Compendium [G.Marciani] - Ingegneria degli Algoritmi

8/12/2019 Compendium [G.Marciani] - Ingegneria degli Algoritmi

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P

pi

P


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Tn

Tn

g(n) O(f(n)) g(n) f(n)

O(f(n)) = {g(n) : c >0 n0 0|g(n) cf(n)n n0}

g(n) (f(n)) g(n) f(n)

(f(n)) = {g(n) : c > 0 n0 0|g(n) cf(n)n n0}

g(n) (f(n)) g(n) f(n)

(f(n)) = {g(n) : c1, c2> 0 n0 0|c1f(n) g(n) c2f(n)n n0}

A

O(f(n)) (f(n)) (f(n)) n r r A n r(n) = O(f(n)) r(n) = (f(n)) r(n) = (f(n)) P O(f(n)) (f(n)) (f(n)) r A P O(f(n)) (f(n)) (f(n)) A P (f(n))

r

O(f(n))

g(n) (f(n)) g(n) O(f(n)) g(n) (f(n))


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log2n n

|I|= O(log n)

T(n)

|I|

c log n |I| c log n


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Tw(n) = max|I|,n{tempo(I)}

Tb(n) = min|I|,n{tempo(I)}

I n P{I}

Tm(n) =

{P{I} tempo(I)}


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n

i

nbi

i f

nbi

logbn

i ai

T(n) =

logb

ni=0

ai f

nbi

n

a

n/b f(n)

n

bi = 1

n = bi

i= logbn

a


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T(n) =

aT

nb

+f(n) se n >1

1 se n= 1

T(n) = (nlogba) f(n) = O(nlogba) > 0 nlogba

f(n)

T(n) = (nlogba log n) f(n) = O(nlogba)

T(n) = (f(n)) f(n) = (nlogba+) > 0af

nb

cf(n)c 0

n

(n) Tamm(n) = O(1)

M(n) = (n)

Tamm(n) = O(1) O(n)


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T(N,A ,r) N A NN

u

v=r (u, v) A

v u (u, v) A

u {v|v= figlio(u)}

r T(N,A ,r) padre(r)

u

u

v {w|padre(w) =padre(v)}


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d

P n M(n) = O(n) P[v] = u (u, v) A v = r =P[v] = NULL

Tw(n) =O(1)

Tw(n) = O(n)

P (n+1) P[0] =NULL P[1] =r P[(d (p 1) + i)] =f iglioi(v) v p d

d n P (n + 1)

v Tw(n) = O(1) v Tw(n) = O(d) v v Tb(n) = O(1)

Tw(n) = O(|P| d)

n


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M(n) = O(n)

NULL

Tw(n) = O(1)

M(n) =O(n) d

M(n) = O(n) d

NULL

S

S=

S

Tw(n) = O(n)

S


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S


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n

Cwt(n) = (n log n)

T nT

T n n! n

f log2f

HT(n) = (log n!) (n log n)

Cw(n) = (HT) = (n log n)

Cm(n) = (HT)

T

n T

n!

2n

ne

n n! 2n

ne

n

1 + 112n 1


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Cw(n) = O(n2)

n

2

= n(n1)2

Cw(n) = (n2)

Cw(n) =

ici

ci

i

ci = n i 1 i [0, n 2]

C(n) =

n2i=0

(n i 1) =n1t=1

t= (n 1)(n 1 + 1)

2 = (n2)


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A n A[0 : n 1]

i

A[i : n1]

A[i] A A[0 :i]

A[i +1 :] i= n 2

Cw(n) = (n2)

Cw(n) =

iterazioni ci

ci

i

ci = n + 1 i [1, n 1]

Cw(n) =

n1i=1

(i+ 1) = (n 1) +n1t=1

t= (n 1)(n 1 + 1)

2 = (n2)


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A n A[0 : n 1] i S[0 : i1] A[pos] A[i] A[i] A[pos 1] A[pos] A[pos:] A

