Membership Query Algorithm Research Articles

Properly Learning Decision Trees in almost Polynomial Time

We give an n O (log log n ) -time membership query algorithm for properly and agnostically learning decision trees under the uniform distribution over { ± 1} n . Even in the realizable setting, the previous fastest runtime was n O (log n ) , a consequence of a classic algorithm of Ehrenfeucht and Haussler. Our algorithm shares similarities with practical heuristics for learning decision trees, which we augment with additional ideas to circumvent known lower bounds against these heuristics. To analyze our algorithm, we prove a new structural result for decision trees that strengthens a theorem of O’Donnell, Saks, Schramm, and Servedio. While the OSSS theorem says that every decision tree has an influential variable, we show how every decision tree can be “pruned” so that every variable in the resulting tree is influential.

P3FA: Unified Unicast/Multicast Forwarding Algorithm for High-Performance Router/Switch

High-performance multicast packet switching technologies are evolving to meet the growing demand for scalability on the Internet and datacenters, etc. Implementing a high-performance switch/router relies on a polynomial-time group membership query algorithm within the Packet Forwarding Engines (PFEs) to determine whether a packet is forwarded through an egress. Among these, Bloom filter (BF)-based and Residue Number System (RNS)-based are being considered as two representatives of the membership query algorithms. However, both approaches suffer from some fatal weaknesses such as false-positive probability and time inefficiencies, especially for a carrier-grade PFE with high port-density features. According to similar properties of the RNS, we propose a simplified forwarding algorithm in this paper, named Per-Port Prime Filter Array (P3FA). The simulation results indicate that the P3FA can significantly improve space efficiencies under specific lower egress-diversities conditions. Under the same space constraints, P3FA improves multicast and unicast time efficiency by 1 to 4 orders of magnitude in the port-density 16–1024 range compared to previous works. Although it comes at the expense of hardware cost, it is still acceptable compared to recently improved previous work.

DH-SVRF: A Reconfigurable Unicast/Multicast Forwarding for High-Performance Packet Forwarding Engines

High-performance multicast-enabled packet forwarding engines (PFEs), as an essential component of high-end switches, use a polynomial-time membership query algorithm to determine which port(s) the data packet should be forwarded. The currently widely used query algorithm is Bloom Filter (BF), which has been proven to have many fatal flaws. Another error-free membership query algorithm includes Scalar-pair Vectors Routing Forwarding (SVRF), Fractional- <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> Scalar-pair Vectors Routing Forwarding (Frac- <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> SVRF), and the Per-Port Prime Filter Array (P <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> FA) also have some shortcomings in space and time efficiencies. In this paper, we proposed a hybrid strategy: Divaricate Heterogeneous SVRF (DH-SVRF) scheme, which based on the P <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> FA and Frac- <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> SVRF, which randomly divides all member ships into <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> groups, and each group has the same structure and is independent of each other to obtain higher time efficiency and space utilization. Finally, we also discussed the selection of the optimal egress-diversity threshold. Through mathematical modeling and simulation, we validate that the proposed DH-SVRF scheme is superior to the SVRF/Frac <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-N</i> SVRF and traditional BF in terms of scalability, space utilization, and time efficiency in specific conditions such as appropriate egress-diversity thresholds.

Fractional-N SVRF Forwarding Algorithm for Low Port-Density Packet Forwarding Engines

High-performance multicast-enabled switches and routers are being constantly developed, which use a polynomial-time group membership query algorithm within the Packet Forwarding Engines (PFEs) to determine whether or not a packet is forwarded through a unique or multiple egress ports. Among these, Bloom filter (BF) and Scalar-pair Vectors Routing and Forwarding (SVRF) are being considered as two representations of the membership query algorithms. However, both approaches suffer from some fatal weaknesses such as space and time inefficiencies, especially for a carrier-grade PFE with high port-density feature. In order to solve these imperfections of SVRF, we propose an improved Fractional-N Scalar-matrix and Vectors Routing and Forwarding (Fractional- N SVRF) scheme based on re-examining the idea of the original SVRF. The Fractional- N SVRF preprocesses a scalar-matrix by dividing an n-element group into N sub-blocks, thus element' keys belonging to distinct sub-blocks and sub-scalars are allowed to reuse relatively smaller identical prime keys. Based on Fractional-N SVRF, membership queries can be partitioned to leverage task parallelism, therefore better performance is achieved in terms of memory consumption and computational complexity.

BioBloom tools: fast, accurate and memory-efficient host species sequence screening using bloom filters.

Large datasets can be screened for sequences from a specific organism, quickly and with low memory requirements, by a data structure that supports time- and memory-efficient set membership queries. Bloom filters offer such queries but require that false positives be controlled. We present BioBloom Tools, a Bloom filter-based sequence-screening tool that is faster than BWA, Bowtie 2 (popular alignment algorithms) and FACS (a membership query algorithm). It delivers accuracies comparable with these tools, controls false positives and has low memory requirements.Availability and implementaion: www.bcgsc.ca/platform/bioinfo/software/biobloomtoolsContact: cjustin@bcgsc.ca or ibirol@bcgsc.caSupplementary information: Supplementary data are available at Bioinformatics online.

