1. .
.
. ¯
234322
8 :
. .
.( ) ¯.
. ¯.
. ¯ ¯.
¯ 1 ¯ 2
( )
, . ¯ ¯ ,
. .(n-tuple) -n , ¯
¯ ¯ ¯ ( ) : (domain)
. ¯
20 ¯
(99999- 10000 ¯)
(31.12.2999 -1.1.1900)
¯–
¯ 3 ¯ 4

2. ( , )
(multiset, bag) - :(relation) prophets - :
. ¯ ¯
{ , - , }- :(attributes ) ¯
prophets -500 52
. , ¯ :(relation scheme) ¯
-550 66
:
-700 55
Prophets(chapters, birth_year, name)
-200 4 Branch(b_name, assets, b_city)
Customer(c_name, street, c_city)
-400 48
Deposit(b_name, account_number, customer_name, balance)
prophets ⊆ string × date × integers Borrow(b_name, loan_number, loan_number, c_name, amount)
¯ 5 ¯ 6
( )
,
¯, ¯ , ¯
. :SQL- , ¯
,
. SELECT name
FROM Prophets
, WHERE birth_year<-500;
( ¯ )
. ¯
. SQL-
.( ¯ C ¯ )
¯ 7 ¯ 8

3. ¯
. ¯ ¯, ,
¯ , , , –× ∩∪ •
) , ¯
– select •
– (projection) –π •
.( ¯
– π A→B • . - ¯ ¯ ,
( ¯ ) –δ •
– join •
¯ ¯ – sort •
,
(? ¯)
¯
– max, min, average, sum •
– count •
¯ 9 ¯ 10
∪ ∩
A1 A2 A3 R ∪ S = {r: r ∈ R or r ∈ S} R ∩ S = {r: r ∈ R and r ∈ S} ¯
a b c A1 A2 A3 R A1 A2 A3
S A1 A2 A3
R
d e f a b c a b c a b c
c b d R∪S d e f a b c c b d
c b d c b d a c e
A1 A2 A3
S c b d R∩S
c b d d a f A1 A2 A3
¯
d a f a b c
¯
¯
¯ 11
c b d ¯ 12

4. × ¯
R × S = {(r1,r2,…,rn,s1,…,sm): r ∈ R and s ∈ S}
R S = {r: r ∈ R and r ∉ S} ¯
R A1 A2 A3 A1 A2 A3 B1 B2
R A1 A2 A3 S A1 A2 A3
a b c a b c a b
a b c a b c a b c a b
a b c
a b c c b e c b d a b
c b d
c b d a c e a b c c b
S B1 B2
a b c c b
A1 A2 A3 a b R×S
RS c b d c b
a b c c b , ¯ ¯
c b d (R.a, S.a ) ¯
( ) (R ) × (S ):
¯ 13 ¯ 14
select σ π projection
.C select
σC(R) = {r ∈ R : r satisfies C} πℓ1, ℓ2,…, ℓn(R)={(rℓ1,rℓ2,…rℓn) : (r1,…,rn) ∈ R}
A1 A2 A3
σA2=b(R)
R R πA1,A3(R) πA2(R)
a b c A1 A2 A3
A1 A2 A3 A1 A3 A2
a b c a b c
a b c a c b
c b d a b c
a b c a c b
c c c c b d
c b d c d b
σa≤A2≤b(R) " [a,b]
¯ 15 ¯ 16

5. join ( ) join
. ¯
¯ ¯ :A = attr(R) ∩ attr(S) ¯ S ,R join
. R S = {r s : r ∈ R, s ∈ S, πA(r) = πA(s) }
:
:
Q(name, address, tel_no) = R(name, address) S(name, tel_no)
A B C A B C D
C D
: join a b c a b c d1
A ¯ S,R c d1
a1 b1 c a1 b1 c d1
r.A = s.A - ¯ r ∈ R, s ∈ S c2 d2
a1 b1 c1 a1 d c2 d2
s- r r s c2 d3
.A ¯ a1 d c2 a1 d c2 d3
s = (Homer Simpson, 555-2121)
r = (Homer Simpson, 123 Fake St. Springfield)
r s = (Homer Simpson, 123 Fake St. Springfield, 555-2121)
¯ 17 ¯ 18
(join) join
. ¯ ' ¯ ¯ R
¯ S R ¯ s r
. ¯ , ' ¯ ¯ S
.A
: ¯ R S
. ¯ , ¯, , ¯ ¯ S R
R S = πS∪(R/S) (σAR=A(ρA→AR(r) × s))
¯ 19 ¯ 20

