Acyclic Joins

A full reducer for S is built “from the inside out.”
1. The empty semijoin program is a full reducer for {T }.
2. A full reducer for {S, T } is given by
T := T ⋉ S;
S := S ⋉ T ;
3. A full reducer for {R, S, T } is given by
S := S ⋉ R;
T := T ⋉ S;
S := S ⋉ T ;
R := R ⋉ S;
7 Order of Joins
Let S be an acyclic database schema. Suppose we have applied a full reducer. We must now join all
relations. Suppose we have removed S 1 , S 2 , . . . , S n in that order. That is, S 1 was the first ear removed,
S 2 was the second ear removed, and so on. Assume that for i ∈ {1, . . . , n − 1}, S i was removed in favor
of T i ∈ {S i+1 , . . . , S n }. In particular, T n−1 = S n . The full reducer in the proof of Theorem 2 is the
following.
T 1 := T 1 ⋉ S 1
T 2 := T 2 ⋉ S 2
.
T n−1 := T n−1 ⋉ S n−1
S n−1 := S n−1 ⋉ T n−1
.
S i := S i ⋉ T i
.
S 2 := S 2 ⋉ T 2
S 1 := S 1 ⋉ T 1
Now we join relations in reverse order, that is,
Result := S n
Result := S n−1 ✶ Result
Result := S n−2 ✶ Result
.
Result := S i ✶ Result
.
Result := S 1 ✶ Result
We argue that the size of Result cannot decrease. When we join S i to S i+1 ✶ · · · ✶ S n , we know that every
tuple of S i joins with some tuple of S i+1 ✶ · · · ✶ S n , because the command S i := S i ⋉ T i in the full reducer
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