Relational Algebra Overview

Consider two relational algebra expressions:

1: π_name(σ_{salary>50000}(Employees))
2: σ_{salary>50000}(π_{name,salary}(Employees))

Do they produce the same result? If so, which should we execute? These questions—of equivalence and optimization—are at the heart of query processing.

Query equivalence is the theoretical foundation that makes query optimization possible. When we know that two expressions are equivalent—they produce identical results for any database state—we can freely substitute one for the other. This freedom to substitute is what allows optimizers to explore vast spaces of alternative execution strategies while guaranteeing correctness.

In this page, we'll formally define query equivalence, explore the equivalence rules that govern relational algebra, understand the conditions under which transformations are valid, and see how these rules enable the optimizations that make databases fast.

Definition: Query Equivalence

Two relational algebra expressions E₁ and E₂ are equivalent, written E₁ ≡ E₂, if and only if:

For every database instance D, the evaluation of E₁ on D produces the same result as the evaluation of E₂ on D.

Formally:

E₁ ≡ E₂  ⟺  ∀D. E₁(D) = E₂(D)

Where:

• D ranges over all possible database instances
• E(D) denotes the relation produced by evaluating expression E on database D
• = denotes set equality (identical tuples)

Key Points:

1. Universal Quantification: Equivalence must hold for ALL possible database states, not just the current one or specific test cases.

2. Set Equality: Results must contain exactly the same tuples. Same tuples in different order are still equal (relations are sets, unordered).

3. Schema Compatibility: Equivalent expressions must produce results with the same schema (column names and types). If schemas differ, expressions aren't equivalent even if the data values match.

Examples:

Equivalent:

σ_{p∧q}(R) ≡ σ_p(σ_q(R)) ≡ σ_q(σ_p(R))

Conjunctive selections can be split and reordered.

NOT Equivalent:

π_A(σ_B(R)) ≢ σ_B(π_A(R))  when B ∉ A

If attribute B isn't in projection list A, we can't filter on it after projection.

Structural vs Semantic Equivalence:

Structural Equivalence: Expressions match syntactically after normalization (essentially the same expression with possible renaming).

Semantic Equivalence: Expressions produce identical results, possibly through completely different computational paths.

Query optimization exploits semantic equivalence—finding syntactically different but semantically identical expressions that execute more efficiently.

Why Equivalence Matters:

1. Optimization Correctness: Transformations are only valid if they preserve equivalence
2. View Substitution: Views can be substituted for their definitions because they're equivalent
3. Constraint Enforcement: Some constraints guarantee equivalences (e.g., key constraints enable certain simplifications)
4. Query Normalization: Multiple user queries can be recognized as equivalent for caching

Selection (σ) has several important equivalence rules that form the foundation of selection pushdown optimization.

Rule 1: Conjunction Splitting (Cascade Rule)

σ_{p₁ ∧ p₂ ∧ ... ∧ pₙ}(R) ≡ σ_{p₁}(σ_{p₂}(...σ_{pₙ}(R)...))

A selection with a conjunctive (AND) predicate can be split into a cascade of selections. This enables pushing individual conditions to different places in the expression tree.

Rule 2: Selection Commutativity

σ_{p₁}(σ_{p₂}(R)) ≡ σ_{p₂}(σ_{p₁}(R))

The order of cascaded selections doesn't matter. This flexibility allows the optimizer to arrange selections in the most beneficial order (e.g., most selective first).

Rule 3: Selection and Projection Commutativity (Conditional)

π_L(σ_p(R)) ≡ σ_p(π_L(R))  only if p references only attributes in L

If the selection predicate uses only attributes that survive projection, we can swap their order. This often enables projecting early to narrow tuples.

Rule 4: Selection and Cartesian Product

σ_p(R × S) ≡ R ⋈_p S  when p references both R and S
σ_p(R × S) ≡ σ_p(R) × S  when p references only R

Selection over product with a join condition IS a theta join. Selection with a predicate referencing only one relation can be pushed to that relation.

Rule 5: Selection and Join

σ_p(R ⋈ S) ≡ σ_{p1}(R) ⋈ σ_{p2}(S) ⋈ (conditions involving both)

Where predicate p can be decomposed into:

• p1: conditions on R's attributes only
• p2: conditions on S's attributes only
• Remaining: conditions involving both R and S

Pushing selections past joins is the most impactful optimization rule, often reducing join input sizes dramatically.

Rule 6: Selection and Union/Intersection/Difference

σ_p(R ∪ S) ≡ σ_p(R) ∪ σ_p(S)
σ_p(R ∩ S) ≡ σ_p(R) ∩ σ_p(S)
σ_p(R − S) ≡ σ_p(R) − S    (note: only pushed to first operand)
           ≡ σ_p(R) − σ_p(S)  (alternative)

Selection distributes over set operations, enabling early filtering.

Projection (π) equivalences enable removing unneeded columns early, reducing data volume and memory usage.