6. .( ) R
.R ( ) = B(R)
. .R = T(R)
. .V(R)=T(δ(R)) " ,R = V(R)
. . ¯ = M
. . ¯
¯ (pipeline) , ¯-
.
,
. ¯
¯ 21 ¯ 22
. -
. max, min, average
Union(relation R, relation S) {
forall r ∈ R { double average(relation R, column A) {
read r; int count = 0;
output r; value sum = 0;
} forall r ∈ R {
forall s ∈ S { read r;
read s; sum = sum + r.A;
output s; count = count + 1;
} }
} return sum / (double) count;
B(R) + B(S) : ¯ }
O(1) : ¯ B(R) : ¯
¯ 23 ¯ 24

7. δ
B(δ(R)) ≤ M - 4 . S
relation delta(relation R) { B(δ (S) ) ≤ M ¯
search_structure = ∅; : -4
( ¯ R )
forall r ∈ R {
read r;
if (r ∉ search_structure) { SR:
insert r into search_structure; . δ(S) ¯
output r; .R
} . S- r ,R- r ¯
} , S ¯
} .
. ¯ ¯ search_structure
¯ 25 ¯ 26
SR SR
relation Difference(relation R, relation S) {
S=(20, 11, 30, 20,11,11) search_structure = ∅;
R=(11,40,20,11, 20, 20, 50, 40) forall s ∈ S {
read s;
11:3, 20:2, 30:1 S if s ∉ search_structure {
insert s into search_structure;
R search_structure[s].count = 0;
11:2, 20:2, 30:1 :11 }
search_structure[s].count ++;
11:2, 20:2, 30:1 :40 }
forall r ∈ R {
11:2, 20:1, 30:1 :20 read r;
if (r ∈ search_structure) and (search_structure[r].count > 0)
11:1, 20:1, 30:1 :11 search_structure[r].count --;
}
11:1, 20:0, 30:1 :20 forall s ∈ search_structure
repeat search_structure[s].count times
11:1, 20:0, 30:1 :20 output s;
¯ 27
} ¯ 28

8. ¯ S ¯ ¯
( ¯) ¯ ¯ ¯ S ¯
, S ¯
: ¯ .(A- ) ¯
B(S) + B(R)
: ¯ .( ¯ ¯) R ¯
B(S) , r.A R- r ¯
(r.A = s.A ¯) s ¯
. r s
¯ 29 ¯ 30
R S R S
A B B C A B B C
a b1 b c1 a b1 b c1
a2 b2 b1 c2 (a,b1,c2) a2 b2 b1 c2
(a,b1,c3)
a1 b1 b1 c3 a1 b1 b1 c3
a6 b3 b3 c4 a6 b3 b3 c4
a4 b3 a4 b3
( ¯ )
a5 b3 a5 b3
( )
¯ 31 ¯ 32

9. R S R S
A B B C A B B C
a b1 b c1 a b1 b c1
--- a2 b2 b1 c2 (a1,b1,c2) a2 b2 b1 c2
(a1,b1,c3)
a1 b1 b1 c3 a1 b1 b1 c3
a6 b3 b3 c4 a6 b3 b3 c4
a4 b3 a4 b3
(a6,b3),(a4,b3),(a5,b3)
a5 b3 a5 b3
(b3,c4)
¯ 33 ¯ 34
¯ ¯ δ(S) ¯ : ¯
¯
:
B(S) + B(R) B(R) O(1) σ, π,
B(R) + B(S) O(1) ∪
. ¯ ¯
B(R) δ(R) δ
B(R) + B(S) δ(S) ×∩
¯ –
¯ 35 ¯ 36

