24,765 research outputs found
Deterministic Edge Connectivity in Near-Linear Time
We present a deterministic near-linear time algorithm that computes the
edge-connectivity and finds a minimum cut for a simple undirected unweighted
graph G with n vertices and m edges. This is the first o(mn) time deterministic
algorithm for the problem. In near-linear time we can also construct the
classic cactus representation of all minimum cuts.
The previous fastest deterministic algorithm by Gabow from STOC'91 took
~O(m+k^2 n), where k is the edge connectivity, but k could be Omega(n).
At STOC'96 Karger presented a randomized near linear time Monte Carlo
algorithm for the minimum cut problem. As he points out, there is no better way
of certifying the minimality of the returned cut than to use Gabow's slower
deterministic algorithm and compare sizes.
Our main technical contribution is a near-linear time algorithm that contract
vertex sets of a simple input graph G with minimum degree d, producing a
multigraph with ~O(m/d) edges which preserves all minimum cuts of G with at
least 2 vertices on each side.
In our deterministic near-linear time algorithm, we will decompose the
problem via low-conductance cuts found using PageRank a la Brin and Page
(1998), as analyzed by Andersson, Chung, and Lang at FOCS'06. Normally such
algorithms for low-conductance cuts are randomized Monte Carlo algorithms,
because they rely on guessing a good start vertex. However, in our case, we
have so much structure that no guessing is needed.Comment: This is the full journal version. Has been accepted for J.AC
Relative multiplexing for minimizing switching in linear-optical quantum computing
Many existing schemes for linear-optical quantum computing (LOQC) depend on
multiplexing (MUX), which uses dynamic routing to enable near-deterministic
gates and sources to be constructed using heralded, probabilistic primitives.
MUXing accounts for the overwhelming majority of active switching demands in
current LOQC architectures. In this manuscript, we introduce relative
multiplexing (RMUX), a general-purpose optimization which can dramatically
reduce the active switching requirements for MUX in LOQC, and thereby reduce
hardware complexity and energy consumption, as well as relaxing demands on
performance for various photonic components. We discuss the application of RMUX
to the generation of entangled states from probabilistic single-photon sources,
and argue that an order of magnitude improvement in the rate of generation of
Bell states can be achieved. In addition, we apply RMUX to the proposal for
percolation of a 3D cluster state in [PRL 115, 020502 (2015)], and we find that
RMUX allows a 2.4x increase in loss tolerance for this architecture.Comment: Published version, New Journal of Physics, Volume 19, June 201
Fast and Deterministic Approximations for k-Cut
In an undirected graph, a k-cut is a set of edges whose removal breaks the graph into at least k connected components. The minimum weight k-cut can be computed in n^O(k) time, but when k is treated as part of the input, computing the minimum weight k-cut is NP-Hard [Goldschmidt and Hochbaum, 1994]. For poly(m,n,k)-time algorithms, the best possible approximation factor is essentially 2 under the small set expansion hypothesis [Manurangsi, 2017]. Saran and Vazirani [1995] showed that a (2 - 2/k)-approximately minimum weight k-cut can be computed via O(k) minimum cuts, which implies a O~(km) randomized running time via the nearly linear time randomized min-cut algorithm of Karger [2000]. Nagamochi and Kamidoi [2007] showed that a (2 - 2/k)-approximately minimum weight k-cut can be computed deterministically in O(mn + n^2 log n) time. These results prompt two basic questions. The first concerns the role of randomization. Is there a deterministic algorithm for 2-approximate k-cuts matching the randomized running time of O~(km)? The second question qualitatively compares minimum cut to 2-approximate minimum k-cut. Can 2-approximate k-cuts be computed as fast as the minimum cut - in O~(m) randomized time?
We give a deterministic approximation algorithm that computes (2 + eps)-minimum k-cuts in O(m log^3 n / eps^2) time, via a (1 + eps)-approximation for an LP relaxation of k-cut
Connectivity Oracles for Graphs Subject to Vertex Failures
We introduce new data structures for answering connectivity queries in graphs
subject to batched vertex failures. A deterministic structure processes a batch
of failed vertices in time and thereafter
answers connectivity queries in time. It occupies space . We develop a randomized Monte Carlo version of our data structure
with update time , query time , and space
for any failure bound . This is the first connectivity oracle for
general graphs that can efficiently deal with an unbounded number of vertex
failures.
We also develop a more efficient Monte Carlo edge-failure connectivity
oracle. Using space , edge failures are processed in time and thereafter, connectivity queries are answered in
time, which are correct w.h.p.
Our data structures are based on a new decomposition theorem for an
undirected graph , which is of independent interest. It states that
for any terminal set we can remove a set of
vertices such that the remaining graph contains a Steiner forest for with
maximum degree
Distributed Edge Connectivity in Sublinear Time
We present the first sublinear-time algorithm for a distributed
message-passing network sto compute its edge connectivity exactly in
the CONGEST model, as long as there are no parallel edges. Our algorithm takes
time to compute and a
cut of cardinality with high probability, where and are the
number of nodes and the diameter of the network, respectively, and
hides polylogarithmic factors. This running time is sublinear in (i.e.
) whenever is. Previous sublinear-time
distributed algorithms can solve this problem either (i) exactly only when
[Thurimella PODC'95; Pritchard, Thurimella, ACM
Trans. Algorithms'11; Nanongkai, Su, DISC'14] or (ii) approximately [Ghaffari,
Kuhn, DISC'13; Nanongkai, Su, DISC'14].
To achieve this we develop and combine several new techniques. First, we
design the first distributed algorithm that can compute a -edge connectivity
certificate for any in time .
Second, we show that by combining the recent distributed expander decomposition
technique of [Chang, Pettie, Zhang, SODA'19] with techniques from the
sequential deterministic edge connectivity algorithm of [Kawarabayashi, Thorup,
STOC'15], we can decompose the network into a sublinear number of clusters with
small average diameter and without any mincut separating a cluster (except the
`trivial' ones). Finally, by extending the tree packing technique from [Karger
STOC'96], we can find the minimum cut in time proportional to the number of
components. As a byproduct of this technique, we obtain an -time
algorithm for computing exact minimum cut for weighted graphs.Comment: Accepted at 51st ACM Symposium on Theory of Computing (STOC 2019
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