A[0 :i] A[i+ 1 :n 1]

i= n 1

Cw(n) = (n2)

A[0]...[n 1]n O(n)

A

i esima A[0 : n i 1] A[ni1] O(ni2)

i esima A[n i 1 : n 1]

O(n 1)

Cw(n) =

ici

ci

i

ci = n i i [0, n 2]

C(n) =

n2i=0

(n i) = n2 n

2 = (n2)


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S

L

L

v chiave(v)

P P[1] i sin(i) = P[2i] des(i) = P[2i+ 1]

n (log n)

n O(n) O(log n)


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Cw(n) = (n log n)

(n + 1) A[0]...[n]

(n 1)

A[1]

A[c] A[1] (c 1)

Cw(n) = O (t1+ nt2) t1 t2

n

k

t2 = O(h) = O(log n)

h= O(log n)

t1 = C1(n) = 2C1n

1

2

+t2 2C1 n

2

+O(log n) = (n)

C(n) = O(n+n log n) = (n log n)

k 1 n

n = 2k1 1< n


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Cw(n) = (n log n)

Cw(n) =t1+ 2Cn2

t1

t1 = O(n)

l1 l2 (l1+ l2) = n

(n 1)

Cw(n) = O(n) + 2Cwn

2

= (n log n)


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(l1 +l2) = n l1 l2 l1 = l2 =

n2

li = 1

l1 l2 (l1 + l2)

Cw(n) = (n2)

Ce(n) = O(n log n)

n l1 = n2 l2 =

n2

Cdw(n) = t1 + Cdw(a) + C

dw(b) t1

a b

a = 0 b= n

1

t1 = (n 1)

Cdw(n) = n 1 + Cdw(n 1) ni=0

in= O(n2)

a Cre (n) =n1

a=0 p (t1 + Cre (a) + Cre (na 1)) p [0, n 1] t1

p= 1

n

t1 = (n 1)Cre (a) =C

re (n a 1)

Cre (n) =

n1a=0

1

n (n 1 + 2Cre (a)) 2n log n= O(n log n)


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n l1 l2 x

i A[i: inf 1] x A[sup+ 1 :f]> x

n [0, k 1]

Tw(n) = O(n + k) k n

A n

A[0 :n1]

Y k A

Tw(n) = t1 + t2 t1

k

t2 n

t1 = O(k)

t2 = O(n)

Tw(n) = O(n+k)

O(n) Tw(n) = O(n2)

n n

k = O(n)

[0, k 1] k = O(nc) = (n)

n


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i A[i] Y[A[i]] j S[j : Y[A[i]] 1] = i

n [1, k] Tw(n) = O(n+k)

A n A[0 : n 1] [1, k] Y k [1, k] A A

n b =

(n)

[1, k]

k = O(nm

)

Tw(n) =O

n

1 + log klog n

n

Y[i]

Tw(n) =O(n+k)

Tw(n) = t1 + ct2 t1 c [1, k] b t2 k= O(nm) m >1 b= (n)

t1 = O(n)

n

c= O(logbk)

[1, k] b k b

t2 = O(n)

O(n+ k) n [1, k] b [0, b 1] O(n+b) b= (n)

Tw(n) =O(n)+O(lognk)O(n) =O(n(1+lognk)) = O

n

1 +

logck

logcn

= O

n

1 +

log k

log n


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A n A[0 : n 1] [1, k] k= O(nm) A b t t b Y b Y[0 : b 1]

Y A A[0 : n 1] t

i (i1)

i

i


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n k k [1, n] Z

n2 n

O(n log n) k= n2

Tw(n) =O(n+k log n) k1

n

A[0]...[n 1]

(k1) (k1) k k

Tw(n) =O(t1+(k1)t2+t3) t1 t2 (k 1) t3

t1 = O(n)

O(n)

t2 = O(log n)

