An official website of the United States government
A constraint satisfaction problem (CSP),\(\textsf {Max-CSP}(\mathcal {F})\), is specified by a finite set of constraints\(\mathcal {F}\subseteq \lbrace [q]^k \rightarrow \lbrace 0,1\rbrace \rbrace\)for positive integersqandk. An instance of the problem onnvariables is given bymapplications of constraints from\(\mathcal {F}\)to subsequences of thenvariables, and the goal is to find an assignment to the variables that satisfies the maximum number of constraints. In the (γ ,β)-approximation version of the problem for parameters 0 ≤ β ≤ γ ≤ 1, the goal is to distinguish instances where at least γ fraction of the constraints can be satisfied from instances where at most β fraction of the constraints can be satisfied. In this work, we consider the approximability of this problem in the context of sketching algorithms and give a dichotomy result. Specifically, for every family\(\mathcal {F}\)and every β < γ, we show that either a linear sketching algorithm solves the problem in polylogarithmic space or the problem is not solvable by any sketching algorithm in\(o(\sqrt {n})\)space. In particular, we give non-trivial approximation algorithms using polylogarithmic space for infinitely many constraint satisfaction problems. We also extend previously known lower bounds for general streaming algorithms to a wide variety of problems, and in particular the case ofq=k=2, where we get a dichotomy, and the case when the satisfying assignments of the constraints of\(\mathcal {F}\)support a distribution on\([q]^k\)with uniform marginals. Prior to this work, other than sporadic examples, the only systematic classes of CSPs that were analyzed considered the setting of Boolean variablesq= 2, binary constraintsk=2, and singleton families\(|\mathcal {F}|=1\)and only considered the setting where constraints are placed on literals rather than variables. Our positive results show wide applicability of bias-based algorithms used previously by [47] and [41], which we extend to include richer norm estimation algorithms, by giving a systematic way to discover biases. Our negative results combine the Fourier analytic methods of [56], which we extend to a wider class of CSPs, with a rich collection of reductions among communication complexity problems that lie at the heart of the negative results. In particular, previous works used Fourier analysis over the Boolean cube to initiate their results and the results seemed particularly tailored to functions on Boolean literals (i.e., with negations). Our techniques surprisingly allow us to get to generalq-ary CSPs without negations by appealing to the same Fourier analytic starting point over Boolean hypercubes.
more »
« less
Joinable Parallel Balanced Binary Trees
In this article, we show how a single function,join, can be used to implement parallelbalanced binary search trees(BSTs) simply and efficiently. Based onjoin, our approach applies to multiple balanced tree data structures, and a variety of functions for ordered sets and maps. We describe our technique as an algorithmic framework calledjoin-based algorithms. We show that thejoinfunction fully captures what is needed for rebalancing trees for a variety of tree algorithms, as long as the balancing scheme satisfies certain properties, which we refer to asjoinabletrees. We discuss four balancing schemes that are joinable: AVL trees, red-black trees, weight-balanced trees, and treaps. We present a variety of tree algorithms that apply to joinable trees, includinginsert,delete,union,intersection,difference,split,range,filter, and so on, most of them also parallel. These algorithms are generic across balancing schemes. Many algorithms are optimal in the comparison model, and we provide a general proof to show the efficiency in work for joinable trees. The algorithms are highly parallel, all with polylogarithmic span (parallel dependence). Specifically, the set-set operationsunion,intersection, anddifferencehave work\( O(m\log (\frac{n}{m}+1)) \)and polylogarithmic span for input set sizes\( n \)and\( m\le n \). We implemented and tested our algorithms on the four balancing schemes. In general, all four schemes have quite similar performance, but the weight-balanced tree slightly outperforms the others. They have the same speedup characteristics, getting around 73\( \times \)speedup on 72 cores (144 hyperthreads). Experimental results also show that our implementation outperforms existing parallel implementations, and our sequential version achieves close or much better performance than the sequential merging algorithm in C++ Standard Template Library (STL) on various input sizes.
