Generate Shortest Bit Sequence Base -2
Given an Integer X Where X Can Be Negative or Positive. Find the Shortest Bit Sequence That Represent X in a Base -2 System. in a Base -2 System, Given an...
Given an integer X where X can be negative or positive. find the shortest bit sequence that represent X in a base -2 system.
In a base -2 system, given an array A of N bits, the represented integer is: sum of A[i]*(-2 power i) for i = 0..N-1
example:
[1,0,1] = 5
[1,0,0,1] = -7
[1,0,0,1,0,1] = -39
so, given X = 18 the algorithm should return [0, 1, 1, 0, 1]
any idea on how to implement such an algorithm.. such that, given an integer X it returns the shortest bit sequence that represent that integer ?
The only thing that I came up with is brute force search.. starting at bit 0 and calculating all possible sums until one of the sums is equal to X.. which does not look nice!
1 Answer
This is not a full solution, but it will probably give a good idea on how to proceed.
Suppose your base-negative-2 representation contains at most N numbers. What would be the range of numbers it represents?
A few examples:
- length 0 or less: 0
- length 1 or less: 0 ... 1
- length 2 or less: -2 ... 1
- length 3 or less: -2 ... 5
- length 4 or less: -10 ... 5
You probably can notice the recursive rule that determines the above; something like this:
Extend the range by adding or subtracting 2n-1 to the appropriate end of the earlier range
You don't even need a "closed" (in a mathematical sense) formula for this, just a recursive implementation. Extend the range until your target numbers falls inside; then generate your representation.
BTW it's also described in Wikipedia.
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