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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| Veröffentlicht in: | Algorithmica Jg. 84; H. 1; S. 37 - 59 |
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| Format: | Journal Article |
| Sprache: | Englisch |
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01.01.2022
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| Abstract | 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. |
|---|---|
| AbstractList | 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. 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)logn. Then, we show how to convert our structure into an efficient f-EFT single-source distance oracle, that can be built in O(fmα(m,n)+fnlog3n) time, has size O(fnlog2n), and in O(|F|2log2n) 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(mlog3n) time an oracle of size O(mlog2n), that reports in O(k2log2n) time the (at most 2k) 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 Ωn1+1k to the size of any f-EFT σ-ASPT with f≥logn and σ<3k+1k+1, that holds if the Erdős’ girth conjecture is true. |
| Author | Gualà, Luciano Bilò, Davide Leucci, Stefano Proietti, Guido |
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| Cites_doi | 10.1137/1.9781611973105.36 10.1006/jagm.2001.1171 10.4230/LIPIcs.ESA.2016.13 10.1007/s00453-015-0006-x 10.1145/1044731.1044732 10.1007/s00453-011-9543-0 10.1145/2888397 10.1145/2767386.2767408 10.1007/s00453-012-9621-y 10.1137/16M1087643 10.4230/LIPIcs.ICALP.2017.127 10.1137/0213024 10.1007/978-3-662-48350-3_15 10.1145/3293611.3331588 10.1137/1.9781611975482.127 10.1016/j.cosrev.2020.100253 10.1016/j.tcs.2015.02.036 10.1007/s00446-015-0252-9 10.1109/FOCS.2012.17 10.1145/502090.502095 10.1007/s00453-002-0988-z 10.4230/LIPIcs.ICALP.2018.72 10.1145/1103963.1103966 10.1145/2591796.2591801 10.1137/0214055 10.1145/2976741 10.1006/jagm.1994.1033 10.1515/crll.1869.70.185 10.4230/LIPIcs.STACS.2018.13 10.1007/BF02189308 10.1145/355541.355562 10.1137/S0097539797327209 10.1007/978-3-030-25209-0 10.1007/s00453-017-0396-z 10.1145/1536414.1536431 10.1145/265910.265914 10.1137/1.9781611973068.56 10.1145/3022730 10.1145/1868237.1868242 10.1007/978-3-319-25258-2_16 |
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| Keywords | 68P05 68W40 Fully-dynamic minimum spanning tree Distance sensitivity oracle Multiple-edge fault-tolerant shortest-path tree |
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| Snippet | 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... 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... |
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| SubjectTerms | Algorithm Analysis and Problem Complexity Algorithms Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Fault tolerance Graph theory Integers Lower bounds Mathematics of Computing Shortest-path problems Theory of Computation Trees (mathematics) |
| Title | Multiple-Edge-Fault-Tolerant Approximate Shortest-Path Trees |
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