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1.
We study the problem of computing the k maximum sum subsequences. Given a sequence of real numbers
and an integer parameter k,
the problem involves finding the k largest values of
for
The problem for fixed k = 1, also known as the maximum sum subsequence problem, has received much attention in the literature
and is linear-time solvable. Recently, Bae and Takaoka presented a
-time algorithm for the k maximum sum subsequences problem. In this paper we design an efficient algorithm that solves the
above problem in
time in the worst case. Our algorithm is optimal for
and improves over the previously best known result for any value of the user-defined parameter k < 1. Moreover, our results
are also extended to the multi-dimensional versions of the k maximum sum subsequences problem; resulting in fast algorithms
as well. 相似文献
2.
We present approximation algorithms for the unsplittable flow problem (UFP) in undirected graphs. As is standard in this line
of research, we assume that the maximum
demand is at most the minimum capacity. We focus on the non-uniform capacity case in which the edge capacities can vary arbitrarily
over the graph. Our results are:
We obtain an
approximation ratio for UFP, where n is the number of vertices,
is the maximum degree, and
is the expansion of the graph. Furthermore, if we specialize to the case where all edges have the same capacity, our algorithm
gives an
approximation.
For certain strong constant-degree expanders considered by we obtain an
approximation for the uniform capacity case.
For UFP on the line and the ring, we give the first constant-factor approximation algorithms.
All of the above results improve if the maximum demand is bounded away from the minimum capacity. The above results either
improve upon or
are incomparable with previously known results for these problems. The main technique used for these results is randomized
rounding followed by greedy alteration, and is inspired by the use of this idea in recent work. 相似文献
3.
For a set of rooted, unordered, distinctly leaf-labeled trees, the NP-hard maximum agreement subtree problem (MAST) asks for
a tree contained (up to isomorphism or homeomorphism) in all of the input trees with as many labeled leaves as possible. We
study the ordered variants of MAST where the trees are uniformly or non-uniformly ordered. We provide the first known polynomial-time
algorithms for the uniformly and non-uniformly ordered homeomorphic variants as well as the uniformly and non-uniformly ordered
isomorphic variants of MAST.
Our algorithms run in time
,
,
, and
, respectively, where n is the number of leaf labels and k is
the number of input trees. 相似文献
4.
Celina M.H. de Figueiredo Guilherme D. da Fonseca Vinicius G.P. de Sa Jeremy Spinrad 《Algorithmica》2006,46(2):149-180
A homogeneous set is a non-trivial module of a graph, i.e. a non-empty,
non-unitary, proper subset of a graph's vertices such that all its elements
present exactly the same outer neighborhood. Given two graphs
the Homogeneous Set Sandwich Problem (HSSP) asks whether there
exists a sandwich graph
which
has a homogeneous set. In 2001 Tang et al. published
an all-fast
algorithm which was recently proven wrong, so that the HSSP's known upper bound would have been reset
thereafter at the former
determined by Cerioli et al. in 1998. We present, notwithstanding, new deterministic
algorithms which have it established at
We give as
well two even faster
randomized algorithms, whose simplicity might
lend them didactic usefulness. We believe that, besides providing efficient
easy-to-implement procedures to solve it, the study of these new approaches
allows a fairly thorough understanding of the problem. 相似文献
5.
Dai, Li, and Wu proposed Rule k, a localized approximation algorithm that attempts to find a small connected dominating
set in a graph. In this paper we consider the "average-case" performance of two closely related versions of Rule k for the
model of random unit disk graphs constructed from n random points in an
square. We show that if
and
then for both versions of Rule k, the expected size of the Rule k dominating set is
as
It follows that, for
in a suitable range, the expected size of the Rule k dominating sets are within a constant factor of the optimum. 相似文献
6.
We use Schnyder woods of 3-connected planar graphs to produce convex straight-line drawings
on a grid of size
The parameter
depends on the Schnyder wood used for the drawing. This parameter is in the range
The algorithm is a refinement of the face-counting algorithm; thus, in particular, the size of the grid is at most
The above bound on the grid size simultaneously matches or improves all previously known bounds for convex drawings, in particular
Schnyder's and the recent Zhang and He bound for triangulations and the Chrobak and Kant bound for 3-connected planar graphs.
The algorithm takes linear time. The drawing algorithm has been implemented and tested. The expected grid size for the drawing
of a random triangulation is close to
For a random 3-connected plane graph, tests show that the expected size of the drawing is
相似文献
7.
The increased availability of data describing biological interactions provides important clues on how complex chains of genes
and proteins interact with each other. Most previous approaches either restrict their attention to analyzing simple substructures
such as paths or trees in these graphs, or use heuristics that do not provide performance guarantees when general substructures
are analyzed. We investigate a formulation to model pathway structures directly and give a probabilistic algorithm to find
an optimal path structure in
time and
space, where n and m are respectively the number of vertices and the number of edges in the given network, k is the number
of vertices in the path structure, and t is the maximum number of vertices (i.e., "width") at each level of the structure.
