2,003 research outputs found
Realizable paths and the NL vs L problem
A celebrated theorem of Savitch [Savitch'70] states that NSPACE(S) is contained in DSPACE(S²). In particular, Savitch gave a deterministic algorithm to solve ST-Connectivity (an NL-complete problem) using O({log}²{n}) space, implying NL (non-deterministic logspace) is contained in DSPACE({log}²{n}). While Savitch's theorem itself has not been improved in the last four decades, several graph connectivity problems are shown to lie between L and NL, providing new insights into the space-bounded complexity classes. All the connectivity problems considered in the literature so far are essentially special cases of ST-Connectivity.
In this dissertation, we initiate the study of auxiliary PDAs as graph connectivity problems and define sixteen different "graph realizability problems" and study their relationships. The complexity of these connectivity problems lie between L (logspace) and P (polynomial time). ST-Realizability, the most general graph realizability problem is P-complete. 1DSTREAL(poly), the most specific graph realizability problem is L-complete. As special cases of our graph realizability problems we define two natural problems, Balanced ST-Connectivity and Positive Balanced ST-Connectivity, that lie between L and NL.
We study the space complexity of SGSLOGCFL, a graph realizability problem lying between L and LOGCFL. We define generalizations of graph squaring and transitive closure, present efficient parallel algorithms for SGSLOGCFL and use the techniques of Trifonov to show that SGSLOGCFL is contained in DSPACE(lognloglogn). This implies that Balanced ST-Connectivity is contained in DSPACE(lognloglogn). We conclude with several interesting new research directions.PhDCommittee Chair: Richard Lipton; Committee Member: Anna Gal; Committee Member: Maria-Florina Balcan; Committee Member: Merrick Furst; Committee Member: William Coo
Electric response of DNA hairpins to magnetic fields
We study the electric properties of single-stranded DNA molecules with
hairpin-like shapes in the presence of a magnetic flux. It is shown that the
current amplitude can be modulated by the applied field. The details of the
electric response strongly depend on the twist angles. The current exhibits
periodicity for geometries where the flux through the plaquettes of the ladder
can be cancelled pairwise (commensurate twist). Further twisting the geometry
and changing its length causes complex aperiodic oscillations. We also study
persistent currents: They reduce to simple harmonic oscillations if the system
is commensurate, otherwise deviations occur due to the existence of closed
paths leading to a washboard shape.Comment: 11 pages, 4 figure
A Framework for Structured Quantum Search
A quantum algorithm for general combinatorial search that uses the underlying
structure of the search space to increase the probability of finding a solution
is presented. This algorithm shows how coherent quantum systems can be matched
to the underlying structure of abstract search spaces, and is analytically
simpler than previous structured search methods. The algorithm is evaluated
empirically with a variety of search problems, and shown to be particularly
effective for searches with many constraints. Furthermore, the algorithm
provides a simple framework for utilizing search heuristics. It also exhibits
the same phase transition in search difficulty as found for sophisticated
classical search methods, indicating it is effectively using the problem
structure.Comment: 18 pages, Latex, 7 figures, further information available at
ftp://parcftp.xerox.com/pub/dynamics/quantum.htm
Arithmetic Branching Programs with Memory
We extend the well known characterization of VPws as the class of polynomials computed by polynomial size arithmetic branching programs to other complexity classes. In order to do so we add additional memory to the computation of branching programs to make them more expressive. We show that allowing different types of memory in branching programs increases the computational power even for constant width programs. In particular, this leads to very natural and robust characterizations of VP and VNP by branching programs with memory. 1
A machine learning approach to constructing Ramsey graphs leads to the Trahtenbrot-Zykov problem.
Attempts at approaching the well-known and difficult problem of constructing Ramsey graphs via machine learning lead to another difficult problem posed by Zykov in 1963 (now commonly referred to as the Trahtenbrot-Zykov problem): For which graphs F does there exist some graph G such that the neighborhood of every vertex in G induces a subgraph isomorphic to F? Chapter 1 provides a brief introduction to graph theory. Chapter 2 introduces Ramsey theory for graphs. Chapter 3 details a reinforcement learning implementation for Ramsey graph construction. The implementation is based on board game software, specifically the AlphaZero program and its success learning to play games from scratch. The chapter ends with a description of how computing challenges naturally shifted the project towards the Trahtenbrot-Zykov problem. Chapter 3 also includes recommendations for continuing the project and attempting to overcome these challenges. Chapter 4 defines the Trahtenbrot-Zykov problem and outlines its history, including proofs of results omitted from their original papers. This chapter also contains a program for constructing graphs with all neighborhood-induced subgraphs isomorphic to a given graph F. The end of Chapter 4 presents constructions from the program when F is a Ramsey graph. Constructing such graphs is a non-trivial task, as Bulitko proved in 1973 that the Trahtenbrot-Zykov problem is undecidable. Chapter 5 is a translation from Russian to English of this famous result, a proof not previously available in English. Chapter 6 introduces Cayley graphs and their relationship to the Trahtenbrot-Zykov problem. The chapter ends with constructions of Cayley graphs Γ in which the neighborhood of every vertex of Γ induces a subgraph isomorphic to a given Ramsey graph, which leads to a conjecture regarding the unique extremal Ramsey(4, 4) graph
Spin tunneling in the Kagom\'e antiferromagnet
The collective tunneling of a small cluster of spins between two degenerate
ground state configurations of the Kagom\'{e}-lattice quantum Heisenberg
antiferromagnet is \mbox{studied}. The cluster consists of the six spins on a
hexagon of the lattice. The resulting tunnel splitting energy is
calculated in detail, including the prefactor to the exponential \exp(- \SSo /
\hbar). This is done by setting up a coherent spin state path integral in
imaginary time and evaluating it by the method of steepest descent. The hexagon
tunneling problem is mapped onto a much simpler tunneling problem, involving
only one collective degree of freedom, which can be treated by known methods.
It is found that for half-odd-integer spins, the tunneling amplitude and the
tunnel splitting energy are exactly zero, because of destructive interference
between symmetry-related -instanton and -instanton tunneling paths.
This destructive interference is shown to occur also for certain larger loops
of spins on the Kagom\'{e} lattice. For small, integer spins, our results
suggest that tunneling strongly competes with \mbox{in-plane}
order-from-disorder selection effects; it constitutes a disordering mechanism
that might drive the system into a partially disordered ground state, related
to a spin nematic.Comment: 38 pages (RevTex), 8 figures upon request PRB921
Mission science value-cost savings from the Advanced Imaging Communication System (AICS)
An Advanced Imaging Communication System (AICS) was proposed in the mid-1970s as an alternative to the Voyager data/communication system architecture. The AICS achieved virtually error free communication with little loss in the downlink data rate by concatenating a powerful Reed-Solomon block code with the Voyager convolutionally coded, Viterbi decoded downlink channel. The clean channel allowed AICS sophisticated adaptive data compression techniques. Both Voyager and the Galileo mission have implemented AICS components, and the concatenated channel itself is heading for international standardization. An analysis that assigns a dollar value/cost savings to AICS mission performance gains is presented. A conservative value or savings of 4.5 million for Galileo, and as much as $7 to 9.5 million per mission for future projects such as the proposed Mariner Mar 2 series is shown
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