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A frozen set is a set that, once created, cannot be changed We can of course rebind the variable that refers to a frozen set to refer to something else, though Frozen sets can only be created using the frozenset data type called as a function With no arguments, frozenset() returns an empty frozen set, with a frozenset argument it returns a shallow copy of the argument, and with any other argument it attempts to convert the given object to a frozenset It does not accept more than one argument Since frozen sets are immutable, they support only those methods and operators that produce a result without affecting the frozen set or sets to which they are applied Table 32 (123 ) lists all the set methods frozen sets support frozensetcopy(), frozensetdifference() (-), frozensetintersection() (&), frozensetisdisjoint(), frozensetissubset() (<=; also < for proper subsets), frozensetissuperset() (>=; also > for proper supersets), frozensetunion() (|), and frozensetsymmetric_difference() (^), all of which are indicated by a in the table If a binary operator is used with a set and a frozen set, the data type of the result is the same as the left-hand operand s data type So if f is a frozen set and s is a set, f & s will produce a frozen set and s & f will produce a set In the case of the == and != operators, the order of the operands does not matter, and f == s will produce True if both sets contain the same items

is the process of reversing the steps taken to achieve a normal form Often, it becomes necessary to violate certain normalization rules to satisfy the real-world requirements of specific queries Let s look at some examples In data models that have a completely normalized structure, there tend to be a great many entities and relationships To retrieve logical sets of data, you often need a great many joins to retrieve all the pertinent information about a given object Logically this is not a problem, but in the physical implementation of a database, joins tend to incur overhead in query processing time For every table that is joined, there is usually a cost to scan the indexes on that table and then retrieve the matching data from each object, combine the resulting data, and deliver it to the end user (for more on indexes and query optimization, see 10) When millions of rows are being scanned and tens or hundreds of rows are being returned, it is costly In these situations, creating a denormalized entity may offer a performance benefit, at the cost of violating one of the normal forms The trade-off is usually a matter of having redundant data, because you are storing an additional physical table that duplicates data being stored in other tables To mitigate the storage effects of this technique, you can often store subsets of data in the duplicate table, clearing it out and repopulating it based on the queries you know are running against it Additionally, this means that you have additional physical objects to maintain if there are schema changes in the original tables In this case, accurate documentation and a managed change control process are the only practices that can ensure that all the relevant denormalized objects stay in sync Denormalization also can help when you re working on reporting applications In larger environments, it is often necessary to generate reports based on application data Reporting queries often return large historical data sets, and when you join various types of data in a single report it incurs a lot of overhead on standard OLTP systems Running these queries on exactly the same databases that the applications are trying to use can result in an overloaded system, creating blocking situations and causing end users to wait an unacceptable amount of time for the data Additionally, it means storing large amounts of historical data in the OLTP system, something that may have other adverse effects, both internally to the database management system and to the physical server resources Denormalizing the data in the database to a set of tables (or even to a different physical database) specifically used for reporting can alleviate the.

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Another consequence of the immutability of frozen sets is that they meet the hashable criterion for set items, so sets and frozen sets can contain frozen sets We will see more examples of set use in the next section, and also in the chapter s Examples section

With our collection of tasks in hand, we can consider the accompanying data decomposition The key data structures are the neighbor list, the atomic coordinates, the atomic velocities, and the force vector Every iteration that updates the force vector needs the coordinates of a neighborhood of atoms The computation of nonbonded forces, however, potentially needs the coordinates of all the atoms, because the molecule being simulated might fold back on itself in unpredictable ways We will use this information to carry out the data decomposition (in the Data Decomposition pattern) and the data sharing analysis (in the Data Sharing pattern)

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