Even for the case t = 1 which corresponds to finding simple paths of length k, our time complexity
is a significant improvement over previous probabilistic approaches. To allow for the analysis of multiple pathway structures,
we further consider a variant of the algorithm that provides probabilistic guarantees for the top suboptimal path structures
with a slight increase in time and space. We show that our algorithm can identify pathway structures with high sensitivity
by applying it to protein interaction networks in the DIP database. 相似文献
8.
Uri Zwick 《Algorithmica》2006,46(2):181-192
We present an
-time algorithm for the All Pairs Shortest Paths (APSP) problem for directed graphs with real edge lengths. This slightly
improves previous algorithms for the problem obtained by Fredman, Dobosiewicz, Han, and Takaoka. 相似文献
9.
The unit ball random geometric graph
has as its vertices n points distributed independently and uniformly in the unit ball in
, with two vertices adjacent if and only if their ℓp-distance is at most λ. Like its cousin the Erdos-Renyi random graph, G has a connectivity threshold: an asymptotic value
for λ in terms of n, above which G is connected and below which G is disconnected. In the connected zone we determine upper
and lower bounds for the graph diameter of G. Specifically, almost always,
, where
is the ℓp-diameter of the unit ball B. We employ a combination of methods from probabilistic combinatorics and stochastic geometry. 相似文献
10.
Sebastian Dörn 《Theory of Computing Systems》2009,45(3):613-628
We present quantum algorithms for the following matching problems in unweighted and weighted graphs with n vertices and m edges:
Our quantum algorithms are faster than the best known classical deterministic algorithms for the corresponding problems.
In particular, the second result solves an open question stated in a paper by Ambainis and Špalek (Proceedings of STACS’06,
pp. 172–183, 2006). 相似文献
• | Finding a maximal matching in general graphs in time . |
• | Finding a maximum matching in general graphs in time . |
• | Finding a maximum weight matching in bipartite graphs in time , where N is the largest edge weight. |
11.
Amitabha Bagchi Ankur Bhargava Amitabh Chaudhary David Eppstein Christian Scheideler 《Theory of Computing Systems》2006,39(6):903-928
We study the problem of how resilient networks are to node faults. Specifically, we investigate the question of how many faults
a network can sustain and still contain a large (i.e., linear-sized) connected component with approximately the same expansion
as the original fault-free network. We use a pruning technique that culls away those parts of the faulty network that have
poor expansion. The faults may occur at random or be caused by an adversary. Our techniques apply in either case. In the adversarial
setting we prove that for every network with expansion
a large connected component with basically the same expansion as the original network exists for up to a constant times
faults. We show this result is tight in the sense that every graph G of size n and uniform expansion
can be broken into components of size o(n) with
faults. Unlike the adversarial case, the expansion of a graph gives a very weak bound on its resilience to random faults.
While it is the case, as before, that there are networks of uniform expansion
that are not resilient against a fault probability of a constant times
it is also observed that there are networks of uniform expansion
that are resilient against a constant fault probability. Thus, we introduce a different parameter, called the span of a
graph, which gives us a more precise handle on the maximum fault probability. We use the span to show the first known results
for the effect of random faults on the expansion of d-dimensional meshes. 相似文献
12.
Graph homomorphism, also called H-coloring, is a natural generalization of graph coloring: There is a
homomorphism from a graph G to a complete graph on k vertices if and only if G is k-colorable. During recent years the topic
of exact (exponential-time) algorithms for NP-hard problems in general, and for graph coloring in particular, has led to
extensive research. Consequently, it is natural to ask how the techniques developed for exact graph coloring algorithms can
be extended to graph homomorphisms. By the celebrated result of Hell and Nesetril, for each fixed simple graph H, deciding
whether a given simple graph G has a homomorphism to H is polynomial-time solvable if
H is a bipartite graph, and NP-complete otherwise. The case where H is the cycle of length 5, is the first NP-hard case different
from graph coloring. We show that for an odd integer
, whether an input graph G with n vertices is homomorphic to the cycle of length k, can be decided in time
. We extend the results obtained for cycles, which are graphs of treewidth two, to graphs of bounded treewidth as follows:
if H is of treewidth at most t, then whether input graph G with n vertices is homomorphic to H can be decided in time
. 相似文献
13.
We show that for arbitrary positive integers
with probability
the gcd of two linear combinations of these integers with rather small random integer coefficients coincides with
This naturally leads to a probabilistic algorithm for computing the gcd of several integers, with probability
via just one gcd of two numbers with about the same size as the initial data (namely the above linear combinations). This
algorithm can be repeated to achieve any desired confidence level. 相似文献
14.
We consider graphs that can be embedded on a surface of bounded genus such that each edge has a bounded number of crossings.
