EJNIKO KAI KAPODISTRIAKO PANEPISTHMIO AJHNWN

TMHMA PLHROFORIKHS KAI THLEPIKOINWNIWN

TOMEAS JEWRHTIKHS PLHROFORIKHS

Algrijmoi kai Poluplokthta

B. Zhsimpoulo

Ajna, 2008

- , , , , , , . .

.

Perieqmena

1 11.1 . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 . . . . . . . . . . . . . . . . . . . . 5

1.2.1 . . . . . . . . . . . . . . . . . . 81.2.2 . . . . . . . . . . . . 121.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 212.1 Divide and Conquer . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.1.1 Merge-Sort . . . . . . . . . . . . . . . . . . . . . . . . . . 212.1.2 Binary Search . . . . . . . . . . . . . . . . . . . . . . . . . 252.1.3 Hanoi . . . . . . . . . . . . . . . . . . . . . 262.1.4 Fibonacci . . . . . . . . . . . . . . . . . . . . 282.1.5 Min-Max . . . . . . . . . . . . . . . . . . . 302.1.6 QuickSort . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.2 . . . . . . . . . . 352.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 362.4 The Master Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.4.1 Master . . . . . . . . . . . . . . . . . . . . . 392.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.5 . . . . . . . . . . . . . . . . . . 402.5.1 k . . . . . . . . . . . . . . . . 412.5.2 . . . . . . . . . . . . . . . . . . . . 412.5.3 , . . . . . . . 422.5.4 . . . . 422.5.5 432.5.6 . . . . . 44

3 473.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2.1 . . . . . . . . . . . . . . 493.2.2 . . . . . . . . . . 493.2.3 . . . . . . . . . . . 50

7

8

3.2.4 . . . . . . . . . . . . . . . . . . . . . . 52

3.3 . . . . . . . . . . . . . . . . . . 55

4 (Hashing) 57

4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2 (Direct Address Tables) . . . 57

4.3 (Hash Tables) . . . . . . . . . . . . . . 59

4.3.1 (Collision Resolution byChaining) . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.3.2 . . . . . . . . . . . . 61

4.4 (Hash Functions) . . . . . . . . . . 63

4.4.1 (Division Method) . . . . . . . 63

4.4.2 (Multiplication Method) . 64

4.5 (Open Addressing) . . . . . . . . . . 65

4.5.1 (Linear Probing) . . . . . . . . . . . 67

4.5.2 (Quadratic Probing) . . . . . . . 67

4.5.3 (Double Hashing) . . . . . . . . 67

4.5.4 68

5 71

5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.2.1 . . . . . . . . . . . . . . . . . . . 75

5.2.2 . . . . . . . . . . . . . . . . . . 75

5.2.3 . . . . . . . . . . . . . . . . . . . . . 75

5.3 . . . . . . . . . . . . . . . . . . . . . 78

5.3.1 . . . . . . . . . . . . . . . . . . . . . . . . 78

5.3.2 . . . . . . . . . . . . . . . . . . . . . . . . 79

5.3.3 . . . . . . . . . . . . . . . . . . . . . . . 80

5.3.4 . . . . . . . . . . . . . . . . . . . . . . . . 80

5.3.5 (subgraph) . . . . . . . . . . . . . . . . . . . 80

5.3.6 . . . . . . . . . . . . . . . . . . . . . . 81

5.3.7 (Chordal Graphs) . . . . . . . . . . . . . 81

5.3.8 (trees) . . . . . . . . . . . . . . . . . . . . . . . . 82

5.4 . . . . . . . . . . . . . . . . . . . . . . . 83

5.4.1 . . . . . . . . . . . . . . . . . . . . 83

5.4.2 . . . . . . . . . . . . . . . . . . 83

5.4.3 (deletion) . . . . . . . . . . . . . . . . . . . . . 84

5.5 . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.5.1 . . . . . . . . . . . . . . . . . . . . . . 85

5.5.2 . . . . . . . . . . . . . . . . . . . . . 87

5.5.3 . . . . . . . . . . . . . . . . . . . . . . . 88

9

6 916.1 . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.1.1 . . . . . . . . 916.1.2 . . . . . . . . . . . . . . 926.1.3 . . . . . . . . . . . . . . . 936.1.4 . . . . . . . . . . 95

6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986.3 . . . . . . . . . . . . . . . . . . . . . . . 1006.4 (disjoint sets) . . . . . . . . . . . . . 1026.5 . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.5.1 Dijkstra . . . . . . . . . . . . . . . . . 1086.5.2 Bellman . . . . . . . . . . . . . . . . . 1126.5.3 Bellman . . . 118

6.6 . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.6.1 -

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.6.2 Prim . . . . . . . . . . . . . . . . . . . . . . . 1246.6.3 Prim (nearest neighbour) . . . . . . . . . . . 1276.6.4 Kruskal . . . . . . . . . . . . . . . . . . . . . 129

6.7 (TSP) . . . . . . 133

7 1377.1 . . . . . . . . . . . . . 1377.2 . . . . . . . . . . . . . . . . . . . . . . . . . 1387.3 . . . . . . . . . . . . . . . . . . . 1417.4 (Discrete Knapsack) . . . . . . 1437.5 (Continuous Knapsack) . . . . . 144

8 1458.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1458.2 . . . . . . . . . . 146

8.2.1 (optimal substructure) . . . . . 1468.2.2 (Overlapping subproblems)148

8.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1488.3.1 . . . . . . . . . . 1498.3.2 . . . . . . . . . . . 1528.3.3 . . . . . . . . . . . . . . . . . . 156

9 1619.1 . . . . . . . . . . . . . . . . . . . . 161

10NP- 16710.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16710.2 P NP . . . . . . . . . . . . . . . 16810.3 NP . . . . . . . . . . . . 170

10.3.1 . . . . . . . . . . . . 171

10

10.4 . . . . . . . . . . . . 17210.5 NP . . . . . . . . . . . . . . . . . . . . . . 17210.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17310.7 . . . . . . . . . . . . . . . . . 174

Keflaio 1

Eisagwg

-. .

- . -, , , .

, , - , :

DNA, 100.000 . 3 , .

, . , - , . , , .

-, , - ( , ...). .

, -. , -

1

2 1.

, , -, , . , , .

; ; ; ; - ; - , .

1.1 Anlush Algorjmwn

- . .

. . :

1. , - .

2. . , . ( ), .

3. , , (.., {0, 1}).

4. . , . .

, . , , ( )

1.1. 3

. , 21000 .

, . - , ... - , . .

, ( ) ( ). , () . ( ), ( ).

- . , , , bits , ... :

n S = (a1, a2, . . . , an) x. x S ;

: . . () 1.1.

(S, x)1. i = 12. while i 6= n+ 1 and ai 6= x do3. i = i+ 14. end while5. if i > n6. return 17. else8. return i9. end if

1.1: x S.

4 1.

; x , 3 ( WHILE IF) ( i = 1). x , 5 3 i- 2i+ 1 i + 1 . , x , 2n ( ) n+ 1 .

( - ). ( ).

Dn d Dn , , :

:C(n) = min{(d), d Dn}

:C(n) = max{(d), d Dn}

:Co(n) =

dDn p(d) (d)

, ( ) . ((d)) d Dn p(d).

, - 1, . . . , n {1, 2, . . . , 2n}. ( ) :

Co(n) =1

2n3 +

1

2n5 + . . .+

1

2n(2n+ 1) +

n

2n(2n+ 3)

=1

2n

ni=1

(2i+ 1) +2n+ 3

2=

3n+ 5

2

( , - ) 3 , 2n + 3 1.5n . , . .

, . :

1.2. 5

a = (ai), b = (bi) Rn.

.

(a, b)1. sp = 02. for i = 1 to n do3. sp = sp+ aibi4. end for5. return sp

1.2: - a b.

n , n n .

C(n) = C(n) = Co(n) = n

, , - . .

1.2 Asumptwtik poluplokthta

n (.. ) , A 100n B n2. n:

n :

= 100 , < 100 > 100

, n A B. A B n. , 106 . n =1000 A 100n 106 = 0.1 B n2 106 = 1 , 10 . n = 10000 100 , ...

n . -

6 1.

f(n)

f(n)

f(n)

cg(n)

cg(n)

c1g(n)

c2g(n)

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

f(n) = (g(n))

n0

n0

n0

T T

T

n

n

n

1.3: O,,

1 , (-) . f(n) () g(n), g(n) g(n) :

f(n) = O(g(n)) c, n0 > 0 f(n) cg(n) n > n0

f(n) = (g(n)) c, n0 > 0 f(n) cg(n) n > n0

f(n) = (g(n)) c1, c2, n0 > 0 0 c1g(n) f(n) 1

Jewrhtik, na algrijmo poluplokthta 10100n enai kaltero ap nan n2; Wstsottoie sunartsei poluplokthta den emfanzontai sthn prxh se mh teqnhto algrijmou,

en akmh ki an uprqoun megle stajer, aut me reuna meinontai drastik.

1.2. 7

c2g(n) n > n0. , f(n) = O(g(n)) f(n) =(g(n)), f(n) = (g(n))

O(g(n)), (g(n)), (g(n)) , , f - , g. , f(n) O(g(n)), f(n) (g(n)) f(n) (g(n)) . :f(n) = O(n) f(n) = O(n2) O(n) = O(n2). , 1.3.

f(n) =3n2 100n+ 6:

3n2 100n+ 6 = O(n2) 3n2 100n+ 6 < 3n2 n > 0.

3n2 100n+ 6 = O(n3) 3n2 100n+ 6 < n3 n > 0.

3n2 100n+ 6 6= O(n) c : 3n2 100n+ 6 < cn n > c.

3n2 100n+ 6 = (n2) 3n2 100n+ 6 > 2.99n2 n > 104.

3n2 100n+ 6 = (n) 3n2 100n+ 6 > (101010)n n > 101010 .

3n2 100n+ 6 6= (n3) c : 3n2 100n+ 6 > cn3 n > n0.

3n2100n+6 = (n2) 3n2100n+6 = (n2) 3n2100n+6 =O(n2)

:

f(n) = o(g(n)) c > 0, n0 f(n) < cg(n) n > n0

f(n) = (g(n)) c > 0, n0 f(n) > cg(n) n > n0

o O . - o , f(n) = o(g(n)), . f(n) = O(g(n)) . , O c . o c . n2 + n 1 = O(n2) = o(n3) 6= o(n2). , f(n) = o(g(n)) f(n) g(n).

8 1.

1.2.1 Klsei Poluplokthta

- , , ...

. A,B,C,D - : TA(n) = O(n), TB(n) = O(n2), TC(n) = O(n3) TD(n) = 2n. , , 1.4.

