We introduce two new metrics of “simplicity” for knight's tours: the number of turns and the number of crossings. We give a novel algorithm that produces tours with 9.25n+O(1) turns and 12n+O(1) crossings on an n×n board, and we show lower bounds of (6−ϵ)n and 4n−O(1) on the respective problems of minimizing these metrics. Hence, our algorithm achieves approximation ratios of 9.25/6+o(1) and 3+o(1). Our algorithm takes linear time and is fully parallelizable, i.e., the tour can be computed in O(n2/p) time using p processors in the CREW PRAM model. We generalize our techniques to rectangular boards, high-dimensional boards, symmetric tours, odd boards with a missing corner, and tours for (1,4)-leapers. In doing so, we show that these extensions also admit a constant approximation ratio on the minimum number of turns, and on the number of crossings in most cases.

We show new applications of the nearest-neighbor chain algorithm, a technique that originated in agglomerative hierarchical clustering. We apply it to a diverse class of geometric problems: we construct the greedy multi-fragment tour for Euclidean TSP in $O(n\log n)$ time in any fixed dimension and for Steiner TSP in planar graphs in $O(n\sqrt{n}\log n)$ time; we compute motorcycle graphs (which are a central part in straight skeleton algorithms) in $O(n^{4/3+\varepsilon})$ time for any $\varepsilon>0$; we introduce a narcissistic variant of the $k$-attribute stable matching model, and solve it in $O(n^{2-4/(k(1+\varepsilon)+2)})$ time; we give a linear-time $2$-approximation for a 1D geometric set cover problem with applications to radio station placement.

Motivation: Gene Ontology (GO) terms are frequently used to score alignments between protein-protein interaction (PPI) networks. Methods exist to measure GO similarity between proteins in isolation, but proteins in a network alignment are not isolated: each pairing is dependent on every other via the alignment itself. Existing measures fail to take into account the frequency of GO terms across networks, instead imposing arbitrary rules on when to allow GO terms. Results: Here we develop NetGO, a new measure that naturally weighs infrequent, informative GO terms more heavily than frequent, less informative GO terms, without arbitrary cutoffs, instead downweighting GO terms according to their frequency in the networks being aligned. This is a global measure applicable only to alignments, independent of pairwise GO measures, in the same sense that the edge-based EC or S3 scores are global measures of topological similarity independent of pairwise topological similarities. We demonstrate the superiority of NetGO in alignments of predetermined quality and show that NetGO correlates with alignment quality better than any existing GO-based alignment measures. We also demonstrate that NetGO provides a measure of taxonomic similarity between species, consistent with existing taxonomic measuresa feature not shared with existing GObased network alignment measures. Finally, we re-score alignments produced by almost a dozen aligners from a previous study and show that NetGO does a better job at separating good alignments from bad ones. Availability and implementation: Available as part of SANA. Contact: [email protected]. Supplementary information: Supplementary data are available at Bioinformatics online.

We introduce a novel method for defining geographic districts in road networks using stable matching. In this approach, each geographic district is defined in terms of a center, which identifies a location of interest, such as a post office or polling place, and all other network vertices must be labeled with the center to which they are associated. We focus on defining geographic districts that are equitable, in that every district has the same number of vertices and the assignment is stable in terms of geographic distance. That is, there is no unassigned vertex-center pair such that both would prefer each other over their current assignments. We solve this problem using a version of the classic stable matching problem, called symmetric stable matching, in which the preferences of the elements in both sets obey a certain symmetry. We show that, for a planar graph or road network with n nodes and k centers, the problem can be solved in O(n √ n log n) time, which improves upon the O(nk) runtime of using the classic Gale--Shapley stable matching algorithm when k is large. Finally, we provide experimental results on road networks for these algorithms and a heuristic algorithm that performs better than the Gale--Shapley algorithm for any range of values of k.

SUMMARY: Every alignment algorithm consists of two orthogonal components: an objective function M measuring the quality of an alignment, and a search algorithm that explores the space of alignments looking for ones scoring well according to M . We introduce a new search algorithm called SANA (Simulated Annealing Network Aligner) and apply it to protein-protein interaction networks using S 3 as the topological measure. Compared against 12 recent algorithms, SANA produces 5-10 times as many correct node pairings as the others when the correct answer is known. We expose an anti-correlation in many existing aligners between their ability to produce good topological vs. functional similarity scores, whereas SANA usually outscores other methods in both measures. If given the perfect objective function encoding the identity mapping, SANA quickly converges to the perfect solution while many other algorithms falter. We observe that when aligning networks with a known mapping and optimizing only S 3 , SANA creates alignments that are not perfect and yet whose S 3 scores match that of the perfect alignment. We call this phenomenon saturation of the topological score . Saturation implies that a measure's correlation with alignment correctness falters before the perfect alignment is reached. This, combined with SANA's ability to produce the perfect alignment if given the perfect objective function, suggests that better objective functions may lead to dramatically better alignments. We conclude that future work should focus on finding better objective functions, and offer SANA as the search algorithm of choice. AVAILABILITY AND IMPLEMENTATION: Software available at http://sana.ics.uci.edu . CONTACT: [email protected]. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.