My inspiration from reading *C++ Primer*, *Effective C++*, *More Effective C++*, *The C++ Standard Library* and some experience from coding.
Include:
* Debug
* C++ Syntax
* Habit && Optimization
* Trick
* Trap
* Reference
My inspiration from reading *C++ Primer*, *Effective C++*, *More Effective C++*, *The C++ Standard Library* and some experience from coding.
Include:
* Debug
* C++ Syntax
* Habit && Optimization
* Trick
* Trap
* Reference
21. 結束
4. 演算法的效能分析
Given a problem, there may be several possible implementations.
Efficiency is the most important consideration including time and
space.
Complexity Theory– to estimate the time and space needed for a
program. It’s machine independent.
Space Complexity of a program – the amount of memory space
needed to complete a program.
Time Complexity of a program – the amount of computation
(computer) time needed to complete a program.
22. 結束
Space Complexity S(p)
S(p)=Sc+Sp(I)
Sc (Fix space requirement) including instruction space,
constants, simple variables, and fix-size structures. The
Sc is independent of the number and size of inputs and
outputs. e.g. int i, sum=0, A[100];
Sp(I) (Variable space requirement) depends on a
particular instance I of a problem. Instance I may be a
function of number, size, or values of inputs and
outputs. e.g. int *n; //score list of a course
23. 結束
Time Complexity T(p)
T(p)=Tc+Tp(I)
Tc (Fix time requirement) compile time, Tc is independent
of any instance of the problem.
Tp(I) (Variable time requirement) execution time, depends
on a particular instance I of a problem.
e.g. matrix multiplication C[ ] [ ] =A[ ] [ ] *B[ ] [ ]
(0,0) 0,1 0,2 0,3 0,n 0,0
1,0
2,0
n,0
(n*) + (n+) => 2n 2n* n*n + an2+bn+c => 2n3 +an2+bn+c
25. 結束
Big-oh的介紹
O(g(n))可視為某演算法在電腦中所需執行時間不會超
過某一常數倍的g(n)，也就是說當某演算法的空間或
時間複雜度(space or time complexity)為O(g(n))(讀成
big-oh of g(n)或order is g(n))。
Definition: The function f(n) is said to be of order at most
g(n) if there are positive constants c and n0 such that
f(n)<=cg(n) for all n, n>=n0.
Therefore, “Big oh” is the smallest upper bound of f(n).
29. 結束
θ(theta)的介紹
是一種比Big-O與Ω更精確複雜度的漸近表示法。定
義如下：
Definition: The function f(n) is θ(g(n)) iff there exists
positive constants c1, c2 and n0 , such that c1 g(n)
<=f(n)<= c2 g(n) for all n, n>=n0.
Therefore, “θ” is both the smallest upper bound and the
greatest lower bound of f(n).
30. 結束
Examples for asymptotic notations
1. 3n + 2 = Ο(n), ∵3n +2 ≦ 4n, for all n ≧ 2, c = 4.
2. 3n + 3 = Ο(n), ∵3n +3 ≦ 4n, for all n ≧ 3, c = 4.
3. 3n + 3 = Ο(n2), ∵3n +2 ≦ 3n2, for all n ≧ 2, c = 3.
(2) is correct. (3) is wrong. “Big oh“ should be a smaller function of n.
4. 10n2 + 4n +2 = Ο(n2), ∵10n2 + 4n +2≦ 11n2, for all n ≧ 5, c = 11.
5. 6．2n + n2 = Ο(2n), ∵6．2n + nn≦ 7．2n, for all n ≧ 4, c = 7.
6. 3n +3 = Ω(n), ∵3n +3 ≧ 3n, for all n ≧ 1, c = 3.
7. 3n +3 = Ω(1), ∵3n +3 ≧ 3, for all n ≧ 1, c = 3.
(6) is correct. (7) is wrong. “Omega” should be a larger function of n.
8. 3n +2 = Θ(n) ∵3n≦3n +2 ≦ 4n, for all n ≧ 2, c1 = 3, c2= 4 , and n0=
2.
31. 結束
Example for S(p): addup values of n
elements in an array called list.
main( ) float addup(float list[ ], int n)
{ {
float list[n], temp = 0.0; float temp = 0.0;
int i; int i;
for( i=0; i < n; i++) for( i=0; i < n; i++)
temp += list[i]; temp += list[i];
return temp; return temp;
} }
Smain(n) = c + n = O(n) Saddup(n) = c = O(1)
Other languages may need to pass the whole array, but addup
passes only addresses of the 1‘st element and the size of array.
32. 結束
Example for T(p): use step count
instead of execution time.
1. on-line step count
float sum(float list[ ], int n) /* calculate the sum of an array */
{
float tmp=0.0; count ++; /* tmp assignment */
int i;
for ( i=0; i <n; i++ ){
count++; /* for loop */
tmp+=list[i]; count++; * adding up */
} count++; /* last time to check for loop and fails */
return tmp; count++; /* for return */
}
=> step count = 2*n +3 => Tsum(n) = O(n)
33. 結束
2. Tabular method
void add( int a[ ][ ], int b[ ][ ], int c[ ][ ], int row, int col)
{ Step per exec. freq. total
int i, j; 0 0 0
for ( i=0; i<row; I++) 1 row +1 row+1
for ( j=0; j<col; j++) 1 row*(col+1) row*col + row
c[i][j] = a[i][j]+b[i][j]; 1 row*col row*col
}
In total, we have step count = 2row*col +2row +1 .
=> Tadd(row, col) = O(row*col)
Thus, if row >>col, one may want to exchange i and j to reduce the step
count and execution time.
34. 結束
3. Execution time Measurement
#include <time.h> 1. Clock 2. Time
Before execution, use start = clock(); start = time(NULL);
After execution, use stop = clock(); stop = time(NULL);
Type return clock_t time_t
result in sec. (stop - start)/ CLK_TCK; difftime(stop, start);
When measuring the execution of a program, we have
CLK_TCK= 18.2, beginclk = 66, stopclk = 193, diffclk = 127, time
= 6.98
begintime=825316157, stoptime = 825316164, difftime = 7
Note: 1.The time is begin at around 1970.
2. Use %ld to print out a long integer
35. 結束
trade-off between program
space and execution time
Example: Two ways to interchange two elements, using a functions or a macro
1. using a function 2. using a macro
void swap (int *x, int *y) #define swap(a,b,t) ((t)=(a), (a)=(b), (b)=(t))
{ …
int temp; swap(p, q, temp);
temp = *x; …
*x = *y; Disadv: duplicated macros may takes lots of space.
*y = temp; Adv: no calls & returns, execution time is reduced.
}
Adv: single copy of function => save space.
Disadv: calls & returns => waste time.
Therefore, a careful evaluation of the trade-offs among various
aspects before the implementation of a solution is important.
