-
2
2
-
(Divide-and-Conquer) (Divide): .(Conquer): .(Combine): () .
(top-down) .
2
-
(Binary Search): : n S x .: n, S[1..n], x: locationout - x S . x
S 0:x , , ! :: x , .: x .: ( )
2
-
(Binary Search): index 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);...
2
-
locationout ? n, S, x . (recursive call) .
2
-
(recursive algorithm) () (tail recursion) - (iterative
algorithm) . (activation records) , (). . (constant factor) () . ML
Scheme .
2
-
: x S[mid] : n (= high - low + 1) while 2 , . : (1) ; (2) x 2 .
() , 1 .
2
-
1: (recurrence) . n > 1 , n = 2k(k 1)
.......
2
-
.: :: n = 1, W(1) = 1 = lg 1 + 1.: 2 (power) n , W(n) = lg n + 1
.: W(2n) = lg(2n) + 1 . ,
2
-
2: - y , n , .
. .
2
n
mid
n/2 - 1
1
n/2
(n-1)/2
1
(n-1)/2
-
n .: : n = 1, .: n > 1, 1 < k < n k , .: n (, ),
2
-
n (, ),
2
-
(Mergesort): n .: n, n S[1..n]: S[1..n]: 27, 10, 12, 20, 25, 13,
15, 22
2
-
:
void 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);}}
2
-
(Merge): .: (1) h, m, (2) U[1..h], V[1..m]: U V S[1..h+m]
2
- :void 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
-
: U[i] V[j] : 2 : h m: i = h, j = m -1 (loop) , h + m -1. ,
W(h,m) = h + m -1.
2
-
: merge : S n: W(h,m) = W(h) + W(m) + (h + m 1) . W(h) U , W(m)
V , h + m - 1 . n 2k(k 1) , . :
2 , .
2
-
n 2 (power) .
. , , n = 2k , . , n = 2k .
2
-
(in-place sort) . . S U V . ? mergesort S U V . merge U V .
mergesort . S n, U V n . . . n , , n ( ). .
2
-
(Mergesort): n .: n, n S[1..n]: S[1..n]:void mergesort2 (index
low, index high) { index mid; if (low < high) { mid = (low +
high) / 2; mergesort2(low, mid); mergesort2(mid+1, high);
merge2(low, mid, high); }} mergesort2(1, n); ...
2
-
(Merge): .: (1) low, mid, high, (2) S[low..high], S[low..mid]
S[mid+1..high] .: S[1..high]
2
- : void merge2(index low, index mid, index high) {index i, j, k;
keytype U[low..high]; // i = low; j = mid + 1; k = low;while
(i
-
(The Master Theorem)a b 1 , f(n) , n T(n) .
T(n) (asymptotic bound) .1. > 0 , 2. , . 3. > 0 , , c <
1 n , , . , .
2
-
a = 9, b = 3, f(n) = n, , = 1 , . 1 , .
a = 1, b = , f(n) = 1, , . 2 , .
2
-
a = 3, b = 4, f(n) = n lg n, , 0.2 , = 1 , . 3 , n , 1 c . , , n
. .
a = 2, b = 2, f(n) = nlgn, , . 3 , n , 1 c . , n c . , , c . .
.
2
-
, k 0 k f(n) . ( ) 4 .
. .
2
-
(Quicksort)1962 (C.A.R. Hoare) (Quicksort) . . (partition
exchange sort) .: 15 22 13 27 12 10 20 25
2
-
: n : n > 0, n S[1..n]: S[1..n]:void quicksort (index low,
index high) {index pivotpoint;if (high > low) {
partition(low,high,pivotpoint); quicksort(low,pivotpoint-1);
quicksort(pivotpoint+1,high);}}
2
- : S .: (1) low,high, (2) low high S : low high S (pivot point),
pivotpoint:void partition (index low, index high, index&
pivotpoint) {index i, j;keytype pivotitem;pivotitem =
S[low];//pivotitem j = low;for(i = low + 1; i
-
: S[i] key : , n = high - low + 1: , T(n) = n - 1.
2
-
: S[i] key : S , n: . ? () , n 0 , n-1 . ,, T(0) = 0, . T(n) =
T(n - 1) + n - 1, n > 0 T(0) = 0
2
-
, T(n) = T(n - 1) + n - 1 T(n - 1) = T(n - 1) + n - 2 T(n - 2) =
T(n - 3) + n - 3 ... T(2) = T(1) + 1 T(1) = T(0) + 0 T(0) = 0
, . ? , .
2
-
n , .: (): n = 0 ,: 0 k < n k ,: pivotpoint p
p 1 .
,
.
2
-
: S[i] key : S , n: . 1 n , . , . p , p [A(p - 1) + A(n - p)], n
- 1, .
2
-
n ,
n n - 1 ,
(1) (2) ,
,
,
, .
,...
2
-
,
. ln n = logen,
, an 2 ln n. ,
2
- (Matrix Multiplication) : n n .: n, n n A B: A B C:void
matrixmult (int n, const number A[][], const number B[][], number
C[][]) { index i, j, k; for (i = 1; i
-
I:: : , n : . II:: : , n : .
2
-
2 2 : : 2 2 A B (product) C,
(Strassen) :
: 8 4 , 7 18 / . ! .
2
-
n n : : n 2 , 4 (submatrix) . n n A B C:
(Strassen) :
2
- : n 2 , n n .: n, n n A B: A B C:void strassen (int n,
n*n_matrix A, n*n_matrix B, n*n_matrix& C) {if (n
-
I: : , n : 1 . ( .)
,
. . 3 1 .
2
-
II: / : , n : 1 .
3 1 .
2
-
(n2) .
.
2
-
n 2 , n : (exponential)
n n , n/c . c . : (nlg n)
2
2 , , , 1999 2 , , , 1999
