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Chap. 10. Exhaustive Search in .NET Integrated 2d Data Matrix barcode in .NET Chap. 10. Exhaustive Search

Chap. 10. Exhaustive Search using barcode implement for visual studio .net control to generate, create data matrix ecc200 image in visual studio .net applications. Planet A k-ary string VS .NET barcode data matrix is a string of digits drawn from {0, 1, . .

. , k 1}. For example, if k = 2, then a k-ary string is simply a binary string.

480. Design an optimal exhaustive search algorithm for generating all k-ary strings of length n in lexicographic order in time O(kn ). For example, if k = 3 and n = 4, then your algorithm should generate the strings 0000, 0001, 0002, 0010, 0011, 0012, 0020, 0021, 0022, 0100, .

. . , 2221, 2222.

481. 482. Prove that your algorithm for Problem 480 is correct.

Analyze the running time of your algorithm for Problem 480.. A minimal-chan ge algorithm for generating k-ary strings generates them in such an order that each string di ers from its predecessor in exactly one place. 483. 484.

485. Design a minimal-change exhaustive search algorithm for generating all k-ary strings of length n. (See also Problem 480.

) Prove that your algorithm for Problem 483 is correct. Analyze the running time of your algorithm for Problem 483..

PERMUTATIONS Consider the p roblem of exhaustively generating all permutations of n things. Suppose A[1..

n] contains n distinct values. The following algorithm generates all permutations of A[1..

n] in the following sense: A call to procedure permute(n) will result in procedure process(A) being called once with A[1..n] containing each permutation of its contents.

1. 2. 3.

4. 5. 6.

7. 486. procedure permute(n) if n = 1 then process(A) else B := A permute(n 1) for i := 1 to n 1 do swap A[n] with A[1] permute(n 1) A := B cyclically shifted one place right.

Prove that thi Data Matrix 2d barcode for .NET s algorithm calls process(A) with A containing each permutation of n entries exactly once..

Sec. 10.2. Permutations 119 What is the running time of this algorithm 487. 488..

Show that the gs1 datamatrix barcode for .NET e ect of line 7 is the same as performing a cyclic shift of A[2..

n]. Use this observation to eliminate the need for the array B. Does this a ect the running time of the algorithm .

Consider the f ollowing algorithm for generating all permutations. To generate all permutations of any n distinct values in A[1..

n], call permute(1). 1. 4.

2. 3. 4.

5. 489. 490.

procedure permute(i) if i = n then process(A) else permute(i + 1) for j := i + 1 to n do swap A[i] with A[j] permute(i + 1) swap A[i] with A[j]. Prove this alg VS .NET barcode data matrix orithm correct. How many swaps are made by this algorithm, as a function of n What is its running time .

The following is a minimal-change algorithm for generating permutations. That is, each permutation generated di ers from the previous one by having just two values swapped. It is known as Heap s algorithm (Heap [35]).

1. 2. 3.

4. procedure permute(n) for i := 1 to n do A[i] := i if n is odd then oddpermute(n) else evenpermute(n) procedure oddpermute(n) if n = 1 then process else evenpermute(n 1) for i := 1 to n 1 do swap A[1] and A[n] evenpermute(n 1) procedure evenpermute(n) if n = 1 then process else oddpermute(n 1) for i := 1 to n 1 do swap A[i] and A[n] oddpermute(n 1). 5. 6. 7.

8. 9..

10. 11. 12.

13. 14..

120 491.. Chap. 10. Exhaustive Search Show that the VS .NET ECC200 algorithm is correct, that is, it generates all permutations in minimal-change order. Show that Heap s algorithm uses the minimum number of swaps, (that is, n! 1 swaps).

What is its running time . COMBINATIONS Consider the p datamatrix 2d barcode for .NET roblem of generating all combinations of r things chosen from n, for some 0 r n. We will, without loss of generality, assume that the n items to be chosen from are the natural numbers 1, 2, .

. . , n.

The following algorithm generates all combinations of r things drawn from n in the following sense: Procedure process(A) is called once with A[1..r] containing each combination.

To use it, call procedure choose(1, r). procedure choose(b, c) comment choose c elements out of b..

n if c = 0 then process(A) else for i := b to n c + 1 do A[c] := i choose(i + 1, c 1). 1. 2. 3.

4. 493. 494.

. Prove that thi VS .NET DataMatrix s algorithm is correct. Prove that the combinations generated by this algorithm are in descending order, that is, when process(A) is called, A[i] > A[i + 1] for all 1 i < n.

Let T (n, r) be the number of assignments to array A. Show that T (n, r) r n . Hence, deduce that the algorithm runs in time O r n .

r r Show that for all 1 r n/2, T (n, r) 2 that the algorithm is optimal for r n/2..
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