• Comparison sort: O(n lg n) • Counting sort: O(n + u) = O(n + 2w) (= O(n) for w = lg n) • Radix sort: O(n · w lg n) (= O(n) for w = O(lg n)) • van Emde Boas: O(n lg w). For w = lg n, this is O(n lg lg n). This can be improved to O(n lg w lg n), making it better than the previous methods (Kirkpatrick and Reisch [7]). • Signature sort: O(n) for w = Ω(lg n). Combined with the previous result for ...

Matroid union and matroid intersection are closely related in the sense that one can be derived from the other. However they are from different perspectives and have different applications. To motivate matroid union theorem we state a well known theorem of Tutte and Nash-Williams on packing disjoint spanning trees in graphs. Theorem 1 (Nash-Williams and Tutte) An undirected multi-graph G = (V, ...

The experts framework consists of n experts who each make predictions every day, and an online algorithm that makes a decision after seeing all the expert predictions. At the end of the day the loss/gain for each expert is revealed. The goal of the online algorithm is to perform not much worse than the single best expert over a long period of time. We analyzed the algorithm where the initial we...

About the notes: These notes serve as written reference for the topics not covered by the papers that are handed out during the lecture. The material contained therein is thus a strict subset of what is relevant for the final exam. This week, the notes discuss the definition of (perfect) zero-knowledge and a proof that the three-move protocols we have encountered so far (graph isomorphism, Fiat...

Numerical similarity measures are used to de­ scribe the relative ranks of the similarity of ob­ jects or cases in many artificial intelligent systems. These measures are usually absolute and context­ independent. On the other hand, humans perceive context-dependent similarity. That is, the ranking of similarity between pairs of objects is varying under a changing context. We consider this prob...

Graph-based codes were introduced in the previous lecture. Definitions as well as important properties of those codes were presented with explicit decoding algorithms left over to this lecture.As mentioned previously, one of the main topics in the near future would be linear time algorithms. A linear time decoding algorithm of the Expander Codes will be discussed in the following part. As the e...

The sequencing of a Hydra genome marked the beginning of a new era in the use of Hydra as a developmental model. Analysis of the genome sequence has led to a number of interesting findings, has required revisiting of previous work, and most importantly presents new opportunities for understanding the developmental biology of Hydra. This review will de-scribe the history of the Hydra genome proj...

Jasmonic acid (JA) and methyl jasm onate (JA-Me, 1) are naturally occurring substances in plants. They are representatives of a group of plant growth regulators [1], The exact quantifica­ tion of JA in plant materials is of great interest for further studies on transport, metabolism and en­ dogenous importance of JA. Using the GC/M SSIM method deuterium-labelled JA should be a good internal sta...

03/15/2012 Lecture 16: CS 880: Complexity of Counting Problems Instructor: Jin-Yi Cai Scribe: Chen Zeng For a symmetric, bipartite matrix A ∈ C, we want to prove the following theorem: Theorem 1. If EVAL(A) is not #P-hard, then there exists an m × m purified bipartite matrix A such that EVAL(A) ≡ EVAL(A). Recall the definition of the purified bipartite matrix : Definition 1. Let A ∈ C be a symm...