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h(c) = h(d) If h(c) = h(d), then c and d must have different values for customer-name However, if h(c) = h(d), we must test c and d to see whether the values in their join attributes are the same, since it is possible that c and d have different customer-names that have the same hash value Figure 139 shows the details of the hash join algorithm to compute the natural join of relations r and s As in the merge join algorithm, tr 1 ts denotes the concatenation of the attributes of tuples tr and ts , followed by projecting out repeated attributes After the partitioning of the relations, the rest of the hash join code performs a separate indexed nested-loop join on each of the partition pairs i, for i = 0, , nh To do so, it rst builds a hash index on each Hsi , and then probes (that is, looks up Hsi ) with tuples from Hri The relation s is the build input, and r is the probe input The hash index on Hsi is built in memory, so there is no need to access the disk to retrieve the tuples The hash function used to build this hash index is different from the hash function h used earlier, but is still applied to only the join attributes In the course of the indexed nested-loop join, the system uses this hash index to retrieve records that will match records in the probe input The build and probe phases require only a single pass through both the build and probe inputs It is straightforward to extend the hash join algorithm to compute general equi-joins The value nh must be chosen to be large enough such that, for each i, the tuples in the partition Hsi of the build relation, along with the hash index on the partition, will t in memory It is not necessary for the partitions of the probe relation to t in memory Clearly, it is best to use the smaller input relation as the build relation If the size of the build relation is bs blocks, then, for each of the nh partitions to be of size less than or equal to M , nh must be at least bs /M More precisely stated, we have.

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/* Partition s */ for each tuple ts in s do begin i := h(ts [JoinAttrs]); Hsi := Hsi {ts }; end /* Partition r */ for each tuple tr in r do begin i := h(tr [JoinAttrs]); Hri := Hri {tr }; end /* Perform join on each partition */ for i := 0 to nh do begin read Hsi and build an in-memory hash index on it for each tuple tr in Hri do begin probe the hash index on Hsi to locate all tuples ts such that ts [JoinAttrs] = tr [JoinAttrs] for each matching tuple ts in Hsi do begin add tr 1 ts to the result end end end Figure 139 Hash join

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to account for the extra space occupied by the hash index on the partition as well, so nh should be correspondingly larger For simplicity, we sometimes ignore the space requirement of the hash index in our analysis

If the value of nh is greater than or equal to the number of page frames of memory, the relations cannot be partitioned in one pass, since there will not be enough buffer pages Instead, partitioning has to be done in repeated passes In one pass, the input can be split into at most as many partitions as there are page frames available for use as output buffers Each bucket generated by one pass is separately read in and partitioned again in the next pass, to create smaller partitions The hash function used in a pass is, of course, different from the one used in the previous pass The system repeats this splitting of the input until each partition of the build input ts in memory Such partitioning is called recursive partitioning A relation does not need recursive partitioning if M > h +1, or equivalently M > n (bs /M ) + 1, which simpli es (approximately) to M > bs For example, consider a memory size of 12 megabytes, divided into 4-kilobyte blocks; it would contain a total of 3000 blocks We can use a memory of this size to partition relations of size 9 million blocks, which is 36 gigabytes Similarly, a relation of size 1 gigabyte requires 250000 blocks, or about 2 megabytes, to avoid recursive partitioning

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