Rule 1: Cascading Projections

π_L₁(π_L₂(R)) ≡ π_L₁(R)  when L₁ ⊆ L₂

Nested projections simplify to the outermost (smallest) projection. Intermediate projections that keep extra columns are wasteful.

Rule 2: Projection and Selection

As noted earlier:

π_L(σ_p(R)) ≡ σ_p(π_{L∪attrs(p)}(R))  followed by π_L

We can project away columns not needed for the final result or the selection predicate. The minimal projection keeps L plus any attributes referenced in p.

Rule 3: Projection and Cartesian Product

π_L(R × S) ≡ π_{L∩attrs(R)}(R) × π_{L∩attrs(S)}(S)  when L = L_R ∪ L_S

We can push projection over product if the projected columns come from separate relations.

Rule 4: Projection and Join

π_L(R ⋈ S) ≡ π_L(π_{L ∪ join_attrs(R)}(R) ⋈ π_{L ∪ join_attrs(S)}(S))

Before joining, project away columns not needed for the join condition or the final result. This narrows tuples, reducing memory and potentially I/O.

Rule 5: Projection and Union

π_L(R ∪ S) ≡ π_L(R) ∪ π_L(S)

Projection distributes over union. This enables projecting before combining, reducing data volume throughout.

Rule 6: Projection and Set Difference

π_L(R − S) ≡ π_L(R) − π_L(S)

Similarly, projection distributes over difference.

Projection Pushdown Limitations:

Projection can't always be pushed:

1. Attributes needed for selection: Must keep columns used in predicates
2. Attributes needed for join: Must keep join columns
3. Grouping attributes: Aggregation requires grouping columns
4. Order by attributes: If ORDER BY is later, need those columns

The optimizer computes the minimum attribute set needed at each tree node and projects away everything else.

Join equivalences are crucial because joins are often the most expensive operations, and reordering can dramatically affect performance.

Rule 1: Join Commutativity

R ⋈ S ≡ S ⋈ R

For natural join and inner join, operand order doesn't affect the result. This enables choosing which relation to use as the "build" side in hash joins, for example.

Rule 2: Join Associativity

(R ⋈ S) ⋈ T ≡ R ⋈ (S ⋈ T)

Join order can be changed without affecting results. This is the foundation of join reordering optimization—finding the order that minimizes intermediate result sizes.

Rule 3: Theta Join Equivalence

R ⋈_θ S ≡ σ_θ(R × S)

Theta join is equivalent to Cartesian product followed by selection. While we'd never execute this way, it helps prove other equivalences.

Rule 4: Join and Selection Commutativity

σ_p(R ⋈ S) ≡ R ⋈_p S  (pushing p into join condition)
σ_{p1∧p2}(R ⋈ S) ≡ σ_{p1}(R) ⋈ σ_{p2}(S) ⋈_{remaining}  (splitting p)

Selections can be pushed into or past joins when predicate attributes allow.

Join Orders and Tree Shapes:

For n tables, the number of different join orders is:

• Left-deep trees only: n!
• All binary trees (bushy): Catalan number C(n-1) × n! (approximately 4^n / (n^1.5))

For 10 tables:

• Left-deep: 3.6 million orderings
• Bushy: billions of orderings

Optimizers use dynamic programming or greedy heuristics to search this space efficiently.

Rule 5: Outer Join Considerations

Outer joins are NOT fully commutative or associative:

R ⟕ S ≢ S ⟕ R  (LEFT vs RIGHT outer join differ)
R ⟕ (S ⟕ T) may ≢ (R ⟕ S) ⟕ T  (depends on join conditions)

Outer join reordering requires careful analysis. Some transformations are valid under specific conditions, but the general rules are more restrictive than for inner joins.

Rule 6: Semi-join and Anti-join

π_{attrs(R)}(R ⋈ S) ≡ R ⋉ S  (semi-join)
R − π_{attrs(R)}(R ⋈ S) ≡ R ▷ S  (anti-join)

Semi-joins keep R tuples that have a match in S; anti-joins keep R tuples that have NO match. These are often more efficient than full joins when only existence is checked.

Set operations (∪, ∩, −) follow classical set theory equivalences, with some unique considerations for relational databases.

Rule 1: Union Commutativity and Associativity

R ∪ S ≡ S ∪ R
(R ∪ S) ∪ T ≡ R ∪ (S ∪ T)

Union is both commutative and associative, like set union in mathematics.

Rule 2: Intersection Commutativity and Associativity

R ∩ S ≡ S ∩ R
(R ∩ S) ∩ T ≡ R ∩ (S ∩ T)

Intersection similarly satisfies both properties.

Rule 3: Set Difference is NOT Commutative

R − S ≢ S − R  (in general)

R − S removes S's tuples from R; S − R removes R's tuples from S. Completely different operations.

Rule 4: Distributive Laws

R ∩ (S ∪ T) ≡ (R ∩ S) ∪ (R ∩ T)
R ∪ (S ∩ T) ≡ (R ∪ S) ∩ (R ∪ T)

Intersection distributes over union and vice versa, following set theory.

