Algorithm Design Exercises

7/31/2019 Algorithm Design Exercises

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Abstract

1 Verification Conditions

Exercise 1 M 9

M mod 9 M 10

10 9

{ 0 M }

q,r := 0 , M+1 ;

{ Invariant: M+1 = 9q+ r }

do 1 0 < r r,q := (r mod 10) + (r 10) , q+ (r 10)

od ;

r := r1

{ 0 r < 9 M = 9q+ r } .


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M = 1367 M = 25679

q r

2

Exercise 2

M m x X y

{ 0 M }

m , x , y := M , X , 0 ;

{ Invariant: 0 m mx+y = MX

Bound function: m }

do m = 0 if even.m m , x := m2 , 2x

2 odd.m m , x , y := (m1)2 , 2x , yx

fi

od

{ y = MX } .

2

2 Saddleback Search

Exercise 3 f g

i j i j f.i f.j. g

m n f.m = g.n


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Exercise 5 M N 0 M 0 N.

k l

0 k N 0 l M Mk = Nl .

M+N

k l Mk Nl kl

2

3 Periodic Functions

Exercise 6 f

i j i < j fi.0 = fj.0 fi

i f f0.0 = 0 fk+1.0 = f.(fk.0) f

p i fi.0 = fi+p.0

f

p:: m :: i,j : i j : fi.0 = fj.0 m i p \ ji .m\n m n n m

k :: n = km

m i j

fm ::

k :: fk.0 = f2k.0 m k p \ k

i j k 2k

f

f

2


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Exercise 7 N 0 1N

d0 , d1 , . . . , dm1dm , dm+1 , . . . , dn1

14

= 0.25000 . . . 25 0

m = 2 n = 3 17

= 0.142857142857 . . . 142857

m = 0 n = 6nm

17

17

= 0.142857142857 . . .

17

1 3 2 6 . . .

7

1N

1N r 1

r,d := (10r) mod N , (10r) N .

r.i r i d.i

d i r.0 = 1 r.(i+1) =(10 r.i) mod N

d.0 d.(i+1) =(10 r.i) N

1N

r

r.N

k : 0 < k : r.N = r.(N+k) .

1N

2

4 Euclids Algorithm


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m n m\n

m n n m k :: n = km

0

m\0 m

Exercise 8 m n m gcd n

p

p \ (m gcd n) p\m p\n .

gcd

0

M N

{ 0 < M 0 < N }

m,n := M,N ;

{ Invariant: 0 < m 0 < n M gcd N = m gcd n }

do m < n n := nm

2 n < m m := mn

od

{ M gcd N = m } .

gcd

gcd

p q

{ 0 < M 0 < N }

m,n,p,q := M,N,M,N ;


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{ Invariant: 0 < m 0 < n M gcd N = m gcd n }

do m < n n,q := nm , p+q

2 n < m m,p := mn , p+q

od

{ M gcd N = m } .

m n p q

mp np mq nq

M N m lcm n

(m lcm n) (m gcd n) = mn .

M N p q

2

5 Quantifiers

Exercise 10 a N i h

k asc.(i,h,k)

asc.(i,h,k) 0 i i+h k j : i j < i+h : a[i] a[j]

asc.(i,h,k) a h ik a len

len.k = h,i : asc.(i,h,k) : h .

len.k k

a

len.0

len.(k+1) = len.k h : asc.(k+1h , h , k+1) : h

2


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6 Miscellaneous

Exercise 11

3 6 2 0 1 4 0 9 4 ,

4 9 0 4 1 0 2 6 3 .

swap

swap(i,j) i j i j

i j

2

Exercise 12

0 0 2 1 0 5 2 ,

0 2 1 0 5 2

0 0 2 0 1 0 5

2

B N


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R

R

N2 N

2

Exercise 13 V N count

k 0 k N x

count(x,k) = j : 0 j < k V[j] = x : 1 .

count xV k

k x e k x

e

{ 0 N }

k , x , e := 0 , a n y , 0 { x }

; { Invariant:

0 k N ke e

count(x,k) e y : y = x : count(y,k) ke

Bound function: Nk }

do k < N body ; k := k+1

od

{ y : y = x : (count(y,N) > N/2) }

body

k , x , e := 0, any , 0


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2

Exercise 14

5 3 8 1 2 7 0 6 4

9

0 3 2 1 0 5 2

4

mex.k

k A 0 i

A[i]

mex.0

a m k s m

k A a

A k a

k A s a

m ks a m

A

0 3 1 1 0 5 2

k = 2 m s 1 a A

k =4 m 2 s 3 a

0 1 1 3 0 5 2


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swap

a swap(i,j)

A[i] A[j] swap(i,i)

a A

(m, k)m m

k

2

Exercise 15 a

N N a

0 1 . . . N1

5 3 8 1 2 7 0 6 4

9 0 1 . . . 8

i j 0 i < j < N a[j] < a[i]

(2, 7) a[2] 8

a[7] 6 a[7] < a[2] 19

a b

a b[i]

i

5 3 6 1 1 3 0 1 0 .