10. nested loops
.( ) :
. for each tuple s in S {
read s;
. for each tuple r in R {
. read r;
. process tuples r and s
}
}
T(S)T(R) : ¯
T(S)B(R) R :
¯ 37 ¯ 38
:
join ¯ .S ¯ M-4 ¯
for each tuple s in S {
read s; :
for each tuple r in R { S M-4
( ¯ )R ¯
read r; R- r ¯
if s and r join to make a tuple t s ¯ ¯ S
output t; r.A = s.A :
} r s
}
T(S)B(R) : ¯ (M-4) + B(R) : ¯
B(S)/(M-4)
 :
¯ 39 ¯ 40

11. :
¯
(M-4) + B(R) : while (S is not exhausted) {
B(S)/(M-4)
 : read the next M-4 blocks of S and put them in a
search structure ordered by A;
: ¯ forall blocks b of R {
read block b;
B(S)/(M-4) ((M-4) + B(R)) = B(S) + B(S) B(R) /(M-4) forall tuples r in b {
≅ B(S) B(R) / M search r.A in the search structure;
forall tuple s found
output s r;
M >> 4 }
, }
? .R }
¯ 41 ¯ 42
.( ) . ¯
. ,( ) S- R
. .R S
.R r- S s ¯ ¯
. ( ) r >s
. .( )S s
.R r r <s
,r=s
. s r
?
¯ 43 ¯ 44

12. ‫היכן השתמשנו‬
?M ‫בהנחה על‬
¯ :
S- R- ¯ ,A S- R-
.|σA=a(S)| > M A a
.R S ¯
.R r- S s ¯ ¯
S s πA(r) >πA(s) 1 B(R) δ
R r πA(r) <πA(s)
1 1 Min, Max
πA(s’) =πA(s) S ¯ ¯
:πA(r’) =πA(s) R r’ ¯ *2 B(S)+B(R) ∩ , ∪, / ( )
r’ s’ ¯ s’ ¯
*2 B(S)+B(R)
B(S) + B(R) : ¯
B(S R) : ¯ ¯ ¯ ¯, ¯ *
¯ 45 ¯ 46
:
.( )
: ¯ ¯
.
. ¯
. . (hash)
¯ ; ¯ ¯
. . ¯
: ¯ ,
, ¯
. ¯
¯ 47 ¯ 48

13. δ
53,42,73,81,22,53,91,61,53,11,92,91 : , ¯
.( ) : , ,
1: 81,91,61,11,91 . k- ¯
2: 42,22,92
3: 53,73,53,53 : , ¯ ¯
: ¯ . ¯ :
¯ ¯ ( ) ¯ M
:( ¯ ) δ k
1: 81,91,61,11,91
2: 42,22,92 ?‫למה צריך יותר ממעבר אחד‬ M ≥ 2k + 2 :
3: 53,73,53,53 ‫למה לא לסלק את הכפולים‬ k ≤ M/2 -1 "
?‫כאשר מנסים להכניסם לדלי‬
¯ 49 ¯ 50
: δ : δ
. ( ) B(R) ¯¯
: ¯ ¯
( ¯, ) B(R) > M
, k= M/2-1 -
. :
, k- ¯ ,
. ¯ : ...
. ¯
.M k- ¯ h-
.( ¯ ¯) δ
h = logk(B(R)/M)
 :
M≥ ⇐ ¯ ¯ ¯
(2h+1)B(R) :
¯ ,B(R)/k ≤ M
M  M
2
B ( R ) ≤ kM = M  − 1 ≈
 2  2
¯
3B(R) , ¯ 51 ¯ 52

14. for all s ∈ S :
write s to bucket BS[h(s)];
2B(S) S
for all r ∈ R
write r to bucket BR[h(r)]; 2B(R) R
B(S) S
. ¯ ¯ BR[i]- BS[i] ,i ¯
B(R) R
for i=1 to number_of_buckets { 3B(S) + 3B(R) ¯"
find the intersection of Bs[i] and Br[i] using the single pass
algorithm and produce the sub-relation Ti
output Ti
}
¯ 53 ¯ 54
¯
¯ A : B ≤ M2 / 2
h(s.A) " S1,…Sk S 3B(R) δ
2B(S) B(Si) ≤ M :i ¯ ¯ 3B(S) + 3B(R) '¯ , ,
2B(R) h(r.A) " R1,…Rk R
(? ¯) Ri Si i=1,..,k
Σi(B(Si)+B(Ri)) (? ) 5- 3 , B
3B(S)+3B(R) ¯"
¯ 55 ¯ 56

