Question

Which of the following schedules is (conflict) serializable? For each serializable schedule, determine the equivalent serial schedules. a. \(r_{1}(X) ; r_{3}(X) ; w_{1}(X) ; r_{2}(X) ; w_{3}(X)\) b. \(r_{1}(X) ; r_{3}(X) ; w_{3}(X) ; w_{1}(X) ; r_{2}(X)\) c. \(r_{3}(X) ; r_{2}(X) ; w_{3}(X) ; r_{1}(X) ; w_{1}(X)\) d. \(r_{3}(X) ; r_{2}(X) ; r_{1}(X) ; w_{3}(X) ; w_{1}(X)\)

Short answer

To provide the solution, construct a precedence graph for each of the schedules 'a', 'b', 'c', and 'd' and look for cycles to determine if they are conflict serializable. For any that are, the equivalent serial schedules can be found from the sequence of transactions in the graph.

Step-by-step solution

Noting transaction operations

First, it's important to note the transaction operations in each schedule. The first schedule 'a' has the operations \[r_{1}(X) ; r_{3}(X) ; w_{1}(X) ; r_{2}(X) ; w_{3}(X)\] in that order. Do the same for schedules 'b', 'c', and 'd'.

Constructing a precedence graph

Next, a precedence graph is constructed for each schedule. This graph represents the operations in a schedule with nodes as the transactions and directed edges from one transaction to another if the first transaction precedes the second in a conflict.

Checking for cycles

Analyzing the precedence graph will show whether the schedule is serializable. If there is a cycle in the precedence graph, the schedule is not conflict serializable. Do this for all schedules.

Finding serial equivalents

For any schedule that is conflict serializable, determine the equivalent serial schedules. This can usually be found by looking at the sequence of the transactions in the precedence graph.

Key concepts

Precedence Graph

When dealing with database transactions, understanding the order of operations is key to maintaining data integrity. A precedence graph (also known as a serialization graph) is an essential tool in database theory used to determine if a given schedule is conflict serializable.

In a precedence graph, each transaction from a schedule is represented as a node. Directed edges then connect these nodes based on the operations' conflict relationships. For instance, if transaction A performs a write operation that precedes a conflicting read or write operation by transaction B on the same data, there is an edge from A to B, indicating A must come before B in any equivalent serial schedule.

Constructing a precedence graph involves inspecting transaction operations for conflicts—combinations of read and write operations on the same data item by different transactions where at least one of the operations is a write. After identifying these conflicts, edges are drawn in the graph to reflect the order of conflicting operations. The absence of cycles in this graph confirms that the transactions are conflict serializable, which means they can be reordered into an equivalent serial schedule without changing the outcome of the operations.

Transaction Operations

Transaction operations are the individual read (denoted as 'r') and write (denoted as 'w') operations performed during transactions within a database. These operations form the basic actions carried out in the execution sequence of transactions. Read operations retrieve a data item, denoted as 'r_{i}(X)', where 'i' represents the transaction number, and 'X' is the data item. Write operations modify a data item, represented as 'w_{i}(X)'.

Understanding these operations is crucial since their order determines the transactions’ interactions and potential conflicts. Conflicts arise when two operations by different transactions act on the same data item and at least one is a write operation. This plays a direct role in the assessment of schedule serializability.

It's important to delineate each operation's occurrence in the schedule explicitly, as evidenced in the original exercise. This step is indispensable for the next stages of serialization analysis, as it forms the basis for constructing the precedence graph and for examining dependencies between transactions.

Serial Schedules

Serial schedules represent a scenario where transactions are executed sequentially, without any overlap in time. This is the simplest form of a schedule but often not the most efficient for systems that require concurrent transactions. Nevertheless, serial schedules are critical as a benchmark for determining the correctness of concurrent schedules.

A schedule is considered serializable if its effect on the database is equivalent to one achieved by some serial execution of the transactions. Conflict serializability, a particular type of serializability, requires that interleaved transactions don't cause any conflicts that would alter the end result of the database state compared to its serial counterpart.

The importance of determining serial equivalents lies in the ability to prove that the non-serial schedule maintains the same level of data integrity as a serial schedule. If a schedule can be shown as conflict serializable, it is ensured to be free of problems like dirty reads, lost updates, or other anomalies associated with concurrent transactions.