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The evaluation of other relational operations also can be parallelized: Selection Let the selection be (r) Consider rst the case where is of the form ai = v, where ai is an attribute and v is a value If the relation r is partitioned on ai , the selection proceeds at a single processor If is of the form l ai u that is, is a range selection and the relation has been rangepartitioned on ai , then the selection proceeds at each processor whose partition overlaps with the speci ed range of values In all other cases, the selection proceeds in parallel at all the processors Duplicate elimination Duplicates can be eliminated by sorting; either of the parallel sort techniques can be used, optimized to eliminate duplicates as soon as they appear during sorting We can also parallelize duplicate elimination by partitioning the tuples (by either range or hash partitioning) and eliminating duplicates locally at each processor Projection Projection without duplicate elimination can be performed as tuples are read in from disk in parallel If duplicates are to be eliminated, either of the techniques just described can be used Aggregation Consider an aggregation operation We can parallelize the operation by partitioning the relation on the grouping attributes, and then com-

The nal example of an op amp circuit we consider is called a summing ampli er or summer This type of ampli er sums up multiple voltages to produce an output voltage Speci cally, it can be shown that the circuit shown in Fig 9-4 produces the output voltage Vout = R f Vb Vc Va + + Ra Rb Rc (97)

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FIGURE 7-9

puting the aggregate values locally at each processor Either hash partitioning or range partitioning can be used If the relation is already partitioned on the grouping attributes, the rst step can be skipped We can reduce the cost of transferring tuples during partitioning by partly computing aggregate values before partitioning, at least for the commonly used aggregate functions Consider an aggregation operation on a relation r, using the sum aggregate function on attribute B, with grouping on attribute A The system can perform the operation at each processor Pi on those r tuples stored on disk Di This computation results in tuples with partial sums at each processor; there is one tuple at Pi for each value for attribute A present in r tuples stored on Di The system partitions the result of the local aggregation on the grouping attribute A, and performs the aggregation again (on tuples with the partial sums) at each processor Pi to get the nal result As a result of this optimization, fewer tuples need to be sent to other processors during partitioning This idea can be extended easily to the min and max aggregate functions Extensions to the count and avg aggregate functions are left for you to do in Exercise 208 The parallelization of other operations is covered in several of the the exercises

We achieve parallelism by partitioning the I/O among multiple disks, and partitioning the CPU work among multiple processors If such a split is achieved without any overhead, and if there is no skew in the splitting of work, a parallel operation using n processors will take 1/n times as long as the same operation on a single processor We already know how to estimate the cost of an operation such as a join or a selection The time cost of parallel processing would then be 1/n of the time cost of sequential processing of the operation We must also account for the following costs: Startup costs for initiating the operation at multiple processors Skew in the distribution of work among the processors, with some processors getting a larger number of tuples than others Contention for resources such as memory, disk, and the communication network resulting in delays Cost of assembling the nal result by transmitting partial results from each processor The time taken by a parallel operation can be estimated as Tpart + Tasm + max(T0 , T1 , , Tn 1 ) where Tpart is the time for partitioning the relations, Tasm is the time for assembling the results and Ti the time taken for the operation at processor Pi Assuming that the tuples are distributed without any skew, the number of tuples sent to each processor

Hence, a summer ampli es each voltage, adds them up, and inverts the output In the special case where Ra = Rb = Rc = R f , then we get a circuit that directly

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