Ni : r : p

i r p

Ni : 0 i < M : a[0] < a[i] = a[0]


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a M

5

k k

k a[0] a[0]

a0

j : 0 j < M : Ni : 0 i < M : a0[i] = j = 1 .

a0a0

k 0 1

{ 0 M j : 0 j < M : a[j] = a0[j] }CountInversions

{ i : 0 i < M : a[i] = Nj : i j < M :a0[j] < a0[i] }

Mk

Mk

Mka0

i : k i < M : Nj : i j < M : a[j] < a[i] = Nj : i j < M :a0[j] < a0[i] ,

k

a0

i : 0 i < k : a[i] = Nj : i j < M : a0[j] < a0[i] .

k 5

5 3 6 1 1 3 0 2 1

5


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5 3 6 1 1

3 0 2 1 ,

0 1 2 3

7 0 6 4 ,

2

Exercise 18

x, y:: ay = bx2 ,

a b

m, n:: 0 m 0 n .

a b

m

n (bm2)/a

n m(an)/b m

1 n 1

2


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1 M r

q

[(1 0 < r) M+1 = 9q + r 0 r1 < 9 M = 9q+ (r1)] .

(1 0 < r) r1 < 9

10 r

0 r1 1 r

1 r M+1 = 9q+ r .

r q1368 0

144 136

18 150

9 151

8 151

M = 1367

r q25680 0

2568 2568

264 2824

30 2850

3 2853

2 2853

M = 25679

r

r

q


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q M Mq

[0 M M

+1 = 90

+M

+1 1 M

+1]

.

[(10 r) 1 r M+1 = 9q+ r 0 r1 < 9 M = 9q+ (r1)] .

K

[10 r 1 r M+1 = 9q+ r r = K (r mod 10) + (r 10) < K] .

[ 10 r 1 r M+1 = 9q+ r

1 (r mod 10) + (r 10)

M+1 = 9(q + (r 10))+ (r mod 10)+ (r 10)

2

2 m

M x X z

[0 M 0 M MX + 0 = MX] .

[(m = 0) 0 m mx+y = MX y = MX] .

[ m = 0 m = K 0 m mx+y = MX even.m

m2 < K


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[ m = 0 m = K 0 m mx+y = MX odd.m

(m1)2 < K

[ m = 0 0 m mx +y = MX even.m

(m2)2x+y = MX

[ m = 0 0 m mx+y = MX odd.m

((m1)2)2x+y x = MX

[0 m mx+y = MX 0 m] ,

[even.m odd.m] .

yx y+x

2

3 m n

i : 0 i < m : j : 0 j : f.i = g.j

j : 0 j < n : i : 0 i : f.i = g.j

g f.m


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(Mm) + (Nn) M N

f.M = g.N

2

4

i,j : m i < M n j < N : |f[i]g[j]|

= { j = n }

i : m i < M : |f[i] g[n]| i,j : m i < M n+1 j < N : |f[i]g[j]|

= { i : m i < M : |f[i] g[n]|

= { i = m }

|f[m] g[n]| i : m+1 i < M : |f[i] g[n]|

= { f[m] g[n] 0

i m+1 i < M f[m] f[i]

f[m] g[n] f[i] g[n] }

f[m] g[n] }

(f[m] g[n]) i,j : m i < M n+1 j < N : |f[i] g[j]| .

i,j : m i < M n j < N : |f[i]g[j]|

= (g[n] f[m]) i,j : m+1 i < M n j < N : |f[i] g[j]| .

m n

m n

d

|f[0] g[0]|

M N

{ i : 0 i < M : j : i j < M : f[i] f[j]

i : 0 i < N : j : i j < N : g[i] g[j] }

m,n,d := 0,0,|f[0] g[0]| ;{ Invariant:

d i,j : m i < M n j < N : |f[i] g[j]|

= i,j : 0 i < M 0 j < N : |f[i] g[j]| }


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do m = M n = N if f.m


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2 Nl < Mk l := l+1

2 Mk = Nl k , l , count := k+1 , l+1 , count+1

fi

od ;

count := count+1

{ count = S.(0,0) = i,j : 0 i N 0 j M Mi = Nj : 1 } .

