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 |
|---|---|
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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Springer US
01.09.2024
Springer Nature B.V |
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| ISSN: | 0178-4617, 1432-0541 |
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| Abstract | 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. |
|---|---|
| AbstractList | 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
$$\mathcal {O}(\log n)$$
O
(
log
n
)
for any fixed
k
. Underlying these algorithms is a method to execute a breadth-first search in
$$\mathcal {O}(k)$$
O
(
k
)
passes and
$$\mathcal {O}(k \log n)$$
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
$$\Omega (n/p)$$
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,
$$\Omega (n \log n)$$
Ω
(
n
log
n
)
bits of memory lower bounds. We also prove a much stronger
$$\Omega (n^2/p)$$
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
$$\mathcal {O}(k^2)$$
O
(
k
2
)
edges) produced as a stream in
$$\text {poly}(k)$$
poly
(
k
)
passes and only
$$\mathcal {O}(k \log n)$$
O
(
k
log
n
)
bits of memory. 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. 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(logn) for any fixed k. Underlying these algorithms is a method to execute a breadth-first search in O(k) passes and O(klogn) 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(nlogn) bits of memory lower bounds. We also prove a much stronger Ω(n2/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 2k vertices (with O(k2) edges) produced as a stream in poly(k) passes and only O(klogn) bits of memory. |
| Author | Oostveen, Jelle J. van Leeuwen, Erik Jan |
| Author_xml | – sequence: 1 givenname: Jelle J. surname: Oostveen fullname: Oostveen, Jelle J. email: j.j.oostveen@uu.nl organization: Department Information and Computing Sciences, Utrecht University – sequence: 2 givenname: Erik Jan surname: van Leeuwen fullname: van Leeuwen, Erik Jan email: e.j.vanleeuwen@uu.nl organization: Department Information and Computing Sciences, Utrecht University |
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| Keywords | Streaming Permutation Vertex cover Graphs Parameter Connectivity Stream Diameter Disjointness Complexity |
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| Snippet | We initiate the investigation of the parameterized complexity of
Diameter
and
Connectivity
in the streaming paradigm. On the positive end, we show that knowing... We initiate the investigation of the parameterized complexity of Diameter and Connectivity in the streaming paradigm. On the positive end, we show that knowing... |
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| SubjectTerms | Algorithm Analysis and Problem Complexity Algorithms Apexes Complexity Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Graph theory Graphs Lower bounds Mathematics of Computing Parameterization Parameters Theory of Computation |
| Title | Parameterized Complexity of Streaming Diameter and Connectivity Problems |
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