Step by Step solution about How to resolve the algorithm Boustrophedon transform step by step in the Python programming language

This Python code explores different number sequences and applies a mathematical operation called the Boustrophedon transform to them. Let's break down the code step by step:

1. Importing Libraries:

   • from math import factorial: Imports the factorial function from the math module.
   • from itertools import islice: Imports the islice function from the itertools module.
   • from sys import setrecursionlimit: Imports the setrecursionlimit function from the sys module.

2. Recursion Limit:

   • setrecursionlimit(1000000): Sets the maximum recursion depth to 1 million to handle deeper recursive calls in subsequent code.

3. Generator Functions:

   • gen_one_followed_by_zeros(): A generator function that yields 1 followed by an infinite sequence of 0s.
   • gen_ones(): A generator function that yields an infinite sequence of 1s.
   • gen_alternating_signs(): A generator function that yields alternating 1s and -1s.
   • gen_primes(): A generator function that yields an infinite sequence of prime numbers.
   • gen_fibonacci(): A generator function that yields an infinite sequence of Fibonacci numbers.
   • gen_factorials(): A generator function that yields an infinite sequence of factorial numbers.

4. compressed Function:

   • compressed(n): Takes an integer n and returns a compressed representation of it as a string, showing the first 20 digits, the last 20 digits, and the total number of digits.

5. boustrophedon Function:

   • boustrophedon(a): This function takes a list a and applies the Boustrophedon transform to it. The Boustrophedon transform involves building a 2D array (or matrix) representing the sequence a and then summing elements along diagonals. The function returns a list containing the sums of these diagonals.
   • The function initializes a cache cache to memoize the results and an output list b to store the transformed sequence. It then iterates over the elements of a and calculates the Boustrophedon transform using a recursive function T.

6. Main Loop:

   • The main loop iterates over a dictionary funcs containing different title-function pairs:
     • Each title corresponds to a specific sequence generated by the corresponding function.
     • The loop generates 1000 elements from each sequence using islice.
     • It then applies the Boustrophedon transform to each sequence and prints the first 15 transformed elements along with the 1000th element in a compressed format.

In summary, this code explores several different number sequences, applies the Boustrophedon transform to them, and prints the results. The Boustrophedon transform creates a 2D representation of a sequence and calculates the sums along diagonals, yielding a new sequence. The code demonstrates the behavior of this transform on various types of number sequences.

# bt_transform.py by Xing216
from math import factorial
from itertools import islice
from sys import setrecursionlimit
setrecursionlimit(1000000)
def gen_one_followed_by_zeros():
    yield 1
    while True:
        yield 0
def gen_ones():
    while True:
        yield 1
def gen_alternating_signs():
    s = 1
    while True:
        yield s
        s *= -1
def gen_primes():
    """Code by David Eppstein, UC Irvine, 28 Feb 2002 http://code.activestate.com/recipes/117119/"""
    D = {}
    q = 2
    while True:
        if q not in D:
            yield q
            D[q * q] = [q]
        else:
            for p in D[q]:
                D.setdefault(p + q, []).append(p)
            del D[q]
        q += 1
def gen_fibonacci():
    a=0
    b=1
    while True:
        yield b
        a,b= b,a+b
def gen_factorials():
    f = 0
    while True:
        yield factorial(f)
        f += 1
def compressed(n):
    strn = str(n)
    return f"{strn[:20]}...{strn[-20:]} ({len(strn)} digits)"
def boustrophedon(a):
    # Copied from the Wren Solution
    k = len(a)
    cache = [None] * k
    for i in range(k): cache[i] = [0] * k
    b = [0] * k
    b[0] = a[0]
    def T(k,n):
        if n == 0: return a[k]
        if cache[k][n] > 0: return cache[k][n]
        cache[k][n] = T(k,n-1) + T(k-1,k-n)
        return cache[k][n]
    for n in range(1,k):
        b[n] = T(n,n)
    return b
funcs = {"One followed by an infinite series of zeros:":gen_one_followed_by_zeros,
         "Infinite sequence of ones:": gen_ones, 
         "Alternating 1, -1, 1, -1:": gen_alternating_signs, 
         "Sequence of prime numbers:": gen_primes, 
         "Sequence of Fibonacci numbers:": gen_fibonacci, 
         "Sequence of factorial numbers:": gen_factorials}
for title, func in funcs.items():
    x = list(islice(func(), 1000))
    y = boustrophedon(x)
    print(title)
    print(y[:15])
    print(f"1000th element: {compressed(y[-1])}\n")