(Ml) + (Nk) 0

0 k N 0 l M 1 2

M k N l

d d = Mk Nl

{ 0 M 0 N }k , l , count , d := 0 , 0 , 0 , 0 ;

{ Invariant: }

do k = N l = M if d < 0 k , d := k+1 , d+M

2 d > 0 l , d := l+1 , dN

2 d = 0 k , l , count , d := k+1 , l+1 , count+1 , d+MN

fi

od ;count := count+1

{ count = S.(0,0) = i,j : 0 i N 0 j M Mi = Nj : 1 } .

2

6 fk.0 k

f2k.0

{ f }

f1,f2,k := 0,0,0 ;{ Invariant: f1 = fk.0 f2 = f2k.0 0 k }

do f1 = f2 f1,f2,k := f.f1,f.(f.f2) , k+1

od ;


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p := k

{ p f }

2

7 m n

j :: d.(m+j+1) = d.(m+n+j+1) j :: r.(m+j) = r.(m+n+j) .

d.(i+1) = d.(i+j+1)

= { d }

(10 r.i) N = (10 r.(i+j)) N

= { 0 < N }

(10 r.i) N N = (10 r.(i+

j)) N N= { p = p N N + p mod N }

10 r.i (10 r.i) mod N = 10 r.(i+j) (10 r.(i+j)) mod N

= { }

10(r.i r.(i+j) ) = (10 r.i) mod N (10 r.(i+j)) mod N

= { }

10(r.i r.(i+j)) = r.(i+1) r.(i+j+1) .

j :: d.(m+j+1) = d.(m+n+j+1)

= { i,j := m+j , n }

j :: 10(r.(m+j) r.(m+n+j)) = r.(m+j+1) r.(m+n+j+1)

= {

c

j :: 10 c.j = c.(j+1)

j :: 10j

c.0 = c.j

.

c.j r.(m+j) r.(m+n+j) }j :: 10j(r.m r.(m+n)) = r.(m+j+1) r.(m+n+j+1)

= { r.(m+j+1) r.(m+n+j+1)


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N N

10j(r.m r.(m+n))

r.m r.(m+n) = 0 }

j :: 0 = r.(m+j) r.(m+n+j)

= { }

j :: r.(m+j) = r.(m+n+j) .

N+1 r.0 r.1 . . . r.N 0

N

j,k : 0 j < k N : r.j = r.k .

r

j,k : 0 j < k N : i : 0 i : r.(i+j) =r.(i+k) .

i i Nj

j,k : 0 j < k N : r.N = r.(Nj+k) .

m := kj

m : 0 < m : r.N = r.(N+m) .

N k

r.N = r.(N+k)

i,r := 0,1 ; do i = N i,r := i+1 , (10r) mod N od

{ r = r.N } ;

k , r := 1 , r mod N ;

do r = r k,r := k+1 , (10r ) mod N od

r = r.i

r = r.N r = r.(N+k) j : 0 < j < k : r.N = r.(N+j) .

r = r k

r 1N

2


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8 m+n

[m < n 0 < m 0 < n M gcd N = m gcd n

0 < m 0 < nm M gcd N = m gcd (nm)

[n < m 0 < m 0 < n M gcd N = m gcd n

0 < mn 0 < n M gcd N = (mn) gcd n

[0 < m 0 < n M gcd N = m gcd n (m < n n < m)

M gcd N = m

[m gcd n = m gcd (nm)]

[m gcd m = m] .

(m < n n < m) m = n

p

p \ (m gcd (nm))

= { }

p\m p\(nm)

= { }

k :: m = kp k :: nm = kp

= { }

k :: m = kp j :: nm = jp

= { }

k :: m = kp j :: n kp = jp= { j := jk }

k :: m = kp j :: n = jp

= { }


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k :: m = kp k :: n = kp

= { }

p\m p\n

= { }

p \ (m gcd n) .

m gcd (nm) p m gcd n p

[m gcd n = m gcd (nm)] .

p

p \ (m gcd m)

= { }

p\m p\m= { }

p\m .

m gcd m p m p

[m gcd m = m] .

c , k :: k \ (cM) k \ (cN) k \ (c m) k \ (cn)

gcd

k :: k \ M k \ N k \ (M gcd N)

m n M gcd N

c , k :: k \ (cM) k \ (cN) k \ (c (M gcd N))

cM gcd cN x

k :: k \ (cM) k \ (cN) k \ x

c (M gcd N) cM gcd cN

k \ (c m) k \ (cn) k \ (c (mn)) k \ (cn)

k c m n n < m


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mq +np = 2MN .