A Unified Unicast and Multicast Routing and Forwarding Algorithm for Software-Defined Datacenter Networks

In this article, we consider a scalability problem associated with software-defined datacenter, of which the unicast/multicast routing states is proven to be NP-complete. We introduce an efficient multiple membership query algorithm, called Scalar-pair Vectors Routing and Forwarding (SVRF), based on the prime theory such as Chinese Remainder Theorem (CRT). Our proposed algorithm simply calculates corresponding output ports of each multicast group by dividing a common scalar-pair with a group-specific key, within pseudo-polynomial time. The result is then used to make a forwarding decision within few cycles through a hardware accelerator. Compared to Bloom filter, our algorithm can achieve remarkable performance in terms of memory consumption, processing time, hardware cost, and 100% delivery accuracy, while applying for a large number of large-scale distinct flows (including unicast and multicast) in a large-scale datacenter networks. Our work may be applied to various research areas of computer science and networking.

Learning unions of [formula omitted]-dimensional rectangles

We consider the problem of learning unions of rectangles over the domain [ b ] n , in the uniform distribution membership query learning setting, where both b and n are “large”. We obtain poly ( n , log b ) -time algorithms for the following classes: • poly ( n log b ) -way Majority of O ( log ( n log b ) log log ( n log b ) ) -dimensional rectangles. • Union of poly ( log ( n log b ) ) many O ( log 2 ( n log b ) ( log log ( n log b ) log log log ( n log b ) ) 2 ) -dimensional rectangles. • poly ( n log b ) -way Majority of poly ( n log b ) - Or of disjoint O ( log ( n log b ) log log ( n log b ) ) dimensional rectangles. Our main algorithmic tool is an extension of Jackson’s boosting- and Fourier-based Harmonic Sieve algorithm [J.C. Jackson, An efficient membership-query algorithm for learning DNF with respect to the uniform distribution, Journal of Computer and System Sciences 55 (3) (1997) 414–440] to the domain [ b ] n , building on work of Akavia et al. [A. Akavia, S. Goldwasser, S. Safra, Proving hard core predicates using list decoding, in: Proc. of the 44th Annual IEEE Symposium on Foundations of Computer Science, FOCS ’03, 2003, pp. 146–156]. Other ingredients used to obtain the results stated above are techniques from exact learning [A. Beimel, E. Kushilevitz, Learning boxes in high dimension, Algorithmica 22 (1/2) (1998) 76–90] and ideas from recent work on learning augmented AC 0 circuits [J.C. Jackson, A.R. Klivans, R.A. Servedio, Learnability beyond AC 0 , in: Proc. of the 34th Annual ACM Symposium on Theory of Computing, STOC ’02, 2002, pp. 776–784] and on representing Boolean functions as thresholds of parities [A.R. Klivans, R.A. Servedio, Learning DNF in time 2 O ̃ ( n 1 / 3 ) , Journal of Computer and System Sciences 68 (2) (2004) 303–318].

An Efficient Membership-Query Algorithm for Learning DNF with Respect to the Uniform Distribution

We present a membership-query algorithm for efficiently learning DNF with respect to the uniform distribution. In fact, the algorithm properly learns with respect to uniform the class TOP of Boolean functions expressed as a majority vote over parity functions. We also describe extensions of this algorithm for learning DNF over certain nonuniform distributions and for learning a class of geometric concepts that generalizes DNF. Furthermore, we show that DNF is weakly learnable with respect to uniform from noisy examples. Our strong learning algorithm utilizes one of Freund's boosting techniques and relies on the fact that boosting does not require a completely distribution-independent weak learner. The boosted weak learner is a nonuniform extension of a parity-finding algorithm discovered by Goldreich and Levin.

Learning Arithmetic Read-Once Formulas

A formula is read-once if each variable appears at most once in it. An arithmetic read-once formula is one in which the operators are addition, subtraction, multiplication, and division. We present polynomial time algorithms for exact learning of arithmetic read-once formulas over a field. We present a membership and equivalence query algorithm that identifies arithmetic read-once formulas over an arbitrary field. We present a randomized membership query algorithm (i.e., a randomized black box interpolation algorithm) that identifies such formulas over finite fields with at least $2n + 5$ elements (where n is the number of variables) and over infinite fields. We also show the existence of nonuniform deterministic membership query algorithms for arbitrary read-once formulas over fields of characteristic 0, and division-free read-once formulas over fields that have at least $2n^{3} + 1$ elements. For our algorithms, we assume we are able to perform efficiently arithmetic operations on field elements and compute square roots in the field. It is shown that the ability to compute square roots is necessary in the sense that the problem of computing $n - 1$ square roots in a field can be reduced to the problem of identifying an arithmetic formula over n variables in that field. Our equivalence queries are of a slightly nonstandard form, in which counterexamples are required not to be inputs on which the formula evaluates to $0/0$. This assumption is shown to be necessary for fields of size $o(n/ \log n)$ in the sense that we prove there exists no polynomial time identification algorithm that uses only membership and standard equivalence queries.