Fibonacci

Motivation

It is important to always reduce the work, then parallelize if possible. In this studio you will build a beautiful, elegant solution to Fibonacci which can be easily parallelized yielding off-the-charts ideal parallelism. Sadly, it performs terribly since the work is exponential. It is far better to build a linear or log time algorithm whether or not it can be parallelized.

We will gain experience with the future construct.

We will convert recurrence relations into code.

Additionally, (and of minor importance) we gain experience working with the BigInteger class.

Background

The fibonacci sequence is a mathematical concept often used in computer science as a means to demonstrate iteration and recursion. Although you should be familiar with it from CSE 131, we will use the fibonacci sequence to demonstrate memoization and dynamic programming. Follow these links for a quick recap on memoization, dynamic programming, and the fibonacci sequence.

• Video:
• Video: Fibonacci Sunflowers

Javadocs

V5

future(body)

interface Future<V>.

get()

BigInteger

BigInteger.ZERO

BigInteger.ONE

bi.add(other)

bi.multiply(other)

Studio (Required)

RecurrenceRelationSequentialFibonacciCalculator

Work:

$O(\Phi ^{n})$

class:	RecurrenceRelationSequentialFibonacciCalculator.java
methods:	fibonacci
package:	fibonacci.studio
source folder:	student/src/main/java

implement a recursive method for:

recurrence relation

$F_{n}=F_{n-1}+F_{n-2}$

seed values

$F_{0}=0$

$F_{1}=1$

RecurrenceRelationParallelFibonacciCalculator

Work:	$O(\Phi ^{n})$
CPL:	$O(n)$
Ideal Parallelism:	${\dfrac {\Phi ^{n}}{n}}$

class:	RecurrenceRelationParallelFibonacciCalculator.java
methods:	fibonacci
package:	fibonacci.studio
source folder:	student/src/main/java

Use futures to add parallelism to the recurrence relation solution.

MemoizationSequentialFibonacciCalculator

Work:

$O(n)$

class:	MemoizationSequentialFibonacciCalculator.java
methods:	fibonacciMemo
package:	fibonacci.studio
source folder:	student/src/main/java

Implement the recurrence relation algorithm but employ memoization for the win.

	@Override
	public BigInteger fibonacci(int n) {
		BigInteger[] memos = new BigInteger[n + 1];
		return fibonacciMemo(memos, n);
	}

Warning: The entire point of memoization is to avoid recalculating the same computation. Be sure to store the value for each calculation as it is made even if that means splitting up a line of code.

DynamicIterativeSequentialFibonacciCalculator

Work:

$O(n)$

class:	DynamicIterativeSequentialFibonacciCalculator.java
methods:	fibonacci
package:	fibonacci.studio
source folder:	student/src/main/java

LinearRecurrenceSequentialFibonacciCalculator

Work:

$O(\log {}n)$

class:	LinearRecurrenceSequentialFibonacciCalculator.java
methods:	fibonacci
package:	fibonacci.studio
source folder:	student/src/main/java

recurrence relation

if n is odd

$k={\dfrac {n+1}{2}}$

$F_{n}=(F_{k}\times F_{k})+(F_{k-1}\times F_{k-1})$

else

$k={\dfrac {n}{2}}$

$F_{n}=((2\times F_{k-1})+F_{k})\times F_{k}$

seed values

$F_{0}=0$

$F_{1}=1$

Fun (Optional)

RecurrenceRelationParallelWithThresholdFibonacciCalculator

Similar to #RecurrenceRelationParallelFibonacciCalculator but with a threshold in place to prevent creating too many tasks for the number of processors.

MemoizationParallelFibonacciCalculator

Create all of your futures [0..N] up front and then call get on memo[n].

RoundPhiToTheNOverSqrt5SequentialFibonacciCalculator

${\dfrac {\Phi ^{n}}{\sqrt {5}}}$

Testing Your Solution

Correctness

class:	FibonacciTestSuite.java
package:	fibonacci.studio
source folder:	testing/src/test/java

Performance

class:	FibonacciTiming.java
package:	fibonacci.studio
source folder:	src/main/java

class:	FibonacciIterations.java
package:	fibonacci.studio
source folder:	src/main/java