4. Summary of Correct Formulas with Annotations
Join Strategy	Formula	Annotations
Page-Oriented NLJ	(\text{Cost} = \text{NPages}(R) + \text{NPages}(R) \times \text{NPages}(S))	Treats each page of R as an outer loop and scans S entirely for each page of R.
Block-Oriented NLJ	(\text{Cost} = \text{NPages}(R) + \lceil \frac{\text{NPages}(R)}{B-2} \rceil \times \text{NPages}(S))	Processes multiple pages of R simultaneously by leveraging B-2 buffer pages for R and S.
Multi-Pass Hash Join	(\text{Cost} = 3 \times (\text{NPages}(R) + \text{NPages}(S)))	Involves three passes: partitioning R and S, building hash tables, and probing.
Optimized Single-Pass Hash Join	(\text{Cost}_{Hash_Optimized} = \text{NPages}(R) + \text{NPages}(S))	Achievable when R fits entirely into memory. Requires B ≥ NPages(R) + 2 buffer pages.
Sort-Merge Join (SMJ)	(\text{Cost}_{SMJ} = 2 \times \text{NPages}(R) \times \text{num_passes}(R) + 2 \times \text{NPages}(S) \times \text{num_passes}(S) + \text{NPages}(R) + \text{NPages}(S))	Accounts for sorting both R and S across multiple passes and then merging them. Requires accurate calculation of num_passes based on buffer size.
Lowest Possible Cost	(\text{Lowest Cost} = \min(\text{Cost}{SMJ}, \text{Cost}{Hash_Optimized}, \text{Cost}{Hash}, \text{Cost}{NLJ}))	Determines the most cost-efficient join strategy based on available buffer pages and relation sizes.
Minimum Buffer Size	(\text{Minimum Buffer} = \text{NPages}(R) + 2)	Ensures that the entire outer relation R can be loaded into memory with additional buffer pages for processing and reading the inner relation S.
5. Key Takeaways for Accurate Cost Calculations
Comprehensive Formula Application:
Sort-Merge Join: Must account for all phases—sorting each relation (with read and write for each pass) and the final merge.
Hash Join: The number of passes and buffer allocation significantly influence the total cost. Optimizing buffer usage can lead to substantial cost reductions.
Buffer Allocation Importance:
Adequate Buffers: Enable holding larger portions or entire relations in memory, reducing the number of required passes and I/O operations.
Minimum Buffer Requirement: Critical to achieving the lowest possible cost. Calculated based on the size of the outer relation and additional buffer pages for processing.
Scenario-Based Adjustments:
Relation Sizes and Buffer Constraints: Influence the choice of the most efficient join strategy. Always adapt formulas based on the specific scenario's buffer availability and relation sizes.
Verification of Pass Calculations:
Number of Passes: Must be accurately determined using logarithmic calculations based on buffer size to ensure all data can be processed efficiently.
Consistent Assumptions:
Relations as Heap Files: Assumes unsorted relations without indexes, influencing the choice of join strategies and their respective costs.
Question 1 1 pts
Q1.1

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

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Question 2 1 pts
Q1.2

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

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Question 3 1 pts
Q1.3

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

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Question 4 1 pts
Q1.4

Hash Join (1 mark)

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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)
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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].WHEN ANSWERING THE QUESTIONS YOU MUST REFERENCE THE FORMULAS GIVEN ABOVE WITHOUT FAIL. SOLVE ALL PARTS OF THE QUESTION WITHOHT DAILY AND CORRECTLY