Database System Concepts
Database System Concepts
7th Edition
ISBN: 9789332901384
Author: Henry F. Korth & S. Sudarshan Abraham Silberschatz
Publisher: MCGRAW-HILL HIGHER EDUCATION
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Chapter 18, Problem 1PE

Explanation of Solution

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 αA
  • Since αα is trivially which means α is considered to be a part of the result.
  • Now, after adding Aα to the resultant there should be existence of functional dependency that is βγ where Aγ and where βresult.
  • 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 Aα+, 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 αA 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 Aα then Aα+ 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 αA and it can be proved in “n” steps or lesser than “n” steps and then it is also Aresult.
  • Now proving αA 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
        • BC and BA then AB
      • Augmentation
        • A is considered to be set of attributes
        • BC, then ABAC will also hold
      • Transitivity
        •  AB and BA, then AC 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 cresult

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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