Question

Consider the three transactions \(T_{1}, T_{2},\) and \(T_{3},\) and the schedules \(S_{1}\) and \(S_{2}\) given below. Draw the serializability (precedence) graphs for \(S_{1}\) and \(S_{2},\) and state whether each schedule is serializable or not. If a schedule is serializable, write down the equivalent serial schedule(s). $$\begin{array}{l} T_{1}: r_{1}(X) ; r_{1}(Z) ; w_{1}(X) \\ T_{2}: r_{2}(Z) ; r_{2}(Y) ; w_{2}(Z) ; w_{2}(Y) \\ T_{3}: r_{3}(X) ; r_{3}(Y) ; w_{3}(Y) ; \\ S_{1}: r_{1}(X) ; r_{2}(Z) ; r_{1}(Z) ; r_{3}(X) ; r_{3}(Y) ; w_{1}(X) ; w_{3}(Y) ; r_{2}(Y) ; w_{2}(Z) ; w_{2}(Y) \\ S_{2}: r_{1}(X) ; r_{2}(Z) ; r_{3}(X) ; r_{1}(Z) ; r_{2}(Y) ; r_{3}(Y) ; w_{1}(X) ; w_{2}(Z) ; w_{3}(Y) ; w_{2}(Y) \end{array}$$

Short answer

Schedule \(S_1\) is serializable with possible equivalent serial schedules being \(T_1\), \(T_2\), \(T_3\) or \(T_1\), \(T_3\), \(T_2\). Schedule \(S_2\) is not serializable and has no equivalent serial schedules.

Step-by-step solution

Constructing the Serializability Graph for \(S_1\)

A serializability graph is built by considering each transaction as a node and creating directed edges between nodes based on read-write conflicts. In \(S_1\), \(T_1\) reads 'X' before \(T_3\) writes 'X', hence \(T_3\) precedes \(T_1\) (an edge from \(T_1\) to \(T_3\)). Similarly, \(T_2\) writes 'Y' before \(T_3\) reads 'Y', hence \(T_3\) also precedes \(T_2\) (an edge from \(T_2\) to \(T_3\)). Since there are no cycles, \(S_1\) is serializable.

Equivalent Serial Schedules for \(S_1\)

A serial schedule is equivalent to the given schedule if it maintains the order of conflicting read and write operations. Based on the constructed graph, possible equivalent serial schedules for \(S_1\) could be: \(T_1\), \(T_2\), \(T_3\) or \(T_1\), \(T_3\), \(T_2\).

Constructing the Serializability Graph for \(S_2\)

Following the same process as in Step 1 to analyze \(S_2\), \(T_1\) reads 'X' before \(T_3\) writes 'X', hence an edge from \(T_1\) to \(T_3\). \(T_2\) writes 'Y' before \(T_3\) reads 'Y', hence an edge from \(T_2\) to \(T_3\). But here, \(T_3\) writes 'Y' before \(T_2\) reads 'Y', creating an edge from \(T_3\) to \(T_2\), thus forming a cycle. Therefore, \(S_2\) is not serializable.

Equivalent Serial Schedules for \(S_2\)

Since \(S_2\) is not serializable, no equivalent serial schedules exist.

Key concepts

Database Transactions

In database systems, a transaction is a logical unit of work that encapsulates a sequence of operations. These operations can include reading or writing data from or into a database. Transactions are vital because they ensure data integrity and consistency, particularly in systems where concurrent access to the data is common.

Transactions adhere to the ACID properties, which are crucial to maintain the consistent state of a database:

• Atomicity: Each transaction is treated as a single unit, meaning it either completes fully or has no effect at all.
• Consistency: Transactions transition the database from one valid state to another, preserving database invariants.
• Isolation: Concurrent transactions occur independently without any interference.
• Durability: Once a transaction is committed, it remains permanent, even in the case of a system failure.

The proper understanding of transactions is critical for managing data integrity and ensuring the seamless operation of a database system, especially when multiple transactions are executed concurrently.

Concurrency Control

Concurrency control is a database management mechanism that handles simultaneous operations without compromising the data's integrity. It ensures that concurrent transactions occur in a way that they do not interfere with each other.

The main purpose of concurrency control is to manage conflicts that arise from concurrent access to resources:

• Read-Write Conflicts: Occur when one transaction reads a data item while another transaction writes to the same data item.
• Write-Write Conflicts: Arise when multiple transactions attempt to write to the same data item.

Methods to achieve concurrency control include locking mechanisms and timestamp ordering, which help preserve the isolation property of transactions. Successful concurrency control ensures that the database maintains consistency and that all transactions commit successfully or do not occur at all, as guaranteed by the atomicity and isolation properties.

Precedence Graphs

Precedence graphs, or serializability graphs, are used to determine whether a schedule of transactions is serializable. In other words, they check if it's possible to reorder the operations in the schedule into a serial schedule, which guarantees the same outcome without concurrency.

To construct a precedence graph, follow these steps:

• Create a node for each transaction involved in the schedule.
• Draw a directed edge from transaction A to transaction B if and only if A's operation conflicts with a later operation in B, creating a dependency.

If the constructed graph is acyclic, the schedule is serializable, meaning its transactions can be reordered without any conflict. Conversely, if the graph contains cycles, the schedule is not serializable, indicating potential for conflicts that cannot be resolved by reordering.

Serializable Schedules

Serializable schedules in database management systems refer to a sequence of transactions whose order of execution results in the same final state of the database as some serial execution of those transactions.

Serializable schedules are important because they simulate the effect of one transaction running at a time, thus maintaining isolation:

• If a schedule is determined to be serializable by analyzing its precedence graph (having no cycles), it guarantees transaction consistency.
• Such a schedule avoids any anomalies that might arise due to concurrent execution, such as lost updates or inconsistent reads.

When a schedule is not serializable, it means there is no way to reorder the transactions to match any serial schedule without conflicts. Hence, ensuring serializability is a key aspect of designing robust database systems that effectively handle concurrency with consistent outcomes.