(, IT, )(Algorithm) Divide and Conquer ()
Computer Algorithmsby Yang-Sae Moon
Divide-and-Conquer?:1805 12 2 - > (15,000 ) Divide ()
ConquerDivide and Conquer
Computer Algorithmsby Yang-Sae Moon
Divide-and-Conquer Divide and Conquer(Divide): .(Conquer):
.(Combine): () .
(top-down) .
Computer Algorithmsby Yang-Sae Moon
: (1/3): n S x .: n, S[1..n], x: locationout - x S , x S 0:x , ,
! :: x , .: x .: ( )
: p. 49 2.1Divide and Conquer
Computer Algorithmsby Yang-Sae Moon
: (2/3)Divide and Conquerindex location (index low, index high)
{index mid;
if (low > high)return 0;// else {mid = (low + high) / 2// (
)if (x == S[mid])return mid;// else if (x < S[mid])return
location(low, mid-1);// elsereturn location(mid+1, high);//
}}locationout = location(1, n);... p. 50 (x == S[mid] x <
S[mid])
Computer Algorithmsby Yang-Sae Moon
: (3/3)Divide and Conquer
Computer Algorithmsby Yang-Sae Moon
: (1/2) ? n, S, x , ?
n, S, x . (recursive call) .?Divide and Conquer
Computer Algorithmsby Yang-Sae Moon
: (2/2) (tail recursion): () (iterative algorithm) . ( p. 10 1.5
) (activation records) , (). . (constant factor) () . .Divide and
Conquer
Computer Algorithmsby Yang-Sae Moon
: x S[mid] : n (= high - low + 1) while , . : (1) ; (2)x . () ,
.Divide and Conquer : (1/7)
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1: (recurrence) . n > 1 , n = 2k(k 1) .Divide and Conquer :
(2/7)
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.: :Induction basis: n = 1, W(1) = 1 = lg 1 + 1.Induction
hypothesis: 2 (power) n , W(n) = lg n + 1 .Induction step: W(2n) =
lg(2n) + 1 . () ,Divide and Conquer : (3/7)
Computer Algorithmsby Yang-Sae Moon
2: n , .
. .Divide and Conquer : (4/7)
N mid n/2 - 11n/2(n-1)/21(n-1)/2
Computer Algorithmsby Yang-Sae Moon
. : Induction basis: n = 1, . Induction hypothesis: n > 1, 1
< k < n k , .Divide and Conquer : (5/7)
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Induction step: 1) n (, ), .Divide and Conquer : (6/7)
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Induction step: 2) n (, ), .Divide and Conquer, n . : (7/7)
Computer Algorithmsby Yang-Sae Moon
(Merge Sort) (1/2): n () .: n, n S[1..n]: () S[1..n]: 27, 10,
12, 20, 25, 13, 15, 22 p. 53 2.2 Divide and Conquer
Computer Algorithmsby Yang-Sae Moon
(2/2)Divide and Conquer
Computer Algorithmsby Yang-Sae Moon
: Divide and Conquervoid mergesort (int n, keytype S[]){const
int h = n/2, m = n - h;keytype U[1..h], V[1..m];
if (n > 1) {copy S[1] through S[h] to U[1] through U[h];copy
S[h+1] through S[n] to V[1] through
V[m];mergesort(h,U);mergesort(m,V);merge(h,m,U,V,S);}}
Computer Algorithmsby Yang-Sae Moon
: Divide and Conquer: .: (1) h, m, (2) U[1..h], V[1..m]: U V
S[1..h+m]
Computer Algorithmsby Yang-Sae Moon
: (1/2)Divide and Conquervoid merge(int h, int m, const keytype
U[], const keytype V[], const keytype S[]){index i, j, k;i = 1; j =
1; k = 1;while (i 0 , , c < 1 n , , ., , .
Computer Algorithmsby Yang-Sae Moon
Master Theorem Examples (1/3) 1: , a = 9, b = 3, f(n) = n, , = 1
, . 1 , .
