There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems. Combinatorial investigations play an important role in this study.

Algorithms in Combinatorial Geometry

Gigantically comprehensive and carefully researched, Security Engineering makes it clear just how difficult it is to protect information systems from corruption, eavesdropping, unauthorized use, and general malice. Better, Ross Anderson offers a lot of thoughts on how information can be made more secure (though probably not absolutely secure, at least not forever) with the help of both technologies and management strategies. His work makes fascinating reading and will no doubt inspire considerable doubt--fear is probably a better choice of words--in anyone with information to gather, protect, or make decisions about. Be aware: This is absolutely not a book solely about computers, with yet another explanation of Alice and Bob and how they exchange public keys in order to exchange messages in secret. Anderson explores, for example, the ingenious ways in which European truck drivers defeat their vehicles' speed-logging equipment. In another section, he shows how the end of the cold war brought on a decline in defenses against radio-frequency monitoring (radio frequencies can be used to determine, at a distance, what's going on in systems--bank teller machines, say), and how similar technology can be used to reverse-engineer the calculations that go on inside smart cards. In almost 600 pages of riveting detail, Anderson warns us not to be seduced by the latest defensive technologies, never to underestimate human ingenuity, and always use common sense in defending valuables. A terrific read for security professionals and general readers alike. --David Wall Topics covered: How some people go about protecting valuable things (particularly, but not exclusively, information) and how other people go about getting it anyway. Mostly, this takes the form of essays (about, for example, how the U.S. Air Force keeps its nukes out of the wrong hands) and stories (one of which tells of an art thief who defeated the latest technology by hiding in a closet). Sections deal with technologies, policies, psychology, and legal matters.

Security Engineering: A Guide to Building Dependable Distributed Systems

Note: Professor Pach's number: [045]; Also in: Facsimile edition: World Classics in Mathematics Series, vol. 28, China Science Press, Beijing, 2006. Mimeographed from 100 Research Problems in Discrete Geometry (with W. O. J. Moser). Reference DCG-BOOK-2008-001 URL: http://www.math.nyu.edu/~pach/research_problems.html Record created on 2008-11-18, modified on 2017-05-12

Research Problems in Discrete Geometry

Preliminary Definitions, Results and Motivation Stable Matching Problems: The Stable Marriage Problem: An Update SM and HR with Indifference The Stable Roommates Problem Further Stable Matching Problems Other Optimal Matching Problems: Pareto Optimal Matchings Popular Matchings Profile-Based Optimal Matchings.

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Algorithmics of Matching Under Preferences

Define the length of a basis of the cycle space of a graph to be the sum of the lengths of all cycles in the basis. An algorithm is given that finds a cycle basis with the shortest possible length in $O(m^3 n)$ operations, where m is the number of edges and n is the number of vertices. This is the first known polynomial-time algorithm for this problem. Edges may be weighted or unweighted. Also, the shortest cycle basis is shown to have at most ${{3(n - 1)(n - 2)} / 2}$ edges for the unweighted case. $O(mn^2 )$ algorithm to obtain a suboptimal cycle basis of length $O(n^2 )$ for unweighted graphs is also given.

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A polynomial-time algorithm to find the shortest cycle basis of a graph

Erdos has defined g(n) as the smallest integer such that any set of g(n) points in the plane, no three collinear, contains the vertex set of a convex n-gon whose interior contains no point of this set. Arbitrarily large sets containing no empty convex 7-gon are constructed, showing that g(n) does not exist for n≥l. Whether g(6) exists is unknown.

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Sets with no empty convex 7-gons

Let $G = ( V,E )$ be a finite undirected graph with n vertices and m edges. A minimum edge dominating set of G is a set of edges D, of smallest cardinality $\gamma ' ( G )$, such that each edge of $E - D$ is adjacent to some edge of D. Let $S( G )$ be the subdivision graph of G and let $T( G )$ be the total graph of G. Let $\alpha ( G )$ be the stability number of G (cardinality of a largest stable set) and let $\alpha _2 ( G )$ be the 2-stability number of G (cardinality of a largest set of vertices in G, no two of which are joined by a path of length 2 or less). The following results are obtained. For any $G,\gamma' ( S ( G ) ) + \alpha _2 ( G ) = n$ and $2\gamma ' ( T ( G ) ) + \alpha ( T ( G ) ) = n + m$ or $n + m + 1$. Also, for any depth-first search tree S of $G,\gamma ' ( S )/2\leqq \gamma ' ( G )\leqq 2\gamma ' (S)$, and these bounds are tight.The edge domination problem is NP-complete for planar bipartite graphs, their subdivision, line, and total graphs, perfect claw-free graphs, and planar cub...

Minimum edge dominating sets

Internet topology information is only made available in aggregate form by standard routing protocols. Connectivity information and latency characteristics must therefore be inferred using indirect techniques. In this paper we consider measurements using a distributed set of measurement points or beacons. We show that computing the minimum number of required beacons on a network under a BGP-like routing policy is NP-hard and at best Ω(log n)-approximable. In the worst case at least (n-1)/3 and at most (n+1)/3 beacons are required for a network with n nodes. We then introduce some observations that allow us to propose a relatively small candidate set of beacons for the current Internet topology. The set proposed has properties with relevant applications for all-paths routing on the public Internet and performance based routing.

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On the number of distributed measurement points for network tomography

An algorithm is given to solve the minimum cycle basis problem for regular matroids. The result is based upon Seymour's decomposition theorem for regular matroids; the Gomory-Hu tree, which is essentially the solution for cographic matroids; and the corresponding result for graphs. The complexity of the algorithm is O((n + m)4), provided that a regular matroid is represented as a binary n×m matrix. The complexity decreases to O((n+m)3.376) using fast matrix multiplication.

J. D. Horton

Papers

A polynomial-time algorithm to find the shortest cycle basis of a graph

Sets with no empty convex 7-gons

Minimum edge dominating sets

On the number of distributed measurement points for network tomography

A Polynomial Time Algorithm to Find the Minimum Cycle Basis of a Regular Matroid