John Peebles

An Extreme Case of Plant-Insect Codiversification: Figs and Fig-Pollinating Wasps

It is thought that speciation in phytophagous insects is often due to colonization of novel host plants, because radiations of plant and insect lineages are typically asynchronous. Recent phylogenetic comparisons have supported this model of diversification for both insect herbivores and specialized pollinators. An exceptional case where contemporaneous plant-insect diversification might be expected is the obligate mutualism between fig trees (Ficus species, Moraceae) and their pollinating wasps (Agaonidae, Hymenoptera). The ubiquity and ecological significance of this mutualism in tropical and subtropical ecosystems has long intrigued biologists, but the systematic challenge posed by >750 interacting species pairs has hindered progress toward understanding its evolutionary history. In particular, taxon sampling and analytical tools have been insufficient for large-scale cophylogenetic analyses. Here, we sampled nearly 200 interacting pairs of fig and wasp species from across the globe. Two supermatrices were assembled: on an average, wasps had sequences from 77% of 6 genes (5.6 kb), figs had sequences from 60% of 5 genes (5.5 kb), and overall 850 new DNA sequences were generated for this study. We also developed a new analytical tool, Jane 2, for event-based phylogenetic reconciliation analysis of very large data sets. Separate Bayesian phylogenetic analyses for figs and fig wasps under relaxed molecular clock assumptions indicate Cretaceous diversification of crown groups and contemporaneous divergence for nearly half of all fig and pollinator lineages. Event-based cophylogenetic analyses further support the codiversification hypothesis. Biogeographic analyses indicate that the present-day distribution of fig and pollinator lineages is consistent with a Eurasian origin and subsequent dispersal, rather than with Gondwanan vicariance. Overall, our findings indicate that the fig-pollinator mutualism represents an extreme case among plant-insect interactions of coordinated dispersal and long-term codiversification. [Biogeography; coevolution; cospeciation; host switching; long-branch attraction; phylogeny.].

Faster Algorithms for Computing the Stationary Distribution, Simulating Random Walks, and More

In this paper, we provide faster algorithms for computing various fundamental quantities associated with random walks on a directed graph, including the stationary distribution, personalized PageRank vectors, hitting times, and escape probabilities. In particular, on a directed graph with n vertices and m edges, we show how to compute each quantity in time Õ(m3/4n + mn2/3), where the Õ notation suppresses polylog factors in n, the desired accuracy, and the appropriate condition number (i.e. the mixing time or restart probability). Our result improves upon the previous fastest running times for these problems; previous results either invoke a general purpose linear system solver on a n × n matrix with m nonzero entries, or depend polynomially on the desired error or natural condition number associated with the problem (i.e. the mixing time or restart probability). For sparse graphs, we obtain a running time of Õ(n7/4), breaking the O(n2) barrier of the best running time one could hope to achieve using fast matrix multiplication. We achieve our result by providing a similar running time improvement for solving directed Laplacian systems, a natural directed or asymmetric analog of the well studied symmetric or undirected Laplacian systems. We show how to solve such systems in time Õ(m3/4n + mn2/3), and efficiently reduce a broad range of problems to solving Õ(1) directed Laplacian systems on Eulerian graphs. We hope these results and our analysis open the door for further study into directed spectral graph theory.

Replacing Mark Bits with Randomness in Fibonacci Heaps

A Fibonacci heap is a deterministic data structure implementing a priority queue with optimal amortized operation costs. An unfortunate aspect of Fibonacci heaps is that they must maintain a “mark bit” which serves only to ensure efficiency of heap operations, not correctness. Karger proposed a simple randomized variant of Fibonacci heaps in which mark bits are replaced by coin flips. This variant still has expected amortized cost O(1) for insert, decrease-key, and merge. Karger conjectured that this data structure has expected amortized cost \(O(\log s)\) for delete-min, where s is the number of heap operations.

Determinant-Preserving Sparsification of SDDM Matrices with Applications to Counting and Sampling Spanning Trees

We show variants of spectral sparsification routines can preserve the totalspanning tree counts of graphs, which by Kirchhoffs matrix-tree theorem, isequivalent to determinant of a graph Laplacian minor, or equivalently, of any SDDM matrix. Our analyses utilizes this combinatorial connection to bridge between statisticalleverage scores / effective resistances and the analysis of random graphsby [Janson, Combinatorics, Probability and Computing 94]. This leads to a routine that in quadratic time, sparsifies a graph down to aboutn^(1.5) edges in ways that preserve both the determinant and the distributionof spanning trees (provided the sparsified graph is viewed as a random object). Extending this algorithm to work with Schur complements and approximateCholesky factorizations leads to algorithms for counting andsampling spanning trees which are nearly optimal for dense graphs.We give an algorithm that computes a (1 +/- δ) approximation to the determinantof any SDDM matrix with constant probability in about n^2 / δ^2 time. This is the first routine for graphs that outperforms general-purpose routines for computingdeterminants of arbitrary matrices. We also give an algorithm that generates in about n^2 / δ^2 time a spanning tree ofa weighted undirected graph from a distribution with total variationdistance of δ from the w-uniform distribution.