That is, C operates on n coordinates x1, . . . , xn and outputs a color. We can see how having such a circuit is equivalent to a unique coloring of the cube: if we want to find the color at a vertex, just give the circuit the coordinates of the vertex, and use the output of the circuit. The Sperner problem then becomes the problem of finding a panchromatic simplex when given a coloring circuit ...

Laws of large numbers imply for a sequence of i.i.d. random variables X1, X2, . . . with mean μ, the sample average, 1 n (X1 + X2 + · · · + Xn), converges to μ as n goes to infinity. Concentration bounds provide a quantitative distance between the sample average and the expectation. In this lecture we review several of these fundamental inequalities. In the next few lectures we will see applica...

Construction 7.1 (Margulis [Mar]) Fix a positive integerM and let [M ] = {1, 2, . . . ,M}. Define the bipartite graph G = (V,E) as follows. Let V = [M ]2∪ [M ]2, where vertices in the first partite set are denoted (x, y)1 and vertices in the second partite set are denoted (x, y)2. From each vertex (x, y)1, put in edges to (x, y)2, (x, x+ y)2, (x, x+ y+1)2, (x+ y, y)2, and (x+ y+ 1, y)2, where a...

1 Overview In this lecture, we discuss dynamic trees, a sophisticated data structure introduced by Sleator and Tarjan. Dynamic trees allow to provide the fastest worst-case running times for many network flow algorithms. In particular, it will allow us to efficiently perform the Cancel operation in the Cancel and Tighten algorithm. Dynamic trees build upon splay trees, which we introduced in th...

The goal is to acquire signals in R that are well approximated by sparse signals with k nonzero components, where k << n. The measurement process can be represented by an m× n matrix A, where m is roughly proportional to k rather than n. The recovery algorithm uses the sketch and a description of the measurement matrix to construct a signal approximation x̂ that has only O(k) nonzero components....

Let x be a vector variable taking values from Zm. We consider a setting where the value of x changes over time through updates of its coordinate values. Every update can be specified in form of a pair (i, a), 1 ≤ i ≤ m and a is an integer, which has a meaning of increasing the value of xi by a. A sequence of such pairs is called a stream in our context. The value of x before the first update is...

Vulnerability analysis using dynamic attack graphs, dynamic binary analysis for mobile code security, incre-mental bisimulation algorithm. Development of algorithms for incremental evaluation of tabled logic programs and its application to program analysis, generation and analysis of multiple counterexamples for error cause localization, model checking security-relevant properties of sequential...

A way to think about what this proof is doing: X dominates the scaled indicator variable T = t · 1X≥t, so we have E[X] ≥ E[T ] = t · Pr[X ≥ t]. Markov’s inequality gives a rather weak bound when applied directly; the random variables we care about are usually much more highly concentrated. Let’s look at a toy example: flip 100 coins. What’s the probability that at least 70 of them come up heads...