Georgia Institute of Technology College of Sciences

In 2006, my coadvisor Xiaoming Huo and his colleague published an annal of statistics paper which designs an asymptotically powerful testing algorithm to detect the potential curvilinear structure in a messy point cloud image. However, such an algorithm involves a membership threshold and a decision threshold which are not well defined in that paper because the distribution of LSP was unknown. Later on, Xiaoming's student Chen, Jihong found some connections between the distribution of LSP and the so-called Erdos-Renyi law. In some sense, the distribution of LSP is just a generalization of the Erdos-Renyi law. However this JASA paper of Chen, Jihong had some restrictions and only partially found out the distribution of LSP. In this talk, I will show the result of the JASA paper is actually very close to the distribution of LSP. However, these is still much potential work to do in order to strengthen this algorithm.

Let X_1, X_2,...,X_n be a sequence of i.i.d random variables with values in a finite alphabet {1,...,m}. Let LI_n be the length of the longest increasing subsequence of X_1,...,X_n. We shall express the limiting distribution of LI_n as functionals of m and (m-1)- dimensional Brownian motions as well as the largest eigenvalue of Gaussian Unitary Ensemble (GUE) matrix. Then I shall illustrate simulation study of these results

Suppose that Amtrak runs a train from Miami, Florida, to Bangor, Maine. The train makes stops at many locations along the way to drop off passengers and pick up new ones. The computer system that sells seats on the train wants to use the smallest number of seats possible to transport the passengers along the route. If the computer knew before it made any seat assignments when all the passengers would get on and off, this would be an easy task. However, passengers must be given seat assignments when they buy their tickets, and tickets are sold over a period of many weeks. The computer system must use an online algorithm to make seat assignments in this case, meaning it can use only the information it knows up to that point and cannot change seat assignments for passengers who purchased tickets earlier. In this situation, the computer cannot guarantee it will use the smallest number of seats possible. However, we are able to bound the number of seats the algorithm will use as a linear function of the minimum number of seats that could be used if assignments were made after all passengers had bought their tickets. In this talk, we'll formulate this problem as a question involving coloring interval graphs and discuss online algorithms for other questions on graphs and posets. We'll introduce or review the needed concepts from graph theory and posets as they arise, minimizing the background knowledge required.

This talk considers the following sequence shufling problem: Given a biological sequence (either DNA or protein) s, generate a random instance among all the permutations of s that exhibit the same frequencies of k-lets (e.g. dinucleotides, doublets of amino acids, triplets, etc.). Since certain biases in the usage of k-lets are fundamental to biological sequences, effective generation of such sequences is essential for the evaluation of the results of many sequence analysis tools. This talk introduces two sequence shuffling algorithms: A simple swapping-based algorithm is shown to generate a near-random instance and appears to work well, although its efficiency is unproven; a generation algorithm based on Euler tours is proven to produce a precisely uniforminstance, and hence solve the sequence shuffling problem, in time not much more than linear in the sequence length.