Question

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Show that the number of nodes of both the fixed-tuple and variable-tuple state-space trees for the sum of subsets problem are exponential in n.

Answer 1

Suppose we find the maximum element in both of the roughly n sized partitions of an n-element n>=1 list. Then in order to find the maximum element of the entire list we simply need to see which of the two maximum elements is the larger, and which of the two minimums is the smaller. We assume that in a1-element list the sole element is both the maximum and the minimum element.

procedure maxmin (L[0...n-1] of numbers) ->(max)

begin

      if(n = = 0)

        return (L[0])

     else if (n = = 1)

       if( L[0] < L[1])

              return (L[1])

        else

              return (L[0])

end

Let be the number of comparisons performed by the maxmin procedure. When n = 0 clearly there are no comparisons. Thus we have T(0) = 0. Similarly, T(1) = 1. Otherwise when n > 1clearly

          T(n) = 2T(n)+1

Since maxmin performs one recursive calls on partitions of roughly the total size of the list and then makes two further comparisons to sort out the max for the entire list.