TA(n)

TB(n)TC(n)

TD(n)

T (n)

n

1.4: - 4

n - . : TA(n) < TB(n) < TC(n) < TD(n) D . , D , .

. 107 (10MIPS) A,B,C,D 0.1, 10 1 . - 1.1, D . D , NP - ( ), (-) .

1.2. 9

0.1 sec 10 min 1 day

TA(n) = O(n) 106 6 109 864 109

TB(n) = O(n2) 103 7.7 104 9.3 104

TC(n) = O(n3) 102 1800 9254

TD(n) = O(2n) 20 32 39

1.1: 4 10 MIPS

NP : n n . . , , ;

- - ( n!) , , , 80 107 , 10112

, ! ( NP), -

. :

. i cij j. , . .

, - ( ) , - 20 20 , . , Hungarian , , 1.95 sec 106

. , -

. Moore ( ) 18 . 15 , 1000 -. n , 15 1.2.

10 1.

O(n) n 1000 nO(n2) n 32 nO(n3) n 10 nO(n4) n 6 nO(2n) n n+ 10O(3n) n n+ 8O(n!) n n+ 7

1.2: 1000 .

1 log(n)

logk(n) k n (, f(n) = an+ b, a, b )nlognn2 n3 nk , k ,

. f(x) = aknk + . . .+ a0an 1 < a < 22n an a > 2n! nn

1.3:

- , .

, - , - ( 1.3). () (). - - .

, . -

1.2. 11

, .

O(n) O(n2) - O(n3) ... : , , ...

, (n) (n2), , .

, . , . , (n) .

. , .

, - , ;

, (f(n)) - O(f(n)) ().

f(n) , .

, , . , n logn . (heapsort) (quicksort) - O(n log n) , - . - O(n log n) , . , (.. ) O(n2) .

, . .

, Ann Bnn

12 1.

Cnn:

cij =n

k=1

aikbkj , 1 i, j n

- 1.5.

(A,B)1. for i = 1 to n do2. for j = 1 to n do3. cij = 04. for k = 1 to n do5. cij = cij + aik bkj6. end for7. end for8. end for9. return C

1.5: n.

n3 ( ) O(n3). 1969 Strassen O(n2.81) Cooper-smith & Winograd 1986 O(n2.379) . 3 .

, (n2) , . Coopersmith & Winograd .

1.2.2 Idithte asumptwtikn sumbolismn

f(n), g(n), h(n) - , :

f(n) = (g(n)) g(n) = (h(n)) f(n) = (h(n)), f(n) = O(g(n)) g(n) = O(h(n)) f(n) = O(h(n)), f(n) = (g(n)) g(n) = (h(n)) f(n) = (h(n)), f(n) = o(g(n)) g(n) = o(h(n)) f(n) = o(h(n)),

1.2. 13

f(n) = (g(n)) g(n) = (h(n)) f(n) = (h(n))

f(n) = (f(n)), f(n) = O(f(n)), f(n) = (f(n))

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

f(n) = O(g(n)) g(n) = (f(n)) f(n) = o(g(n)) g(n) = (f(n))

cO(f(n)) = O(f(n)) O(g(n)) +O(g(n)) = O(g(n)) O(g1(n) +O(g2(n)) = O(max{g1(n), g2(n)}) O(g1(n)) O(g2(n)) = O(g1(n) g2(n))

, . , :

1.

limn+

f(n)

g(n)= a 6= 0

f(n) = (g(n))

2.

limn+

f(n)

g(n)= 0

f(n) = O(g(n)) g(n) = (f(n)) ( f(n) 6= (g(n)))3.

limn+

f(n)

g(n)=

g(n) = O(f(n)) f(n) = (g(n)) ( g(n) 6= (f(n)))

14 1.

:

f(n) =(n2 + logn)(n 1)

n+ n2

(n).:

limn+

(n2+logn)(n1)n+n2

n= lim

n+n3 n2 + n logn log n

n3 + n2= 1

f(n) = (n). 2

1.2.3 Asksei

log n 2, logn =log2 n.

1. T1(n) = O(f(n)) T2(n) = O(g(n)) - P1 P2 P . P2 P1 P ;

: , - , :

T (n) = max{T1(n), T2(n)} = max{O(f(n)), O(g(n))}

: T1(n) = O(n2), T2(n) = O(n

3), T3(n) = O(n log n)

T (n) = max{O(n2), O(n3), O(n log n)} = O(n3)

2. O -:

1. 2 logn 4n+ 3n logn2. 2 + 4 + . . .+ 2n

3. 2 + 4 + 8 + . . .+ 2n

. :

1. 2 logn 4n+ 3n logn = O(n logn)2. 2 + 4 + . . .+ 2n = 2(1 + 2 + . . .+ n) = 2

ni=1 i = n(n+ 1) = O(n

2)

3. 2 + 4 + 8 + . . .+ 2n = 2 2n121 = 2 2n 2 = O(2n)

1.2. 15

3. a > 1 f(n) = O(loga(n)). f(n) =O(log n). f(n) = O(loga(n)) , c n n0:f(n) c loga(n). loga(n) = lognlog a f(n) clog a logn f(n) c logn c = clog a n n0. f(n) = O(log n).

4. log(n!) = O(n log n). n! = n(n 1) . . . 2 < n . . . n = nn , log(n!) < n logn, log(n!) = O(n log n).

5. -:

1. for i = 1 to n do2. for j = 1 to i+12 do3. x = x+ 14. end for5. end for

, -. (n ), . :

T (n) =

ni=1

i+ 1

2=

1

2

( ni=1

i+

ni=1

1)=

1

2

(n(n+ 1)2

+ n)= O(n2)

6. 3n 6= O(2n)

limn+

2n

3n= 0

3n = (2n)

7. ( 1.6) (...) .

m n. q, r m = qn + r, q 1, r 0. (m,n)=(n, r),

16 1.

- (m,n Z)1. repeat2. r = m mod n3. if r 6= 0 then4. m = n5. n = r6. end if7. until r = 08. return n

1.6: ... m n.

:

m = q0n+ r0 , 0 r0 < n, q0 = q, r0 = rn = q1r0 + r1 , 0 r1 < r0r0 = q2r1 + r2 , 0 r2 < r1r1 = q3r2 + r3 , 0 r3 < r2

. . . . . .

k . m,n, r0, r1, . . . rk rk = 0 rk1 =(m,n). O(1), O(k).

() ri max then5. max = ai6. else if ai < min then7. min = ai8. end if9. end for

2.8:

2.1. DIVIDE AND CONQUER 31

, 3(n1), for. n 1, (n).

2.9 .

MinMax (A, i, j,min,max)1. if i j 1 then2. if ai < aj then3. max = aj4. min = ai5. else6. max = ai7. min = aj8. end if9. else10. m = (i+ j)/211. MinMax(A, i,m,min1,max1)12. MinMax(A,m, j,min2,max2)13. min = fmin(min1,min2)14. max = fmax(max1,max2)15. end if

2.9:

- :

i j 1 i = j i = j 1

fmax() fmin() () .

:

T (n) =

{2 if n = 2 or n = 1,

2T (n2 ) + 3 if n > 2.

32 2.

(. 2.1):

T (n) = 2T (n

2) + 3

= 2[2T (n

22) + 3] + 3 = 22T (

n

22) + 2 3 + 3

= 22[2T (n

23) + 3] + 2 3 + 3 = 23T ( n

23) + 22 3 + 2 3 + 3

= . . .

= 2k1T (n

2k1) + 2k2 3 + . . .+ 2 3 + 3

n = 2k

T (n) =n

2 T (2) + [2

k1 12 1 ] 3

= n+ 3 2k1 3= n+ 3

n

2 3

= 5n

2 3

-, Fibonacci -.

2.1.6 QuickSort

Quicksort C. A. R. Hoare, (n2) n . , Quicksort , : (nlogn) (nlogn) . , .

QuickSort, MergeSort, Divide-and-Conquer. Divide-and-Conquer 3 A[p . . . r] :

Divide: A[p . . . r] , , A[p . . . q1] A[q+1 . . . r], A[p . . . q1] A[q] A[q + 1 . . . r]. , A[q] A[p . . . r].

Conquer: A[p . . . q 1] A[q + 1 . . . r] QuickSort.

2.1. DIVIDE AND CONQUER 33

Combine: . A[p . . . r] QuickSort.

2.10 QuickSort.

QuickSort (A, p, r)1. if p < r then2. q =Partition(A, p, r)3. QuickSort (A, p, q 1)4. QuickSort (A, q + 1, r)5. end if

2.10: Quicksort

(Partition) (pivot) : , 2.11.

Partition (A, p, r)1. x = A[r]2. i = p 13. for j = p to r 1 do4. if A[j] x5. i = i+ 16. swap(A[i], A[j])7. end if8. end for9. swap(A[i+ 1], A[r])10. return i+ 1

2.11: .

QuickSort . . , QuickSort MergeSort. , QuickSort .

34 2.

n 2 n 1 0 . . (n) .

T (n) = T (n 1) + T (0) + (n)= T (n 1) + (n)= (n2)

, , T (n) = (n2). .

, Partition - n/2, - n/2 n/2 1. QuickSort . T (n) 2T (n/2)+(n), 2 Master Theorem T (n) = O(nlogn).

v (= 1

n). Tn n 2 :

T0 = T1 = 0

Tn = n+ 1 +1

n

nk=1

(Tk1 + Tnk)

. :

Tn = n+ 1 +2

n

nk=1

Tk1 nTn = n2 + n+ 2n

k=1

Tk1

n 1 :

Tn1 = n+2

n 1n1k=1

Tk1 (n 1)Tn1 = n2 n+ 2n1k=1

Tk1

:

nTn = (n+ 1)Tn1 + 2n

2.2. 35

n(n+ 1) :

Tnn+ 1

=Tn1n

+2

n+ 1=

c

3+

nk=3

2

k + 1

:

Tnn+ 1

2n

k=1

1

k 2

n1

1

xdx = 2 lnn

:

Tn = 1.38n logn

2.2 Asumptwtik prosggish merikn ajro-

ismtwn

, . , .

(. 2.12):

q+1p

f(x)dx q

i=p

f(i) qp1

f(x)dx

Hn = 1 +1

2+ . . .+

1

n

n- . f(x) = 1

x

n+12

1

xdx

ni=2

1

i n

1

1

xdx

, n-

ln(n+ 1) ln(2) + 1 Hn ln(n) + 1

Hn = (lnn).

36 2.

f(x)

x

p qp 1 q 1

2.12: .

