Explain meet in the middle time complexity [closes #57]

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Antti H S Laaksonen 2017-12-10 11:25:35 +02:00
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@ -749,10 +749,10 @@ because $S_A$ contains the sum $6$,
$S_B$ contains the sum $9$, and $6+9=15$.
This corresponds to the solution $[2,4,9]$.
The time complexity of the algorithm is $O(2^{n/2})$,
because both lists $A$ and $B$ contain about $n/2$ numbers
and it takes $O(2^{n/2})$ time to calculate the sums of
their subsets to lists $S_A$ and $S_B$.
After this, it is possible to check in
$O(2^{n/2})$ time if the sum $x$ can be created
from $S_A$ and $S_B$.
We can implement the algorithm so that
its time complexity is $O(2^{n/2})$.
First, we generate \emph{sorted} lists $S_A$ and $S_B$,
which can be done in $O(2^{n/2})$ time using a merge-like technique.
After this, since the lists are sorted,
we can check in $O(2^{n/2})$ time if
the sum $x$ can be created from $S_A$ and $S_B$.