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return x As can be expected, makeset is a constant-time operation On the other hand, find follows parent pointers to the root of the tree and therefore takes time proportional to the height of the tree The tree actually gets built via the third operation, union, and so we must make sure that this procedure keeps trees shallow Merging two sets is easy: make the root of one point to the root of the other But we have a choice here If the representatives (roots) of the sets are r x and ry , do we make rx point to ry or the other way around Since tree height is the main impediment to computational ef ciency, a good strategy is to make the root of the shorter tree point to the root of the taller tree.

To write messages directly to the system logger (syslogd on UNIX and the Event Viewer on Windows), use the Zend_Log_Writer_Syslog writer, as shown in the following example. Note the use of the setFacility() method, which is used to set the log event s facility, or category, for the UNIX system logger.

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Here is a brief explanation of what each of these arguments does: The --with-apxs2 argument tells PHP where to find Apache s APXS (APache eXtenSion) script. This script simplifies the task of building and installing loadable modules for Apache. The --with-zlib argument tells PHP to activate compression (zip) features, which are used by different PHP services. The --with-gd argument tells PHP to activate image-processing features. The --with-mysqli argument activates PHP s MySQLi extension and tells PHP to use the new MySQL native driver (mysqlnd) as the MySQL client library. The --with-pdo-mysql argument activates PHP s MySQL PDO driver and tells PHP to use the MySQL native driver as the MySQL client library.

This way, the overall height increases only if the two trees being merged are equally tall Instead of explicitly computing heights of trees, we will use the rank numbers of their root nodes which is why this scheme is called union by rank procedure union(x, y) rx = find(x) ry = find(y) if rx = ry : return if rank(rx ) > rank(ry ): (ry ) = rx else: (rx ) = ry if rank(rx ) = rank(ry ) : rank(ry ) = rank(ry ) + 1 See Figure 56 for an example By design, the rank of a node is exactly the height of the subtree rooted at that node This means, for instance, that as you move up a path toward a root node, the rank values along the way are strictly increasing Property 1 For any x, rank(x) < rank( (x)).

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Replacing CAVE with a published algorithm. Likely candidates include the Secure Hash Algorithm (SHA-1). The new algorithm will also increase key size from 56 bits to 128 bits. SHA-1 was originally designed by the United States National Security Agency and is widely reviewed and understood. For a new authentication process, NIST is considering adopting 3GPP s AKA scheme, but asymmetric systems are also being considered.

A root node with rank k is created by the merger of two trees with roots of rank k 1 It follows by induction (try it!) that Property 2 Any root node of rank k has at least 2 k nodes in its tree This extends to internal (nonroot) nodes as well: a node of rank k has at least 2 k descendants After all, any internal node was once a root, and neither its rank nor its set of descendants has changed since then Moreover, different rank-k nodes cannot have common descendants, since by Property 1 any element has at most one ancestor of rank k Which means Property 3 If there are n elements overall, there can be at most n/2 k nodes of rank k This last observation implies, crucially, that the maximum rank is log n.

If you want more information about Solution Explorer, you can always do a search in the Help system and product documentation. Before trying to perform a search, please read the next section; you ll learn a lot about all the information that is at your disposal.

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