O(1) +O(1) O(log n) = O(log n)

t3 = O(1)

k 1

k


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Tw(n) = O(n2) Te(n) = O(n)

k

5

Tw(n) = O(t1) + T(n1) t1 n1

t1 = O(n)

O(n)

n1 = (n 1)

Tw(n) =O(n) + T(n 1) T(n) = O(n2)

Te(n) = t1+

i=|partizionamento|PTw(i) t1 P

i

t1 = n 1

(n 1)

P = 1n2

= 2

n

i n2

i n 1 n2

i

P = 1n2

Te(n) = n 1 +n1i=n

2

2

n Tw(i)

Te(n) 4n Te(n) = O(n)

k


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Tw(n) = O(n)

5 k

T(n) = t1+ T(n1) + T(n2) t1 5 n1 n2

t1 6n

5=

6n

5 + 1 =

11n

5

6

n5

n1 = n

5

n5

n2 = 7n

10 + 3

12n5= n

10

n10

1 3

n2 = n 3

n

10 1

=

7n

10 + 3

T(n) = 11n

5 +T

n

5

+T

7n

10 + 3

T(n) 22n= O(n)

T(n) (cn 4c)

T(n) 11n5

+cn

5 + 1

4c+c

7n

10 + 3

4c=

11

5 +

9c

10

n 4c

11

5 +

9c

10

n 4c (cn 4c) c 22

T(n) 22n T(n) = O(n)


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v elem(v)

chiave(v)

chiavi(sotto albero sinistro(v)) chiave(v)

chiavi(sotto albero destro(v)) chiave(v)

T(h) = O(h) v v Tsin(v) Tdes(v)

S

T(h) = O(h)


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c S

S = Tsin(sin(c))

S = Tdes(des(c))

S = Tdes(sin(c))

S= Tsin(des(c))

S

S

n

h(n) = O(log n)

T(n) = O(log n)

T(n) =O(log n)

v

(v) 2

h(n) = O(log n)

Tfib h

h(n) = (log n) nh = 1 + nh1+ nh2 T h n Tfib

h nfib

h(n) = h(nfib) = (log n)

nfib n

h(n) = O(log n)


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T(n) = O(log n)

v (v)2

n T(n) = O(log n)

v

S[v] = max(Tsin(v))

M[v] = max(Tcent(v)) grad(v) = 3

n f h

2h+1 1 n (3h+1 1)

2

2h f3h

h(n) = (log n)

T(n) = O(1) grad(v) > 3 w v v v w S M v w

v

T(n) = O(1) grad(v) < 2 l v grad(l) = 3 l v grad(l) = 2 v l v S M v l

T(n) = (log n) k S[v] Tsin(v) S[v] k M[v] Tcent(v) Tdes(v)

T(n) = (log n)


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T(n) = O(log n) v u k S[u] M[u] v u v S M u u v u u

T(n) =O(log n) v u v v S M u u

u

v

u

u

u

u

n

T(n) = O(log n)

t 2

kr 1 kr 2t 1

kv t 1 kv 2t 1

kv+ 1

ci i v chiavej v ci chiavej ci+1 i [1, kv+ 1], j [1, kv]

(2t 1) (t 1)

2t t2

T(n) = O(log n)


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T(n) = (t) kv 2t t 1 t t

T(n) = (t) kv t 2

T(n) = O(log n) v kj k k kj v kj1 kj v

T(n) = O(t logtn)

v u u v u u

v

u

u

T(n) = O(t logtn) u v

v

u

u

u

T(n) = O(t1

t2) t1

t2

t1 = O(log t)

(t)

t2 = O(logtn)

(t)

h logtn+ 1

2 h= O(logtn)

T(n) = O(logt logtn)

logab = logcb

logc

a

T(n) = O(log n)

T(n) = O(log n)


42/42

u u

t n T(n) =O(t logtn) T(n) = O(log n) T(n) = O(logtn)

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Compendium [G.Marciani] - Ingegneria degli Algoritmi