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If T29 uses the tuple newly inserted by T30 in computing sum(balance), then T29 read a value written by T30 Thus, in a serial schedule equivalent to S, T30 must come before T29 If T29 does not use the tuple newly inserted by T30 in computing sum(balance), then in a serial schedule equivalent to S, T29 must come before T30 The second of these two cases is curious T29 and T30 do not access any tuple in common, yet they con ict with each other! In effect, T29 and T30 con ict on a phantom tuple If concurrency control is performed at the tuple granularity, this con ict would go undetected This problem is called the phantom phenomenon To prevent the phantom phenomenon, we allow T29 to prevent other transactions from creating new tuples in the account relation with branch-name = Perryridge To nd all account tuples with branch-name = Perryridge , T29 must search either the whole account relation, or at least an index on the relation Up to now, we have assumed implicitly that the only data items accessed by a transaction are tuples However, T29 is an example of a transaction that reads information about what tuples are in a relation, and T30 is an example of a transaction that updates that information Clearly, it is not suf cient merely to lock the tuples that are accessed; the information used to nd the tuples that are accessed by the transaction must also be locked The simplest solution to this problem is to associate a data item with the relation; the data item represents the information used to nd the tuples in the relation Transactions, such as T29 , that read the information about what tuples are in a relation would then have to lock the data item corresponding to the relation in shared mode Transactions, such as T30 , that update the information about what tuples are in a relation would have to lock the data item in exclusive mode Thus, T29 and T30 would con ict on a real data item, rather than on a phantom Do not confuse the locking of an entire relation, as in multiple granularity locking, with the locking of the data item corresponding to the relation By locking the data item, a transaction only prevents other transactions from updating information about what tuples are in the relation Locking is still required on tuples A transaction that directly accesses a tuple can be granted a lock on the tuples even when another transaction has an exclusive lock on the data item corresponding to the relation itself The major disadvantage of locking a data item corresponding to the relation is the low degree of concurrency two transactions that insert different tuples into a relation are prevented from executing concurrently A better solution is the index-locking technique Any transaction that inserts a tuple into a relation must insert information into every index maintained on the relation We eliminate the phantom phenomenon by imposing a locking protocol for indices For simplicity we shall only consider B+ -tree indices As we saw in 12, every search-key value is associated with an index leaf node A query will usually use one or more indices to access a relation An insert must insert the new tuple in all indices on the relation In our example, we assume that there is an index on account for branch-name Then, T30 must modify the leaf containing the key Perryridge If T29 reads the same leaf node to locate all tuples pertaining to the Perryridge branch, then T29 and T30 con ict on that leaf node.

The McGraw Hill Companies, 2001

The index-locking protocol takes advantage of the availability of indices on a relation, by turning instances of the phantom phenomenon into con icts on locks on index leaf nodes The protocol operates as follows: Every relation must have at least one index A transaction Ti can access tuples of a relation only after rst nding them through one or more of the indices on the relation A transaction Ti that performs a lookup (whether a range lookup or a point lookup) must acquire a shared lock on all the index leaf nodes that it accesses A transaction Ti may not insert, delete, or update a tuple ti in a relation r without updating all indices on r The transaction must obtain exclusive locks on all index leaf nodes that are affected by the insertion, deletion, or update For insertion and deletion, the leaf nodes affected are those that contain (after insertion) or contained (before deletion) the search-key value of the tuple For updates, the leaf nodes affected are those that (before the modi cation) contained the old value of the search-key, and nodes that (after the modi cation) contain the new value of the search-key The rules of the two-phase locking protocol must be observed Variants of the index-locking technique exist for eliminating the phantom phenomenon under the other concurrency-control protocols presented in this chapter

And sine is related to the exponential via sin = e j e j 2j (713)

CHAPTER 4: