Parameterized Complexity of Streaming Diameter and Connectivity Problems
We initiate the investigation of the parameterized complexity of Diameter and Connectivity in the streaming paradigm. On the positive end, we show that knowing a vertex cover of size k allows for algorithms in the Adjacency List (AL) streaming model whose number of passes is constant and memory is O...
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| Vydáno v: | Algorithmica Ročník 86; číslo 9; s. 2885 - 2928 |
|---|---|
| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
New York
Springer US
01.09.2024
Springer Nature B.V |
| Témata: | |
| ISSN: | 0178-4617, 1432-0541 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | We initiate the investigation of the parameterized complexity of
Diameter
and
Connectivity
in the streaming paradigm. On the positive end, we show that knowing a vertex cover of size
k
allows for algorithms in the Adjacency List (AL) streaming model whose number of passes is constant and memory is
O
(
log
n
)
for any fixed
k
. Underlying these algorithms is a method to execute a breadth-first search in
O
(
k
)
passes and
O
(
k
log
n
)
bits of memory. On the negative end, we show that many other parameters lead to lower bounds in the AL model, where
Ω
(
n
/
p
)
bits of memory is needed for any
p
-pass algorithm even for constant parameter values. In particular, this holds for graphs with a known modulator (deletion set) of constant size to a graph that has no induced subgraph isomorphic to a fixed graph
H
, for most
H
. For some cases, we can also show one-pass,
Ω
(
n
log
n
)
bits of memory lower bounds. We also prove a much stronger
Ω
(
n
2
/
p
)
lower bound for
Diameter
on bipartite graphs. Finally, using the insights we developed into streaming parameterized graph exploration algorithms, we show a new streaming kernelization algorithm for computing a vertex cover of size
k
. This yields a kernel of 2
k
vertices (with
O
(
k
2
)
edges) produced as a stream in
poly
(
k
)
passes and only
O
(
k
log
n
)
bits of memory. |
|---|---|
| Bibliografie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0178-4617 1432-0541 |
| DOI: | 10.1007/s00453-024-01246-z |