Directed acyclic graphs of locks:

The notions up to now introduce can be generalized to work for directed acyclic graphs (DAGs) of resources rather than simply hierarchies of resources. A tree is a easy DAG. The key observation is that too implicitly or explicitly locks a node one must lock all the parents of the node in the DAG as well as so by induction lock all ancestors of the node. Particularly to lock a sub graph one should implicitly or explicitly lock all ancestors of the sub graph in the appropriate mode (for a tree there is only one parent). To give an illustration of a non-hierarchical structure imagine the locks are organized as:

Figure: a non-hierarchical lock graph.

We postulate that areas are ‘physical’ notions as well as that file, indices and records are logical notions. The database is a collection of areas and every area is a collection of files as well as indices. Every file has some corresponding indices in the same area. Every record belongs to some file and to its corresponding indices. A record is including of field values and some field is indexed by the index associated with the file containing the record. The file provides a sequential access path to the records and the index gives an associative access path to the records based on field values. Because individual fields are never locked they do not appear in the lock graph.

To write the record A in file P with index I:

lock data base                     with mode =IX
lock area containing F            with mode =IX
lock file F                             with mode =IX
lock index I                           with mode =IX
lock record R                         with mode = X

Note that every path to record R is locked. On the other hand one could lock F and I in exclusive mode thereby implicitly locking R in exclusive mode.

To give a more absolute explanation we observe that a node can be locked explicitly (by requesting it) or implicitly (by appropriate explicit locks on the ancestors of the node) in one of five modes: IS, IX. S, SIX, X. Nevertheless the definition of implicit locks and the protocols for setting explicit locks have to be extended for DAG's as follows:

A node is absolutely granted in S mode to a transaction if at least one of its parents is (implicitly or explicitly) granted to the transaction in S, SIX or X mode. By induction that signifies that at least one of the node's ancestors should be explicitly granted in S, SIX, or X mode to the transaction. A node is absolutely granted in X mode if all of its parents are (implicitly or explicitly) granted to the transaction in X mode. By induction this is equal to the condition that all nodes in some cut set of the collection of all paths leading from the node to the roots of the graph are explicitly granted to the transaction in X mode as well as all ancestors of nodes in the cut set are explicitly granted in IX or SIX mode.

By examination of the biased order of modes (see figure above) a node is implicitly granted in IS mode if it is implicitly granted in S mode as well as a node is implicitly granted in IS, IX, S and SIX mode if it is implicitly granted in X mode.

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