# Minimal MapReduce Algorithms

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2 l = window(o) o window sum = 55 window max = 2 Figure 1: Sliding aggregates dataset S at any moment. This effectively prevents partition skew, where some machines are forced to handle considerably more than m objects, as isamajor cause of inefficiencyinmapreduce [36]. Second, bounded net-traffic guarantees that, the shuffle phase of each round transfers at most O(m t) = O(n) words of network traffic overall. The duration of the phase equals roughly the time for a machine to send and receive O(m) words, because the data transfers to/from different machines are in parallel. Furthermore, this property is also useful when one wants to make an algorithm stateless for the purpose of fault tolerance, as discussed in Section 2.1. The third property constant round is not new, as it has been the goal of many previous MapReduce algorithms. Importantly, this and the previous properties imply that there can be only O(n) words of network traffic during the entire algorithm. Finally, optimal computation echoes the very original motivation of MapReduce to accomplish a computational task t times faster than leveraging only one machine. Contributions. The core of this work comprises of neat minimal algorithms for two problems: Sorting. TheinputisasetS ofnobjectsdrawnfromanordered domain. When the algorithm terminates, all the objects must have been distributed across the t machines in a sorted fashion. That is, we can order the machines from 1 to t such that all objects inmachine iprecede those inmachine j forall1 i < j t. Sliding Aggregation. The input includes asets ofnobjectsfromanordereddomain,whereevery object o S is associatedwithanumeric weight aninteger l n and a distributive aggregate function AGG(e.g., sum, max, min). Denote by window(o) the set of l largest objects in S not exceeding o. The window aggregate of o is the result of applying AGG tothe weights of the objects in window(o). The sliding aggregation problem is to report the window aggregate of every object ins. Figure 1 illustrates an example where l = 5. Each black dot represents an object in S. Some relevant weights are given on top of the corresponding objects. For the object o as shown, its window aggregate is 55 and 2 for AGG = sum and max, respectively. The significance of sorting is obvious: a minimal algorithm for this problem leads to minimal algorithms for several fundamental database problems, including ranking, group-by, semi-join and skyline, as we willdiscuss inthis paper. The importance of the second problem probably deserves a bit more explanation. Sliding aggregates are crucial in studying time series. For example, consider a time series that records the Nasdaq index in history, with one value per minute. It makes good senses to examine moving statistics, that is, statistics aggregated from a sliding window. For example, a 6-month average/maximum with respect to a day equals the average/maximum Nasdaq index in a 6-month period ending on that very day. The 6-month averages/maximums of all days can be obtained by solving a sliding aggregation problem(note that an average can be calculated bydividing a window sum bythe period length l). SortingandslidingaggregationcanbothbesettledinO(nlogn) time on a sequential computer. There has been progress in developing MapReduce algorithms for sorting. The state of the art is TeraSort [5], which won the Jim Gray s benchmark contest in 29. TeraSort comes close to being minimal when a crucial parameter is set appropriately. As will be clear later, the algorithm requires manual tuning of the parameter, an improper choice of which can incur severe performance penalty. Sliding aggregation has also been studied in MapReduce by Beyer et al. [6]. However, as explained shortly, the algorithm is far from being minimal, and is efficient only whenthe window length l is short the authors of [6] commented that this problem is notoriously difficult. Technical Overview. This work was initialized by an attempt to justify theoretically why TeraSort often achieves excellent sorting time with only 2 rounds. In the first round, the algorithm extracts a random sample set S samp of the input S, and then picks t 1 sampled objects as the boundary objects. Conceptually, these boundary objects divide S into t segments. In the second round, eachofthetmachinesacquiresalltheobjectsinadistinctsegment, andsorts them. The sizeof S samp isthe key toefficiency. IfS samp is too small, the boundary objects may be insufficiently scattered, which can cause partition skew in the second round. Conversely, an over-sized S samp entails expensive sampling overhead. In the standard implementation of TeraSort, the sample size is left as a parameter, although it always seems to admit a good choice that gives outstanding performance [5]. In this paper, we provide rigorous explanation for the above phenomenon. Our theoretical analysis clarifies how to set the size ofs samp toguarantee theminimalityof TeraSort. Inthemeantime, we also remedy a conceptual drawback of TeraSort. As elaborated later, strictly speaking, this algorithm does not fit in the MapReduce framework, because it requires that (besides network messages) the machines should be able to communicate by reading/writing a common (distributed) file. Once this is disabled, the algorithm requires one more round. We present an elegant fix so that the algorithm still terminates in 2 rounds even by strictly adhering to MapReduce. Our findings of TeraSort have immediate practical significance, given the essential role of sorting in a large number of MapReduce programs. Regarding sliding aggregation, the difficulty lies in that l is not a constant, but can be any value up to n. Intuitively, when l m, window(o) is so large that the objects in window(o) cannot be found on one machine under the minimum footprint constraint. Instead, window(o) would potentially span many machines, making it essential to coordinate the searching of machines judiciously to avoid a disastrous cost blowup. In fact, this pitfall has captured the existing algorithm of [6], whose main idea is to ensure that every sliding window be sent to a machine for aggregation (various windows may go to different machines). This suffers from prohibitive communication and processing cost when the window length l is long. Our algorithm, on the other hand, achieves minimality with a novel idea of perfectly balancing the input objects across the machines while still maintaining their sorted order. Outline. Section 2 reviews the previous work related to ours. Section 3 analyzes TeraSort and modifies it into a minimal

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