Given a list of integers $[a_1, a_2, \dots a_n]$, I want to find the number of $n$-tuples $(x_1,\dots,x_n)$ of integers such that the following three conditions are satisfied:
- $x_1 \oplus x_2 \oplus \dots \oplus x_n = 0$.
- $x_k \in [0,a_k]$ for each $k$.
- $x_k=a_k$ for at least one $k$.
For instance, if the given list is [1,2,3], the answer is 4, and the specific solutions are
[0,2,2],[1,0,1],[1,1,0], and[1,2,3].
A brute-force approach would just compute the Cartesian product of all $x_k$s or simply count through the whole number space starting with [0,0,0]. But how can I tackle this problem with a more sophisticated algorithm? What paradigm is a good choice? Dynamic programming? If I only consider condition 2, then I could think of denoting the sub-problem by
$$E[n,i]$$
Where $n$ defines the number of integers and $i$ the current slot. Also given the caps $[m_1, m_2, \dots,m_n]$ then the number of combinations could be computed with:
$$ \begin{align*} E[0,0] & = 0 \\ E[0,i] & = 1 + m_i \\ E[n,i] & = (1 + m_i) \cdot E[n, i-1] \\ E[n,i] & = \text{invalid if } i + 1 > n \end{align*} $$
But how can I integrate the XOR constraint here?
On the other hand it seems this is a classical Constraint Satisfaction Problem which could be backtracked.
[0, 0, 0]and[0, 1, 1], both, do not have one number slot, which was kept untouched. There is no $x_k$ in it, which sill has the original value $a_k$. $\endgroup$ – Mahoni Feb 3 '13 at 12:40