Analyzing Volcano Operators

• CPU Used
• Memory Bounds
• Disk IO Used

Databases are usually IO- or Memory-bound

Memory Bounds

• Constant
• Scales with output
• Scales with part of the input
• Worse

Core Question: Do we have enough memory to use this operator?

Disk IO

IO measured in:

• Number of Tuples
• Number of Data Pages (absolute size)

Select ($\sigma(R)$)

Select ($\sigma(R)$)

Memory Required?

Constant!

IOs added?

None! (Can "inline" into cost of $R$)

Example

Example, assume $R$ is 100 tuples.

How many IOs do we need to compute $Q := R$

How many IOs do we need to compute $Q := \sigma(R)$

Project ($\pi(R)$)

Project ($\pi(R)$)

Memory Required?

Constant!

IOs added?

None!

Example

Example, assume $R$ is 100 tuples.

How many IOs do we need to compute $Q := \pi(R)$

How many IOs do we need to compute $Q := \pi(\sigma(R))$

Projection and Selection do not add IO.

Union ($R \cup S$)

Union ($R \cup S$)

Memory Required?

Constant!

IOs added?

None!

Cross ($R \times S$)

Cross ($R \times S$)

Memory Required?

It depends

IOs added?

It depends

Cross ($R \times S$)

How do you "reset" $S$?

"Materialize" S into memory

No extra IOs (but $O(|S|)$ memory)

Rerun the entire iterator

$(|R|-1) \cdot \texttt{cost}(S)$ extra tuples read

"Materialize" S onto disk

$|S|$ tuples written

$(|R|-1) \cdot |S|$ extra tuples read

This can get very expensive

Example

Example, assume $R$ and $S$ are both 100 tuples.

How many IOs do we need to compute $Q := R \cup S$?

1. Getting an Iterator on $R$: 100 tuples
2. Getting an Iterator on $S$: 100 tuples
3. Getting an Iterator on $R \cup S$ using the above iterators: 0 extra tuples

Example

Example, assume $R$ is 20 tuples and $S$ is 100 tuples.

How many IOs do we need to compute $Q := R \times S$?

1. Getting an Iterator on $R$: 20 tuples
2. Getting an Iterator on $S$: 100 tuples
3. Getting an Iterator on $R \times S$ using the above iterators:

• Memory: 0 extra tuples
• Replay: $(|R|-1) \times \texttt{cost}(S) = 19 \times 100 = 1900$ extra tuples
• Cache: $|R| \times |S| = 20 \times 100 = 2000$ extra tuples

Best Total Cost $100 + 20 + 1900 = 2010$

Example

Example, assume $R$ is 20 tuples and $S$ is 100 tuples,
and $c$ filters out 90% of tuples.

How many IOs do we need to compute $Q := R \times \sigma_c(R \times S)$

1. Getting an Iterator on $\sigma_c(R \times S)$: 2010 tuples
2. Getting an Iterator on $R$: 20 tuples
3. Getting an Iterator on $R \times \sigma_c(R \times S)$ using the above iterators:

• Memory: 0 extra tuples
• Replay: $(|R|-1) \times \texttt{cost}(\sigma_c(R \times S)) = 19 \times 2010 = 38190$ extra tuples
• Cache: $|R| \times (0.1 \times (|R| \times |S|)) = 20 \times 200 = 4000$ extra tuples

Best Total Cost $2010 + 20 + 4000 = 6030$

Can we do better with cartesian product
(and joins)?

Problem: We need to evaluate rhs iterator
once per record in lhs

Preloading Data

Better Solution: Load both lhs and rhs records in blocks.

                    def apply_cross(lhs, rhs):
                      result = []

                      while r_block = lhs.take(100):
                        while s_block = rhs.take(100):
                          for r in r_block:
                            for s in s_block: 
                              result += [r + s]
                        rhs.reset()

                      return result
    

Block-Nested Loop Join

Block-Nested Loop ($R \times S$)

(with $\mathcal B$ as the block size for $R$)

(and with caching $S$ to disk)

Memory Required?

$O(\mathcal B)$

IOs added?

$|S|$ tuples written.

$(\frac{|R|}{\mathcal B} - 1) \cdot |S|$ tuples read.

In-memory caching is a special case of block-nested loop with $\mathcal B = |R|$

Does the block size for $S$ matter?

How big should the blocks be?

As big as possible!

... but more on that later.

Cross Product

Problem: Naively, any tuple matches any other

Join Conditions

Solution: First organize the data

Hash Functions

• A hash function is a function that maps a large data value to a small fixed-size value
  • Typically is deterministic & pseudorandom
• Used in Checksums, Hash Tables, Partitioning, Bloom Filters, Caching, Cryptography, Password Storage, …
• Examples: MD5, SHA1, SHA2
  • MD5() part of OpenSSL (on most OSX / Linux / Unix)
• Can map h(k) to range [0,N) with h(k) % N (modulus)

Hash Functions

$$h(X) \mod N$$

• Pseudorandom output between $[0, N)$
• Always the same output for a given $X$

1-Pass Hash Join

1-Pass Hash Join

Limited Queries

Only supports join conditions of the form $R.A = S.B$

Moderate-High Memory

Keeps 1 full relation in memory

Low Added IO Cost

Only requires 1 scan over each input.

Alternative: Build an in-memory tree (e.g., B+Tree) instead of a hash table!

Limited Queries

Also supports $R.A \geq S.B$, $R.A > S.B$

Moderate-High Memory

Keeps 1 full relation in memory

Low Added IO Cost

Only requires 1 scan over each input.

2-Pass Hash Join

2-Pass Hash Join

Limited Queries

Only supports join conditions of the form $R.A = S.B$

Low Memory

Never need more than 1 pair of partitions in memory

High IO Cost

$|R| + |S|$ tuples written out

$|R| + |S|$ tuples read in

Why is it important that the hash function is pseudorandom?

What if the data is already organized (e.g., sorted) in a useful way?

Sort/Merge Join

Sort/Merge Join

Limited Queries

Only supports join conditions of the form $R.A = S.B$

Low Memory

Only needs to keep ~2 rows in memory at a time (not counting sort).

Low Added IO Cost

No added IO! (not counting sort).

Recap: Joins

Block-Nested Join

Moderate Memory, Moderate IO, High CPU

In-Memory Index Join (e.g., 1-Pass Hash)

High Memory, Low IO

Partition Join (e.g., 2-Pass Hash)

High IO, Low Memory

Sort/Merge Join

Low IO, Low Memory (But need sorted data)