# Precedence graph

A precedence graph, also named conflict graph and serializability graph, is used in the context of concurrency control in databases.

The precedence graph for a schedule S contains:

• A node for each committed transaction in S
• An arc from Ti to Tj if an action of Ti precedes and conflicts with one of Tj's actions.

## Precedence graph example

$D = \begin{bmatrix} T1 & T2 & T3 \\ R(A) & & \\ & R(A) & \\ &R(A) \\ W(A) & & \\ &W(A) & \\ & & W(A)\\ & W(A) \\ \end{bmatrix}$

or

$D = R1(A)$ $W2(A)$ $Com.2$ $W1(A)$ $Com.1$ $W3(A)$ $Com.3$

A precedence graph of the schedule D, with 3 transactions. As there is a cycle (of length 2; with two edges) through the committed transactions T1 and T2, this schedule (history) is not Conflict serializable.

## Testing Serializability with Precedence Graph

The drawing sequence for the precedence graph:-

1. For each transaction Ti participating in schedule S, create a node labelled Ti in the precedence graph. So the precedence graph contains T1, T2, T3
2. For each case in S where Ti executes a write_item(X) then Tj executes a read_item(X), create an edge (Ti --> Tj) in the precedence graph. This occurs nowhere in the above example, as there is no read after write.
3. For each case in S where Ti executes a read_item(X) then Tj executes a write_item(X), create an edge (Ti --> Tj) in the precedence graph. This will bring to front a directed graph from T1 to T2.
4. For each case in S where Ti executes a write_item(X) then Tj executes a write_item(X), create an edge (Ti --> Tj) in the precedence graph. It creates a directed graph from T2 to T1, T1 to T3, and T2 to T3.
5. The schedule S is serializable if the precedence graph has no cycles. As T1 and T2 constitute a cycle, then we cannot declare S as serializable or not and serializability has to be checked using other methods.