs, having the same size but guaranteeing a larger stretch factor of
3
(
f
+
1
)
, plus an additive term of
(
f
+
1
)
log
n
. Then, we show how to convert our structure into an efficient
f
-EFT single-source distance oracle, that can be built in
O
(
f
m
α
(
m
,
n
)
+
f
n
log
3
n
)
time, has size
O
(
f
n
log
2
n
)
, and in
O
(
|
F
|
2
log
2
n
)
time is able to report a
(
2
|
F
|
+
1
)
-approximate distance from the source to any node in
G
-
F
. Moreover, our oracle can return a corresponding approximate path in the same amount of time plus the path’s size. The oracle is obtained by tackling another fundamental problem, namely that of updating a minimum spanning forest (MSF) of
G
following a
batch
of
k
simultaneous modification (i.e., edge insertions, deletions and weight changes). For this problem, we build in
O
(
m
log
3
n
)
time an oracle of size
O
(
m
log
2
n
)
, that reports in
O
(
k
2
log
2
n
)
time the (at most 2
k
) edges either exiting from or entering into the MSF. Finally, for any integer
k
≥
1
, we complement all our results with a lower bound of
Ω
n
1
+
1
k
to the size of any
f
-EFT
σ
-ASPT with
f
≥
log
n
and
σ
<
3
k
+
1
k
+
1
, that holds if the Erdős’ girth conjecture is true. |
|---|---|
| Bibliografie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0178-4617 1432-0541 |
| DOI: | 10.1007/s00453-021-00879-8 |