2.3 Dndra Apfash

. . - . . - . 2.13 -.

n n! , n . h h log2(n!). n! Stirling

n! =(2n)

(ne

)n(1 +

1

12n+ o(

1

n)

)

2.4. THE MASTER THEOREM 37

a1 > a2

a2 > a3

1, 2, 3 a1 > a3

1, 3, 2 3, 1, 2

a1 > a3

2, 1, 3 a2 > a3

2, 3, 1 3, 2, 1

2.13: .

log2(n!) = log2

((2n)

(ne

)n(1 +

1

12n+ o(

1

n)

))=

n log2 n n log2 e+1

2log2 n+

1

2log2 2 + log2

(1 +

1

12n+ o

(1

n

))

h = (n logn). ,

n . n , (n logn) . , . QuickSort , .

2.4 The Master Theorem

Master Master Theorem

38 2.

T (n) = aT (n

b) + f(n)

a 1 b > 1. .

.

lgn+ k = (lg n) + k

lg(n) = log2(n) ( )

lnn = loge(n) ( )

lgk n = (lg n)k ()

lg lg n = lg(lg n) ()

a > 0, b > 0, c > 0 :

a = blogb a

n = 2log2 n

logc(ab) = logc a+ logc b

logb an = n logb a

logb a =logc a

logc b

logb1

a= logb a

logb a =1

loga b

alogb c = clogb a

, - .

f(n) f(n) = O(nk) k.

1 . , a b , a > 1, :

limn

nb

an= 0 nb = o(an).

f(n) f(n) = O(logk n) k.

2.4. THE MASTER THEOREM 39

- . , k b :

limn

logk n

nb= 0 logk(n) = o(nb).

2.4.1 To Master jerhma

T (n) = aT (n

b) + f(n)

a 1 b > 1 f(n) . n

b

, nb n

b n

b.

1. T (n) :

1. f(n) = O(nlogb a) > 0 T (n) = (nlogb a).

2. f(n) = (nlogb a) T (n) = (nlogb a lg n).

3. f(n) = (nlogb a+) > 0 af(nb) cf(n)

c < 1 n T (n) = (f(n)).

f(n) nlogb a. , - . , 1, nlogb a T (n) = (nlogb a). , 3, T (n) = (f(n)). , 2, , T (n) = (nlogb a lg n) = (f(n) lgn).

. , f(n) nlogb a, . f(n) nlogb a

n > 0. , , f(n) nlogb a, af(n

b)

cf(n). .

, - f(n). 1 2 f(n) nlogb a, . , 2 3 f(n) nlogb a,

40 2.

. f(n) , 3, .

2.4.2 Paradegmata

Master .

1. T (n) = 9T (n3 ) + n. a = 9, b = 3 f(n) = n. nlogb a =nlog3 9 = (n2). f(n) = O(nlog3 9) = 1, 1 T (n) = (n2).

2. T (n) = T (2n3 ) + 1.

a = 1, b = 32 f(n) = 1. nlogb a = nlog3/2 1 = n0 =

1. f(n) = (nlogb a) = (1), 2 T (n) = (lgn).

3. T (n) = 3T (n4 ) + n logn. a = 3, b = 4 f(n) = n logn nlogb a = nlog4 3 = O(n0.793). f(n) =(nlog4 3+) 0.2. af(n

b) = 3(n4 ) log(

n4 ) 34n logn =

cf(n) c = 34 . 3 T (n) = (n logn).

4. T (n) = 2T (n2 ) + n logn. a = 2, b = 2 f(n) = n logn. nlogb a = n. n logn > n n logn

n= logn

n . f(n) , n, nlogb a = n 3 .

2.5 Anadromik grammik exissei

1. f . f - T (p), T (p) ...

2. p T (n). :

T (n) = f(T (n 1)) 1 T (n) = f(T (n 1), . . . , T (n k)) k k T (n) = f({T (p); p < n})

2.5. 41

2.5.1 Grammik anadrom txh k

k

T (n) = f(n, T (n 1), . . . , T (n k)) + g(n)

k 1 , f T (i) n k i n 1 g n.: . an - n n 1 1 . (. 2.14)

an = 2an1 + 1 (n 1, k = 1, a0 = 1)

2.14:

2.5.2 Anadrom diamersewn

:

T (n) = aT (n

b) + d(n)

a, b d(n) . .: Mergesort. - Mergesort T (n)

T (n) = 2T (n

2) + n 1 n 2

T (1) = 0

n 1 .

42 2.

2.5.3 Anadrom plrei, grammik poluwnumik

:

T (n) = f(n, T (n 1), T (n 2), . . . , T (0)) + g(n) f T (i) g(n) .: . n

bn =

n1k=0

bkbn1k

b0 = 1

: . Quicksort

Cn = (n+ 1) +2

n

n1k=1

Ck, n 2

C0 = C1 = 0

2.5.4 Paradegmata Grammikn Anadromikn Exissewn

1. , ( n)

an = an1 + 2n, n 1a0 = 1

:

an = an1 + 2n

an1 = an2 + 2n1

an2 = an3 + 2n2

. . .

a2 = a1 + 22

a1 = a0 + 21

:

an = a0 +

ni=1

2i, n 1

an = 1 +2n+1 22 1

an = 2n+1 1

2.5. 43

2. .

:

an = 2an1 + 1 (20)an1 = 2an2 + 1 (21)an2 = 2an3 + 1 (22). . .

a2 = 2a1 + 1 (2n2)a1 = 2a0 + 1 (2n1)

:

an = 2na0 +

n1i=0

2i

an = 2n +

2n 12 1

an = 2n+1 1, n 0

2.5.5 Grammik Anadromik Exissei me stajero

suntelest

k :

un + a1un1 + a2un2 + . . .+ akunk = b(n)

ai .

un + a1un1 + a2un2 + . . .+ akunk = 0

.

k :

un = rn.

:

rk + a1rk1 + . . .+ ak = 0

k r1, r2, . . . , rk - :

un = 1rn1 + 2r

n2 + . . .+ kr

nk

i u1, . . . , uk1.

44 2.

: :

Fn = Fn1 + Fn2, n 2F0 = 0

F1 = 1

: :

Fn Fn1 Fn2 = 0.

un = rn -

:

rn rn1 rn2 = 0rn2(r2 r 1) = 0

r2 r 1 = 0. :

r1 =1 +

5

2

r2 =15

2

:

Fn = 1rn1 + 2r

n2

: F0 = 0 1 = 2 F1 = 1 1r1 + 2r2 = 1 1r1 1r2 = 1 1 =

1r1r2 2 =

1r2r1

1 =1

1+

52 1

5

2

=15

2 = 15

, :

Fn =15[(1 +

5

2)n (1

5

2)n]

2.5.6 Exissei metatrepmene se grammik anadrom

:

T (n) = aT (n

b) + d(n)

2.5. 45

n 2, a, b T (1) = 1. n a n

b, d(n)

. , n = bk

T (bk) = aT (bk1) + d(bk).

tk = T (bk) :

tk = atk1 + d(bk)

t0 = 1. , :

a(n)un = b(n)un1 + c(n).

f(n) =

n1i=1 a(i)ni=1 b(i)

vn = vn1 + f(n)c(n)

vn = a(n)f(n)un -

un =1

a(n)f(n)(a(0)f(0)u0 +

ni=1

f(i)c(i))

tk = ak +

k1j=0

ajd(bkj)

, k = logb n alogb n = nlogb a

T (n) = nlogb a +

(logb n)1j=0

ajd(n

bj).

: (. 2.1.1 - Merge Sort):

T (n) = 2T (n

2) + n 1 n 2

T (1) = 1

46 2.

: a = b = 2 d(n) = n 1

T (n) = n+

k1j=0

2j((n

2j) 1)

= n+ kn (2k 1) k = log2 n

T (n) = n log2 n+ 1.

sn.

anTn = bnTn1 + cn.

:snanTn = snbnTn1 + sncn.

sn snbn = sn1an1 un = snanTn

un = un1 + sncn

un = s0a0T0 +

nk=1

skck = s1b1T0 +

nk=1

skck

Tn =unsnan

=s1b1T0 +

nk=1 skck

snan

Tn =1

snan(s1b1T0 +n

k=1 skck).

: (. 2.1.3 Hanoi):

Tn = 2Tn1 + 1

: un = 1, bn = 2 sn = 2n

Tn =1

2n(0 +n

k=1 skck)= 2n(s1 + . . .+ sn)

= 2n 1

Keflaio 3

Swro

3.1 Dndra

( ), . . . , . 0. . .

. - , ( 0) (20), ( 1) 2 (21) ... i 2i .

, , ( 2i). :

, - , .

: i/2, i, 2i i 2i+1. m n

log2m n m 1

47

48 3.

1

2

2

3

3

4

4 5 6

7

78

8

9

9

10

10

10

14

16 20

21

22

3.1:

.

3.2 H dom swro

: - . . . :

. , . , , .

( (heap) ). n . :

3.2. 49

1

1

2

2

2

2

3

3

3

3

4

4

4

4

5

5

6

6

7

7

7

7

8

8

8

8

9

9

9

9

10 10

10

10

10

10

14

14

16

16

ia[i]

7 = 2 3 + 1

3.2:

3.2.

, O(log n).

3.2.1 Ulopohsh me pnaka en swro

: . 3.2.

3.2.2 Eisagwg en nou stoiqeou sto swr

, . , . . - 3.3

( ), , ,

50 3.

- (u : )1. n = n+ 12. k = n, a[k] = u3. while a[k/2] < a[k]4. swap (a[k], a[k/2])5. k = k/26. end while

3.3: PSfrag replaement

1

2

2

3

3

4

4 5 6

7

78

8

9

9

10

10

10

14

16

PSfrag replaement

1

2

2

3

34

4

5 6 7

78

8

9

9

10

10

10

11

14

15

16

PSfrag replaement

1

2

2

3

3

4

4 5 6

7

78

8

9

9

10

1010

11

14

15

16

PSfrag replaement

1

2

2

3

34

4

5 6 7

78

8

9

9

1010

10

11

14

15

16

3.4:

O(log n). 3.4 10 .

3.2.3 Diagraf en stoiqeou ap to swr

1 , . , . , ,

3.2. 51

3.5.

- ( )1. a[1] = a[n]2. n = n 1, i = 13. do4. l = 2 i, r = 2 i+ 15. if l n and a[l] > a[i] then max = l6. else max = i7. if r n and a[r] > a[max] then max = r8. if i 6= max then swap (a[i], a[max]), i = max9. else break10. while (i < n)

3.5: PSfrag replaement

1

2

2

3

34

4

5 6 7

78

8

9

9

1010

10

11

14

15

16

PSfrag replaement

1

2

2

3

34

4

5 6 7

78

8

9

9

10

10

10

14

15

PSfrag replaement

1

2

2

3

34

4

5 6 7

78

8

9

9

10

1010

14

15

PSfrag replaement

1

2

2

3

34

4

5 6 7

78

8

9

9

10

10

10

14

15

3.6:

( ), , , O(log n). 3.6 11 .

