Robert T. Schweller

Complexities for Generalized Models of Self-Assembly

In this paper, we extend Rothemund and Winfrees examination of the tile complexity of tile self-assembly [6]. They provided a lower bound of Ω(log <i>N</i>/log log <i>N</i>) on the tile complexity of assembling an <i>N</i> × <i>N</i> square for almost all <i>N</i>. Adleman et al. [1] gave a construction which achieves this bound. We consider whether the tile complexity for self-assembly can be reduced through several natural generalizations of the model. One of our results is a tile set of size <i>O</i>(√log <i>N</i>) which assembles an <i>N</i> × <i>N</i> square in a model which allows flexible glue strength between non-equal glues (This was independently discovered in [3]). This result is matched by a lower bound dictated by Kolmogorov complexity. For three other generalizations, we show that the Ω(log <i>N</i>/log log <i>N</i>) lower bound applies to <i>N</i> × <i>N</i> squares. At the same time, we demonstrate that there are some other shapes for which these generalizations allow reduced tile sets. Specifically, for thin rectangles with length <i>N</i> and width <i>k</i>, we provide a tighter lower bound of Ω(<i>N</i>(1/<i>k</i>)/<i>k</i>) for the standard model, yet we also give a construction which achieves <i>O</i>(log <i>N</i>/log log <i>N</i>) complexity in a model in which the temperature of the tile system is adjusted during assembly. We also investigate the problem of verifying whether a given tile system uniquely assembles into a given shape, and show that this problem is NP-hard.

Reversible sketches for efficient and accurate change detection over network data streams

Traffic anomalies such as failures and attacks are increasing in frequency and severity, and thus identifying them rapidly and accurately is critical for large network operators. The detection typically treats the traffic as a collection of flows and looks for heavy changes in traffic patterns (<i>e.g.</i>, volume, number of connections). However, as link speeds and the number of flows increase, keeping per-flow state is not scalable. The recently proposed sketch-based schemes [14] are among the very few that can detect heavy changes and anomalies over massive data streams at network traffic speeds. However, sketches do not preserve the key (<i>e.g.</i>, source IP address) of the flows. Hence, even if anomalies are detected, it is difficult to infer the culprit flows, making it a big practical hurdle for online deployment. Meanwhile, the number of keys is too large to record. To address this challenge, we propose efficient <i>reversible hashing</i> algorithms to infer the keys of culprit flows from sketches without storing any explicit key information. No extra memory or memory accesses are needed for recording the streaming data. Meanwhile, the heavy change detection daemon runs in the background with space complexity and computational time sublinear to the key space size. This short paper describes the conceptual framework of the reversible sketches, as well as some initial approaches for implementation. See [23] for the optimized algorithms in details. comment We further apply various emph IP-mangling algorithms and emph bucket classification methods to reduce the false positives and false negatives. Evaluated with netflow traffic traces of a large edge router, we demonstrate that the reverse hashing can quickly infer the keys of culprit flows even for many changes with high accuracy.

Reducing tile complexity for self-assembly through temperature programming

We consider the tile self-assembly model and how tile complexity can be eliminated by permitting the temperature of the self-assembly system to be adjusted throughout the assembly process. To do this, we propose novel techniques for designing tile sets that permit an arbitrary length m binary number to be encoded into a sequence of O(m) temperature changes such that the tile set uniquely assembles a supertile that precisely encodes the corresponding binary number. As an application, we show how this provides a general tile set of size O(1) that is capable of uniquely assembling essentially any n X n square, where the assembled square is determined by a temperature sequence of length O(log n) that encodes a binary description of n. This yields an important decrease in tile complexity from the required Ω(log n/log log n) for almost all n when the temperature of the system is fixed. We further show that for almost all n, no tile system can simultaneously achieve both o(log n) temperature complexity and O(log n/log log n) tile complexity, showing that both versions of an optimal square building scheme have been discovered. This work suggests that temperature change can constitute a natural, dynamic method for providing input to self-assembly systems that is potentially superior to the current technique of designing large tile sets with specific inputs hardwired into the tileset.

Reversible sketches: enabling monitoring and analysis over high-speed data streams

A key function for network traffic monitoring and analysis is the ability to perform aggregate queries over multiple data streams. Change detection is an important primitive which can be extended to construct many aggregate queries. The recently proposed sketches are among the very few that can detect heavy changes online for high speed links, and thus support various aggregate queries in both temporal and spatial domains. However, it does not preserve the keys (e. g., source IP address) of flows, making it difficult to reconstruct the desired set of anomalous keys. To address this challenge, we propose the reversible sketch data structure along with reverse hashing algorithms to infer the keys of culprit flows. There are two phases. The first operates online, recording the packet stream in a compact representation with negligible extra memory and few extra memory accesses. Our prototype single FPGA board implementation can achieve a throughput of over 16 Gb/s for 40-byte packet streams (the worst case). The second phase identifies heavy changes and their keys from the representation in nearly real time. We evaluate our scheme using traces from large edge routers with OC-12 or higher links. Both the analytical and experimental results show that we are able to achieve online traffic monitoring and accurate change/intrusion detection over massive data streams on high speed links, all in a manner that scales to large key space size. To the best of our knowledge, our system is the first to achieve these properties simultaneously.

