Proof: Consider the set of attributes be α . F is considered to be the set of functional dependencies (FDs). The set of attributes is considered to be functionally determined by α under F and is defined to be close of α under F ( α + ) . Using the Figure 7.19 and Figure 7.8: The algorithm provided in the figure 7.19 will correctly compute the ( α + ) as shown below: Computing what will happen after adding A to result. Proving α → A Since α → α is trivially which means α is considered to be a part of the result. Now, after adding A ∉ α to the resultant there should be existence of functional dependency that is β → γ where A ∈ γ and where β ⊆ result . Or else the value of the “fdcount” will not become zero, then the related of condition of the algorithm will not change to be the value “false”. When A ∈ α + , then it will added to the result for sure. Proving this by induction by utilizing the Armstrong’s axiom on the length of proof on α → A are shown below: The add- in procedure gets called with the argument say “c”, then all the attribute present in the “C” gets added to the result. Similarly in addition to it when the functional dependency “fdcount” value is zero, then all the attributes of its tail is surely added to the resultant too. Utilizing the induction to the base case that is when A ∈ α then A ∈ α + is considered to be true because the very first call that is made to the add-in procedure is made with the argument α . Assume that α → A and it can be proved in “n” steps or lesser than “n” steps and then it is also A ∈ result . Now proving α → A in n+1 steps; The last step of proof can be obtained by applying one of the Armstrong axioms. The below are three Armstrong’s axiom and they are Reflexivity A is considered to be set of attributes B ⊆ C and B ⊆ A then A → B Augmentation A is considered to be set of attributes B → C, then AB → AC will also hold Transitivity A → B and B → A , then A → C can also be hold. This indicates that the A was already present in the result and also in the nth step itself. Or else by induction we can obtain that c ⊆ result Comparison: The algorithm present in the figure 7.19 is comparatively more efficient when compared with that of the figure 7.8 as shown below: Each and every functional dependency gets scanned only one in the main program. The resultant array size that appears is proportional to the size of the functional dependencies. The recursive call that is made to the procedure add-in is also considered to be linear to the size that it appears. Therefore, the time taken for the overall algorithm is linear to the size of the functional dependencies. The algorithm presented in the figure 7.8 requires quadratic time because The loop gets performed based the times of functional dependencies(FD’s) Also for each loop the FD gets scanned again and again.

7th Edition

ISBN: 9780078022159

Author: Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher: McGraw-Hill Education

Expert Solution & Answer

Chapter 18, Problem 1PE

Proof:

Consider the set of attributes be $α$ .

F is considered to be the set of functional dependencies (FDs).

The set of attributes is considered to be functionally determined by $α$ under F and is defined to be close of $α$ under $F(α+)$ .

Using the Figure 7.19 and Figure 7.8:

The algorithm provided in the figure 7.19 will correctly compute the $(α+)$ as shown below:

Computing what will happen after adding A to result.

Proving $α→A$

Since $α→α$ is trivially which means $α$ is considered to be a part of the result.

Now, after adding $A∉α$ to the resultant there should be existence of functional dependency that is $β→γ$ where $A∈γ$ and where $β⊆result$ .

Or else the value of the “fdcount” will not become zero, then the related of condition of the algorithm will not change to be the value “false”.

When $A∈α+$ , then it will added to the result for sure.

Proving this by induction by utilizing the Armstrong’s axiom on the length of proof on $α→A$ are shown below:

The add- in procedure gets called with the argument say “c”, then all the attribute present in the “C” gets added to the result.

Similarly in addition to it when the functional dependency “fdcount” value is zero, then all the attributes of its tail is surely added to the resultant too.

Utilizing the induction to the base case that is when $A∈α$ then $A∈α+$ is considered to be true because the very first call that is made to the add-in procedure is made with the argument $α$ .

Assume that $α→A$ and it can be proved in “n” steps or lesser than “n” steps and then it is also $A∈result$ .

Now proving $α→A$ in n+1 steps;

The last step of proof can be obtained by applying one of the Armstrong axioms.

The below are three Armstrong’s axiom and they are

Reflexivity

A is considered to be set of attributes

$B⊆C and B⊆A then A→B$

Augmentation

A is considered to be set of attributes

$B→C, then AB→AC$ will also hold

Transitivity

$A→B and B→A$ , then $A→C$ can also be hold.

This indicates that the A was already present in the result and also in the nth step itself.

Or else by induction we can obtain that $c⊆result$

Comparison:

The algorithm present in the figure 7.19 is comparatively more efficient when compared with that of the figure 7.8 as shown below:

Each and every functional dependency gets scanned only one in the main program.

The resultant array size that appears is proportional to the size of the functional dependencies.

The recursive call that is made to the procedure add-in is also considered to be linear to the size that it appears.

Therefore, the time taken for the overall algorithm is linear to the size of the functional dependencies.

The algorithm presented in the figure 7.8 requires quadratic time because

The loop gets performed based the times of functional dependencies(FD’s)

Also for each loop the FD gets scanned again and again.

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