52 3.

3.2.4 Kataskeu swro

. -. 1, 2, 3, 4. :

1

1 1

1 11

11

2 2 2

2

2

2

2

3

3

3

3

3

4

4

4

1 2 3

4

3.7:

, . .

- ( a n )1. for i = 1 to n do2. - (a[i])3. end for

3.8:

( ), , i, O(log i) . n, :

3.2. 53

O(log 1) +O(log 2) + . . .+O(log n) = O(n logn)

- . , a , , . , - .

- ( a n , i)1. l = 2 i2. r = 2 i+ 13. if l n and a[l] > a[i]4. max = l5. else6. max = i7. if r n and a[r] > a[max]8. max = r9. if i 6= max then10. swap (a[i], a[max])11. - (a, max)

3.9: . , .

- ( a n )1. for i = n/2 to 1 do2. - (a, i)3. end for

3.10:

- . ( , n/2 1) - 3.10.

54 3.

3.11 , [5, 6, 1, 4, 8, 7, 3, 2, 9],

- O(h) h, -

log nh=0

n2h+1

O(h) = O

(n

log nh=0

h

2h

)

k=0

kxk =x

(1 x)2

x = 1/2 k=0

k

2k= 2

O(n 2) =O(n).

1

1

2

2

344

5

5

6

6

77

8

8

9

9

9

1

1

1

2

2

3

4

4

5

5

6

67

78

8

9

9

9

1

1

2

2

3

4

4

5

5

6

67

78

8

9

9

9

2

1

1

2

2

3

4

4

5

5

6

6

7

78

8

9

9

9

3.3. 55

1

1

2

2

34

4

5

5

6

6

7

7

8

8

9

9

9

3

1

1

2

2

3

4

4

5

56

6

7

78

8

99

9

1

1

2

2

34

4

5

5

66

7

7

8

8

9

99

4

1

1

2

2

3

4

4

5

56

6

7

78

8

9

9

9

3.11:

3.3 Taxinmhsh me th bojeia swro

, . , . . 3.12

HeapSort ( a n )1. - (a)2. for i = 1 to n3. a[i] = -( )4. -( )

3.12: . -( ) .

56 3.

: 6, 9, 2, 7, 4, 5, 8.

: (. 3.13).

24 56

7 8

9

3.13:

- . 3.14 .

O(n) O(n log n), i- O(log i) . O(n log n).

PSfrag replaement

24 56

7 8

9

PSfrag replaement

24

5

6

7

8

PSfrag replaement

2 4

56

7

PSfrag replaement

2

4 5

6

PSfrag replaement

24

5

PSfrag replaement

2

4

PSfrag replaement

2

3.14: - .

Keflaio 4

Katakermatism

(Hashing)

4.1 Eisagwg

- INSERT (), SEARCH () DELETE() . , - , , , - . -. (Hash table) . (n), . -, O(1).

4.2 Pnake 'Amesh Dieujunsiodthsh (Direct

Address Tables)

U = {0, 1, . . . ,m1} m . , . 4.1 .

57

58 4. (HASHING)

0

1

2

3

4

5

6

7

8

9

k0

k1 k2

k2k3

k3

k4k5

k5

k6k7

k8k8

k9

U

T

4.1: - . U = {0, . . . , 9} . K = {2, 3, 5, 8} - . NIL.

, , T [0, . . . ,m 1]. U . T [k] = NIL, k k.

O(1).DIRECT-ADDRESS-SEARCH(T,k)

return T[k]

DIRECT-ADDRESS-INSERT(T,x)

T[key[x]]x

DIRECT-ADDRESS-DELETE(T,x)

T[key[x]]NIL

4.3. (HASH TABLES) 59

0

1 = h(k4)

2 = h(k1)

3

4

5 = h(k2) = h(k5)

6

m 2 = h(k3)m 1

k1

k2

k3

k4

k5

U

T

4.2: h T . k2 k5 , .

4.3 Pnake Katakermatismo (Hash Tables)

. U T |U | . K U .

, , (|K|) , O(1). , .

k h(k) T . h : U {0, 1, . . . ,m 1} (hash function) 4.2 |K| m .

. (collision). h ., |K| > m, .

60 4. (HASHING)

k1

k1

k2

k2

k3 k3

k4

k4

k5k5

k6k6

k7 k7

k8

k8

U

T

4.3: . - T [j] j.

- (collision resolution by chaining), , (open addressing) ( 4.5).

4.3.1 Eplush Sugkrosewn me Lste (Collision Res-olution by Chaining)

4.3 , , .

.CHAINED-HASH-SEARCH(T,k)

k T[h(k)]

CHAINED-HASH-INSERT(T,x)

x T[h(key[x])]

4.3. (HASH TABLES) 61

CHAINED-HASH-DELETE(T,x)

x T[h(key[x])]

4.3.2 Anlush Katakermatismo me Lste

, O(1) ( ). , , O(1). x , , , .

, n m , (n). , h T .

m , (Simple Uniform Hash-ing). .

(load factor) a T a = n/m. j = 0, 1, . . . ,m1, T [j] nj

n = n0 + n1 + . . .+ nm1, (4.1)

nj E[nj ] = a = n/m. h O(1).

k nh(k). . , k .

2. - , (1 + a), .

. T [h(k)], E[nh(k)] = a, . - a , h(k), (1 + a). 2

, . , . (1 + a).

62 4. (HASHING)

3. - , (1 + a), .

. n . - x x . x x, - . n x , 1 x . xi i- - , i = 1, 2, . . . , n ki = key[xi]. ki kj , (indicator) Xij = I{h(ki) = h(kj)}. Pr{h(ki) =h(kj)} = 1/m E[Xij ] = 1/m.

E

1n

ni=1

1 + n

j=i+1

Xij

==1

n

ni=1

1 + n

j=i+1

E [Xij ]

=1

n

ni=1

1 + n

j=i+1

1

m

=1 +1

nm

ni=1

(n i)

=1 +1

nm

(ni=1

nni=1

i

)

=1 +1

nm

(n2 n(n+ 1)

2

)=1 +

n 12m

=1 +a

2 a2n

.

, h (2 + a/2 a/2n) = (1 + a). 2

n = O(m) a = n/m = O(m)/m =O(1). .

4.4. (HASH FUNCTIONS) 63

4.4 Sunartsei Katakermatismo (Hash Func-tions)

- . h. - , () , .

h - m . , .

- k, 0 k < 1. h(k) = km -.

, , - - .

. , , {N} = {0, 1, 2 . . .} . . - . , (112,116), ASCII p = 112 t = 116. 128 (112 128) + 116 = 14452. .

4.4.1 H Mjodo th Diaresh (Division Method)

h(k) = k mod m.

m = 4 k = 100 h(k) = 4. m 2, 2 h(k) p k.

. n = 2000 3

64 4. (HASHING)

m = 701 , 2000/3 2.

, , [0, . . . , N1], .

h(x) = (x1Bl1 + x2Bl2 + . . .+ xl) mod N,

N , l - x B 2 (128, 256 ). ..B = N = 256 h(x) = x1 .

h (x : string, l : integer)1. r = xl, p = 12. for i = l 1 to 1 do3. p = p B4. r = r + xi p5. end for6. r = r mod N7. return r

4.4.2 H Pollaplasiastik Mjodo (Multiplication Method)

. - k A 0 < A < 1 kA. m .

h(k) = m(kA mod 1).

m . 2, m = 2p, p N, . w bits k . A s/2w, s 0 < s < 2w. 4.4 k w- s = A 2w. 2w- r12w+r0, r1 r0 . p- p r0. A , (

5 1)/2.

4.5. (OPEN ADDRESSING) 65

r1 r0

A 2w

k

w bits

h(k) p bits

4.4: . w- - w- s = A 2w. p w- h(k).

4.5 Anoiqt Dieujunsiodthsh (Open Address-ing)

. -. , , . . - a, , 1.

, , (probe), . . , - ( 0) .

h : U {0, 1, . . . ,m 1} {0, 1, . . . ,m 1}.

k, (probe sequence) h(k, 0), h(k, 1), . . . , h(k,m1) 0, 1, . . . ,m1, .

66 4. (HASHING)

. .

HashInsert (T, k)1. i = 02. repeat3. j = h(k, i)4. if T [j] = NIL then5. T [j] = k6. return j7. else8. i = i+ 19. until i = m10. error hash table overflow

k . NIL, k . .

HashSearch (T, k)1. i = 02. repeat3. j = h(k, i)4. if T [j] = k then5. return j6. else7. i = i+ 18. until i = m or T [j] = NIL9. return NIL

, DELETED - . a, - .

- (Uniform Hashing), -. - m! 0, 1, . . . ,m 1 .

4.5. (OPEN ADDRESSING) 67

.

-: . 0, 1, . . . ,m 1 - , m2 m! . - .

4.5.1 Grammik Dierenhsh (Linear Probing)

h : U {0, 1, . . . ,m 1}, , i = 0, 1, . . . ,m 1

h(k, i) = (h(k) + i) mod m.

, , - (primary clustering). -, - . , i , (i+ 1)/m.

4.5.2 Tetragwnik Dierenhsh (Quadratic Probing)

h(k, i) = (h(k) + c1i+ c2i2) mod m,

h , c1, c2 6= 0 i = 0, 1, . . . ,m 1. , c1, c2 m. - , h(k1, 0) = h(k2, 0), . , (secondary clustering), , m .

4.5.3 Dipl Katakermatism (Double Hashing)

- .

h(k, i) = (h1(k) + ih2(k)) mod m,

68 4. (HASHING)

0 1 2 3 4 5 6 7 8 9 10 11 12

14 506979 7298

4.5: . 13 h1(k) = k mod 13, h2(k) = 1 + (k mod 11). 14 1(mod 13) 14 3( 11), 14 9, 1 5.

h1 h2 . 4.5 .

h2(k) m . m 2 h2 . m h2 m. , m

h1(k) = k mod m,

h2(k) = 1 + (k mod m),

m m (.. m 1). k = 123456,m = 701 m = 700, h1(k) = 80 h2(k) = 257. 80 257 (modulom) .

- (m2) ( (h1(k), h2(k))) (m) - .

4.5.4 Anlush tou Katakermatismo Anoiqt Dieu-

junsiodthsh

, , a = n/m n m . a 1 - . .

4.5. (OPEN ADDRESSING) 69

.

4. - a = n/m < 1, 1/(1 a), - .

. , , , . X Ai, i = 1, 2, . . ., i- . {X i} A1A2 Ai1. Pr{X i} Pr{A1A2 Ai1}.

