Data Structures and Algorithms in Java[1].pdf - Fulvio Frisone

This claim is true at the beginning of the first iteration of the loop, since there are
no elements among the first 0 in A (this kind of a trivially true claim is said to
hold vacuously). In iteration i, we compare element x to element A[i] and return
the index i if these two elements are equal, which is clearly correct and completes
the algorithm in this case. If the two elements x and A[i] are not equal, then we
have found one more element not equal to x and we increment the index i. Thus,
the claim S i will be true for this new value of i; hence, it is true at the beginning
of the next iteration. If the while-loop terminates without ever returning an index
in A, then we have i = n. That is, S n is true—there are no elements of A equal to x.
Therefore, the algorithm correctly returns —1 to indicate that x is not in A.
4.4. Exercises
For source code and help with exercises, please visit
java.datastructures.net.
Reinforcement
R-4.1
Give a pseudo-code description of the O(n)-time algorithm for computing the
power function p(x, n). Also, draw the recursion trace of this algorithm for the
computation of p(2,5).
R-4.2
Give a Java description of Algorithm Power for computing the power function
p(x, n) (Code Fragment 4.3).
R-4.3
Draw the recursion trace of the Power algorithm (Code Fragment 4.3, which
computes the power function p(x,n)) for computing p(2,9).
R-4.4
Analyze the running time of Algorithm BinarySum (Code Fragment 3.34) for
arbitrary values of the input parameter n.
R-4.5
Graph the functions 8n, 4nlogn, 2n 2 , n 3 , and 2 n using a logarithmic scale for the
x- and y-axes , that is, if the function value f(n) is y, plot this as a point with x-
coordinate at log n and y-coordinate at log y.
R-4.6
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