How to resolve the algorithm Boustrophedon transform step by step in the Java programming language
How to resolve the algorithm Boustrophedon transform step by step in the Java programming language
Table of Contents
Problem Statement
A boustrophedon transform is a procedure which maps one sequence to another using a series of integer additions. Generally speaking, given a sequence:
(
a
0
,
a
1
,
a
2
, … )
{\displaystyle (a_{0},a_{1},a_{2},\ldots )}
, the boustrophedon transform yields another sequence:
(
b
0
,
b
1
,
b
2
, … )
{\displaystyle (b_{0},b_{1},b_{2},\ldots )}
, where
b
0
{\displaystyle b_{0}}
is likely defined equivalent to
a
0
{\displaystyle a_{0}}
. There are a few different ways to effect the transform. You may construct a boustrophedon triangle and read off the edge values, or, may use the recurrence relationship: The transformed sequence is defined by
b
n
=
T
n , n
{\displaystyle b_{n}=T_{n,n}}
(for
T
2 , 2
{\displaystyle T_{2,2}}
and greater indices). You are free to use a method most convenient for your language. If the boustrophedon transform is provided by a built-in, or easily and freely available library, it is acceptable to use that (with a pointer to where it may be obtained).
If your language supports big integers, show the first and last 20 digits, and the digit count of the 1000th element of each sequence.
Let's start with the solution:
Step by Step solution about How to resolve the algorithm Boustrophedon transform step by step in the Java programming language
Boustrophedon Transform in Java
The code you provided implements the Boustrophedon transform, which is a mathematical function that converts an integer sequence into a new sequence with a zigzag pattern. The transform is defined as follows:
T(k, n) = S(k) + T(k - 1, k - n)
where:
- T(k, n) is the Boustrophedon transform of the kth term in the original sequence, using the nth zigzag
- S(k) is the original sequence at index k
- n is the number of rows or columns in the zigzag pattern
Implementation Details
The BoustrophedonTransform class contains the main method and several helper methods:
listPrimeNumbersUpTo: Computes a list of prime numbers up to the given limit using the Sieve of Eratosthenes.fibonacci: Computes the nth Fibonacci number using a recursive algorithm with memoization.factorial: Computes the factorial of the given number using a loop with memoization.compress: Formats a BigInteger into a string with a maximum length specified by thesizeparameter.display: Prints the title and the first 15 elements of the transformed sequence, followed by the 1,000th element.
BoustrophedonIterator Class
The BoustrophedonIterator class represents an iterator that generates the Boustrophedon transform. It has the following methods:
- Constructor: Initializes the iterator with the original sequence to be transformed.
next: Computes and returns the next element in the transform.transform: Computes the transform for the givenkandnvalues using memoization.
Usage
In the main method, the code demonstrates the Boustrophedon transform for several different sequences:
- One followed by an infinite series of zeros (A000111)
- An infinite series of ones (A000667)
- Alternating 1, -1, 1, -1 (A062162)
- Sequence of prime numbers (A000747)
- Sequence of Fibonacci numbers (A000744)
- Sequence of factorial numbers (A230960)
For each sequence, it prints the first 15 elements and the 1,000th element of the transform.
Overall
This code provides a comprehensive implementation of the Boustrophedon transform and uses different sequences to demonstrate its application. The use of memoization optimizes the computation of the transform, making it efficient even for large integers.
Source code in the java programming language
import java.math.BigInteger;
import java.util.ArrayList;
import java.util.HashMap;
import java.util.List;
import java.util.Map;
import java.util.function.Function;
public final class BoustrophedonTransform {
public static void main(String[] args) {
listPrimeNumbersUpTo(8_000);
Function<Integer, BigInteger> oneOne = x -> ( x == 0 ) ? BigInteger.ONE : BigInteger.ZERO;
Function<Integer, BigInteger> alternating = x -> ( x % 2 == 0 ) ? BigInteger.ONE : BigInteger.ONE.negate();
display("One followed by an infinite series of zeros -> A000111", oneOne);
display("An infinite series of ones -> A000667", n -> BigInteger.ONE);
display("(-1)^n: alternating 1, -1, 1, -1 -> A062162", alternating);
display("Sequence of prime numbers -> A000747", n -> primes.get(n));
display("Sequence of Fibonacci numbers -> A000744", n -> fibonacci(n));
display("Sequence of factorial numbers -> A230960", n -> factorial(n));
}
private static void listPrimeNumbersUpTo(int limit) {
primes = new ArrayList<BigInteger>();
primes.add(BigInteger.TWO);
final int halfLimit = ( limit + 1 ) / 2;
boolean[] composite = new boolean[halfLimit];
for ( int i = 1, p = 3; i < halfLimit; p += 2, i++ ) {
if ( ! composite[i] ) {
primes.add(BigInteger.valueOf(p));
for ( int a = i + p; a < halfLimit; a += p ) {
composite[a] = true;
}
}
}
}
private static BigInteger fibonacci(int number) {
if ( ! fibonacciCache.keySet().contains(number) ) {
if ( number == 0 || number == 1 ) {
fibonacciCache.put(number, BigInteger.ONE);
} else {
fibonacciCache.put(number, fibonacci(number - 2).add(fibonacci(number - 1)));
}
}
return fibonacciCache.get(number);
}
private static BigInteger factorial(int number) {
if ( ! factorialCache.keySet().contains(number) ) {
BigInteger value = BigInteger.ONE;
for ( int i = 2; i <= number; i++ ) {
value = value.multiply(BigInteger.valueOf(i));
}
factorialCache.put(number, value);
}
return factorialCache.get(number);
}
private static String compress(BigInteger number, int size) {
String digits = number.toString();
final int length = digits.length();
if ( length <= 2 * size ) {
return digits;
}
return digits.substring(0, size) + " ... " + digits.substring(length - size) + " (" + length + " digits)";
}
private static void display(String title, Function<Integer, BigInteger> sequence) {
System.out.println(title);
BoustrophedonIterator iterator = new BoustrophedonIterator(sequence);
for ( int i = 1; i <= 15; i++ ) {
System.out.print(iterator.next() + " ");
}
System.out.println();
for ( int i = 16; i < 1_000; i++ ) {
iterator.next();
}
System.out.println("1,000th element: " + compress(iterator.next(), 20) + System.lineSeparator());
}
private static List<BigInteger> primes;
private static Map<Integer, BigInteger> fibonacciCache = new HashMap<Integer, BigInteger>();
private static Map<Integer, BigInteger> factorialCache = new HashMap<Integer, BigInteger>();
}
final class BoustrophedonIterator {
public BoustrophedonIterator(Function<Integer, BigInteger> aSequence) {
sequence = aSequence;
}
public BigInteger next() {
index += 1;
return transform(index, index);
}
private BigInteger transform(int k, int n) {
if ( n == 0 ) {
return sequence.apply(k);
}
Pair pair = new Pair(k, n);
if ( ! cache.keySet().contains(pair) ) {
final BigInteger value = transform(k, n - 1).add(transform(k - 1, k - n));
cache.put(pair, value);
}
return cache.get(pair);
}
private record Pair(int k, int n) {}
private int index = -1;
private Function<Integer, BigInteger> sequence;
private Map<Pair, BigInteger> cache = new HashMap<Pair, BigInteger>();
}
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