Description
The Problem To Be Solved: The Grindelwald Gaggle
At the Hogwarts School of Witchcraft and Wizardry, significant time is spent studying the mystical properties of various sequence of integers. These include the Grindelwald Gaggle — a
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sequence of numbers G0, G1, G2, G3, . . . defined as follows: For every integer i ≥ 0,
Gi =
1 if i = 0,
2 if i = 1,
3 if i = 2,
4 if i = 3,
2Gi−1 − 2Gi−3 + Gi−4 if i ≥ 4 and i is even,
Gi−1 + 3Gi−2 − 5Gi−3 + 2Gi−4 if i ≥ 4 and i is odd.
Consequently, Hermione Granger is interested in the following computational problem:
Grindelwald Gaggle Computation
Precondition: A non-negative integer n is given as input.
Postcondition: The n
th Grindelwald number, Gn, is returned as output.
A Simple — But Slow — Algorithm for This Computation
Hermione has written a Java program that implements the algorithm shown in Figure 1 on
page 3.
1. Give a reasonably short proof that the function f(n) = n is a bound function for this
recursive algorithm.
2. Prove that the sGrin algorithm correctly solves the “Grindelwald Gaggle Computation”
problem.
3. Write a Java program, SGrindelwald.java, that satisfies the following properties:
• It is part of the cpsc331.assignment1 package.
• Integer inputs and outputs are represented using Java’s int data type.
• It includes a method, sGrin, which receives an integer n as input and which computes the n
th Grindelwald number if n ≥ 0, using the algorithm which has now been
analyzed, but which actually solves a more general computational problem than the
one given above:
Extended Grindelwald Gaggle Computation
Precondition: An integer n is given as input.
Postcondition: If n ≥ 0 then the n
th Grindelwald number, Gn, is returned as output. An IllegalArgumentException is thrown otherwise.
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integer sGrin (integer n) {
// Assertion: A non-negative integer n has been given as input.
1. if (n == 0) {
2. return 1
3. } else if (n == 1) {
4. return 2
5. } else if (n == 2) {
6. return 3
7. } else if (n == 3) {
8. return 4
9. } else if ((n mod 2) == 0) {
10. return 2 × sGrin(n − 1) − 2 × sGrin(n − 3) + sGrin(n − 4)
} else {
11. return sGrin(n − 1) + 3 × sGrin(n − 2) − 5 × sGrin(n − 3)
+2 × sGrin(n − 4)
}
}
Figure 1: A Slow Algorithm for the “Grindelwald Gaggle Computation” Problem
The method should not be public but should be accessible by other classes in the
cpsc331.assignment1 package.
• The main method should read its input from the command line. If either the number
of inputs is incorrect, or the input is not an integer, then it should return the message
Gadzooks! One integer input is required.
If there is a single integer input, but it is negative, then the main method should
return the message
Gadzooks! The integer input cannot be negative.
Otherwise —- when given a non-negative integer input n – it should report the
corresponding Grindelwald number Gn.
A few sample runs of the program should therefore look like the following.
> java cpsc331.assignment1.SGrindelwald 0
> 1
3
> java cpsc331.assignment1.SGrindelwald 1
> 2
> java cpsc331.assignment1.SGrindelwald 2
> 3
> java cpsc331.assignment1.SGrindelwald 3
> 4
> java cpsc331.assignment1.SGrindelwald 4
> 5
> java cpsc331.assignment1.SGrindelwald -1
> Gadzooks! The input integer cannot be negative.
> java cpsc331.assignment1.SGrindelwald
> Gadzooks! One integer input is required.
> java cpsc331.assignment1.SGrindelwald xyz
> Gadzooks! One integer input is required.
The following files are available and can be used before you submit your program for
assessment:
• test sGrin.java: A program that can be executed using JUnit to perform unit
tests of the method sGrin. See Java Development Exercise #5 for information
about how to use this.
• test SGrindelwald.sh: A Unix shell script that can be used to perform testing
of the main method. See Java Development Exercise #6 for information about how
to use this.
4. Let TsGrin(n) be the number of steps included in the execution of the algorithm, sGrin,
shown in Figure 1 on input n, for a non-negative integer n — assuming that the uniform
cost criterion is used to define this and the only steps counted are the numbered steps
shown in Figure 1.
Give a recurrence for TsGrin(n).
Note: If your answer is correct then you should be able to use your recurrence to show
that TsGrin(0) = 2, TsGrin(1) = 3, TsGrin(2) = 4, TsGrin(3) = 5, TsGrin(4) = 16, and
TsGrin(5) = 30.
5. Use your recurrence to prove that TsGrin(n) ≥
3
2
n
for every non-negative integer n.
Thus the number of steps executed by the sGrin algorithm is at least exponential in its
input.
An Asymptotically Faster Algorithm for This Computation
After a discovery of what is considered when solving the above problem, Ms. Granger realized
that this algorithm is far too slow to be used on Muggles computers to compute the n
th Grindelwald number, Gn, when n is very large. Indeed, one should not even try to use it to compute
G1000.