Pr{A1 A2 Ai1} == Pr{A1} Pr{A2|A1} Pr{A3|A1 A2} Pr{Ai1|A1 A2 Ai2}. n m Pr{A1} = n/m. j >

1, j- , j 1 (n j + 1)/(m j + 1). (n (j 1)) (m (j 1)) , , . n < m (n j)/(m j) < n/m j 0 j < m, i 1 i m,

Pr{X 1} = nm n 1m 1

n i+ 2m i+ 2

( nm

)i1= ai1.

E[X ] =i=1

Pr{X i} i=1

ai1 =i=0

ai =1

1 a.

2

1 + a + a2 + a3 + . , - . a . a2 ...

a , O(1). ,

70 4. (HASHING)

1/(1 0.5) = 2, 90% 1/(1 0.9) = 10.

4 .

1. - a 1/(1a) , .

. , a < 1. . 1/(1 a).2

.

5. - a < 1,

1

aln

1

1 a ,

.

. k k . - k (i+1)- , k 1/(1 i/m) = m/(m i). n :

1

n

n1i=0

m

m i =m

n

n1i=0

1

m i =1

a(Hm Hmn),

Hi =i

j=1 1/j i- .

1

a(Hm Hmn) = 1

a

mk=mn+1

1

k 1a

mmn

1

xdx =

1

aln

m

m n =1

aln

1

1 a .2

Keflaio 5

Grfoi

5.1 Genik

, - , , . . - , . .

:

.

1 ( - ). a, b, c D, E,

71

72 5.

F . : ( ) 3 , 3 , ;

a

b

c

D E F

5.1: 3 3

5.1 8 () . . . .

2 ( ). A,B,C - (, ) M N . : A = 4, B = 7, C = 6, : M = 8 N = 9.

- . : - , .

5.1. 73

2

3

3

4

4

5

5

6

7

8

9

A

B

C

M

N

5.2:

3 ( ). , 5.3.

(. 5.2). - . - . - .

. : :

7MBps pc 1 pc 4 6MBps pc 3 pc 5 10MBps pc 2 pc 3 5MBps pc 4 pc 1

;

.

74 5.

5.3:

. .

5.2 Orismo

(nodes) (edges). , G = (V,E), V E .

1. G = (V,E) V E V V , G.

n = |V | m = |E|. 2. n .

3. (G) = |E|n2

4. . . .

5.2. 75

5.2.1 Kateujunmenoi grfoi

5. (directed graph) G = (X,U), X U () , - ( ).

. (i, j) i j, j i.

(i, j) i j. i (tail) j (head). (i, j) i j. (i, j) i j.

6. (indegree) di i . (outdegree) d+i i . (degree) di i i (di=d

+i +d

i ).

5.2.2 Mh kateujunmenoi grfoi

7. (undirected graph) G = (V,E), V E , .

, .

( -) , - . , .

8. (degree) di i .

9. ( ) , e , w(e) . - . .

5.2.3 Kkloi kai monoptia

= (x, y) x y . : () = x () = y, .

76 5.

10. G = (X,U) X = (x1, x2, ..., xn) U =(u1, u2, ..., um). (path) f G x1, u1,x2, u2, . . . , xk (ui = [xi, xi+1]), . x1 xk. k 1, k -. p, p .

:

11. f G = (X,U) 1, 2, . . . , p : i, 1 i < p (i+1) = (i).

. .

4 ().

1

2

3

4

5

6

a1

a2a3

a4

a5

a6

a7

a8

a9

5.4: 6 9

5.4 :

: = 1 7 8 9 6 5 8

: = 1 7 8 9 6

5.2. 77

: = 1 3 4 5

f , (f) 1 , (f) p.

(cycle) , (x1 = xk) () = (). (simple cycle) . .

5 ( - - ).

1

2

3

4

5

a1 a2

a3

a4a5

a6

5.5: 5

5.5 :

: = 6 1 4 3 2 56 1 5

: = 6 1 4 3 2 5

: = 1 5 6

De Bruijn

De Bruijn 0 1. f g:

78 5.

f = xh

g = hy x y 0 1 h 1 (. 5.6).

000

001

010

011

100

101

110

111

5.6: De Bruijn = 3

{2, 3, . . . , n}. p q p q (. 5.7).

5.3 Eidik Kathgore Grfwn

5.3.1 Plrei Grfoi

12. n (Kn) , .

5.3. 79

2

3

4 6

7

8

89

10

11

12

5.7:

K3 K4

5.8: 3 4

5.8 3 4 .

5.3.2 Dimere Grfoi

13. (bipartite) G = (V,E) (V1,V2) V1 V2 = , V1 V2 = V V1 V2 (. 5.9).

80 5.

x1

x2

y1

y2

y3

5.9:

14. G(V,E) . V1 V2, v V1, d(v) = |V2| ( Ka,b).

5.10 K2,3 K4,4.

5.3.3 Kanoniko Grfoi

15. - G =(V,E) , . v V d(v) = .

5.3.4 Eppedoi grfoi

16. - , . 5.1 K3,3 .

5.3.5 Upogrfoi (subgraph)

G = (X,U). G G = (X , U ), X X U U .

5.3. 81

u1

u2

u3

u4

u5

K2,3 K4,4

5.10:

G = (X,U) G = (X,U ), U U . G G .

5.11 G

G G

G.

5.3.6 Sunektiko grfoi

i, j i j.

17. (connected) - , .

, - .

5.3.7 Qordiko Grfoi (Chordal Graphs)

. .

G = (V,E) (chordal) G, 3 .

18. G 3 .

82 5.

x1x1

x1

x2

x2 x2

x3

x3x3

G = (X,U)

G = (X,U ) G = (X,U )

5.11: G

G G

G

5.3.8 Dntra (trees)

19. G .

G . . .

(root) . 1 , . 1 .

6. T n , -

1. T

5.4. 83

2. T

3. T n 1 4. T n 1 5. T e T+e

6. T

5.4 Prxei epi twn grfwn

5.4.1 Isomorfism grfwn

G1 = (V1, E1) G2 = (V2, E2) f : V1 V2 [f(v1), f(v2)] E2 (v1, v2) E1. v1, v2 G1 f(v1), f(v2) G2. , . 5.12 . 5.13 3- . 3- 3- 3- .

AA B

B

C

CX

X

Y

Y

Z Z

5.12:

5.4.2 Sumplrwma en Grfou

G = (V,E) G = (V , E) =(V, V 2 E) V = V , E E. G

84 5.

v1

v1

v1

v2

v2

v2

v3

v3

v3

v4

v4

v4

v5

v5

v5

v6

v6

v6

v7

v7

v7

v8

v8

v8

5.13:

G. G G .

5.4.3 Diagraf (deletion)

G = (V,E) v V . G v G = (V v,E adj(v)), adj(v) - v. G G v .

G = (V,E) e E. e G G =(V,E e). G G

5.5. 85

e.

5.5 Apojkeush Grfwn

. .

5.5.1 Pnaka Geitnash

G = (V,E), n = |V |,m = |E|. M n n , M G. M(i, j) 1 vi vj 0.

M(i, j) =

{1 (vi, vj) E0 (vi, vj) / E

(adjacency matrix) M(i, j) = M(j, i) n2 M(i, i) = 0. G = (V,E) M G = (V,E) ( ), . G (G = (V,E,W )) wij W (vi, vj) :

M(i, j) =

{wij (vi, vj) E0 (vi, vj) / E

6 ( ). G =(X,U), |X | = 6, |U | = 9 5.14. .

1 2

3

4

5

6

M =

0 1 1 0 0 00 0 1 1 0 10 0 0 0 0 10 0 0 0 1 00 1 0 0 0 00 0 0 1 0 0

5.14: M

86 5.

7 ( ). - G(X,U,W ), |X | = 5, |U | = 8, 5.15. .

1

2

3 4 51

2

37

11

13

19

25 M =

0 25 13 1 07 0 19 0 00 0 0 0 00 11 0 2 00 0 0 3 0

5.15: M

G, M . Mp, p- M , Mp(i, j) - G, p, xi xj . xi, xj . M p, 1 p n ( M Mp1) (i, j) Mp. O(n3) (O(n2.81) - Strassen) n , O(n4). 8 ( ).

G(X,U), |X | = 4, |U | = 4, 5.16. M M M2.

M2(1, 3) = 2 ( ) 1 3.

n2

bits . (O(1)). O(n) , 1. - . , , n2 bit-s. (.. 3)

5.5. 87

M =

0 1 0 10 0 1 00 0 0 00 0 1 0

1

2

3

4

M2 =

0 0 2 00 0 0 00 0 0 00 0 0 0

5.16: , M M2

0.

xi xj , , j, O(n). - xi xj j, O(n).

5.5.2 Pnaka Prsptwsh

G = (V,E), n = |V | m = |E|. B n m , B G B G. B(i, j) 1 ej vi 0.

B(i, j) =

{1 ej vj)0

(incidence matrix) - nm .

9 ( ). G = (V,E), |V | = 6, |E| = 9, 5.17.

88 5.

v1

v2

v3v4 v5

e1

e2e3

e4e5

e6

5.17: e1, e2, . . . , e6

:

B =

e1 e2 e3 e4 e5 e6v1 0 0 1 1 1 0v2 1 0 1 0 0 1v3 0 0 0 0 1 1v4 1 1 0 1 0 0v5 0 1 0 0 0 0

O(nm) bits . () . .

5.5.3 Lste Geitnash

G = (V,E), n = |V |,m = |E|, 1 . , . :

1. . .

1

Epigraf enai mia tim sunjw arijmhtik pou prosdiorzei monadik kje kmbo sto

grfo

5.5. 89

2. . .

10 ( ). G =(X,U), |X | = 6, |U | = 9 5.18. .

1

2

3

4

5

6

M =

0 1 1 0 0 00 0 1 1 0 10 0 0 0 0 10 0 0 0 1 00 1 0 0 0 00 0 0 1 0 0

5.18: M

5.19 5.20.

- . G 20000 5, - 20000 . 6 .

(n+m) bits (n+2m) . O(m) . .

90 5.

1

2

2

2 3

3

3

4

4

4 5

5

6

6

6

5.19:

0

0

1

1

1

1

22

2

2

2

33

3

3

3

3

44

4

4

4

5

5

5

5

66

6

6

6

6

7

7

7

7

8

8

8

9

9

9

10

10

10

w13w24w23w26w36w45w52w64w12

head

succ

weight

5.20:

Keflaio 6

Algrijmoi Grfwn

6.1 Exerenhsh Grfwn

( ) , .

1. , - s.

2. , - . t

3. .

4. .

5. .

6. (bipartite) .