15. . ¯
.( )
. a≤x≤b C(x) ¯ σC :
. .
. :R S ¯ :
. .R- S- ¯
O(T(S)) : ¯
. R ¯ ¯
¯ , S :
. ,R
¯ 57 ¯ 58
¯
.Y ,S S(X,Y) R(Y,Z) , ¯
. ¯ ,
r∈R ¯ , ¯ ¯
S- r.Y .
s.Y = r.Y s∈S ¯
. s r : ¯
¯ .(' ¯ , , )
.B(R) :R
S T(S)/V(S,Y) r.Y ¯ :( ¯ ) ¯
(S- Y ¯ V(S,Y) ¯)
T(R) T(S)/V(S,Y) :¯" .
¯ 59 ¯ 60

16. : , ¯ ¯ ¯ π name,addr(σ title =“Gone…”, gender = F(MovieStar StarsIn))
:
πname,addr(σtitle=“gone with the wind” and gender = F (MovieStar
StarsIn))
: ¯ πname,addr πname,addr
MovieStar(name, addr, gender), StarsIn(title, year, name)
σ title=“Gone…” and gender = F
" " ¯
. ¯
σ gender = F σ title=“Gone…”
SELECT name, addr
FROM MovieStar, StarsIn
MovieStar StarsIn MovieStar StarsIn
WHERE title=“gone with the wind” and
gender=F and
MovieStar.name=StarsIn.name; π name,addr (σ gender = F(MovieStar) σ title=“Gone…” (StarsIn))
¯ 61 ¯ 62
: ,
: ,
(R S) T=R (S T) . ¯ ¯
: .
R S= S R
: σ title =“Gone…”, gender = F(MovieStar StarsIn)) =
σC(R ∩ S) = σC(R) ∩ S = R ∩ σC(S) = σC(R) ∩ σC(S) σ gender = F(MovieStar) σ title=“Gone…” (StarsIn)
: (? ) σ gender = F
σ C(R S) = σC(R) S=R σC(S) = σC(R) σC(S) σ title=“Gone…”
. ¯ ¯ ¯
¯ 63 ¯ 64

17. : ¯
Movie(title, year, studio) , StarsIn(title, year, name) :
π name,studio (σ year = 1996(Movie) StarsIn) ,
. ¯ ¯
πname,studio πname,studio
? π- ¯
, ¯
σ year=1996 StarsIn σ year=1996 σ year=1996 .
. ¯ ,
Movie Movie StarsIn
π name,studio (σ year = 1996(Movie) σ year = 1996 (StarsIn))
¯ 65 ¯ 66
δ ¯
, δ ... ,
. ¯¯ ¯ , ¯-
, ¯
δ(R S) = δ (δ(R) S) ,
¯
= δ (R δ(S))
.
= δ(R) δ(S)
, - ,¯"
, δ, ¯ . ¯ ¯
. ¯ ¯
, n R :π ¯
n⋅f π(R) , f
¯ 67 ¯ 68

