README.md

Dynamic Programming

All challenges are found here

1. Max Array Sum challenge

The solution is very simple , just initiate 2 zero sums

• $excl is the maximum sum excluding the previous element
• $incl is the maximum sum including the the previous element At the end of the loop return max of incl and excl. More info (here)[https://www.geeksforgeeks.org/maximum-sum-such-that-no-two-elements-are-adjacent/]

2. Decibinary Numbers challenge

This is very hard and complicated question. Here is how I thought about it. According to the question description, a number in the decibinary system should be interpreted with the following equation.

2016_decibinary = 2*2³ + 0 * 2² + 1 * 2¹+ 6 * 2⁰ = (24)_decimal

But there is a problem , 2008 in decibinary also can be interpreted as 24 in decimal.

So we are going to order based on their decimal value , then based on decibinary value

Here is the list of first 20 number.

x	Decibinary	Decimal
1	0	0
2	1	1
3	2	2
4	10	2
5	3	3
6	11	3
7	4	4
8	12	4
9	20	4
10	100	4
11	5	5
12	13	5
13	21	5
14	101	5
15	6	6
16	14	6
17	22	6
18	30	6
19	102	6
20	110	6

I noticed that every number in decimal can be interpreted in many forms in decibinary system, the first one is the decimal value , and the last one is the binary value.

So if we want to know all the forms of number 5 .

    protected function findDebicinaryRepresentation($x)
    {
        $xforms = [$x];
        $digits = 2;
        $divisor = 2;

        while ($divisor <= $x) {
            $xforms = array_merge($xforms, $this->decibinaryFormsOfXInDigits($x, $digits));
            $divisor *= 2;
            $digits += 1;
        }

        return $xforms;
    }

I am trying to find all possible form of using 2 digits in decibinary , then 3 digits . There is no form using 4 digits at the minimum value is 8 which is greater than 5

Finding decibinary value for more than 2 digits can be done recurisvely , take a look at the possible forms of 7.

x	Decibinary	Decimal
21	7	7
22	15	7
23	23	7
24	31	7
25	103	7
26	111	7

For indices 25 and 26 you will find the first digit is 1 that represents 1*2<sup>2</sup> = 4, so the remainer is 3 . You will find that 03 and 11 are the possible decibinary values of 3.

That's how decibinaryFormsOfXInDigits is implemented.

    protected function decibinaryFormsOfXInDigits($x, $digits = 1)
    {
        $forms = [];
        $base = pow(2, $digits - 1);

        for ($i = 1; $i * $base <= $x; $i++) {
            $remainer = $x - ($i * $base);
            if ($digits > 2) {
                $lowerForms = $this->decibinaryFormsOfXInDigits($remainer, $digits - 1);
                foreach ($lowerForms as $f) {
                    $forms[] = (int) ($i . str_pad($f, $digits - 1, '0', STR_PAD_LEFT));
                }
            }
            $forms[] = $i * pow(10, $digits - 1) + ($x - ($i * $base));
        }
        sort($forms);
        return $forms;
    }

The rest is lookup table to find the decibinary value at index $x.

Improvements needed

• Figure better solution to find the value of $x instead of loading all the lookup table till we find $x