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A Linear-Time Algorithm for a Special Case of Disjoint Set Union ; CU-CS-261-82
AbstractThis paper presents a linear-time algorithm for the special case of the disjoint set union problem in which the structure of the unions (defined by a “union tree”) is known in advance. The algorithm executes an intermixed sequence of m union and find operations on n elements in O(m+n) time and O(n) space. This is a slight but theoretically significant improvement over the fastest known algorithm for the general problem, which runs in O(mα(m+n, n)+n) time and O(n) space, where a is a functional inverse of Ackermann's function. Used as a subroutine, the algorithm gives similar improvements in the efficiency of algorithms for solving several other problems, including two-processor scheduling, matching on convex graphs, finding nearest common ancestors off-line, testing a flow graph for reducibility, and finding two disjoint directed spanning trees. The algorithm obtains its efficiency by combining the fast algorithm for the general problem with table look-up on small sets, and requires a random access machine for its implementation. The algorithm extends to the case in which single-node additions to the union tree are allowed. The extended algorithm is useful in finding maximum cardinality matchings in nonbipartite graphs
Approximating Minimum Cost Connectivity Orientation and Augmentation
We investigate problems addressing combined connectivity augmentation and
orientations settings. We give a polynomial-time 6-approximation algorithm for
finding a minimum cost subgraph of an undirected graph that admits an
orientation covering a nonnegative crossing -supermodular demand function,
as defined by Frank. An important example is -edge-connectivity, a
common generalization of global and rooted edge-connectivity.
Our algorithm is based on a non-standard application of the iterative
rounding method. We observe that the standard linear program with cut
constraints is not amenable and use an alternative linear program with
partition and co-partition constraints instead. The proof requires a new type
of uncrossing technique on partitions and co-partitions.
We also consider the problem setting when the cost of an edge can be
different for the two possible orientations. The problem becomes substantially
more difficult already for the simpler requirement of -edge-connectivity.
Khanna, Naor, and Shepherd showed that the integrality gap of the natural
linear program is at most when and conjectured that it is constant
for all fixed . We disprove this conjecture by showing an
integrality gap even when
Sampling and Recovery of Pulse Streams
Compressive Sensing (CS) is a new technique for the efficient acquisition of
signals, images, and other data that have a sparse representation in some
basis, frame, or dictionary. By sparse we mean that the N-dimensional basis
representation has just K<<N significant coefficients; in this case, the CS
theory maintains that just M = K log N random linear signal measurements will
both preserve all of the signal information and enable robust signal
reconstruction in polynomial time. In this paper, we extend the CS theory to
pulse stream data, which correspond to S-sparse signals/images that are
convolved with an unknown F-sparse pulse shape. Ignoring their convolutional
structure, a pulse stream signal is K=SF sparse. Such signals figure
prominently in a number of applications, from neuroscience to astronomy. Our
specific contributions are threefold. First, we propose a pulse stream signal
model and show that it is equivalent to an infinite union of subspaces. Second,
we derive a lower bound on the number of measurements M required to preserve
the essential information present in pulse streams. The bound is linear in the
total number of degrees of freedom S + F, which is significantly smaller than
the naive bound based on the total signal sparsity K=SF. Third, we develop an
efficient signal recovery algorithm that infers both the shape of the impulse
response as well as the locations and amplitudes of the pulses. The algorithm
alternatively estimates the pulse locations and the pulse shape in a manner
reminiscent of classical deconvolution algorithms. Numerical experiments on
synthetic and real data demonstrate the advantages of our approach over
standard CS
Fault-Tolerant Shortest Paths - Beyond the Uniform Failure Model
The overwhelming majority of survivable (fault-tolerant) network design
models assume a uniform scenario set. Such a scenario set assumes that every
subset of the network resources (edges or vertices) of a given cardinality
comprises a scenario. While this approach yields problems with clean
combinatorial structure and good algorithms, it often fails to capture the true
nature of the scenario set coming from applications.
One natural refinement of the uniform model is obtained by partitioning the
set of resources into faulty and secure resources. The scenario set contains
every subset of at most faulty resources. This work studies the
Fault-Tolerant Path (FTP) problem, the counterpart of the Shortest Path problem
in this failure model. We present complexity results alongside exact and
approximation algorithms for FTP. We emphasize the vast increase in the
complexity of the problem with respect to its uniform analogue, the
Edge-Disjoint Paths problem
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