18. ¯
σ ¯ σ
T(σA≤x(R))
A ¯ ¯ :( ¯ ) :
V(R,A) = T(δ(π A(R))) E(T(σA≤x(R))) = P(A ≤ x) T(R)
E(T(σA=x(R))) = T(R) / V(R,A) :
:A = [a,b]
x ,R ¯ E(T(σA≤x(R))) = (x-a)/(b-a) T(R)
.( ) σ-
: ( V(R,A) = 2) a,b ¯ A ¯ , : a1< a2 < ⋅ ⋅ ⋅ <an ¯ A= {a1,…,an}
P(π (R) = a)= 0.9, P(π (R) = b)= 0.1 E(T(σA≤ai(R))) = (i / n) T(R)
A A
,a,b x-
E(T(σA=x(R))) = ½ T(σA=a(R)) + ½ T(σA=b(R))
: , -
= ½ (0.9 T(R)) + ½ (0.1 T(R)) = T(R) /2 T(σA≤x(R)) = ⅓T(R)
= T(R) / V(R,A) T(σA>x(R)) = ⅔T(R)
¯ 69 ¯ 70
2 ¯ δ ¯
? T(σ x≤A≤y (R)) R(A1,…, An) ¯
σ x≤A≤y (R) = σ x≤A (σ A≤y (R)) ,T(δ(R)) = V(R)
. V(R) ¯-
" 2/9-
: " " V(R , Ai ) = V(π Ai (R ))
T(σ x≤A≤y(R)) = ⅔ T(σ A≤y (R)) = ⅔ ⅓ T(R) = (2/9)T(R)
1 ≤ V(R ) ≤ ∏ V(R , A )
1≤ i ≤ n
i
: ¯
 1 
V(R ) = min  ∏ V(R , Ai ), T (R )
1≤i ≤ n 2 
¯ 71 ¯ 72

19. S(Y,Z) R(X,Y) ¯ S(Y,Z) R(X,Y) ¯
: :
¯ ¯ Y ¯ ¯ Y
p(y1)≥… ≥ p(yn) - ¯ Y ¯ p(y1)≥… ≥ p(yn) Y ¯
y1,…,yn Y ¯ y1,…,yn Y ¯
yi∈ V(Y,R) ⇐ yi+1∈ V(Y,R) , ¯ yi∈ V(Y,R) ⇐ yi+1∈ V(Y,R) , ¯
πY(S) ⊆ πY(R) V(S,Y) ≤ V(R,Y) ¯
: πX(R S) = πX(R) : ¯
e, t, a, o, …, q, z ¯ R ¯ ¯
( z- , e) ¯ S ¯ ¯ ,
. " ¯ Y
e,t,a ∈ V(Y,R) o ∈ V(Y,R)
¯ 73 ¯ 74
S(Y,Z) R(X,Y) ¯
:¯ V(S,Y) ≤ V(R,Y) R(X,Y) S(Y,Z) U(Z,W) :
1/ V(R,Y) r∈R, s∈S
R(X,Y) S(Y,Z) U(Z,W)
R- T(R)/ V(R,Y) s 
E(T(R S)) = T(S)⋅T(R)/V(R,Y)  T(R)=1000 T(S)=2000 T(U)=5000
V(R,Y) = 20 V(S,Y) = 50
,V(S,Y)>V(R,Y)
E(T(R S)) = T(S)⋅T(R)/V(S,Y) V(S,Z) = 100 V(U,Z) = 500
: ¯ :R(X,Y) S(Y,Z) ,
E(T(R S)) = T(S)⋅T(R)/max{V(S,Y),V(R,Y)} E(T(R S)) = T(S) ⋅ T(R)/max{V(S,Y),V(R,Y)}
= 2,000 × 1,000 / 50 = 40,000
¯ 75 ¯ 76

20. T((R(X,Y) S(Y,Z)) U(Z,W)) ¯ T(R(X,Y) (S(Y,Z)) U(Z,W))) ¯
T((R S) U) T (S U) = T(S) × T(U) / max{V(S, Z), V(U, Z) }
= T (R S) T(U) /max{V (R S, Z), V(U, Z) } = 2,000 × 5,000 / 500 = 20,000
¯ ¯
V (R S, Z) = V(S,Z) = 100 V (S U, Y) = V(S,Y) = 50
V(U, Z) = 500 T(R (S U)) = T(R)×T(S U)/max{V(R,Y),V(S U,Y)}
T(U) = 5,000 = 1,000×20,000/max{20, 50} = 400,000
:¯
T ((R S) U) = 40,000 × 5,000 / max{100, 500} ¯ ,(R S) U=R (S U) ¯
= 400,000
¯ 77 ¯ 78
¯ ¯
A={a1<…<an} R(day, month, weather)
ai∈A .T(πweather(σ month=Jan(R)) πweather( σ month=July(R)))
ni =T(σR(A)=ai(R))
"¯ 31 π weather(σJuly(R)) πweather(σJan(R))
:( )σ . T = 31 × 31 / 4 ≈ 240 ¯ ¯
σ R (A)<x (R ) = ∑
ai ≤ x
ni
Weather Jan July
Snow 10 0
,
. ¯ ¯ ¯ , : Rain 15 1 ¯
. ¯ " , ¯ ¯ Fair 5 10
sunny 1 20
.T = 15 × 1 + 5 × 10 + 1 × 20 = 85
¯ 79 ¯ 80