We prove that many optimization problems, including maximum
independent set, minimum vertex cover, minimum dominating set and many others, admit polynomial time approximation schemes
when restricted to such graphs. This extends previous results by Baker and Eppstein to a much broader class of graphs. We
also prove that for the considered class of graphs, there are balanced separators of size
where n is a number of vertices in the graph. On the negative side, we prove that it is intractable to recognize the graphs
embeddable in the plane with at most one crossing per edge. 相似文献
15.
We consider the management of FIFO buffers for network switches providing differentiated services. In each time step, an arbitrary
number of packets arrive and only one packet can be sent. The buffer can store a limited number of packets and, due to the
FIFO property, the sequence of sent packets has to be a subsequence of the arriving packets. The differentiated service model
is abstracted by attributing each packet with a value according to its service level. A buffer management strategy can drop
packets, and the goal is to maximize the sum of the values of sent packets.
For only two different packet values, we introduce the account strategy and prove that this strategy achieves an optimal competitive
ratio of
if the buffer size tends to infinity and an optimal competitive ratio of
for arbitrary buffer sizes. For general packet values, the simple preemptive greedy strategy (PG) is studied. We show that
PG achieves a competitive ratio of
which is the best known upper bound on the competitive ratio of this problem. In addition, we give a lower bound of
on the competitive ratio of PG which improves the previously known lower bound. As a consequence, the competitive ratio of
PG cannot be further improved significantly.
Supported by the DFG grant WE 2842/1. A preliminary version of this paper appeared in Proceedings of the 14th Annual European
Symposium on Algorithms (ESA), 2006. 相似文献
16.
Martin Ziegler 《Theory of Computing Systems》2007,41(1):177-206
By the sometimes so-called Main Theorem of Recursive Analysis, every computable real function is necessarily continuous. We
wonder whether and which kinds of hypercomputation allow for the effective evaluation of also discontinuous
. More precisely the present work considers the following three super-Turing notions of real function computability: - relativized
computation; specifically given oracle access to the Halting Problem
or its jump
; - encoding input
and/or output y = f(x) in weaker ways also related to the Arithmetic Hierarchy; - nondeterministic computation. It turns
out that any
computable in the first or second sense is still necessarily continuous whereas the third type of hypercomputation provides
the required power to evaluate for instance the discontinuous Heaviside function. 相似文献
17.
We consider the problem of testing the commutativity of a black-box group specified by its k generators. The complexity (in
terms of k) of this problem was first considered by Pak, who gave a randomized algorithm involving O(k) group operations.
We construct a quite optimal quantum algorithm for this problem whose complexity is in
. The algorithm uses and highlights the power of the quantization method of
Szegedy. For the lower bound of
, we give a reduction from a special case of Element Distinctness to our problem. Along the way, we prove the optimality of
the algorithm of Pak for the randomized model. 相似文献
18.
We consider the problem of computing a minimum cycle basis in a directed graph G with m arcs and n vertices. The arcs of G
have non-negative weights assigned to them. In this problem a {-1,0,1} incidence vector is associated with each cycle and
the vector space over
generated by these vectors is the cycle space of G. A set of cycles is called a cycle basis of G if it forms a basis for
its cycle space. A cycle basis where the sum of weights of the cycles is minimum is called a minimum cycle basis of G. This
paper presents an
algorithm, which is the first polynomial-time algorithm for computing a minimum cycle basis in G. We then improve it to
an
algorithm. The problem of computing a minimum cycle basis in an undirected graph has been well studied. In this problem
a {0,1} incidence vector is associated with each cycle and the vector space over
generated by these vectors is the cycle space of the graph. There are directed graphs in which the minimum cycle basis has
lower weight than any cycle basis of the underlying undirected graph. Hence algorithms for computing a minimum cycle basis
in an undirected graph cannot be used as black boxes to solve the problem in directed graphs. 相似文献
19.
Abstract We obtain a multivariate extension of a classical result of Schoenberg on cardinal spline interpolation. Specifically, we
prove the existence of a unique function in
, polyharmonic of order p on each strip
,
, and periodic in its last n variables, whose restriction to the parallel hyperplanes
,
, coincides with a prescribed sequence of n-variate periodic data functions satisfying a growth condition in
. The constructive proof is based on separation of variables and on Micchelli’s theory of univariate cardinal
-splines.
Keywords: cardinal
-splines, polyharmonic functions, multivariable interpolation
Mathematics Subject Classification (2000): 41A05, 41A15, 41A63 相似文献
20.
Klaus Meer 《Theory of Computing Systems》2007,41(1):107-118
In [10] it was recently shown that
that is the existence of transparent long proofs for
was established. The latter denotes the class of real number decision problems verifiable in polynomial time as introduced
by Blum et al. [6]. The present paper is devoted to the question what impact a potential full real number
theorem
would have on approximation issues in the BSS model of computation. We study two natural optimization problems in the BSS
model. The first, denoted by MAX-QPS, is related to polynomial systems; the other, MAX-q-CAP, deals with algebraic circuits.
Our main results combine the PCP framework over
with approximation issues for these two problems. We also give a negative approximation result for a variant of the MAX-QPS
problem. 相似文献