Happily, Hermione was familiar with one of the Muggles arts that you will learn about in
CPSC 413 — Dynamic Programming.
1 Another algorithm for the “Grindelwald Gaggle Computation” problem, making use of this technique, is shown in Figure 2 on page 6. This algorithm
is not recursive; instead, it creates and accesses an array.
6. State a loop invariant for the while loop at lines 15–19 of this algorithm.
For full marks (or even very many part marks) your answer should have the following
properties:
• It is correct. That is it really is a loop invariant for this while loop.
• It is complete enough to be used to establish the partial correctness of this algorithm.
• It is not necessary to make any guesses about the value of Gn, for n ≥ 4, in order
to see that this is the case.
7. Prove that your answer for the previous question really is a loop invariant for the while
loop in this algorithm.
8. Use this to prove that this algorithm is partially correct.
9. State a bound function for the while loop in this algorithm and prove that your answer
is correct.
10. Prove that if this algorithm is executed when the precondition for the “Grindelwald Gaggle
Computation” problem is satisfied, and the while loop is reached and executed, then
the execution of the loop eventually ends.
11. Using what has been proved so far, complete a proof that the fGrin algorithm correctly
solves the “Grindelwald Gaggle Computation” problem.
12. Let TfGrin(n) be the number of steps executed by the algorithm fGrin when executed
on input n, for a natural number n — as defined using the Uniform Cost Criterion (and
1Ms. Granger wonders why Muggles insist on giving such confusing names to simple ideas — like storing a
solution for an instance of a problem and reusing this solution, instead of solving the same instance of the problem
over and over again, as if you hadn’t already solved it before!
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integer fGrin (integer n) {
// Assertion: A non-negative integer n has been given as input.
1. if (n == 0) {
2. return 1
3. } else if (n == 1) {
4. return 2
5. } else if (n == 2) {
6. return 3
7. } else if (n == 3) {
8. return 4
} else {
9. int[] G := new int[n + 1]
10. G[0] := 1
11. G[1] := 2
12. G[2] := 3
13. G[3] := 4
14. int i := 4
15. while (i ≤ n) {
16. if (i mod 2 == 0) {
17. G[i] := 2 × G[i − 1] − 2 × G[i − 3] + G[i − 4]
} else {
18. G[i] := G[i − 1] + 3 × G[i − 2] − 5 × G[i − 3] + 2 × G[i − 4]
}
19. i := i + 1
}
20. return G[n]
}
}
Figure 2: Another Algorithm for the “Grindelwald Gaggle Computation” Problem
counting only the numbered steps shown in Figure 2) — so that, for example, TfGrin(0) =
2, because the steps at lines 1 and 2 are carried out when this algorithm is executed on
input 0.
Using techniques introduce in this course, give an upper bound for TfGrin(n), as a func6
tion of n, that is as precise as you can. Your final answer should be “in closed form”, so
that it does not include any summations or recurrences.
13. Write a Java program, FGrindelwald.java, that satisfies the following properties:
• It is part of the cpsc331.assignment1 package.
• Integer inputs and outputs are represented using Java’s int data type.
• It includes a method, fGrin, which receives an integer n as input and which computes the n
th Grindelwald number if n ≥ 0, using the algorithm which has now been
analyzed, but which solves the more general “Extended Grindelwald Gaggle Computation” problem that has also been defined. The method should not be public but
should be accessible by other classes in the cpsc331.assignment1 package.
• The main method should read its input from the command line. If either the number
of inputs is incorrect, or the input is not an integer, then it should return the message
Gadzooks! One integer input is required.
If there is a single integer input, but it is negative, then the main method should
return the message
Gadzooks! The integer input cannot be negative.
Otherwise —- when given a non-negative integer input n – it should report the
corresponding Grindelwald number Gn.
A few sample runs of the program should therefore look like the following.
> java cpsc331.assignment1.FGrindelwald 0
> 1
> java cpsc331.assignment1.FGrindelwald 1
> 2
> java cpsc331.assignment1.FGrindelwald 2
> 3
> java cpsc331.assignment1.FGrindelwald 3
> 4
> java cpsc331.assignment1.FGrindelwald 4
> 5
> java cpsc331.assignment1.FGrindelwald -1
> Gadzooks! The input integer cannot be negative.
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> java cpsc331.assignment1.FGrindelwald
> Gadzooks! One integer input is required.
> java cpsc331.assignment1.FGrindelwald xyz
> Gadzooks! One integer input is required.
The following files are available and can be used before you submit your program for
assessment:
• test fGrin.java: A program that can be executed using JUnit to perform unit
tests of the method sGrin.
• test FGrindelwald.sh: A Unix shell script that can be used to perform testing
of the main method.
Finding a Closed Form
14. Find an expression for the n
th Grindelwald number, as a function of n, in closed form —
so that it does not include a summation or recurrence — and prove that your answer is
correct.
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