Randomized Self-assembly for Approximate Shapes

In this paper we design tile self-assembly systems which assemble arbitrarily close approximations to target squares with arbitrarily high probability. This is in contrast to previous work which has only considered deterministic assemblies of a single shape. Our technique takes advantage of the ability to assign tile concentrations to each tile type of a self-assembly system. Such an assignment yields a probability distribution over the set of possible assembled shapes. We show that by considering the assembly of close approximations to target shapes with high probability, as opposed to exact deterministic assembly, we are able to achieve significant reductions in tile complexity. In fact, we restrict ourselves to constant sized tile systems, encoding all information about the target shape into the tile concentration assignment. In practice, this offers a potentially useful tradeoff, as large libraries of particles may be infeasible or require substantial effort to create, while the replication of existing particles to adjust relative concentration may be much easier. To illustrate our technique we focus on the assembly of n×nsquares, a special case class of shapes whose study has proven fruitful in the development of new self-assembly systems.

ASYNCHRONOUS SIGNAL PASSING FOR TILE SELF-ASSEMBLY: FUEL EFFICIENT COMPUTATION AND EFFICIENT ASSEMBLY OF SHAPES

In this paper we demonstrate the power of a model of tile self-assembly based on active glues which can dynamically change state. We formulate the Signal-passing Tile Assembly Model (STAM), based on the model of Padilla et al. [24] to be asynchronous, allowing any action of turning a glue on or off, attaching a new tile, or breaking apart an assembly to happen in any order. Within this highly generalized model we provide three new solutions to tile self-assembly problems that have been addressed within the abstract Tile Assembly Model and its variants, showing that signal passing tiles allow for substantial improvement across multiple complexity metrics. Our first result utilizes a recursive assembly process to achieve tile-type efficient assembly of linear structures, using provably fewer tile types than what is possible in standard tile assembly models. Our second system of signal-passing tiles simulates any Turing machine with high fuel efficiency by using only a constant number of tiles per computation step. Our third system assembles the discrete Sierpinski triangle, demonstrating that this pattern can be strictly self-assembled within the STAM. This result is of particular interest in that it is known that this pattern cannot self-assemble within a number of well studied tile self-assembly models. Notably, all of our constructions are at temperature 1, further demonstrating that signal-passing confers the power to bypass many restrictions found in standard tile assembly models.

Reverse Hashing for High-Speed Network Monitoring: Algorithms, Evaluation, and Applications

A key function for network traffic monitoring and analysis is the ability to perform aggregate queries over multiple data streams. Change detection is an important primitive which can be extended to construct many aggregate queries. The recently proposed sketches (Krishnamurthy, 2003) are among the very few that can detect heavy changes online for high speed links, and thus support various aggregate queries in both temporal and spatial domains. However, it does not preserve the keys (e.g., source IP address) of flows, making it difficult to reconstruct the desired set of anomalous keys. In an earlier abstract we proposed a framework for a reversible sketch data structure that offers hope for efficient extraction of keys (Schweller, 2004). However, this scheme is only able to detect a single heavy change key and places restrictions on the statistical properties of the key space. To address these challenges, we propose an efficient reverse hashing scheme to infer the keys of culprit flows from reversible sketches. There are two phases. The first operates online, recording the packet stream in a compact representation with negligible extra memory and few extra memory accesses. Our prototype single FPGA board implementation can achieve a throughput of over 16 Gbps for 40-byte-packet streams (the worst case). The second phase identifies heavy changes and their keys from the representation in nearly real time. We evaluate our scheme using traces from large edge routers with OC-12 or higher links. Both the analytical and experimental results show that we are able to achieve online traffic monitoring and accurate change/intrusion detection over massive data streams on high speed links, all in a manner that scales to large key space size. To the best of our knowledge, our system is the first to achieve these properties simultaneously.

Strong Fault-Tolerance for Self-Assembly with Fuzzy Temperature

We consider the problem of fault-tolerance in nanoscale algorithmic self-assembly. We employ a standard variant of Winfree’s abstract Tile Assembly Model (aTAM), the two-handed aTAM, in which square “tiles” – a model of molecules constructed from DNA for the purpose of engineering self-assembled nanostructures – aggregate according to specific binding sites of varying strengths, and in which large aggregations of tiles may attach to each other, in contrast to the seeded aTAM, in which tiles aggregate one at a time to a single specially designated “seed” assembly. We focus on a major cause of errors in tile-based self-assembly: that of unintended growth due to “weak” strength-1 bonds, which if allowed to persist, may be stabilized by subsequent attachment of neighboring tiles in the sense that at least energy 2 is now required to break apart the resulting assembly, i.e., the errant assembly is stable at temperature 2. We study a common self-assembly benchmark problem, that of assembling an n×n square using O(log n) unique tile types, under the two-handed model of self-assembly. Our main result achieves a much stronger notion of fault-tolerance than those achieved previously. Arbitrary strength-1 growth is allowed, however, any assembly that grows sufficiently to become stable at temperature 2 is guaranteed to assemble into the correct final assembly of an n×n square. In other words, errors due to insufficient attachment, which is the cause of errors studied in earlier papers on fault-tolerance, are prevented absolutely in our main construction, rather than only with high probability and for sufficiently small structures, as in previous fault tolerance studies.

Self-assembly with geometric tiles

In this work we propose a generalization of Winfrees abstract Tile Assembly Model (aTAM) in which tile types are assigned rigid shapes, or geometries, along each tile face. We examine the number of distinct tile types needed to assemble shapes within this model, the temperature required for efficient assembly, and the problem of designing compact geometric faces to meet given compatibility specifications. We pose the following question: can complex geometric tile faces arbitrarily reduce the number of distinct tile types to assemble shapes? Within the most basic generalization of the aTAM, we show that the answer is no. For almost all n at least

Linear Time Probabilistic Algorithms for the Singular Haplotype Reconstruction Problem from SNP Fragments

\Omega(\sqrt{\log n})