Precedence Graph

Precedence Graph, also known as a Serializability Graph, is a directed graph used in database systems to determine the serializability of a transaction schedule. The graph represents the order of operations across transactions and helps identify potential conflicts that could prevent a schedule from being equivalent to a serial schedule.

Key Concepts

• Nodes: Represent individual transactions in the schedule.
• Edges: Indicate a precedence (or dependency) between transactions due to conflicting operations.
• Cycle Detection: The presence of a cycle in the precedence graph implies that the schedule is not serializable.

Construction of a Precedence Graph

To construct a precedence graph:

1. Represent each transaction as a node.
2. Add a directed edge from Transaction T1 to Transaction T2 if:
   • T1 performs an operation on a data item before T2 and both operations conflict.
   • A conflicting operation involves:
     • Read-Write (RW) conflict.
     • Write-Read (WR) conflict.
     • Write-Write (WW) conflict.
3. Check for cycles:
   • If there is a cycle in the graph, the schedule is not serializable.
   • If no cycles exist, the schedule is serializable.

Example

Consider the following schedule:

• T1: Read(A), Write(A)
• T2: Read(A), Write(B)
• T3: Write(A), Read(B)

Steps to Construct the Graph

Step	Action	Explanation
1	Add nodes T1, T2, T3	Each transaction is represented as a node.
2	Add edge T1 → T3	T1 writes to A, and T3 reads A (WR conflict).
3	Add edge T3 → T2	T3 writes to A, and T2 reads A (WR conflict).
4	Add edge T2 → T1	T2 writes to B, and T1 reads B (WR conflict).

Result

The precedence graph contains the following edges:

• T1 → T3
• T3 → T2
• T2 → T1

Since there is a cycle (T1 → T3 → T2 → T1), the schedule is not serializable.

Applications of Precedence Graphs

Precedence graphs are widely used in:

• Concurrency Control: Verifying serializability of transaction schedules.
• Deadlock Detection: Identifying cycles in wait-for graphs, a variant of precedence graphs.
• Database Design: Ensuring consistency and preventing anomalies in concurrent transactions.

Advantages

• Clarity: Provides a clear visualization of transaction dependencies.
• Cycle Detection: Offers an effective way to check serializability.
• Conflict Resolution: Helps identify and resolve transaction conflicts.

Limitations

• Scalability: Constructing and analyzing precedence graphs can become complex for schedules with a large number of transactions.
• Static Nature: Only applicable to fixed schedules, not dynamic transaction processing.

See Also

• Transaction Management
• Serializability
• Conflict Serializability
• Dependency Graph
• Concurrency Control
• Deadlock Detection