Rule 5: De Morgan's Laws

U − (R ∪ S) ≡ (U − R) ∩ (U − S)
U − (R ∩ S) ≡ (U − R) ∪ (U − S)

Where U is a "universal" relation. These enable alternative formulations for complex set expressions.

Rule 6: Selection Distribution (Revisited)

σ_p(R ∪ S) ≡ σ_p(R) ∪ σ_p(S)
σ_p(R ∩ S) ≡ σ_p(R) ∩ σ_p(S)
σ_p(R − S) ≡ σ_p(R) − σ_p(S)

Selection distributes over all set operations, enabling pushdown before set combination.

Rule 7: Intersection as Join

R ∩ S ≡ R ⋈ S  (when R and S have the same schema)

Intersection of union-compatible relations is equivalent to natural join on all attributes. This enables using join algorithms for intersection.

Rule 8: Difference through Anti-Join

R − S ≡ R ▷ S  (anti-join; tuples in R with no match in S)

For union-compatible relations, set difference equals anti-join. This may enable more efficient execution using anti-join algorithms.

Beyond operation-specific rules, several general principles govern equivalence in relational algebra.

Rule 1: Rename Equivalence

ρ_S(ρ_R(T)) ≡ ρ_S(T)

Sequential renames collapse—only the final name matters.

Rule 2: Rename and Other Operators

Rename generally commutes with most operators, modulo adjusting predicate/column references:

ρ_S(σ_p(R)) ≡ σ_{p[renamed]}(ρ_S(R))
ρ_S(R ⋈ T) ≡ ρ_S(R) ⋈ T  (adjusting references)

Rule 3: Aggregation Decomposition

For certain aggregates, we can compute partial results then combine:

SUM(R ∪ S) ≡ SUM(R) + SUM(S)  (for disjoint R, S)
COUNT(R ∪ S) ≡ COUNT(R) + COUNT(S)  (for disjoint R, S)

This enables parallel aggregation with final merge.

Rule 4: Redundant Operation Elimination

σ_true(R) ≡ R
π_{all-attrs}(R) ≡ R
R ∪ ∅ ≡ R
R ⋈ R ≡ R  (for relations with keys)

Redundant or identity operations can be eliminated.

Rule 5: Constant Folding and Simplification

σ_{false}(R) ≡ ∅
σ_{A=A}(R) ≡ R  (assuming no nulls)
R ⋈_{false} S ≡ ∅

Predicates with constant truth values can be simplified.

Rule 6: Constraint-Based Equivalences

With knowledge of constraints, additional equivalences hold:

-- If K is a key of R:
π_K(R ⋈ R) ≡ π_K(R)

-- If FK references PK:
R ⋈ S ≡ R  (if the join is automatically satisfied by FK)

-- If NOT NULL constraint exists:
σ_{A IS NOT NULL}(R) ≡ R

Optimizers use constraint knowledge for semantic optimization.

Understanding equivalence rules is one thing; applying them systematically to find optimal plans is another. Let's examine how optimizers use these rules.

The Search Space

Given a query's initial expression tree, equivalence rules define a search space:

• Each rule application produces an equivalent tree
• Multiple rules can be applied in various orders
• The number of equivalent trees can be enormous

For a query joining 10 tables, there may be millions of equivalent join orderings alone.

Search Strategies

Exhaustive Enumeration:

Generate all equivalent expressions, cost each, return the cheapest.

• Guaranteed optimal
• Feasible only for small queries
• Used: System R, early relational optimizers

Dynamic Programming:

Build optimal plans for subexpressions, combine into full plans.

• Avoids recomputing equivalent subplans
• Polynomial complexity for many query patterns
• Used: Most modern optimizers

Heuristic/Rule-Based:

Apply transformation rules in a fixed order without exhaustive search.

• Fast but may miss optimal plan
• Useful for simple queries or as a first pass
• Used: Often combined with cost-based search

Cost-Based Selection

After generating equivalent plans, the optimizer estimates costs:

1. Cardinality Estimation: How many tuples does each operator produce?
2. I/O Cost: How many disk pages read/written?
3. CPU Cost: How many comparisons, computations?
4. Memory Usage: Does the plan fit in available memory?
5. Network Cost: (For distributed) How much data transferred?

The plan with lowest total cost wins.

Practical Considerations:

• Time Limits: Optimization can't take longer than execution; budgets limit search
• Statistics: Cost estimation depends on statistics (histograms, row counts)
• Hints: Users can influence optimizer choices when it makes mistakes
• Plan Caching: Prepared statements reuse optimized plans

We've explored query equivalence—the theoretical foundation of query optimization. Let's consolidate the key insights:

Module Complete

With this page, we've completed our overview of relational algebra. You now understand:

• Relational algebra as a procedural query language
• The types of operations and their categorization
• The closure property that enables composition
• Expression trees as the query representation
• Query equivalence as the foundation of optimization

In the following modules, we'll dive deeper into specific operators—selection, projection, joins, and more—exploring their semantics, implementation algorithms, and optimization techniques in exhaustive detail.