Maintaining Discrete Probability Distributions in Practice

A classical problem in random number generation is the sampling of elements from a given discrete distribution. Formally, given a set of indices $S = \{1, \dots, n\}$ and sequence of weights $w_1, \dots, w_n \in \mathbb{R}^+$, the task is to provide samples from $S$ with distribution $p(i) = w_i / W$ where $W = \sum_j w_j$. A commonly accepted solution is Walker's Alias Table, which allows for each sample to be drawn in constant time. However, some applications correspond to a dynamic setting, where elements are inserted or removed, or weights change over time. Here, the Alias Table is not efficient, as it needs to be re-built whenever the underlying distribution changes. In this paper, we engineer a simple data structure for maintaining discrete probability distributions in the dynamic setting. Construction of the data structure is possible in time $O(n)$, sampling is possible in expected time $O(1)$, and an update of size $\Delta$ can be processed in time $O(\Delta n / W)$. As a special case, we maintain an urn containing $W$ marbles of $n$ colors where with each update $O(W / n)$ marbles can be added or removed in $O(1)$ time per update. To evaluate the efficiency of the data structure in practice we conduct an empirical study. The results suggest that the dynamic sampling performance is competitive with the static Alias Table. Compared to existing more complex dynamic solutions we obtain a sampling speed-up of up to half an order of magnitude.Comment: ALENEX 202

Uniform Generation of Temporal Graphs with Given Degrees

Uniform sampling from the set $\mathcal{G}(\mathbf{d})$ of graphs with a given degree-sequence $\mathbf{d} = (d_1, \dots, d_n) \in \mathbb N^n$ is a classical problem in the study of random graphs. We consider an analogue for temporal graphs in which the edges are labeled with integer timestamps. The input to this generation problem is a tuple $\mathbf{D} = (\mathbf{d}, T) \in \mathbb N^n \times \mathbb N_{>0}$ and the task is to output a uniform random sample from the set $\mathcal{G}(\mathbf{D})$ of temporal graphs with degree-sequence $\mathbf{d}$ and timestamps in the interval $[1, T]$. By allowing repeated edges with distinct timestamps, $\mathcal{G}(\mathbf{D})$ can be non-empty even if $\mathcal{G}(\mathbf{d})$ is, and as a consequence, existing algorithms are difficult to apply. We describe an algorithm for this generation problem which runs in expected time $O(M)$ if $\Delta^{2+\epsilon} = O(M)$ for some constant $\epsilon > 0$ and $T - \Delta = \Omega(T)$ where $M = \sum_i d_i$ and $\Delta = \max_i d_i$. Our algorithm applies the switching method of McKay and Wormald $[1]$ to temporal graphs: we first generate a random temporal multigraph and then remove self-loops and duplicated edges with switching operations which rewire the edges in a degree-preserving manner

Patent Institutions: Shifting Interactions Between Legal Actors

This contribution to the Research Handbook on Economics of Intellectual Property Rights (Vol. 1 Theory) addresses interactions between the principal legal institutions of the U.S. patent system. It considers legal, strategic, and normative perspectives on these interactions as they have evolved over the last 35 years. Early centralization of power by the U.S. Court of Appeals for the Federal Circuit, newly created in 1982, established a regime dominated by the appellate court\u27s bright-line rules. More recently, aggressive Supreme Court and Congressional intervention have respectively reinvigorated patent law standards and led to significant devolution of power to inferior tribunals, including newly created tribunals like the USPTO\u27s Patent Trial and Appeals Board. This new era in institutional interaction creates a host of fresh empirical and normative research questions for scholars. The contribution concludes by outlining a research agenda

Gene duplication in the family Salmonidae 111. Linkage between two duplicated loci coding for aspartate aminotransferase in the cutthroat trout (\u3ci\u3eSalmo clarki\u3c/i\u3e)

The genetic control of the supernatant form of aspartate aminotransferase (AAT) was studied in the cutthroat trout (Salmo clarki) through a series of experimental matings. 509 individuals of eight families were examined to determine (1) the number of loci, (2) the mode of inheritance (i.e. disomic or tetrasomic), and (3) the linkage relationship of the loci involved. The variation observed is controlled by a duplicated locus resulting from a presumed tetraploid event of an ancestral salmonid. The inheritance experiments revealed .the presence of two disomic loci rather than a single tetrasomic locus. indicating that disomy has been reestablished for the chromosomes carrying the AAT loci. The two families in which linkage between these loci could be tested displayed significant nonrandom segregation between these loci with an estimated frequency of recombination of 30.6x,. These results are discussed in regard to the proposed evolution of tetraploidy in the family Salmonidae