more »
« less
- Award ID(s):
- 1910030
- PAR ID:
- 10479472
- Publisher / Repository:
- ACM
- Date Published:
- Journal Name:
- ACM Transactions on Parallel Computing
- Volume:
- 9
- Issue:
- 2
- ISSN:
- 2329-4949
- Page Range / eLocation ID:
- 1 to 41
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
This paper introduces a new data-structural object that we call the tiny pointer. In many applications, traditional\(\log n\)-bit pointers can be replaced with\(o(\log n)\)-bit tiny pointers at the cost of only a constant-factor time overhead and a small probability of failure. We develop a comprehensive theory of tiny pointers, and give optimal constructions for both fixed-size tiny pointers (i.e., settings in which all of the tiny pointers must be the same size) and variable-size tiny pointers (i.e., settings in which the average tiny-pointer size must be small, but some tiny pointers can be larger). If a tiny pointer references an item in an array filled to load factor\(1-\delta\), then the optimal tiny-pointer size is\(\Theta(\log\log\log n+\log\delta^{-1})\)bits in the fixed-size case, and\(\Theta(\log\delta^{-1})\)expected bits in the variable-size case. Our tiny-pointer constructions also require us to revisit several classic problems having to do with balls and bins; these results may be of independent interest. Using tiny pointers, we apply tiny pointers to five classic data-structure problems. We show that:A data structure storing\(n\)\(v\)-bit values for\(n\)keys with constant-factor time modifications/queries can be implemented to take space\(nv+O(n\log^{(r)}n)\)bits, for any constant\(r\gt0\), as long as the user stores a tiny pointer of expected size\(O(1)\)with each key—here,\(\log^{(r)}n\)is the\(r\)-th iterated logarithm.Any binary search tree can be made succinct, meaning that it achieves\((1+o(1))\)times the optimal space, with constant-factor time overhead, and can even be made to be within\(O(n)\)bits of optimal if we allow for\(O(\log^{*}n)\)-time modifications—this holds even for rotation-based trees such as the splay tree and the red-black tree.Any fixed-capacity key-value dictionary can be made stable (i.e., items do not move once inserted) with constant-factor time overhead and\((1+o(1))\)-factor space overhead.Any key-value dictionary that requires uniform-size values can be made to support arbitrary-size values with constant-factor time overhead and with an additional space consumption of\(\log^{(r)}n+O(\log j)\)bits per\(j\)-bit value for an arbitrary constant\(r\gt0\)of our choice.Given an external-memory array\(A\)of size\((1+\varepsilon)n\)containing a dynamic set of up to\(n\)key-value pairs, it is possible to maintain an internal-memory stash of size\(O(n\log\varepsilon^{-1})\)bits so that the location of any key-value pair in\(A\)can be computed in constant time (and with no IOs). In each case tiny pointers allow for us to take a natural space-inefficient solution that uses pointers and make it space-efficient for free.more » « less
-
Given a weighted, ordered query set\(Q\)and a partition of\(Q\)into classes, we study the problem of computing a minimum-cost decision tree that, given any query\(q\in Q\), uses equality tests and less-than tests to determine\(q\)'s class. Such a tree can be faster and smaller than a conventional search tree and smaller than a lookup table (both of which must identify\(q\), not just its class). We give the first polynomial-time algorithm for the problem. The algorithm extends naturally to the setting where each query has multiple allowed classes.more » « less
-
Stream processing, which involves real-time computation of data as it is created or received, is vital for various applications, specifically wireless communication. The evolving protocols, the requirement for high-throughput, and the challenges of handling diverse processing patterns make it demanding. Traditional platforms grapple with meeting real-time throughput and latency requirements due to large data volume, sequential and indeterministic data arrival, and variable data rates, leading to inefficiencies in memory access and parallel processing. We present Canalis, a throughput-optimized framework designed to address these challenges, ensuring high-performance while achieving low energy consumption. Canalis is a hardware-software co-designed system. It includes a programmable spatial architecture, Flux Stream Processing Unit (FluxSPU), proposed by this work to enhance data throughput and energy efficiency. FluxSPU is accompanied by a software stack that eases the programming process. We evaluated Canalis with eight distinct benchmarks. When compared to CPU and GPU in mobile SoC to demonstrate the effectiveness of domain specialization, Canalis achieves an average speedup of 13.4\(\times\)and 6.6\(\times\), and energy savings of 189.8\(\times\)and 283.9\(\times\), respectively. In contrast to equivalent ASICs of the benchmarks, the average energy overhead of Canalis is within 2.4\(\times\), successfully maintaining generalizations without incurring significant overhead.more » « less
-
Current computing device authentication often presents accessibility barriers for people withupper extremity impairments (UEI). In this article, we present a framework calledAccessible image-Association-based Authentication for Computing devices (A3C), a novel recognition-based graphical authentication framework specifically designed for people with UEI to authenticate to their computing devices. A3C requires users to provide a set of primary images the user knows that are recognizable to them and subsequently associate each primary image with a secondary image. To evaluate the efficacy of the A3C framework, we instantiated the framework by implementing a version of A3C calledA3C-FA, which uses images of faces of people the user knows as the primary image and animal images as the secondary image. We then performed three studies to evaluate A3C-FA: a shoulder-surfing attack study (N\(=\)319), a close-adversary attack study (N\(=\)268), and a usability study with people with UEI (N\(=\)14). We found that A3C was robust against both shoulder-surfing and close-adversary attacks. We also performed a detailed study to evaluate the accessibility of A3C-FA. Our participants reported that A3C-FA was more usable and more secure than the authentication approaches with which they were familiar. Based on these findings, we suggest four areas of future research to further improve the design of the A3C framework.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1145/3512769
