The Floyd-Warshall algorithm for all-pair shortest paths computation is based on

A complete binary min-heap is made by including each integer in $$[1,1023]$$ exactly once. The depth of a node in the heap is the length of the path from the root of the heap to that node. Thus, the root is at depth $$0.$$ The maximum depth at which integer $$9$$ can appear is ___________.

The given diagram shows the flowchart for a recursive function $$A(n).$$ Assume that all statements, except for the recursive calls, have $$O(1)$$ time complexity. If the worst case time complexity of this function is $$O\left( {{n^\alpha }} \right),$$ then the least possible value (accurate up to two decimal positions) of $$\alpha $$ is ____________ .

Let $${A_1},{A_2},{A_3},$$ and $${A_4}$$ be four matrices of dimensions $$10 \times 5,\,\,5 \times 20,\,\,20 \times 10,$$ and $$10 \times 5,\,$$ respectively. The minimum number of scalar multiplications required to find the product $${A_1}{A_2}{A_3}{A_4}$$ using the basic matrix multiplication method is ______________.