Introduction to Algorithms (third edition) practice 2.1-1 ~ 2.1-4

2.1-1

(a) 31-"41"-59-26-41-58

(b) 31-41-"59"-26-41-58

(c) 31-41-59-"26"-41-58 (31, 41, 59 move right one position, insert 31 at the original 26 position)

(d) 26-31-41-59-"41"-58 (59 move right one position, insert 41 at the original 59 position)

(e) 26-31-41-41-59-"58" (59 move right one position, insert 58 at the original 59 position)

(f) 26-31-41-41-58-59

The number in "" "is temporarily stored in key.

2.1-2

insertion-SORT (A)    for j = 2 to a.length        = A[j]        = j-1 while           and A[i] < key            a[i+1] = A[i]            = i-1        a[i+1] = key

2.1-3

We Use the loop invariants to help us understand what an algorithm is correct.

Try to prove it , maybe it's wrong!

Initialize: proves that the loop is invariant before the first iteration. When the input size is 1 (array length), the array is composed of only a single element a[1], after an equal judgment, the output either takes i = 1 (this number is exactly equal to V), or take V = NIL (this number is not v), the loop invariant form.

Hold: proves that each iteration keeps the loop invariant. Assuming that the input scale is n, the loop is invariant, that is, either the subscript I (a[i] = = v) is taken in the array n, or it is not found (v = NIL). Then add an element, make the scale of n+1, easy to know, the size of n since the establishment (can get a useful nature, conclusion), scale n+1 at most one time equal judgment (that is, in the original array does not take the case of I), and finally can obtain useful conclusions. The iteration (in this case, size + 1) will keep the loop invariant.

Termination: Finally, the study of what happened at the end of the cycle. The condition that causes the iteration to terminate is I > a.length=n or take subscript I, if the former so that V = NIL, this time has been a useful conclusion, so the algorithm is correct.

2.1-4

Input: An array of length n, a, B, storing an n-bit binary integer, respectively. (the highest bit must be 1, the other bit is 0 or 1)

Output: An array of length n+1 C that stores the sum of the binary integers stored with a and B.

Pseudo code:

ADD (A, B, C)      for 1 To a.length         = A[k] + b[k]        if2            c[k+1] = c[k+11              2

The correctness of this algorithm can also be understood by cyclic invariant.

Introduction to Algorithms (third edition) practice 2.1-1 ~ 2.1-4