Matt Dube Doctoral Student, Spatial Information Science and Engineering Predicates, Joins, and Algebra, Oh my!

Wednesday’s Lecture Projection (unary) Horizontal reduction Selection (unary) Vertical reduction Renaming (unary) Making attributes kosher Cartesian Product (binary or more) Most expensive operator, consisting of all possible combinations Union (binary or more) Linker Difference (binary or more) What’s mine is only mine

Today Relations as first order language predicates Efficiencies in combination Algebra Join

Projection Projection takes a tuple and reduces its number of attributes dog(fido,dalmatian,bob). Suppose we would like a table that stored only the owner’s name and the dog’s name. petname(A,B):-dog(B,C,A). This is a projection.

Selection Selections act upon particular criteria dog(fido,dalmatian,bob). Suppose we want to return anyone who owns a dalmatian and what the name of the dog is dalmatian(A,B):-dog(A,dalmatian,B). This is a selection. Note the constant for the variable.

Renaming Renaming changes the name of an attribute or a relation. dog(fido,dalmatian,bob). Suppose I now want to call all dogs hippos. hippo(A,B,C):-dog(A,B,C). This is a renaming. It makes more sense with an attribute if you consider attributes spilled out in binary listings.

Cartesian Product Cartesian products bind two different relations together, maintaining all attributes dog(fido,dalmatian,bob). coffee(starbucks,brazil). Suppose we want something to give us data about all coffee and dogs javadog(A,B,C,D,E):-coffee(A,B),dog(C,D,E). This is a Cartesian product. Note that it is an AND statement and needs all variables.

Union Unions link different relations together in similar strucutres dog(fido,dalmatian,bob). cat(mittens,coon,matt). Suppose we want to know all pets. Dogs and cats are pets. pet(A,B,C):-dog(A,B,C) ∨ cat(A,B,C). This is a union of the dog relation and the cat relation.

Difference Differences show what is in one set, but not the other dog(fido,dalmatian,bob). cat(fido,dalmatian,bob). dog(rover,pitbull,joe). Suppose I want to find dogs who don’t share qualities with a cat. onlydog(A,B,C):-dog(A,B,C),cat(A,B,C),!. (fail it) onlydog(A,B,C):-dog(A,B,C). Only unique dogs are produced here.

Combination of Operations All of our operators produce relations Remember, some of the operators need to refine the output to have a relation output (think projection of a non-key) Since all operators produce relations, this system of operators is a closed system Closed systems take in inputs of a particular form (in our case relation) and output that form in return. Since the system is closed, we can string together operators.

Equivalent Combinations π a 1,…,a n ( σ A (R)) = σ A ( π a 1,…,a n (R)) σ A (R – P) = σ A (R) – σ A (P) σ A ∧ B (R) = σ A ( σ B (R)) = σ B ( σ A (R)) σ A ∨ B (R) = σ A (R) ∪ σ B (R) σ A (R x P) = σ B ∧ C ∧ D (R x P) = σ D ( σ B (R) x σ C (P))

Optimization π a 1,…,a n ( σ A (R)) = σ A ( π a 1,…,a n (R)) σ A (R – P) = σ A (R) – σ A (P) σ A ∧ B (R) = σ A ( σ B (R)) = σ B ( σ A (R)) σ A ∨ B (R) = σ A (R) ∪ σ B (R) σ A (R x P) = σ B ∧ C ∧ D (R x P) = σ D ( σ B (R) x σ C (P)) Project, then select. Difference, then select. Union, then select. Break apart complex selections

Algebra From the arabic Al-jabr, meaning “reunion” Algebras are mathematical structures formed off of operators You are familiar with an algebra from your studies in mathematics (+, -, *, /), which can be boiled down to simply + and * Algebras are closed systems under their operators and also have identities for every operator (+ 0, - 0, * 1, / 1)

Relational Algebra What is the identity form for each operator? Projection Project all of the attributes Selection Select the entire key structure Renaming Rename the attribute to the same name Cartesian Product Cross with an empty relation Union Union with a subset of the original relation Difference Difference with a mutually exclusive set

Why is an Algebra important? Algebras (reunions) are what allows for operations to be strung together An example of an important consequence: operational efficiency You have seen the relational algebra equivalencies, but there is an easier example that you already know.

Here’s the test: What are the key properties of an algebra? Closed under operations Every operation has an identity state Why is an algebra important for a system? Stringing together operators Efficient processes (think the distributive property) Operational equivalence

Joins Joins are what we would call a higher level operation Higher level operation? Think of an exponent (successive multiplications of terms) Why can we use higher level operations? (You already know the answer…) Most important operator for some key reasons Done by imposing a condition on a pair of attributes

Types of Joins Inner Joins Equi-Joins Natural Joins Outer Joins

Inner Join Selection on a Cartesian Product Selection involves some sort of criteria between a member of one relation and a member of the other (similar to the keys for the Cartesian product)

Inner Join a1a2 abc45 cde52 b1b2 45zyx 23ijk σ A.a2>B.b1 (A x B) AB

Inner Join a1a2b1b2 ab c 45 zyx ab c 4523ijk cd e 5245zyx cd e 5223ijk σ A.a2>B.b1 (A x B) A x B

Inner Join a1a2b1b2 ab c 45 zyx ab c 4523ijk cd e 5245zyx cd e 5223ijk σ A.a2>B.b1 (A x B) A x B

Inner Join a1a2b1b2 ab c 45 zyx ab c 4523ijk cd e 5245zyx cd e 5223ijk σ A.a2>B.b1 (A x B) A x B

Inner Join a1a2b1b2 ab c 4523ijk cd e 5245zyx cd e 5223ijk σ A.a2>B.b1 (A x B)

Equi-Join Another selection on a Cartesian product As the name implies, this has to do with two of the attributes being equal

Equi-Join a1a2 abc45 cde52 b1b2 45zyx 23ijk σ A.a2=B.b1 (A x B) AB

Equi-Join a1a2b1b2 ab c 45 zyx ab c 4523ijk cd e 5245zyx cd e 5223ijk σ A.a2=B.b1 (A x B) A x B

Equi-Join a1a2b1b2 ab c 45 zyx σ A.a2=B.b1 (A x B)

Natural Join σ (A x B) attribute_1 and attribute_2 must have same name only one column with this attribute shows up in the result (i.e., a projection follows) Selection on a Cartesian product (Equi-Join) followed by a projection

Natural Join xxyy abc45 cde52 yyzz 45zyx 23ijk σ A.yy=B.yy (A x B) AB

Natural Join xxyy zz ab c 45 zyx ab c 4523ijk cd e 5245zyx cd e 5223ijk σ A.a2=B.b1 (A x B) A x B

Natural Join xxyy zz ab c 45 zyx π ( σ A.a2=B.b1 (A x B))

Natural Join xxyyzz abc45zyx π ( σ A.a2=B.b1 (A x B))

Outer Joins An outer join is a join which allows for NULL values to be involved Three types Left Outer Join Everything in the left relation will be present, regardless if found in B or not Right Outer Join Everything in the right relation will be present, regardless if found in A or not Full Outer Join Both left and right outer joins

Summary Converted operators into a first order language parallel Showed how to combine operators for efficiency Defined the concept of algebra Showed the different types of joins