a b c d

amp +bmq+ cnp + dnq

amp + bmq+ cnp + dnq

= a(mn)(p+q) +b(mn)q+ cn(p+q) +dnq

a d 0 b c 1

amp + bmq+ cnp + dnq

= amp +bm(p+q) + c(nm)p +dn(p+q)

a d 0 b c 1m = n = M gcd N

true

= { m = n = M gcd N }

(M gcd N) (p+q) = 2MN

= { (m lcm n) (m gcd n) = mn }

(M gcd N) (p+q) = 2 (M lcm N) (M gcd N)

{ }

(p+q)2 = M lcm N .

p q M

N

2

10

len.0

= { len } h,i : asc.(i,h,0) : h

= { asc }

h,i : 0 i i+h 0 j : i j < i+h : a[i] a[j] : h


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= { 0 i i+h 0 0 = i = h

}

h,i : 0 = i = h j : 0 j < 0 : a[i] a[j] : h

= { }

h,i : 0 = i = h: h

= { }

0 .

len.(k+1)

= { len }

h,i : asc.(i , h , k+1) : h

= { asc }

h,i : 0 i i+h k+1 j : i j < i+h : a[i] a[j] : h

= { i+h= k+1

len }

len.k h,i : 0 i i+h= k+1 j : i j < i+h : a[i] a[j] : h

= { }

len.k h : 0 k+1h k+1 j : k+1h j < k+1 : a[i] a[j] : h

= { asc }

len.k h : asc.(k+1h , h , k+1) : h .

2

11 j k

N

{ 0 N i : 0 i < N : a[i] = a0[i] }

j,k := 0 , N1{ Invariant:

0 j < N 0 k < N j = N1k

i : 0 i < j k i < N : a[i] = a0[N1i]


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i : j i k : a[i] = a0[i]

Bound function: kj }

; do j < k swap(j,k) ; j,k := j+1 , k1

od

{ i : 0 i < N : a[i] = a0[N1i] }

j k

j k j k

2

12

R (c, n) c

n c

n

{ 0 < M 0 < N }

c,n,k := 0,AnyValue,0 ;

{ Invariant: c n B[0..k) }

do k < N if c = 0 n = B[k] c := c+1

2 c = 0 n = B[k] c := c1

2 c = 0 c,n := 1,B[k]

fi ;

k := k+1

od

{ c n B[0..N) }

N2

N2

c

n c


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n

2

13 k

0 k N V[k] = x count(x,k) 1

e := e+1 V[k] = xcount(x,k) ke < e ke e

V[k] = x ke = e ke e

count(x,k) ke count(x,k) e ke e

x V[k]

if V[k] = x e := e+1

2 V[k] = x ke < e skip

2 V[k] = x ke = e x , e := V[k] , e+1

fi

inv

count N V k e z

[0 N 0 0 N 00 0 y : y = any : count(y,0) 00] .

[(k < N) inv y :y = x : (count(y,k) > k/2)] .

[ k < N inv V[k] = x

0 k+1 N (k+1)(e+1) e+1

y : y = x : count(y,k) (k+1)(e+1)


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[ k < N inv V[k] = x ke < e

0 k+1 N (k+1)e e

y : y = x : count(y,k) (k+1)e

[ k < N inv V[k] = x ke = e

0 k+1 N (k+1)(e+1) e+1

y : y = V[k] : count(y,k) (k+1)(e+1)

k

[Nk = K N(k+1) < K] .

count

[ 0 N

0 N y : y = any : j : 0 j < 0 V[j] =y : 1 0

y :y = any : j : 0 j < 0 V[j] =y : 1 0

= { 0 j < 0 false }

y :y = any : 0 0

= { 0 0 true true }true

[0 N 0 N true]

true

(k < N) inv { }

k = N ke e y : y = x : count(y,k) ke

= { }


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k = N N/2 e y : y = x : count(y,N) Ne

{ }

y : y = x : count(y,N) NN/2

= { N = N/2+N/2 }

y : y = x : count(y,N) N/2

= { > }

y : y = x : (count(y,N) > N/2)

2

14

mex.k = n : i : 0 i < k : n = A[i] : n

mex.0

= { }

n : i : 0 i < 0 : n = A[i] : n

= { }

n : false : n

= { n }

0 .

