Revised Error Explanation

The mistake occurred because the initial formula did not account for scenarios where the buffer size allows the entire outer relation to be loaded into memory. This led to an overestimation of the Hash Join cost. To prevent this error, it's crucial to modify the join cost formulas to consider the available buffer size.
Modified Join Cost Formulas Incorporating Buffer Size

To accurately calculate the cost of different join strategies based on buffer availability, the formulas need to be adjusted to account for the number of buffer pages (B) available. Below are the revised formulas:
1. Nested Loops Join (NLJ)

a. Tuple-oriented NLJ
[ \text{Cost} = NPages(R) + NTuples(R) \times NPages(S) ]

b. Page-oriented NLJ
[ \text{Cost} = NPages(R) + (NPages(R) \times NPages(S)) ]

c. Block-oriented NLJ
[ \text{Cost} = NPages(R) + \left\lceil \frac{NPages(R)}{B - 2} \right\rceil \times NPages(S) ]
2. Hash Join

The Hash Join cost depends on whether the buffer size is sufficient to hold the entire outer relation R:

    If ( B > NPages(R) ):
    [ \text{Cost} = 2 \times (NPages(R) + NPages(S)) ]

    Else:
    [ \text{Cost} = 3 \times (NPages(R) + NPages(S)) ]

Explanation:

    Sufficient Buffer (( B > NPages(R) )):
        First Pass: Read all pages of R into memory and build the hash table.
        Second Pass: Scan all pages of S and probe the hash table.

    Insufficient Buffer (( B \leq NPages(R) )):
        First Pass: Partition both R and S into smaller chunks.
        Subsequent Passes: Process each partition, leading to additional I/O costs.

3. Sort-Merge Join (SMJ)

[ \text{Cost}_{SMJ} = NPages(R) + NPages(S) + 2 \times NPages(R) \times \text{num_passes}(R) + 2 \times NPages(S) \times \text{num_passes}(S) ]

Assumption: The number of passes for sorting is pre-determined (e.g., 2 passes).

Final Adjustments to Prevent Future Errors

To ensure accurate cost evaluations in future scenarios:

    Incorporate Buffer Size in All Join Formulas: Always consider the available buffer pages when calculating join costs, especially for algorithms like Hash Join and Sort-Merge Join that are sensitive to buffer availability.

    Conditional Formulas Based on Buffer Availability: Adapt the cost formulas to have conditional branches based on whether the buffer size meets certain thresholds relative to the relation sizes.

    Validate Assumptions: Before applying a formula, verify if its underlying assumptions (e.g., buffer size sufficiency) hold true for the given problem context.



Q1 Introduction  (Total marks for Q1: 5 marks)

Consider two relations A and B. A has 80,000 tuples, and B has 100,000 tuples. Both relations store 100 tuples per page. Consider the following SQL statement:

SELECT *

  FROM A INNER JOIN B

    ON A.a = B.a;

We wish to evaluate an equijoin between A and B, with an equality condition A.a = B.a. There are 102 buffer pages available for this operation. Both relations are stored as (unsorted) heap files. Neither relation has any indexes built on it.

Consider the alternative join strategies described below and calculate the cost of each alternative. Evaluate the algorithms using the number of disk I/O's (i.e., pages) as the cost. 

    Page-oriented Nested Loops Join. Consider A as the outer relation. (1 mark)
    Block-oriented Nested Loops Join. Consider A as the outer relation. (1 mark)
    Sort-Merge Join. Assume that Sort-Merge Join can be done in 2 passes. (1 mark)
    Hash Join (1 mark)
    Assuming that no indexes are built on any of the two relations, and assuming that sufficient buffer space is available:
        What would be the lowest possible cost to perform this query ? (0.5 mark)
        What would be the minimum buffer size required to achieve this cost from 5A? (0.5 mark)

 
Flag question: Question 1
Question 1 1 pts

Q1.1

Page-oriented Nested Loops Join. Consider A as the outer relation. (1 mark)

[do not give any decimals]
 
Flag question: Question 2
Question 2 1 pts

Q1.2

Block-oriented Nested Loops Join. Consider A as the outer relation. (1 mark)

[do not give any decimals]
 
Flag question: Question 3
Question 3 1 pts

Q1.3

Sort-Merge Join. Assume that Sort-Merge Join can be done in 2 passes. (1 mark)

[do not give any decimals]
 
Flag question: Question 4
Question 4 1 pts

Q1.4

Hash Join (1 mark)

[do not give any decimals]
 
Flag question: Question 5
Question 5 0.5 pts

Q1.5A

Assuming that no indexes are built on any of the two relations, and assuming that sufficient buffer space is available:

        What would be the lowest possible cost to perform this query ? (0.5 mark)

[do not give any decimals]
 
Flag question: Question 6
Question 6 0.5 pts

Q1.5B

 Assuming that no indexes are built on any of the two relations, and assuming that sufficient buffer space is available:

       B. What would be the minimum buffer size required to achieve this cost from Q1.5A? (0.5 mark)

[do not give any decimals]

USING THE ABOVE FORMULAS, SOLVE THE QUESITONS IN FULL WITHOUT FAIL. YOU MUST REFERENCE THE ABOVE FORMULAS WHEN SOLVING THE QUESTIONS