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Register a new userOptimal (Randomized) Parallel Algorithms in the Binary-Forking Model
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Added by
Yihan Sun
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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In this paper we develop optimal algorithms in the binary-forking model for a variety of fundamental problems, including sorting, semisorting, list ranking, tree contraction, range minima, and ordered set union, intersection and difference. In the binary-forking model, tasks can only fork into two child tasks, but can do so recursively and asynchronously. The tasks share memory, supporting reads, writes and test-and-sets. Costs are measured in terms of work (total number of instructions), and span (longest dependence chain). The binary-forking model is meant to capture both algorithm performance and algorithm-design considerations on many existing multithreaded languages, which are also asynchronous and rely on binary forks either explicitly or under the covers. In contrast to the widely studied PRAM model, it does not assume arbitrary-way forks nor synchronous operations, both of which are hard to implement in modern hardware. While optimal PRAM algorithms are known for the problems studied herein, it turns out that arbitrary-way forking and strict synchronization are powerful, if unrealistic, capabilities. Natural simulations of these PRAM algorithms in the binary-forking model (i.e., implementations in existing parallel languages) incur an $Omega(log n)$ overhead in span. This paper explores techniques for designing optimal algorithms when limited to binary forking and assuming asynchrony. All algorithms described in this paper are the first algorithms with optimal work and span in the binary-forking model. Most of the algorithms are simple. Many are randomized.
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In this paper we show that many sequential randomized incremental algorithms are in fact parallel. We consider algorithms for several problems including Delaunay triangulation, linear programming, closest pair, smallest enclosing disk, least-element lists, and strongly connected components. We analyze the dependences between iterations in an algorithm, and show that the dependence structure is shallow with high probability, or that by violating some dependences the structure is shallow and the work is not increased significantly. We identify three types of algorithms based on their dependences and present a framework for analyzing each type. Using the framework gives work-efficient polylogarithmic-depth parallel algorithms for most of the problems that we study. This paper shows the first incremental Delaunay triangulation algorithm with optimal work and polylogarithmic depth, which is an open problem for over 30 years. This result is important since most implementations of parallel Delaunay triangulation use the incremental approach. Our results also improve bounds on strongly connected components and least-elements lists, and significantly simplify parallel algorithms for several problems.
The general adwords problem has remained largely unresolved. We define a subcase called {em $k$-TYPICAL}, $k in Zplus$, as follows: the total budget of all the bidders is sufficient to buy $k$ bids for each bidder. This seems a reasonable assumption for a typical instance, at least for moderate values of $k$. We give a randomized online algorithm, achieving a competitive ratio of $left(1 - {1 over e} - {1 over k} right)$, for this problem. We also give randomized online algorithms for other special cases of adwords. Another subcase, when bids are small compared to budgets, has been of considerable practical significance in ad auctions cite{MSVV}. For this case, we give an optimal randomized online algorithm achieving a competitive ratio of $left(1 - {1 over e} right)$. Previous algorithms for this case were based on LP-duality; the impact of our new approach remains to be seen. The key to these results is a simplification of the proof for RANKING, the optimal algorithm for online bipartite matching, given in cite{KVV}. Our algorithms for adwords can be seen as natural extensions of RANKING.
Given $n$ colored balls, we want to detect if more than $lfloor n/2rfloor$ of them have the same color, and if so find one ball with such majority color. We are only allowed to choose two balls and compare their colors, and the goal is to minimize the total number of such operations. A well-known exercise is to show how to find such a ball with only $2n$ comparisons while using only a logarithmic number of bits for bookkeeping. The resulting algorithm is called the Boyer--Moore majority vote algorithm. It is known that any deterministic method needs $lceil 3n/2rceil-2$ comparisons in the worst case, and this is tight. However, it is not clear what is the required number of comparisons if we allow randomization. We construct a randomized algorithm which always correctly finds a ball of the majority color (or detects that there is none) using, with high probability, only $7n/6+o(n)$ comparisons. We also prove that the expected number of comparisons used by any such randomized method is at least $1.019n$.
We study the complexity of quantum query algorithms that make p queries in parallel in each timestep. This model is in part motivated by the fact that decoherence times of qubits are typically small, so it makes sense to parallelize quantum algorithms as much as possible. We show tight bounds for a number of problems, specifically Theta((n/p)^{2/3}) p-parallel queries for element distinctness and Theta((n/p)^{k/(k+1)} for k-sum. Our upper bounds are obtained by parallelized quantum walk algorithms, and our lower bounds are based on a relatively small modification of the adversary lower bound method, combined with recent results of Belovs et al. on learning graphs. We also prove some general bounds, in particular that quantum and classical p-parallel complexity are polynomially related for all total functions f when p is small compared to fs block sensitivity.
We study emph{parallel} online algorithms: For some fixed integer $k$, a collective of $k$ parallel processes that perform online decisions on the same sequence of events forms a $k$-emph{copy algorithm}. For any given time and input sequence, the overall performance is determined by the best of the $k$ individual total results. Problems of this type have been considered for online makespan minimization; they are also related to optimization with emph{advice} on future events, i.e., a number of bits available in advance. We develop textsc{Predictive Harmonic}$_3$ (PH3), a relatively simple family of $k$-copy algorithms for the online Bin Packing Problem, whose joint competitive factor converges to 1.5 for increasing $k$. In particular, we show that $k=6$ suffices to guarantee a factor of $1.5714$ for PH3, which is better than $1.57829$, the performance of the best known 1-copy algorithm textsc{Advanced Harmonic}, while $k=11$ suffices to achieve a factor of $1.5406$, beating the known lower bound of $1.54278$ for a single online algorithm. In the context of online optimization with advice, our approach implies that 4 bits suffice to achieve a factor better than this bound of $1.54278$, which is considerably less than the previous bound of 15 bits.
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