{ A }

m,k,done := 0,0,false ;

{ Invariant: 0 k N m = mex.k (done m = mex.N)

Bound function: k }

do done k < N if m = A[k] m := m+1

2 m < A[k] done := true2 m > A[k] skip

fi ;

k := k+1


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od

{ m = mex.N }

{ 0 N }

m,s,k := 0,0,0 ;

{ Invariant:

0 m s k N m = mex.k

i : 0 i < s : a[i] < m

i : s i < k : a[i] > m

Bound function: (m, k) }

do k < N if m = A[k] swap(s,k) ; m,s,k := m+1 , s+1 , s+1

2 m < A[k] k := k+1

2 m > A[k] swap(s,k) ; s,k := s+1 , k+1

fi

od

{ m = mex.N }

2

15

a[m..n) a

m n count.i count0.i

a[i] a0[i]

count.i = Nj : i j < M : a[i] < a[j] ,

count0.i = Nj : i j < M :a0[i] < a0[j] ,

permutation.a[k..M) ,

i : k i < M : count.i = count0.i ,

i : 0 i < k : a[i] = count0.i .


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k := 0

k = M

a[k] = count.k .

k 1

k

i 0 i k

k < i < M a[i] < a[k] k < i < M a[i] > a[k] k < i < M

a[i] = a[k] a[i] 1 i

k < i < M a[i] > a[k]

2 1

2 1

{ 0 M j : 0 j < M : a[j] = a0[j]

permutation.a[0..M) }

k := 0

{ Invariant:

0 k M

Bound function: Mk }


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; do k = M k := k+1

; innerloop

od

{ j : 0 j < M : a[j] = count0.j }

innerloop

j := k+1

; do j < M if a[j] < a[k] skip

2 a[j] > a[k] a[j] := a[j]1

fi

; j := j+1

od

2

18 g

g.m =

bm2

a

1

2

.

g.m m 0 1 2

g.(m+1) > g.m +1 .

{ true }

m,n := 0,0 ;

{ Invariant: n = g.m }

do g.(m+1) g.m +1 plot.(m,n) ;

m,n := m+1 , g .(m+1)od .

g g.(m+1) g.m


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g.(m+1) g.m

= { n = g.m

g.m n

g }b(

m+1)

2

a 1

2

n

= { }b(m+1)2

a

1

2 n

= { }

0 an b(m+1)2 + a

2

= { h = an b(m+1)2 + a

2 }

0 h .

g.(m+1) g.m +1

g.(m+1) g.m + 1

= { }b(m+1)2

a

1

2 n+1

= { }

0 an b(m+1)2 + 3a

2

= { h = an b(m+1)2 + a2

}

0 h+a .

h

{ true }

m , n , h := 0 , 0 , a

2 b ;

{ Invariant:

n = g.m

h = an b(m+1)2 + a

2

(a h < 0 g.(m+1) g.m = 1) }

do 0 h+a plot.(m,n) ;


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if 0 h { g.(m+1) = g.m }

skip

2 (0 h) { g.(m+1) = g.m +1 }

n,h := n+1 , p

fi ;

m,h := m+1 , q

od .

p q p q

h = an b(m+1)2 + a

2 .

k

{ h = k b(m+1)2 }

m,h := m+1 , h b(2m+ 3)

{ h = k b(m+1)2 } .

k

{ h = an + k }

n,h := n+1 , h+a

{ h = an + k } .

p q

{ true }

m , n , h := 0 , 0 , a

2 b ;

{ Invariant:

n = g.m

h = an b(m+1)2 + a

2

((2a) h < 0 g.(m+1) g.m = 1) }do 0 h+a plot.(m,n) ;

if 0 h { g.(m+1) = g.m }

skip


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2 (0 h) { g.(m+1) = g.m +1 }

n,h := n+1 , h+a

fi ;

m,h := m+1 , h b(2m +3)

od .

n

m

f

f.n =

an

b

1

2

.

{ n = g.m }

m := f.n ;

{ Invariant: m = f.n }

do true plot.(m,n) ;

m,n := f.(n+1) , n+1

od .

f.(n+1) f.n

f.(n+1) f.n

= { m = f.n

m

g }a(n+1)

b

1

2

m

= { }a(n+1)

b

1

2 m

= { }a(n+1) b (m2+ m+

1

4)

= { }

0 b (m2+ m) a(n+1) b

4 .


36/36

h

m

{ n = g.m }

m := f.n ;

h := b (m2+ m) a(n+1) b

4 ;

{ Invariant:

m = f.n

h = b (m2+ m) a(n+1) b

4

(f.(n+1) f.n 0 h) }

do true plot.(m,n) ;

if 0 h { f.(n+1) = f.n }

skip

2 (0 h) { f.(n+1) = f.n +1 }

m,h := m+1 , h+ 2b(m+1)

fi ;

n,h := n+1 , h a

od .

bm b m

h

2

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Algorithm Design Exercises