2: a = 1, b = 3/2, f(n) = 1, , . 2 , .Divide and Conquer
Computer Algorithmsby Yang-Sae Moon
Master Theorem Examples (2/3) 3: a = 3, b = 4, f(n) = n lg n, ,
0.2 , = 1 , . 3 , n , 1 c . , , n . .
Divide and Conquer
Computer Algorithmsby Yang-Sae Moon
Master Theorem Examples (3/3) 4: a = 2, b = 2, f(n) = nlgn, , .
3 , n , 1 c . , n c . , , c . , Master Theorem . Master Lemma
.Divide and Conquer
Computer Algorithmsby Yang-Sae Moon
The Master Lemma , k 0 k f(n) .Master Lemma : 4 .
Master Theorem(Lemma) . Divide and Conquer
Computer Algorithmsby Yang-Sae Moon
(Quick Sort) (1/3)1962 (C.A.R. Hoare) (Quicksort) . (partition
exchange sort) .Divide and Conquer
Computer Algorithmsby Yang-Sae Moon
, . ? 1: , , (pivot item) , , . 2: , , .Divide and Conquer(Quick
Sort) (2/3)
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: 15, 22, 13, 27, 12, 10, 20, 25 ( p. 60 2.2 )Divide and
Conquer(Quick Sort) (3/3)
Computer Algorithmsby Yang-Sae Moon
: n () : n > 0, n S[1..n]: S[1..n]:Divide and Conquer void
quicksort (index low, index high){index pivotpoint;if (high >
low)
{partition(low,high,pivotpoint);quicksort(low,pivotpoint-1);quicksort(pivotpoint+1,high);}}
Computer Algorithmsby Yang-Sae Moon
: S .: (1) low, high (2) low high S : (1) S pivotpoint (2) S (
): (the next page)Divide and Conquer (1/2)
Computer Algorithmsby Yang-Sae Moon
Divide and Conquer (2/2)void partition (index low, index high,
index& pivotpoint){index i, j;keytype pivotitem;pivotitem =
S[low]; // pivotitem j = low;for(i = low + 1; i 0 (T(0) = 0,
otherwise)
Computer Algorithmsby Yang-Sae Moon
Divide and Conquer (Worst Case) (3/5) , . n(n1)/2 . , ? , ,
.
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Divide and Conquer (Worst Case) (4/5) n , .: (): n = 0 ,: 0 k
< n k ,:
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Divide and Conquer (Worst Case) (5/5) p n - 1 .
,
. .
Computer Algorithmsby Yang-Sae Moon
Divide and Conquer (Average Case) (1/4), Worst Case Complexity ,
., ? Average Case Complexity .
Computer Algorithmsby Yang-Sae Moon
Divide and Conquer (Average Case) (2/4): S[i] key : S , n: . 1 n
, . p 1/n, p [A(p - 1) + A(n - p)], n - 1, .
Computer Algorithmsby Yang-Sae Moon
Divide and Conquer (Average Case) (3/4)(): n ,n n - 1 ,(1) (2) ,
,
, , .
, .
Computer Algorithmsby Yang-Sae Moon
Divide and Conquer (Average Case) (4/4)(): , .
. , ln n = logen, , an 2 ln n. , A(n) .
Computer Algorithmsby Yang-Sae Moon
Divide and Conquer (1/4) ? ?
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Divide and Conquer (2/4) .mn (m by n) m n .
nxn . .
Computer Algorithmsby Yang-Sae Moon
Divide and Conquer : A+B = C = [ci,j] = [ai,j+bi,j] where A =
[ai,j] and B = [bi,j] (3/4)
Computer Algorithmsby Yang-Sae Moon
Divide and Conquer :The organization of the paper is poor.
(2/4)
Computer Algorithmsby Yang-Sae Moon
Divide and Conquer: n n .: n, n n A B: A B C: (1/3)void
matrixmult (int n, const number A[][], const number B[][], number
C[][]){index i, j, k;for (i = 1; i