- -. . (d(v)), d(v) v. (d+(v))

6.1.1 Genik morf en algorjmou exerenhsh

G = (V,E) s V , (source). s, G s.

91

92 6.

(. 6.1) s , 1 . - (marked) (unmarked). s - . i , j (i, j) j ( j s i (i, j)). (i, j) (admissible) (inadmissible). s . j (i, j) i (predecessor) j.

mark , List order .

Search (G, s)1. mark false2. Mark[s] = True3. List = {s}4. List 6= 5. i List6. i 7. List = List {i}8. 9. j i10. j 11. Mark[j] = True12. List = List {j}

6.1: G s

6.1.2 Exerenhsh Prta Kat Plto

List , List . (. 6.2) - .

1

Didoqo tou s lgetai kje kmbo pou brsketai se monopti me arq ton kmbo s

6.1. 93

List. - . s 1 , 2 .... (Breadth-First Search). :

ni=1

(d+(i) + 1

)=

ni=1

(d+(i)

)+ n = n+m.

(m + n). m > n - (m).

Breadth First Search (G, s)1. mark false2. Mark[s] = True3. s List4. List 6= 5. i List6. j i7. Mark[j] = True8. j List

6.2: Breadth First Search s

11 ( ). 6.5 - 6.3. a, (. 6.4). a . b, d , . b . 6.4. , , s = a (. 6.5).

6.1.3 Exerenhsh Prta Kat Bjo

List , List . - .

94 6.

a

b

c

d

e

f g

h

6.3: 8

a

a

b

bc

c

c

d

d

d

e

ef

fg

g

g

h

h

h

6.4:

s . (Depth First Search) 6.6. () (m+ n).

12 ( ). 6.8 6.3. a - . (

6.1. 95

g

a

b

d

c e

f

h

6.5:

) ( b) . b c . d c d . d . . d c, . . 6.7.

, (. 6.8).

6.1.4 Efarmog twn algorjmwn exerenhsh

, , father n. father[i] = 0, i [1, n] father[s] = s. fathers[j] = i - , i, j, - ( ). father . - (father[n]) (father[1]), .

96 6.

Depth First Search (G, s)1. mark false2. Mark[s] = True3. s List4. List 6= 5. i List6. i 7. i List8. 9. j i10. j 11. Mark[j] = True12. j List

6.6: Depth First Search

a a a a a a a aa

b b b b b b b b

c c c c c c c

d e e e e e

f f f f

g h h

6.7:

G = (V,E) s V . distance n, distance[i] = , i [1, n], distance[s] = 0. distance[j] = distance[i] + 1

6.1. 97

a

b

c

d

e

f g

h

6.8:

j i. , .

(s, v) s v. s v (s, v) = . .

1. G = (V,E) s V . (u, v) E

(s, v) (s, u) + 1.

. u s v. s v s u (u, v), . u s (s, u) = .

7. G = (V,E) Breadth First Search (BFS) s V . BFS v V s distance[v] = (s, v) v V . v 6= s s s v s father[v] (father[v], v).

.

98 6.

() . j i, .

: , G = (V,E).

6.2 Sunektikthta

20. () ( block). .

, - , , - . (. 6.9), (. 6.10) (. 6.11).

a

b

c

d

6.9:

21. (u, v) u v v u. , u v ().

(.. 6.9).

6.2. 99

a

b

c

de

f

g

h

6.10: 2

a

b c

d e

f

g

6.11:

, p > 1 - , , - . 6.12 .

. 6.13 -, , - . false. - . -. - . (m+ n) ().

100 6.

1

2

3

45

6.12:

Breadth First Graph Traversal1. mark false2. i = 1 i = n3. mark[vi] = false BFS(vi)

6.13:

6.3 Topologik Taxinmhsh

X , v X u X . (schedule) , .

6.3. 101

G = (X,U) X (u, v) U u , v. , G , , . 6.14 .

v1v1

v2

v2

v3

v3v4

v4

v5

v5

1 2 3 4

....

6.14: .

( ), - , , . .

Layer , - Sorted. 6.14 Layer [1, 3, 2, 4, 2], Sorted [v1, v3, v5, v2, v4].

, , :

(u, v) U Layer[u] < Layer[v], .

v , u 1(v) :Layer[u] Layer[v] 1.

: Q, .

102 6.

, Q s , G. 6.15.

(G = (X,U))1. inDeg 2. NLayer = 0 / /3. i = 1 / Sorted /4. Q = 5. for x = 1 to n6. if inDeg[x] = 0 then EnQueue(Q, x)7. end for8. while Q 6= do9. NLayer = NLayer + 110. for iter = 1 to |Q|11. Dequeue(Q, x) / x /12. Sorted[i] = x, i = i+ 113. Layer[x] = NLayer14. for k = head[x] to head[x+ 1] 115. y = succ[k]16. inDeg[y] = inDeg[y] 117. if inDeg[y] = 0 then EnQueue(Q, y)18. end for19. end for20. end while

6.15:

inDeg ( 1) O(m), - O(m) ().

6.4 Prxei se asndeta snola (disjoint set-

s)

( ) . disjoint-set . - (representative) , . :

Find(x): x

6.4. (DISJOINT SETS) 103

x.

Union(x, y): / x y , . x y.

Make-Set(x): x( ).

1 2

3 4

5

6

78

9null

Find(5) = 6

Find(7) = 9

Union(6, 9) = 9

6.16: Union Find.

Union Find 6.16. - 5 () 7 () Union Find : Find(5) = 6, Find(7) = 9 Union(6, 9) = 9 .

. 6.17.

6.18 .

104 6.

a b

c d

e f

g

h

i

j

6.17: :{a, b, c, d}, {e, f, g}, {h, i}, {j}.

Connected-Components(G)1. for each v V2. Make-Set(v)3. for each (u, v) E4. a = Find(u)5. b = Find(v)6. if a 6= b then Union(a, b)7. end for

6.18:

6.19 Connected-Components.

Pleur Sullog ap asndeta snola

arqik snola {a} {b} {c} {d} {e} {f} {g} {h} {i} {j}(b, d) {a} {b, d} {c} {e} {f} {g} {h} {i} {j}(e, g) {a} {b, d} {c} {e, g} {f} {h} {i} {j}(a, c) {a, c} {b, d} {e, g} {f} {h} {i} {j}(h, i) {a, c} {b, d} {e, g} {f} {h, i} {j}(a, b) {a, b, c, d} {e, g} {f} {h, i} {j}(e, f) {a, b, c, d} {e, f, g} {h, i} {j}(b, c) {a, b, c, d} {e, f, g} {h, i} {j}

6.19: 6.17.

, , - ( Find ) .

6.5. 105

Union Union . Find, . Find O(log n) ().

6.5 Elqista Monoptia

, , .

- . , , , .. - .

G = (V,E,W ) , V , E W w(i, j) [i, j] .

p = u0, u1, . . . , uk, ui V : w(p) =

ki=1 w(ui1, ui). s

t , ps,t = s, . . . , t .

:

- (Single Source Shortest Paths): s .

, Dijkstra Bellman.

- (All Pairs Shortest Paths): s V t V ps,t.

, Floyd-Warshall O(n3) Johnson O(n2 logn).

-, .

6.20. -, s 6.21.

106 6.

s = 1

1

1

2

2

22

2

23

3

3

3

3

4

4

55

6

7

7

t = 8

6.20: G = (V,E) s t

s = 1

1

1

2

2

2

22

2

3

3

3

3

3

3

4

4

55

6

7

7

t = 8

6.21: G = (V,E)

. s t w13+w36+w67+w78 = 7

- .

6.5. 107

3 7 3 7. . . :

s t pst =

s, u1, . . . , ui, . . . , uj, . . . , uk, t. - pij = ui, . . . , uj ui uj :, () s t : w(ps,i)+w(pi,j) + w(pj,t).

s tui uj

pij

pij

6.22: pij pij

pi,j w(pi,j) < w(pi,j), ps,t (.

6.22) , pi,j pi,j , ps,t .

:

: , .

, . , 6.23, s t, .

s t . .

: . , . . ,

108 6.

s

u

v

t

1

1 1

1

12

6.23: s u v t

. . . |V |1 , , |V | 1 .

6.5.1 O algrijmo tou Dijkstra

, . , E.W.Dijkstra 1954, . , s V , . ( ) .

FIXE[x], x = 1 . . . n . , FIXE[i] = TRUE i . i (FIXE[i] = FALSE) , V [i] ( -) i, , P [i] () i . i P [i] = 0 V [i] = +.

6.5. 109

, - . , P, V FIXE .

Dijkstra(G = (V,E,W ), s V )1. FIXE FALSE2. P 03. V 4. V [s] = 0, P [s] = s5. repeat6. x V [x]7. if V [x]

110 6.

1 2 3 4 5 6 7 8

1 FIXE 1 0 0 0 0 0 0 0P 1 1 1 0 0 0 0 0V 0 1 2 + + + + +

2 FIXE 1 1 0 0 0 0 0 0P 1 1 1 2 2 0 0 0V 0 1 2 4 4 + + +

3 FIXE 1 1 1 0 0 0 0 0P 1 1 1 2 2 3 0 0V 0 1 2 4 4 4 + +

4 FIXE 1 1 1 1 0 0 0 0P 1 1 1 2 2 3 4 0V 0 1 2 4 4 4 7 +

5 FIXE 1 1 1 1 1 0 0 0P 1 1 1 2 2 3 4 0V 0 1 2 4 4 4 7 +

6 FIXE 1 1 1 1 1 1 0 0P 1 1 1 2 2 3 6 6V 0 1 2 4 4 4 6 8

7 FIXE 1 1 1 1 1 1 1 0P 1 1 1 2 2 3 6 7V 0 1 2 4 4 4 6 7

8 FIXE 1 1 1 1 1 1 1 1P 1 1 1 2 2 3 6 7V 0 1 2 4 4 4 6 7

6.1: Dijkstra 6.20

O(nlogn+m) (m = O(n)) O(nlogn), O(n2) (m = O(n2)).

6.5. 111

1

11

11

111

2

2

22

2

2

2

2

2

2

2

2

2 2

3

33

3 3

3

3

3

3

3

3

3

3

3

3

3

3

44

5

7

1(1, 0)

1(1, 0)

1(1, 0)

1(1, 0)1(1, 0)

1(1, 0)1(1, 0)1(1, 0)

2(1, 1)

2(1, 1)2(1, 1)

2(1, 1)2(1, 1)

2(1, 1)2(1, 1)2(1, 1)

3(1, 2)

3(1, 2)3(1, 2)

3(1, 2)3(1, 2)

3(1, 2)3(1, 2)3(1, 2)

4(2, 4)

4(2, 4)4(2, 4)

4(2, 4)4(2, 4)

4(2, 4)4(2, 4)

5(2, 4)

5(2, 4)5(2, 4)

5(2, 4)5(2, 4)

5(2, 4)5(2, 4)

6(3, 4)

6(3, 4)6(3, 4)

6(3, 4)6(3, 4)

6(3, 4)

7(4, 7) 7(4, 7)

7(6, 6)

7(6, 6)7(6, 6) 8(6, 8)

8(7, 7)

8(7, 7)

6.25: Dijkstra. , s = 1.

