Final answer:
The time complexity of the given recurrence relation is derived by analyzing its recursive calls and the work done in each call. It can be expressed as a sum of the work done at each level in the recursion tree, which is determined by the size of the input. The time complexity can be further simplified to obtain an exact expression using logarithmic properties and algebraic identities.
Step-by-step explanation:
The time complexity of the given recurrence relation T(n) = 12T(n/3) + n³ can be determined by analyzing its recursive calls and the work done in each call. In each recursive call, the input size is divided by 3, and the recurrence relation is applied 12 times. Additionally, each call involves a constant amount of work represented by n³. Therefore, the time complexity can be expressed as:
T(n) = 12T(n/3) + n³
Expanding this recurrence relation using the iterative tree method, we can observe that the work done in each level of recursion is n³, n³/3, n³/9, n³/27, and so on. The number of levels in the recursion tree is determined by the value of n in the base case (which is 1 in this case).
Therefore, the time complexity of the given recurrence relation is represented by the sum of the work done at each level, which can be calculated as:
At the 1st level: n³ + n³ + n³ + ... (12 times) = 12n³
At the 2nd level: (n/3)³ + (n/3)³ + (n/3)³ + ... (12 times) = 12(n/3)³
At the 3rd level: (n/9)³ + (n/9)³ + (n/9)³ + ... (12 times) = 12(n/9)³
...
At the i-th level: (n/3^i)³ + (n/3^i)³ + (n/3^i)³ + ... (12 times) = 12(n/3^i)³
The number of levels in the recursion tree is determined by the value of n in the base case (which is 1 in this case). Therefore, to find the time complexity, we need to consider the number of levels in the recursion tree. Since the size of the input is divided by 3 in each recursive call until it reaches the base case of size 1, we can express the number of levels as log base 3 of n:
log₃(n)
Each level of recursion involves 12 recursive calls, and the amount of work done in each level is given by 12(n/3^i)³. Therefore, the total work done can be calculated by summing up the work done at each level in the recursion tree:
T(n) = 12n³ + 12(n/3)³ + 12(n/9)³ + ... + 12(n/3^log₃(n))³
Simplifying this expression, we can observe that the sum of a geometric series is:
a(1 - r^k)/(1 - r)
Where a represents the first term, r represents the common ratio, and k represents the number of terms. In this case, the first term is 12n³, the common ratio is 1/3, and the number of terms is log₃(n) + 1. Substituting these values, we get:
T(n) = 12n³ * ((1 - (1/3)^(log₃(n) + 1))/(1 - 1/3))
This expression can be further simplified using logarithmic properties and algebraic identities to obtain an exact expression for the time complexity of the given recurrence relation.