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The DBMS catalog stores the following statistical information about database relations: nr , the number of tuples in the relation r br , the number of blocks containing tuples of relation r lr , the size of a tuple of relation r in bytes fr , the blocking factor of relation r that is, the number of tuples of relation r that t into one block V (A, r), the number of distinct values that appear in the relation r for attribute A This value is the same as the size of A (r) If A is a key for relation r, V (A, r) is nr
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The last statistic, V (A, r), can also be maintained for sets of attributes, if desired, instead of just for individual attributes Thus, given a set of attributes, A, V (A, r) is the size of A (r) If we assume that the tuples of relation r are stored together physically in a le, the following equation holds: nr br = fr Statistics about indices, such as the heights of B+ -tree indices and number of leaf pages in the indices, are also maintained in the catalog If we wish to maintain accurate statistics, then, every time a relation is modi ed, we must also update the statistics This update incurs a substantial amount of overhead Therefore, most systems do not update the statistics on every modi cation Instead, they update the statistics during periods of light system load As a result, the statistics used for choosing a query-processing strategy may not be completely accurate However, if not too many updates occur in the intervals between the updates of the statistics, the statistics will be suf ciently accurate to provide a good estimation of the relative costs of the different plans The statistical information noted here is simpli ed Real-world optimizers often maintain further statistical information to improve the accuracy of their cost estimates of evaluation plans For instance, some databases store the distribution of values for each attribute as a histogram: in a histogram the values for the attribute are divided into a number of ranges, and with each range the histogram associates the number of tuples whose attribute value lies in that range As an example of a histogram, the range of values for an attribute age of a relation person could be divided into 0 9, 10 19, , 90 99 (assuming a maximum age of 99) With each range we store a count of the number of person tuples whose age values lie in that range Without such histogram information, an optimizer would have to assume that the distribution of values is uniform; that is, each range has the same count
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The size estimate of the result of a selection operation depends on the selection predicate We rst consider a single equality predicate, then a single comparison predicate, and nally combinations of predicates A = a (r): If we assume uniform distribution of values (that is, each value appears with equal probability), the selection result can be estimated to have nr /V (A, r) tuples, assuming that the value a appears in attribute A of some record of r The assumption that the value a in the selection appears in some record is generally true, and cost estimates often make it implicitly However, it is often not realistic to assume that each value appears with equal probability The branch-name attribute in the account relation is an example where the assumption is not valid There is one tuple in the account relation for each account It is reasonable to expect that the large branches have more accounts than smaller branches Therefore, certain branch-name values appear with greater probability than do others Despite the fact that the uniform-
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