2,061 research outputs found
Shortest Reconfiguration of Sliding Tokens on a Caterpillar
Suppose that we are given two independent sets I_b and I_r of a graph such
that |I_b|=|I_r|, and imagine that a token is placed on each vertex in |I_b|.
Then, the sliding token problem is to determine whether there exists a sequence
of independent sets which transforms I_b into I_r so that each independent set
in the sequence results from the previous one by sliding exactly one token
along an edge in the graph. The sliding token problem is one of the
reconfiguration problems that attract the attention from the viewpoint of
theoretical computer science. The reconfiguration problems tend to be
PSPACE-complete in general, and some polynomial time algorithms are shown in
restricted cases. Recently, the problems that aim at finding a shortest
reconfiguration sequence are investigated. For the 3SAT problem, a trichotomy
for the complexity of finding the shortest sequence has been shown, that is, it
is in P, NP-complete, or PSPACE-complete in certain conditions. In general,
even if it is polynomial time solvable to decide whether two instances are
reconfigured with each other, it can be NP-complete to find a shortest sequence
between them. Namely, finding a shortest sequence between two independent sets
can be more difficult than the decision problem of reconfigurability between
them. In this paper, we show that the problem for finding a shortest sequence
between two independent sets is polynomial time solvable for some graph classes
which are subclasses of the class of interval graphs. More precisely, we can
find a shortest sequence between two independent sets on a graph G in
polynomial time if either G is a proper interval graph, a trivially perfect
graph, or a caterpillar. As far as the authors know, this is the first
polynomial time algorithm for the shortest sliding token problem for a graph
class that requires detours
On the Use of Suffix Arrays for Memory-Efficient Lempel-Ziv Data Compression
Much research has been devoted to optimizing algorithms of the Lempel-Ziv
(LZ) 77 family, both in terms of speed and memory requirements. Binary search
trees and suffix trees (ST) are data structures that have been often used for
this purpose, as they allow fast searches at the expense of memory usage.
In recent years, there has been interest on suffix arrays (SA), due to their
simplicity and low memory requirements. One key issue is that an SA can solve
the sub-string problem almost as efficiently as an ST, using less memory. This
paper proposes two new SA-based algorithms for LZ encoding, which require no
modifications on the decoder side. Experimental results on standard benchmarks
show that our algorithms, though not faster, use 3 to 5 times less memory than
the ST counterparts. Another important feature of our SA-based algorithms is
that the amount of memory is independent of the text to search, thus the memory
that has to be allocated can be defined a priori. These features of low and
predictable memory requirements are of the utmost importance in several
scenarios, such as embedded systems, where memory is at a premium and speed is
not critical. Finally, we point out that the new algorithms are general, in the
sense that they are adequate for applications other than LZ compression, such
as text retrieval and forward/backward sub-string search.Comment: 10 pages, submited to IEEE - Data Compression Conference 200
Complexity of Token Swapping and its Variants
In the Token Swapping problem we are given a graph with a token placed on
each vertex. Each token has exactly one destination vertex, and we try to move
all the tokens to their destinations, using the minimum number of swaps, i.e.,
operations of exchanging the tokens on two adjacent vertices. As the main
result of this paper, we show that Token Swapping is -hard parameterized
by the length of a shortest sequence of swaps. In fact, we prove that, for
any computable function , it cannot be solved in time where is the number of vertices of the input graph, unless the ETH
fails. This lower bound almost matches the trivial -time algorithm.
We also consider two generalizations of the Token Swapping, namely Colored
Token Swapping (where the tokens have different colors and tokens of the same
color are indistinguishable), and Subset Token Swapping (where each token has a
set of possible destinations). To complement the hardness result, we prove that
even the most general variant, Subset Token Swapping, is FPT in nowhere-dense
graph classes.
Finally, we consider the complexities of all three problems in very
restricted classes of graphs: graphs of bounded treewidth and diameter, stars,
cliques, and paths, trying to identify the borderlines between polynomial and
NP-hard cases.Comment: 23 pages, 7 Figure
The Complexity of Change
Many combinatorial problems can be formulated as "Can I transform
configuration 1 into configuration 2, if certain transformations only are
allowed?". An example of such a question is: given two k-colourings of a graph,
can I transform the first k-colouring into the second one, by recolouring one
vertex at a time, and always maintaining a proper k-colouring? Another example
is: given two solutions of a SAT-instance, can I transform the first solution
into the second one, by changing the truth value one variable at a time, and
always maintaining a solution of the SAT-instance? Other examples can be found
in many classical puzzles, such as the 15-Puzzle and Rubik's Cube.
In this survey we shall give an overview of some older and more recent work
on this type of problem. The emphasis will be on the computational complexity
of the problems: how hard is it to decide if a certain transformation is
possible or not?Comment: 28 pages, 6 figure
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