Suppose a database schedule S involves transactions T1, ....Tn. Construct the precedence graph of S with vertices representing the transactions and edges representing the conflicts. If S is serializable, which one of the following orderings of the vertices of the precedence graph is guaranteed to yield a serial schedule?a)Topological orderb)Depth-first orderc)Breadth-first orderd)Ascending order of transaction indicesCorrect answer is option 'A'. Can you explain this answer?

Question

Answer

Precedence Graph:

A precedence graph is a directed graph that represents the order in which transactions in a database schedule should be executed. In the graph, each transaction is represented by a vertex, and there is an edge between two vertices if there is a conflict between the corresponding transactions. A conflict occurs when two transactions have conflicting operations on the same data item, such as a read followed by a write, or two writes.

Orderings of Vertices:

To determine if a schedule is serializable, we need to find an ordering of the vertices of the precedence graph that guarantees a serial schedule. A serial schedule is one in which transactions are executed one after another, without any interleaving.

There are multiple orderings in which the vertices of the precedence graph can be arranged. Let's analyze the given options:

a) Topological order: In a topological order, the vertices are arranged in such a way that if there is an edge from vertex A to vertex B, then A appears before B in the ordering. This ordering ensures that there are no cycles in the graph, and hence the schedule is serializable.

b) Depth-first order: In a depth-first order, the vertices are arranged in the order in which they are visited during a depth-first traversal of the graph. This ordering may or may not guarantee a serial schedule, as it depends on the structure of the graph.

c) Breadth-first order: In a breadth-first order, the vertices are arranged in the order in which they are visited during a breadth-first traversal of the graph. Similar to the depth-first order, this ordering may or may not guarantee a serial schedule.

d) Ascending order of transaction indices: This ordering arranges the vertices based on the transaction indices in ascending order. However, the transaction indices may not necessarily reflect the conflicts between transactions, and hence this ordering may not guarantee a serial schedule.

Guaranteed Serial Schedule:

The only ordering that is guaranteed to yield a serial schedule is the topological order. This is because a topological order ensures that there are no cycles in the graph, which means there are no conflicting dependencies between transactions. Therefore, transactions can be executed one after another in the topological order, resulting in a serial schedule.

Conclusion:

In conclusion, the topological order of the vertices in the precedence graph is guaranteed to yield a serial schedule. This is because the topological order ensures that there are no conflicting dependencies between transactions, allowing them to be executed one after another without any interleaving.

Answer

The topological ordering for graphs get many applications where the nature of the problem is ordered sequential processing. Few of the problems in this category are as follows:

Dynamic Linking/Loading of programs while building and execution.

Deciding the pre-requisites in a course structure.

Job Scheduling in Processors or Assembly Line Processing.

Building an Alien Dictionary from given words.
pravin pandit

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