Meander, Folding and Arch Statistics

The statistics of meander and related problems are studied as particular realizations of compact polymer chain foldings. This paper presents a general discussion of these topics, with a particular emphasis on three points: (i) the use of a direct recursive relation for building (semi) meanders (ii) the equivalence with a random matrix model (iii) the exact solution of simpler related problems, such as arch configurations or irreducible meanders.Comment: 82 pages, uuencoded, uses harvmac (l mode) and epsf, 26+7 figures include

On the Memory-Hardness of Data-Independent Password-Hashing Functions

We show attacks on five data-independent memory-hard functions (iMHF) that were submitted to the password hashing competition. Informally, an MHF is a function which cannot be evaluated on dedicated hardware, like ASICs, at significantly lower energy and/or hardware cost than evaluating a single instance on a standard single-core architecture. Data-independent means the memory access pattern of the function is independent of the input; this makes iMHFs harder to construct than data-dependent ones, but the latter can be attacked by various side-channel attacks. Following [Alwen-Blocki\u2716], we capture the evaluation of an iMHF as a directed acyclic graph (DAG). The cumulative parallel pebbling complexity of this DAG is a good measure for the cost of evaluating the iMHF on an ASIC. If n denotes the number of nodes of a DAG (or equivalently, the number of operations --- typically hash function calls --- of the underlying iMHF), its pebbling complexity must be close to n^2 for the iMHF to be memory-hard. We show that the following iMHFs are far from this bound: Rig.v2, TwoCats and Gambit can be attacked with complexity O(n^{1.75}); the data-independent phase of Pomelo (a finalist of the password hashing competition) and Lyra2 (also a finalist) can be attacked with complexity O(n^{1.83}) and O(n^{1.67}), respectively. For our attacks we use and extend the technique developed by [Alwen-Blocki\u2716], who show that the pebbling complexity of a DAG can be upper bounded in terms of its depth-robustness

Parallel and Distributed Algorithms for a Class of Graph-Related Computational Problems.

There exist at least two models of parallel computing, namely, shared-memory and message-passing. This research addresses problems in both these types of systems, and proposes efficient parallel (Shared-Memory Model) and distributed (message-passing) algorithms for a variety of graph related computational problems. In part I, we design algorithms for three generic problems in distributed systems: set manipulation, network structure recognition and facility placement. We present optimal distributed algorithms for recognizing rectangular-mesh networks. The time and message complexity of our algorithm is linear in the number of nodes in the network. We also lay the foundation for the recognition of 2-reducible, outer-planar and cactus graphs. These algorithms have a message complexity of O(kn), where, k is the number of isolated two degree nodes in the network. We introduce the problem of reliable r-domination and design unified optimal distributed algorithms for the total, reliable and independent r-domination on trees. The time and message complexity of our algorithm is O(n), where n is the number of nodes in the tree. In the domain of set manipulation we design optimal algorithms for determining the intersection of sets in a distributed environment, where each processor is assumed to have its own set. The time and message complexity of our set intersection algorithm is O(mn), where m is the cardinality of the smallest set. In part II of our research we design optimal algorithms for r-domination and efficient parallel algorithms for the p-center problems on trees. We also present an optimal algorithm for computing the maximum independent set on intervals i the EREW-PRAM model. The r-domination problem on trees can now be solved in O(logn)time with O(n/logn) processors using the EREW-PRAM model. A parallel algorithm for range searching is developed using the concept of distributed data structures. We show that O(logn) search time can be effected for a range search on n 3-dimensional points using (2.log\sp2n-14.logn + 8) processors. Our algorithm can easily be generalized for the case of d-dimensional range search. (Abstract shortened with permission of author.)

Ahlfors circle maps and total reality: from Riemann to Rohlin

This is a prejudiced survey on the Ahlfors (extremal) function and the weaker {\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e. those (branched) maps effecting the conformal representation upon the disc of a {\it compact bordered Riemann surface}. The theory in question has some well-known intersection with real algebraic geometry, especially Klein's ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a gallery of pictures quite pleasant to visit of which we have attempted to trace the simplest representatives. This drifted us toward some electrodynamic motions along real circuits of dividing curves perhaps reminiscent of Kepler's planetary motions along ellipses. The ultimate origin of circle maps is of course to be traced back to Riemann's Thesis 1851 as well as his 1857 Nachlass. Apart from an abrupt claim by Teichm\"uller 1941 that everything is to be found in Klein (what we failed to assess on printed evidence), the pivotal contribution belongs to Ahlfors 1950 supplying an existence-proof of circle maps, as well as an analysis of an allied function-theoretic extremal problem. Works by Yamada 1978--2001, Gouma 1998 and Coppens 2011 suggest sharper degree controls than available in Ahlfors' era. Accordingly, our partisan belief is that much remains to be clarified regarding the foundation and optimal control of Ahlfors circle maps. The game of sharp estimation may look narrow-minded "Absch\"atzungsmathematik" alike, yet the philosophical outcome is as usual to contemplate how conformal and algebraic geometry are fighting together for the soul of Riemann surfaces. A second part explores the connection with Hilbert's 16th as envisioned by Rohlin 1978.Comment: 675 pages, 199 figures; extended version of the former text (v.1) by including now Rohlin's theory (v.2