ADBMS Tut 2 Solution

Okay, let’s solve these database query optimization problems, using MathJax for all equations and variables:

1. Join Cost Calculation

Given:

• Relation $R(A,B)$ with $|R|=10,000$ tuples.
• Relation $S(B,C)$ with $|S|=50,000$ tuples.
• Memory can hold $M=100$ tuples.

We’ll assume a simple nested loop join where R is the outer relation and S is the inner relation.

Assumptions:

• Data is stored in blocks.
• A block can contain M = 100 tuples.

Calculations:

• Number of blocks for R: $B_R=\lceil |R|/M\rceil =\lceil 10,000/100\rceil =100$ blocks.
• Number of blocks for S: $B_S=\lceil |S|/M\rceil =\lceil 50,000/100\rceil =500$ blocks.

Cost of Nested Loop Join:

• Cost = B_R + |R| * B_S = 100 + 10,000 * 500 = 5,000,100 block transfers.

If we also consider the cost to write the result. The size of result is atmost 10,000 * 50,000 = 500,000,000 tuples, which will take at most 5,000,000 blocks. So the total cost is 5,000,100 + 5,000,000 = 10,000,100

Answer: The cost of a nested loop join is approximately 10,000,100 block transfers (including writing the result).

2. Indexing Cost

Given:

• A B+ tree index.
• 1,000,000 records.
• Tree has 4 levels.

Answer: To retrieve a record using a B+ tree index, we need to traverse the tree from the root to a leaf node. Since the tree has 4 levels, we need 4 disk accesses.

3. Query Execution Time

Given:

• Full table scan takes 500ms.
• Index scan takes 50ms.

Answer: The index scan is 500ms / 50ms = 10 times faster than the full table scan.

4. Cost of Sorting

Given:

• Sorting 1,000,000 rows takes O(n \log n) time, where n = 10^6.

Calculation:

• Time ∝ n \log n = 10^6 \log (10^6) = 10^6 * 6 \log (10). Since we’re estimating the time and not calculating the exact time we can ignore the constant factor: Time ∝ 6 * 10^6

Answer: The time is proportional to 6 * 10^6. Without knowing the constant factor, we can only say that the time required is proportional to 6,000,000.

5. Selection Query Cost

Given:

• 5,000,000 records.
• 10% of records satisfy the condition.
• Sequential scan takes 0.1ms per record.

Calculation:

• Number of records satisfying the condition: 5,000,000 * 0.10 = 500,000 records.
• Total time: 500,000 * 0.1ms = 50,000ms = 50s.

Answer: The query takes 50 seconds.

6. Partitioning Benefit Calculation

Given:

• 10,000,000 rows.
• 10 partitions.
• Query searches records in one partition.

Calculation:

• Rows per partition: 10,000,000 / 10 = 1,000,000 rows.
• Data scanning reduced by a factor of 10.

Answer: Data scanning is reduced by a factor of 10, so only 1/10th of the data (1,000,000 rows) needs to be scanned.

7. Materialized View Storage Cost

Given:

• Materialized view stores 1GB.
• Recalculating the query takes 500MB of temporary storage.

Answer: The materialized view uses 1GB - 500MB = 500MB of extra storage compared to recalculating.

8. I/O Cost for Index Lookup

Given:

• 3 disk accesses.
• Each disk access takes 5ms.

Calculation:

• Total retrieval time: 3 * 5ms = 15ms.

Answer: The total retrieval time is 15ms.

9. Query Optimization Decision

Given:

• Plan A: 200ms
• Plan B: 150ms
• Plan C: 120ms

Answer: Plan C should be selected because it has the lowest estimated cost (120ms).

10. Advantages of Using Indexes in Query Evaluation

• Faster Data Retrieval: Indexes allow the database to locate specific rows much faster than scanning the entire table.
• Improved Query Performance: Queries that use indexes generally execute much faster, especially for large tables.
• Reduced I/O Operations: Indexes reduce the number of disk I/O operations required to retrieve data.

11. Differences between Nested Loop Join and Sort-Merge Join

Feature	Nested Loop Join	Sort-Merge Join
Algorithm	Iterates through each row of the outer table and searches the inner table for matching rows.	Sorts both tables on the join key, then merges the sorted tables.
Efficiency	Can be inefficient for large tables.	More efficient for large tables, especially when tables are already sorted or if indexes on the join keys exist.
Memory Usage	Generally requires less memory.	May require significant memory for sorting.
Best Use Case	Small tables or when an index exists on the inner table.	Large tables or when tables are already sorted on the join keys.

12. Role of Query Execution Plans in DBMS and Analysis with EXPLAIN ANALYZE

• Role: A query execution plan is a sequence of steps that a database system will take to execute a query. It shows how the database will access tables, use indexes, perform joins, and apply other operations.
• Analysis with EXPLAIN ANALYZE:
  • EXPLAIN ANALYZE is a command in many database systems (like PostgreSQL) that not only shows the execution plan but also executes the query and provides actual runtime statistics for each step in the plan.
  • This allows you to:
    • Identify performance bottlenecks.
    • See the actual number of rows processed at each step.
    • Compare the estimated costs with the actual costs.
    • Determine if indexes are being used effectively.
    • Refine the query or database schema to improve performance based on the analysis.