Question

Prove that the basic two-phase locking protocol guarantees conflict serializability of schedules. (Hint: Show that, if a serializability graph for a schedule has a cycle, then at least one of the transactions participating in the schedule does not obey the two-phase locking protocol.)

Short answer

The basic Two-Phase Locking protocol guarantees conflict serializability because a cycle cannot exist in its serializability graph, meaning the schedule is conflict serializable. If a cycle exists, it indicates that at least one transaction in the schedule is not following the Two-Phase Locking protocol.

Step-by-step solution

Understand Two-Phase Locking Protocol

The Two-Phase Locking protocol has two phases, the Expanding phase and Shrinking phase. In the first phase (Expanding), a transaction may obtain locks, but cannot release any lock. In the second phase (Shrinking), it may release locks, but cannot obtain any new locks. This means once a transaction releases its first lock, it is in the second phase and cannot request more locks.

Analyze Serializability Graph

A serializability graph is a directed graph where nodes represent transactions and an edge from node T1 to T2 means that T1 is scheduled before T2. A schedule is considered conflict serializable if its serializability graph is acyclic.

Finding Cycle in Serializability Graph

For any two transactions T1 and T2, if T1 must wait for T2 (T1 -> T2 in the graph), it means T1 has requested for a data item that T2 has locked. If there is a cycle, it means there is a conflicting operation. Therefore, for every pair of transactions (T1, T2) that T1 -> T2, if T2 releases the lock before T1, then T1 and T2 are not following the Two-Phase Locking protocol.

Conclude the Proof

If a schedule follows the Two-phase Locking protocol, it is guaranteed to be conflict serializable, because a cycle cannot exist in its serializability graph. This is because, once a transaction releases its first lock (enters the shrinking phase), it cannot request a new lock, avoiding the possibility of a cycle. Hence, if a serializability graph for a schedule has a cycle, then at least one of the transactions participating in the schedule does not obey the two-phase locking protocol.

Key concepts

Conflict Serializability

Conflict serializability is a fundamental concept in database management systems, ensuring that a schedule (or sequence of transactions) operates without leading to conflicts that would arise if these transactions occurred concurrently.
When a schedule is conflict serializable, it means that the schedule can be rearranged into a serial order — where transactions proceed one after the other without overlapping — without causing any discrepancies in the result.
Essentially, conflict serializability guarantees that even though transactions might be interleaved in execution, their effects remain as if they were conducted serially.

• Transactions are equivalent in terms of their net effect on the database.
• No cycles in the serializability graph, ensuring a clear order of operations.

This property is crucial for maintaining data integrity and consistency when multiple users access the database simultaneously.

Serializability Graph

A serializability graph, also known as a precedence graph, is a tool used to determine whether a schedule is conflict serializable.
It is constructed by representing each transaction as a node and creating directed edges between these nodes to show dependencies based on conflicting operations.

• An edge from node T1 to T2 signifies that T1's operation must precede T2's operation due to a conflicting data access.
• This graph enables us to visually assess potential conflicts.

The absence of cycles in a serializability graph confirms the conflict serializability of a schedule.
If a cycle exists, it indicates the presence of circular dependencies, which means that no serial order can be imposed without causing conflicts. Acyclic graphs are therefore necessary to ensure that the schedule can be serialized into a non-conflicting order.

Expanding Phase

The expanding phase is the first of the two phases in the Two-Phase Locking (2PL) protocol. In this phase, a transaction is allowed to acquire locks on data items, enabling it to gain the necessary permissions it needs to execute without any issues.
However, during this period, it is not allowed to release any locks.
This is called the lock-acquisition period and its strict limitation is designed to ensure that critical sections of transactions do not conflict with others.

• Transactions can only acquire new locks.
• Lock releases are prohibited.

Once a transaction completes the expanding phase and releases its first lock, it enters the shrinking phase.
This rule maintains the order in which operations acquire data access, thereby preventing potential deadlocks or race conditions.

Shrinking Phase

The shrinking phase follows the expanding phase in the Two-Phase Locking protocol.
This phase is characterized by the transaction being able to release locks it no longer needs. However, it is strictly prohibited from acquiring any new locks.
This prohibition safeguards against creating circular waiting scenarios that could lead to cycles in the serializability graph.

• Transactions release locks gradually, thereby reducing their hold on resources.
• Acquisition of additional locks is not permitted.

The transition from the expanding to the shrinking phase marks a turning point for a transaction. Once this occurs, the transaction is committed to completing its remaining operations without seeking additional access lock grants.
This critical control helps enforce serializability and prevent cycles in the graph, maintaining the integrity of the overall transaction scheduling.