21. ¯, ¯ ¯
w0 < w1<…< wm , ¯ ¯
ni = T(σwi<R(A)≤w(i+1)(R)) .( ) ¯
:σ
σ R(A)≤ x (R) = ∑n
ai ≤ x
i ?
.
: (percentile) , ,
{x ∈ R : x ≤ a i } i : i ¯ ¯ a1,…,a99 . ¯ R(X,Y) S(Y,Z) U(Z,W) :
= (R S) U) .
T ( R) 100
E(T(R S)) = 40,000 :
m a x {i : x ≤ a i } : ¯ R (S U) .
σ R (A ) ≤ x (R ) = T (R ) E(T(S U)) = 20,000 :
100 '
¯ 81 ¯ 82
Pipeling
,(S U) R - :
R S U ¯ . ¯
.(R )
S U
:pipelining .
U
= R ≠ R
¯ R S S U U S
. ¯
. ¯ pipeline .
. ¯ , ¯
¯ 83 ¯ 84

22. , ¯ :pipeline R(X,Y) S(Y,Z) U(Z,W)
T(R)=1000 T(S)=2000 T(U)=5000
Bushy
E(T(R S)) = T(S)T(R)/max{V(S,Y),V(R,Y)}
= 2,000 × 1,000 / 50 = 40,000
U R
( ) n !n
: R S S U
, •
Estimated cost = 40,000 Estimated cost = 20,000
. ¯¯ ¯ •
¯ 85 ¯ 86
¯ (1) ¯ : ¯
W(A,B) X(B,C) Y(C,D) Z(D,E) . R1=W X
T 100 200 300 400 . R1
: " R2 = R1 Y
V A: 20 B: 50 C: 50 D: 40
; ¯ R1
V B: 60 C: 100 D: 50 E: 100 ; Y
W X Y Z ;R1 Y
.
((W X) Y) Z: : " R2 Z
" ; R2
333 :R1 = W X ; Z
1000 :R2 = (W X) Y ;R2 Z
333 + 1000 = 1333 : .
¯ 87 ¯ 88

23. (2) ¯ : ¯ (3) ¯ : ¯
¯ , ¯ ¯ ,R2 = R1 Y
. "¯ 10 20 ¯ ¯ ¯ .
? . R1
: R1=W X ¯ , 8
. 100/10 W .8/2 = 4
.( ) X ¯ W ¯ . 333/10 = 34 R1
. 20 – 10 - 2 = 8 . 2×300/10 = 60 - Y
. ¯ R1 Y- ,Y- R1 ¯
. 34 + 30 = 64 ¯
100/10 = 10 :W : . 158 :¯"
200/10 = 20 :X , ¯ Y
30 : ¯" . (300/10)/4 = 8
.R1 ¯
¯ 89 ¯ . 10 —
90
(4) ¯ : ¯ (5) ¯ : ¯
. ,R2 Z ¯"
. 10- , 5- R2 30 R1
. 1000/10 = 100 ¯ 158 R2
. 2×400/10 = 80 . 5 Z 100 R2
80 Z
:Z ¯ 140
.R2 , ¯ ¯ . 508
. ¯ ¯ ¯ ¯ , 40/5 = 8 ¯ Z ¯
.R2 , ¯ ¯
. ¯
.Z R2 ¯
. 40 + 100 = 140 ¯" . ¯ ¯ ¯" , :
¯ 91 ¯ 92

24. ¯
¯ ¯
, .( )
.
¯ query optimizer ¯ " ( )
. ¯ ,
( , ¯) , ¯ ¯
(evaluation) .
¯ . ¯
(? ? ) ¯
(? ) ¯ ¯
?pipelining
( )
¯ 93 ¯ 94