Possible combinations of binary - php

Question

The problem statement is as following:
A particular kind of coding which we will refer to as "MysteryCode" is a binary system of encoding in which two successive values, differ at exactly one bit, i.e. the Hamming Distance between successive entities is 1. This kind of encoding is popularly used in Digital Communication systems for the purpose of error correction.
LetMysteryCodes(N)represent the MysteryCode list for N-bits.
MysteryCodes(1) = 0, 1 (list for 1-bitcodes,in this order)
MysteryCodes(2) = 00, 01, 11, 10 (list for 2-bitcodes,in this order)
MysteryCodes(3) =000, 001, 011, 010,110, 111, 101, 100 (list for 3-bitcodes,in this order)
There is a technique by which the list of (N+1) bitcodescan be generated from (N)-bitcodes.
Take the list of N bitcodesin the given order and call itList-N
Reverse the above list (List-N), and name the new reflected list: Reflected-List-N
Prefix each member of the original list (List-N) with 0 and call this new list 'A'
Prefix each member of the new list (Reflected-List-N) with 1 and call this new list 'B'
The list ofcodeswith N+1 bits is the concatenation of Lists A and B.
A Demonstration of the above steps: Generating the list of 3-bitMysteryCodesfrom 2-BitMysteryCodes
2-bit list ofcodes:00, 01, 11, 10
Reverse/Reflect the above list:10, 11, 01, 00
Prefix Old Entries with 0:000, 001, 011, 010
Prefix Reflected List with 1:110, 111, 101, 100
Concatenate the lists obtained in the last two steps:000, 001, 011, 010, 110, 111, 101, 100
Your Task
Your task is to display the last N "MysteryCodes" from the list of MysteryCodes for N-bits. If possible, try to identify a way in which this list can be generated in a more efficient way, than iterating through all the generation steps mentioned above.
More efficient or optimized solutions will receive higher credit.
Input Format
A single integer N.
Output Format
N lines, each of them with a binary number of N-bits. These are the last N elements in the list ofMysteryCodesfor N-bits.
Input Constraints 1 = N = 65
Sample Input 1
1
Sample Output 1
1
Explanation for Sample 1
Since N = 1, this is the (one) last element in the list ofMysteryCodesof 1-bit length.
Sample Input 2
2
Sample Output 2
11
10
Explanation for Sample 2 Since N = 2, these are the two last elements in the list ofMysteryCodesof 2-bit length.
Sample Input 3
3
Sample Output 3
111
101
100
$listN = 25;
$bits = array('0','1');
//check if input is valid or not
if(!is_int($listN))
{
echo "Input must be numeric!";
}
if($listN >= 1 && $listN <=65){
if($listN == 1){
echo '1'; exit;
}
ini_set('memory_limit', -1);
for($i=1; $i<=($listN - 1); $i++){
$reverseBits = array_reverse($bits);
$prefixBit = preg_filter('/^/', '0', $bits);
$prefixReverseBits = preg_filter('/^/', '1', $reverseBits);
$bits = array_merge($prefixBit, $prefixReverseBits);
unset($prefixBit, $prefixReverseBits, $reverseBits);
}
$finalBits = array_slice($bits, -$listN);
foreach($finalBits as $k=>$v){
echo $v."\n";
}
}
else{
echo "Invalid input!";
}
I have tried above solution, but didnt worked for input greater than 20.
for eg. If the input is 21, I got "Couldnt allocate memory" error.
It will be great if somebody figure out the optimized solutions...

Answer 1

The numbers follow a pattern which I transformed to below code.
Say given number is N
then create a N x N matrix and fill it's first column with 1's
and all other cells with 0's
Start from rightmost column uptil 2nd column.
For any column X start from bottom-most row and fill values like below:
Fill 2^(N - X + 1)/2 rows with 0's.
Fill 2^(N - X + 1) rows with 1's and then 0's alternatively.
Repeat step 2 till we reach topmost row.
Print the N x N matrix by joining the values in each row.
<?php
$listN = 3;
$output = [];
for ($i = 0; $i < $listN; $i++) {
$output[$i] = [];
for ($j = 0; $j < $listN; $j++) {
$output[$i][$j] = 0;
}
}
$output[$listN - 1][0] = 1;
for ($column = 1; $column < $listN; $column++) {
$zeroFlag = false;
for ($row = $listN - 1; $row >= 0;) {
$oneZero = 1;
if (!$zeroFlag) {
for ($k = 1; $k <= pow(2, $column) / 2 && $row >= 0; $k++) {
$output[$row][$listN - $column] = 0;
$row--;
$zeroFlag = true;
}
}
for ($k = 1; $k <= pow(2, $column) && $row >= 0; $k++) {
$output[$row][$listN - $column] = $oneZero;
$row--;
}
$oneZero = 0;
for ($k = 1; $k <= pow(2, $column) && $row >= 0; $k++) {
$output[$row][$listN - $column] = $oneZero;
$row--;
}
}
}
for ($i = 0; $i < $listN; $i++) {
$output[$i][0] = 1;
}
for ($i = 0; $i < $listN; $i++) {
print(join('', $output[$i]));
print("\n");
}