Chapter 4, Problem 5R

Explanation of Solution

Running time of algorithm “BinarySum()”:

• The algorithm of “BinarySum()” calculates the sum of the elements in the array using binary recursion.
  • The recursive call to “BianrySum()” is made for half of the value of “n”.
  • The maximum number of function instances is “${\text{1+log}}_{\text{2}}\text{n}$”.
  • This algorithm “BinarySum()” uses an additional amount of space that is propositional to the value “${\text{1+log}}_{\text{2}}\text{n}$”  and therefore, the time complexity of this algorithm is “O(n)”.

Explanation:

• The recurrence relation is as follows,

$\begin{array}{c}\text{T}\left(\text{n}\right)\text{=2T}\left(\frac{\text{n}}{\text{2}}\right)\text{+O}\left(\text{1}\right)\\ {\text{=2}}^{\text{2}}\text{T}\left(\frac{\text{n}}{{\text{2}}^{\text{2}}}\right)\text{+O}\left(\text{1}\right)\text{.[1+2]}\\ {\text{=2}}^{\text{3}}\text{T}\left(\frac{\text{n}}{{\text{2}}^{\text{3}}}\right)\text{+O}\left(\text{1}\right){\text{.[2}}^{\text{0}}{\text{+2}}^{\text{1}}{\text{+2}}^{\text{2}}\text{]}\\ \text{=}\mathrm{....}\end{array}$