Determine a recurrence relation for the divide-and-conquer sum-computation algorithm. The problem is computing the sum of n numbers. This algorithm divides the problem into two instances of the same problem: to compute the sum of the first ⌊n/2⌋ numbers and compute the sum of the remaining ⌊n/2⌋ numbers. Once each of these two sums is computed by applying the same method recursively, we can add their values to get
Determine a recurrence relation for the divide-and-conquer sum-computation algorithm. The problem is computing the sum of n numbers. This algorithm divides the problem into two instances of the same problem: to compute the sum of the first ⌊n/2⌋ numbers and compute the sum of the remaining ⌊n/2⌋ numbers. Once each of these two sums is computed by applying the same method recursively, we can add their values to get
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Determine a recurrence relation for the divide-and-conquer sum-computation
![### Recurrence Relations
In this image, we see four different recurrence relations, which are commonly used in the analysis of algorithms to describe the runtime of recursive functions. Each recurrence relation expresses the runtime \( T(n) \) of an algorithm in terms of the runtime on a smaller input size, often \( T(n/2) \). Here we detail each of these recurrence relations:
1. **Recurrence Relation 1**:
\[
T(n) = T(n/2) + 2
\]
This relation suggests that the runtime for input size \( n \) is equivalent to the runtime for half the input size \( n/2 \) plus a constant time of 2.
2. **Recurrence Relation 2**:
\[
T(n) = T(n/2) + 1
\]
This relation is similar to the previous one but adds a constant time of 1 instead.
3. **Recurrence Relation 3**:
\[
T(n) = 2T(n/2) + 1
\]
This relation indicates that the runtime for size \( n \) is equal to twice the runtime of the input size \( n/2 \) plus a constant time of 1.
4. **Recurrence Relation 4**:
\[
T(n) = 2T(n/2) + 2
\]
Here, the runtime for input size \( n \) is twice the runtime for size \( n/2 \), plus a constant time of 2.
These recurrence relations help in determining the time complexity of algorithms, particularly those that employ a divide-and-conquer approach. By solving these recurrences using techniques such as the Master Theorem, Iteration Method, or Recurrence Tree Method, one can determine the overall asymptotic complexity of the algorithm.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc8241d13-b770-4dad-946c-6942db4056c7%2F9b8b5fc0-64e6-492a-9e2f-d9e376367d5a%2Fnmff307_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Recurrence Relations
In this image, we see four different recurrence relations, which are commonly used in the analysis of algorithms to describe the runtime of recursive functions. Each recurrence relation expresses the runtime \( T(n) \) of an algorithm in terms of the runtime on a smaller input size, often \( T(n/2) \). Here we detail each of these recurrence relations:
1. **Recurrence Relation 1**:
\[
T(n) = T(n/2) + 2
\]
This relation suggests that the runtime for input size \( n \) is equivalent to the runtime for half the input size \( n/2 \) plus a constant time of 2.
2. **Recurrence Relation 2**:
\[
T(n) = T(n/2) + 1
\]
This relation is similar to the previous one but adds a constant time of 1 instead.
3. **Recurrence Relation 3**:
\[
T(n) = 2T(n/2) + 1
\]
This relation indicates that the runtime for size \( n \) is equal to twice the runtime of the input size \( n/2 \) plus a constant time of 1.
4. **Recurrence Relation 4**:
\[
T(n) = 2T(n/2) + 2
\]
Here, the runtime for input size \( n \) is twice the runtime for size \( n/2 \), plus a constant time of 2.
These recurrence relations help in determining the time complexity of algorithms, particularly those that employ a divide-and-conquer approach. By solving these recurrences using techniques such as the Master Theorem, Iteration Method, or Recurrence Tree Method, one can determine the overall asymptotic complexity of the algorithm.
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