112 6.

- . , . , - . a (. 6.26)

aa a

a

bb b

b

c

c cc

2

2

2 2

1

1

1 1

33

3

6.26: Dijkstra . a c b a b 1 a b 1 .

6.5.2 O algrijmo tou Bellman

Bellman . , .

, n1 . Bellman 1, 2, . . . , n 1 - s.

6.5. 113

, , - , .

- 6.27.

a

b

c d

e

f

g hs

1

1

1

2

2

2

22

23

3

3

4

5

6

7

7

8

6.27: 9 s

- ( 6.28).

a

b

c

s 3

4

7

6.28: 1.

114 6.

a, b, c . 7, 3, 4 . 2 ( 6.29).

a

b

c d

e

gs

1

1

2

3

3

4

7

8

6.29: 2.

, . a , s, b, a 5, b c 3 4 , d, e, g 8, 11, 72. 3( 6.30).

2

Parathrome ti h etikta tou e enai 8 pou enai to suntomtero monopti me 2 akm kai

qi 6 pou enai to monopti me 3 akm

6.5. 115

PSfrag

a

b

c d

e

f

g hs

1

1

2

2

2

2

2

3

3

3

467

7

8

6.30: 3.

a, b, c . e - , s, b, a, e 6. d, g , f, h 10 13. 4, . . . , 8.

, , 1, 2, . . . , n 1 () . . :

. 2 d, e, g. .

. - e 3. .

, .

, k . :

Vk(y) = minx{Vk1(y), Vk1(x) +W (x, y)}

116 6.

x y. k s y :

k 1 k 1 x y [x, y].

6.31:

s

yVk1[y]

Vk1[x1]

Vk1[x2]x1

x2

6.31: s y Vk1[y], Vk1[x1]+W (x1, y) Vk1[x2]+W (x2, y)

V [k][j] j(j = 1 . . . n) k(k = 1 . . . n1) . , +. P () P [i] i. , 6.32.

k ( stable). ( ) k+1, . . . , n 1. k .

u, V [n 1][u] = +, .

- 6.27. (, )

6.5. 117

Bellman(G = (V,E,W ), s V )1. P [i], i = 1 . . . n 02. V [0][i], i = 1 . . . n +3. V [0][s] = 0, P [s] = s, k = 04. repeat5. k = k + 16. for i = 1 to n7. V [k][i] = V [k 1][i]8. stable = TRUE9. for y = 1 n10. for each x y11. if V [k 1][x] 6= + and V [k][y] > V [k 1][x] +W [x, y]12. V [k][y] = V [k 1][x] +W [x, y]13. P [y] = x14. stable = FALSE15. end if16. end for17. end for18. until stable or k = n 1

6.32: Bellman G s

. k (V [k][y]) = P (y) y. .

H ektlesh tou algorjmou sto pardeigma

s a b c d e f g h

k = 0 s, 0 s, + s, + s, + s, + s, + s, + s, + s, +k = 1 s, 0 s, 7 s, 3 s, 4 s, + s, + s, + s, + s, +k = 2 s, 0 b,5 s, 3 s, 4 c,7 a,8 s, + b,11 s, +k = 3 s, 0 b, 5 s, 3 s, 4 c, 7 a,6 e,10 e, 9 d, 13k = 4 s, 0 b, 5 s, 3 s, 4 c, 7 a, 6 e, 8 e, 9 g,11k = 5 s, 0 b, 5 s, 3 s, 4 c, 7 a, 6 e, 8 e, 9 g, 11

, 1, . . . , n 1

. , . -, , O(d(y)) O(nm). (n2),

118 6.

. (n).

, . .. h (. 6.27 ) h g e a b s. d d c s. O(n).

-

Dijkstra (: 6.25). . n 1 , , . n .

6.5.3 O algrijmo tou Bellman gia grfou qwrkklo

, ( ) Bellman. ( 6.33):

Layer 3Layer 0 Layer 1 Layer 2

aa

bb

cc

dd

ee

ss

112

22

2

3

3

3

3

3

3

5

5

5

5

99

6.33: ( )

- , , .

6.6. 119

- Bellman : Sorted G , - (Sorted[i], Sorted[j]) i < j. i < j (Sorted[i], Sorted[j]) Layer[Sorted[i]] Layer[Sorted[j]].

Bellman-Acyclic(G = (V,E,W ), s V )1. P [i], i = 1 . . . n 02. V [i], i = 1 . . . n +3. V [s] = 0, P [s] = s4. for each x s5. V [x] =W [s, x]6. for i = 1 n7. x = Sorted[i]8. if Layer[x] > Layer[s] then9. for each y x V [x] +W [x, y] < V [y]10. V [y] = V [x] +W [x, y]11. P [y] = x12. end for13. end for14. end for

6.34: Bellman G. sorted .

, - . - ( sorted), . - , , O(m).

6.6 Dntra Epikluyh

- G = (V,E,W ), E V . . (pins) (, ...) . n -

120 6.

n1 . . .

, G = (V,E,W ), V -, E (u, v) E w(u, v) u v. E E w(E) =

(u,v)E w(u, v) .

.

22. : G = (V,E)

. n n1 . .

6.35: Web

n G n.

E . (spanning tree).

6.6. 121

23. T G = (V,E) E T = (V,E) ., G .

v1v1

v2 v2v3 v3

v4 v4v5 v5

v6 v6

6.36: G = (V,E)

6.6.1 Prblhma eresh en dntrou epikluyh elqis-

tou kstou

G = (V,E,W ) . e E w(e).

24. (minimum cost s-panning tree) T = (V,E) W (T ) =

eE w(e).

6.37 21 15 . 1, ( n 1).

122 6.

v1

v1

v1

v2

v2

v2

v3

v3

v3

v4

v4

v4v5

v5

v5

v6

v6

v6

1

1

12

2

3

3

4

4

4

5

5 5

55

5

6

6

6

6

T1: = 21

T2: = 15

6.37: G = (V,E,W ) T1 T2 W (T1) = 21 W (T2) = 15

() (). -.

. - 6.38. . . , . , .

. (. 6.39). - .: (minimum spanning tree) .

- , ,

6.6. 123

6.38: . .

v1

v1

v1

v2

v2

v2

v3

v3

v3v4

v4

v4

v5

v5

v5

v6

v6

v6

v7

v7

v71

1

1

22

22

22

3

3

3

3

3

4

6 66

6 6

66

7

8

T1: = 20

T2: = 20

6.39: G = (V,E,W )

124 6.

. 1875 .

8. (Arthur Cayley, 1875) , nn2, n .

n 1 2 3 4 5 6 7 8 10 1 1 3 16 125 1296 16807 262144 108

2

. - . ( Prim Kruskal) .

9. F = (F1, F2, . . . , Fk) Fi = (Vi, Ei) G = (V,E,W ) (u, v) V1. , - , , (u, v).

: T =ki=1 Ei F .

Ai T . A = (V,E) T (u, v) ,

(A) (Ai) (6.1) (u, v) E c. c (x, y) x V1 y V \V1. Wxy > Wuv (x, y) / T . (x, y) (u, v) A = (V,E

(u, v)\(x, y))

:

(A) < (A)

6.1.

6.6.2 Algrijmo Prim

Prim A A . . A A

6.6. 125

uv

F1F2

F3

x

y

wxy

wuv

6.40: F1, F2, F3 G. (u, v) V1 G.

A. . (greedy) , .

Prim (G = (V,E,W ))1. T = {u} /* u V */2. for i = 2 to |V | do3. T T {v}, v T4. end for

6.41: Prim G.

T v, T , w(u, v) u T . T . 6.42 - v5.

126 6.

15.

v1v1

v1 v1

v1v1

v2v2

v2v2

v2v2

v3v3

v3 v3

v3v3

v4v4

v4 v4

v4v4

v5

v5v5

v5v5

v5

v6 v6

v6v6

v6

1 1

1

11

1

22

2 2

22

33

3

33

3

44

4

44

4

55

5555

55

5555

5

55

5

5 5

66

66

66

66

6

6

6

6 6

66

6

6

1 2

3 4

5 6

6.42: Prim v5.

Prim

6.6. 127

G = (V,E) |V | = n |E| = m di i . i-,1 i n 1, :

d1d1 + d2d1 + d2 + d3. . .d1 + d2 + d3 + . . .+ di. . .d1 + d2 + d3 + . . .+ di + . . .+ dn1(n 1)d1 + (n 2)d2 + . . .+ (n i)di + . . .+ 2dn2 + dn1 n

n1i=1 di = O(mn)

6.6.3 Algrijmo Prim (nearest neighbour)

Prim O(mn). - O(n2). near[x], dist[x], x V :

near[x] =

x, x T0, x / T, y T : (x, y) Ey, y T

dist[x] =

Wx,y, near[x] = y

+, near[x] = 00, near[x] = x

v , . dist[v] .

128 6.

Prim (G = (V,E,W ))1. VT = {s} /* s V */2. near[s] = s, ET = 3. for every node i (s) do4. near[i] = s; dist[i] = ws,i5. end for6. for every node j / {s} (s) do7. near[j] = 0; dist[j] = +8. end for9. while |VT | < n do10. u V VT dist[u]11. if dist[u] = then graph disconnected, exit12. VT = VT {u} ET = ET {(u, near[u])}13. for x (V VT ) do14. if wux < dist[x] then15. dist[x] wux; near[x] u16. end if17. end for18. end while

6.43: Prim G.

. 6.44 -, . v4 dist[v4] = 14. v4 v3 v3 .

Prim (nearest neigh-bour) O(n2). n 1 O(n+ di) , di i (. 6.45).:

G n 1 ,

i < n1 (x, y) x y w(x, y)

6.6. 129

v1v1

v2v2

v3v3

v4v4 v5v5

v6v6

v7v7 v8v8

14

15 1616

2828

3939

near[v4] = v5, dist[v4] = 14

near[v3] = v1, dist[v3] = 28

near[v4] = v4, dist[v4] = 0

near[v3] = v4, dist[v3] = 16

6.44: v4 v3 . v3 v4.

, Kruskal.

6.6.4 Algrijmo Kruskal

n , . - , . - ( ) ( ).

