# 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

Example 2

Example 1:

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

or

Example 2:

${\displaystyle D=R1(A)}$ ${\displaystyle R2(B)}$ ${\displaystyle W2(A)}$ ${\displaystyle Com.2}$ ${\displaystyle W1(A)}$ ${\displaystyle Com.1}$ ${\displaystyle W3(A)}$ ${\displaystyle Com.3}$

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 results in a directed edge 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. This results in directed edges from T2 to T1, T1 to T3, and T2 to T3.
5. The schedule S is conflict 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.