235 research outputs found
Sorting and Selection with Imprecise Comparisons
In experimental psychology, the method of paired comparisons was proposed as a means for ranking preferences amongst n elements of a human subject. The method requires performing all (n2) comparisons then sorting elements according to the number of wins. The large number of comparisons is performed to counter the potentially faulty decision-making of the human subject, who acts as an imprecise comparator.
We consider a simple model of the imprecise comparisons: there exists some δ> 0 such that when a subject is given two elements to compare, if the values of those elements (as perceived by the subject) differ by at least δ, then the comparison will be made correctly; when the two elements have values that are within δ, the outcome of the comparison is unpredictable. This δ corresponds to the just noticeable difference unit (JND) or difference threshold in the psychophysics literature, but does not require the statistical assumptions used to define this value.
In this model, the standard method of paired comparisons minimizes the errors introduced by the imprecise comparisons at the cost of (n2) comparisons. We show that the same optimal guarantees can be achieved using 4 n 3/2 comparisons, and we prove the optimality of our method. We then explore the general tradeoff between the guarantees on the error that can be made and number of comparisons for the problems of sorting, max-finding, and selection. Our results provide close-to-optimal solutions for each of these problems.Engineering and Applied Science
Transformers Meet Directed Graphs
Transformers were originally proposed as a sequence-to-sequence model for
text but have become vital for a wide range of modalities, including images,
audio, video, and undirected graphs. However, transformers for directed graphs
are a surprisingly underexplored topic, despite their applicability to
ubiquitous domains, including source code and logic circuits. In this work, we
propose two direction- and structure-aware positional encodings for directed
graphs: (1) the eigenvectors of the Magnetic Laplacian - a direction-aware
generalization of the combinatorial Laplacian; (2) directional random walk
encodings. Empirically, we show that the extra directionality information is
useful in various downstream tasks, including correctness testing of sorting
networks and source code understanding. Together with a data-flow-centric graph
construction, our model outperforms the prior state of the art on the Open
Graph Benchmark Code2 relatively by 14.7%.Comment: 29 page
Sorting from Crowdsourced Comparisons using Expert Verifications
We introduce a novel noisy sorting model motivated by the Just Noticeable
Difference (JND) model from experimental psychology. The goal of our model is
to capture the low quality of the data that are collected from crowdsourcing
environments. Compared to other celebrated models of noisy sorting, our model
does not rely on precise data-generation assumptions and captures crowdsourced
tasks' varying levels of difficulty that can lead to different amounts of noise
in the data. To handle this challenging task, we assume we can verify some of
the collected data using expert advice. This verification procedure is costly;
hence, we aim to minimize the number of verifications we use.
We propose a new efficient algorithm called CandidateSort, which we prove
uses the optimal number of verifications in the noisy sorting models we
consider. We characterize this optimal number of verifications by showing that
it is linear in a parameter , which intuitively measures the maximum number
of comparisons that are wrong but not inconsistent in the crowdsourcing data
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