130 6.

yx

v1 v2

v3dx

6.45: x near dist

Kruskal (G = (V,E,W ))1. T 2. while (|T | < n 1) and (E 6= ) do3. e =smallest edge in E4. E = E {e}5. if T {e} has no cycle then6. T T {e}7. end if8. end while9. if (|T | < n 1) then10. write graph disconnected

. (. 6.46) - . 15.

6.6. 131

v1v1

v1 v1

v1v1

v1

v2v2

v2 v2

v2v2

v2

v3v3

v3 v3

v3v3

v3

v4v4

v4 v4

v4v4

v4

v5

v5v5

v5

v5

v5

v5

v6 v6

v6v6

v6

v6

v6

1 1

1

11

1

1

22

2 2

22

2

33

3

33

3

3

44

4

44

4

4

55

5555

55

5555

5

55

5

5 5

5

5 5

66

66

66

66

6

6

6

6 6

66

6

6

6

6

6

(v1, v3) : 1(v4, v6) : 2(v2, v5) : 3(v3, v6) : 4(v1, v4) : 5(v3, v4) : 5(v2, v3) : 5(v1, v2) : 6(v3, v5) : 6(v5, v6) : 6

1 2

3 4

5,6

6.46: Kruskal. 5 (v1, v4)

132 6.

Kruskal O(m lg n), m - n , - Union Find. O(m lgm). O(lg n) m , O(m lgm) +O(m lg n) , m < n2, O(m lg n) .

:

, ( , 8).

Kruskal - - Prim .

: Prim O(n2) Kruskal O(m lg n), m . , m = O(n2) , m = O(n). - O(n2) Prim O(n2 logn) Kruskal. Prim . Prim - O(n2) Kruskal O(n log n).

6.7. (TSP)133

6.7 DEEK kai to prblhma tou Plandiou

Pwlht (TSP)

G = (V,E,W ) , n. (Traveling Salesman Problem) ( Hamilton) .

6.47 v1, v2, v3, v5, v4, v1 15. v1, v5, v2, v3, v4, v1 13.

v1

v1

v1

v2

v2

v2

v3

v3v3

v4

v4

v4

v5

v5

v5

1

1

1

1

1 1 2

22

2

2

2

22

2

3

3

3

3

3

3

44

4

4

4

4

5

55

6.47: . .

6.48 K4. . TSP .

. - . B(I) , I (instance) TSP. - B(I) :

vi G. T G =(V \{vi}, E), E = E {(vi, v)} v (vi). K(T )

134 6.

v1 v1v2 v2

v3 v3v4 v4

2 2

3 3

4 4

55

66

8 8

= 21 = 17

6.48: 21 17 K4

.

(vi, vk) (vi, vl),k, l 6= i, wil wik.

B(I) = K(T ) + wil + wik: {v1, v2, . . . , vi, . . . , vn, v1} Copt. vi, T ( vi+1, . . . , v1, v2, . . . , vi1). :

(T ) (T ).

6.50 A K5 K4 16. AE,AD,AC,AB AE AC wAE = 2 wAC = 4. B(I) TSP 22.

6.7. (TSP)135

v1

v2

vi

vi1

vi+1

vn1

vn

B(I) = (T ) +Wik +Wil

(T ) +Wi,i1 +Wi,i+1= ( )

= Copt

.

6.49: TSP. vi Kn1

E

E

E

A B

B

B

C

C

C

D

D

D

7

7

7

2 8

8

84

4

4

4

5

5

5

6

6

6

9

9

G = (V,E)

A

K(T ) = 16

B(I) = K(T ) + wAE + wAC = 16 + 2 + 4 = 22

6.50: K5. K4 = K5 {A}. K4 16.

136 6.

Keflaio 7

'Aplhstoi Algrijmoi

( - , ), -, .

(greedy algorithm) - . , . , , .

( Prim Kruskal Dijkstra ).

, . , ( ) .

7.1 Genik morf en 'Aplhstou Algrij-

mou

E e E v(e) ( (v(e) N).

F E, C, C(F ) = true. F (feasible solution) .

137

138 7.

F E,

eFv(e)

( ) ( ).

F . e E ( ), . E. 7.1.

Greedy (E : )1. F = 2. while E 6= do3. x E4. E = E {x}5. if F {x} feasible then6. F = F {x}7. end if8. end while9. return F

7.1:

5 , x , .

Kruskal (. 7.2).

E E ( ). , . , .

6.6 . .

7.2 Anjesh en prou

( - ), P = {P1, . . . , Pn}

7.2. 139

Kruskal (G = (V,E,W ))1. T = 2. while (|T | < n 1) and (E 6= ) do3. e =smallest edge in E4. E = E {e}5. if T {e} has no cycle then6. T T {e}7. end if8. end while9. if (|T | < n 1) then10. write graph disconnected

7.2: Kruskal G = (V,E).

. Pi, (ai, Ti) , ai - Ti . .

7.3. - {P3, P4, P5} . {P1, P2} (maximal), .

a1 a2

a3 a4 a5

T1 T2

T3 T4 T5

7.3: .

(. 7.4) , . Pi

140 7.

Pj , 1 j < i.

- (P : )1. (P )2. F = 3. for i = 1 to n4. Pi5. if F {Pi} feasible then6. F = F {Pi}7. end if8. end while9. return F

7.4:

. -, , . 7.3 .

. 7.5. {P1, P2, P3}, - {P4, P5}.

a1 a2 a3

a4 a5

T1 T2 T3

T4 T5

7.5: .

, 7.3 [P1, P3, P4, P2, P5]

7.3. 141

7.5 [P1, P4, P2, P5, P3]. , .

-.

F = {x1, x2, . . . , xk, . . . , xp}

Opt = {y1, y2, . . . , yk, . . . , yq}

(q p). (k 1)- k- , :

x1 = y1, x2 = y2, . . . , xk1 = yk1, xk 6= yk

xk xk yk. yk xk , xk . Opt :

Opt = {x1, x2, . . . , xk1, xk, yk+1, . . . , yq}

yk+1, . . . , yq p = q .

7.3 Apojkeush arqewn se dskou

n l1, l2, . . . , ln - L. ,

ni=1 li 2L.

. . 7.6 .

142 7.

- (l : , L : )1. -(l)2. i = 13. for j = 1 to 24. sum = 05. while sum+ li L6. i j7. sum = sum+ li8. i = i+ 19. end while10. end for

7.6: -

, 7.7, 10 4 2, 4, 5, 6

6

5

5

2

2

4

4

Greedy

Optimum

7.7: . 3 4.

, . NP -,

7.4. (DISCRETE KNAPSACK) 143

() .

7.4 To Diakrit Prblhma Sakidou (Dis-

crete Knapsack)

. n X = {x1, x2, . . . , xn} a1, a2, . . . , an. ci . , b. , zi {0, 1} i, :

max

ni=1

cizi

:ni=1

aizi b

-, ci/ai . 7.8.

Discrete-Knapsack (X : , a, c : , b : )1. -(ci/ai)2. S = 3. for i = 1 to n4. if b ai5. S = S {xi}6. b = b ai7. end if8. end for

7.8:

. : 3 {x1, x2, x3} a = (k + 1, k, k)

144 7.

c = (k+2, k, k). 2k. ci/ai x1, x2, x3

k + 2

k + 1>k

k=k

k

x1 k+2 x2, x3 2k. , k 100%.

NP, - -. , .

7.5 To Suneq Prblhma Sakidou (Contin-uous Knapsack)

, . zi [0, 1].

ci/ai . 7.9

Continuous-Knapsack (X : , a, c : , b : )1. -(ci/ai)2. for i = 1 to n zi = 03. i = 13. while b ai do4. zi = 15. b = b ai6. i = i+ 17. end while8. zi = b (ci/ai)

7.9:

Keflaio 8

Dunamik

Programmatism

8.1 Eisagwgik

1 (dynamic programming), , -, .

, , . - - . , -, , .

, , , - - (overlapping subproblems). , , .

, .

1. .

2. .

3. .

4. .

1

O ro programmatism prorqetai ap ma susthmatik mjodo pou qrhsimopoie pnake

(tabloid method) kai den sqetzetai me thn anptuxh kdika se kpoia glssa programmatismo

145

146 8.

, .

8.2 Basik stoiqea tou dunamiko program-

matismo

, - . . .

8.2.1 Bltista diaspmenh dom (optimal substructure)

, , . :

.

, , .

, - , :

1. , . - . , , p u v. u v, u p p v.

2. , , .

3. , . , p - u v,

8.2. 147

. u p p v.

4. . , .

- , (bottom-up). , - , . - , . - .

- . (independency) -, . , - , .

- . , w u v, u v u w w v. 8.1.

q

st

r

8.1:

148 8.

q t, p1 : q r p2 : r t. p1 q t s r, t s. -, . .

8.2.2 Epikaluptmena upoproblmata (Overlapping sub-problems)

- .

, , , , . - , -, . :

, , - . , ( ).

- -. - , . .

, - () . , .

8.3 Efarmog

( ) . , , .

, -, .

8.3. 149

, Bellman , .

8.3.1 Qronodromolghsh gramm paragwg

.

, n . - Sij . S1j S2j . aij , Sij . ei, , , i, i {1, 2} xi i .

. , , , , . . tij Sij , j = 1, 2, . . . , n 1.

:

.

8.2 . - 1 2.

Entrance

e1

e2

x1

x2

a1,1 a1,2 a1,3 a1,n1a1,n

a2,1 a2,2 a2,3 a2,n1 a2,n

t1,1 t1,2 t1,3 t1,n1

t2,1 t2,2 t2,3t2,n1

S1,1 S1,2 S1,3 S1,n1 S1,n

S2,1 S2,2 S2,3 S2,n1 S2,n

Exit

8.2:

150 8.

1:

S1j .

j = 1 ( ), . (j = 2, 3, . . . , n) 8.2. S1j1 S2j1, t2j1.

:

S1j S1j1, S1j1 . , , -, S1j1 S1j S1j , .

S1j S2j1, S2j1 .

:

S1j - S1j1 S2j1.

, S1j :

S1j1 S1j

S2j1 S1j t2j1

, S2j

S2j1 S2j

S1j1 S2j t1j1

2:

, - , - .

8.3. 151

f 2 n, fij (i = 1, 2 j = 1, 2, . . . , n) Sij .

:

f = min{f1n + x1, f2n + x2} (8.1)

, S1n, S2n.

f .

:

f1j =

{e1 + a11 j = 1min{f1j1 + a1j , f2j1 + t2j1 + a1j} j 2 (8.2)

:

f2j =

{e2 + a21 j = 1min{f2j1 + a2j , f1j1 + t1j1 + a2j} j 2 (8.3)

, i , ei + ai1.

.

(8.2) (8.3) aij , . ,