We reconsider a recently published algorithm (Dalkilic et al.) for merginglists by way of the perfect shuffle. The original publication gave onlyexperimental results which, although consistent with linear execution time onthe samples tested, provided no analysis. Here we prove that the timecomplexity, in the average case, is indeed linear, although there is anOmega(n^2) worst case. This is then the first provably linear time mergealgorithm based on the use of the perfect shuffle. We provide a proof ofcorrectness, extend the algorithm to the general case where the lists are ofunequal length and show how it can be made stable, all aspects not included inthe original presentation and we give a much more concise definition of thealgorithm.
We establish a refined search tree technique for the parameterized DOMINATING SET problem on planar graphs. Here, we are given an undirected graph and we ask for a set of at most k vertices such that every other vertex has at least one neighbor in this set. We describe algorithms with running times O(8kn) and O(8kk+n3), where n is the number of vertices in the graph, based on bounded search trees. We describe a set of polynomial time data-reduction rules for a more general “annotated” problem on black/white graphs that asks for a set of k vertices (black or white) that dominate all the black vertices. An intricate argument based on the Euler formula then establishes an efficient branching strategy for reduced inputs to this problem. In addition, we give a family examples showing that the bound of the branching theorem is optimal with respect to our reduction rules. Our final search tree algorithm is easy to implement; its analysis, however, is involved.
We consider the problem of assigning radii to a given set of points in theplane, such that the resulting set of circles is connected, and the sum ofradii is minimized. We show that the problem is polynomially solvable if aconnectivity tree is given. If the connectivity tree is unknown, the problem isNP-hard if there are upper bounds on the radii and open otherwise. We giveapproximation guarantees for a variety of polynomial-time algorithms, describeupper and lower bounds (which are matching in some of the cases), providepolynomial-time approximation schemes, and conclude with experimental resultsand open problems.
In this paper, we consider a facility location problem to find a minimum-sum coverage of n points by disks centered at a fixed line. The cost of a disk with radius r has the form of a nondecreasing function f(r)=rα for any α⩾1. The goal is to find a set of disks in any Lp-metric such that the disks are centered on the x-axis, their union covers the points, and the sum of the cost of the disks is minimized. Alt et al.  presented an algorithm running in O(n4logn) time for any α>1 in any Lp-metric. We present a faster algorithm that runs in O(n2logn) time for any α>1 and any Lp-metric.
Recent times have seen quite some progress in the development of ‘efficient’ exponential-time algorithms for NP-hard problems. These results are also tightly related to the so-called theory of fixed parameter tractability. In this incomplete, personally biased survey, we reflect on some recent developments and prospects in the field of fixed parameter algorithms.