Brent-Kung Parallel Prefix
The Brent-Kung Algorithm is an efficient parallel prefix computation algorithm that minimizes the number of computation steps and communication overhead.
Brent-Kung Algorithm for Prefix Calculation
The Brent-Kung Algorithm is a parallel prefix computation algorithm, commonly used in designing efficient hardware circuits for binary addition, such as parallel adders. It is particularly advantageous due to its logarithmic depth and optimized wiring, making it suitable for scalable and high-speed designs.
Key Features of the Brent-Kung Algorithm
Divide-and-Conquer Approach: The algorithm splits the prefix computation into two main phases:
Reduction Phase: Propagates partial results up the tree.
Propagation Phase: Combines results back down the tree to compute the final prefix values.
Logarithmic Depth: The computation depth is (O(\log n)), where (n) is the number of elements.
Optimized Fan-Out: Reduces the number of connections per node, improving circuit layout efficiency.
Scalability: Ideal for applications requiring prefix computations on large inputs, such as parallel processors or digital circuits.
Algorithm Overview
The Brent-Kung algorithm works on a binary tree structure, performing two phases:
1. Reduction Phase
The algorithm computes partial results in a binary tree structure, where each node depends on the results of its children. This phase calculates intermediate prefix sums in a bottom-up manner.
For an array (A = [a_1, a_2, ..., a_n]):
Compute pairwise partial prefixes: [ P_i = A_1 \oplus A_2 \oplus \ldots \oplus A_i ] ((\oplus) represents the associative operation, e.g., addition, XOR).
2. Propagation Phase
After the reduction phase, the intermediate results are propagated back down the tree to calculate the final prefixes for all elements. Each node updates its value using the results of its parent.
Example Walkthrough
Consider an array (A = [a_1, a_2, a_3, a_4]). The steps are as follows:
Reduction Phase:
Compute intermediate prefixes for pairs of inputs:
Level 1: Combine ([a_1, a_2]) and ([a_3, a_4]).
Level 2: Combine the results of Level 1 to create the root.
Propagation Phase:
Back-propagate results from the root to calculate final prefixes for all elements.
The computation graph is structured to minimize depth and maximize parallelism.
Complexity Analysis
Time Complexity: (O(log n)), as the algorithm has a logarithmic number of levels.
Space Complexity: (O(n)), for storing intermediate results.
Brent-Kung Parallel Prefix Network
Below is the visual representation of the Brent-Kung prefix :
Let’s go step by step to clarify the down-sweep phase of the Brent-Kung Algorithm, focusing on an inclusive prefix sum example. I'll show how the up-sweep results are used to calculate the final prefix sum.
Inclusive Prefix Sum
We compute the prefix sum such that: [ P[i] = A[0] + A[1] + .... + A[i] ]
Input Example
Let the input array be: [ A = [a_0, a_1, a_2, a_3, a_4, a_5, a_6, a_7] ]
We'll use the Brent-Kung Algorithm to compute the inclusive prefix sum step by step.
Step 1: Up-Sweep (Reduce Tree)
In this stage, we compute partial sums, building a binary reduction tree.
Level 0: Combine adjacent pairs:
[ a_1 = a_0 + a_1, a_3 = a_2 + a_3, a_5 = a_4 + a_5, a_7 = a_6 + a_7 ]
Result: [ [a_0, a_1, a_2, a_3, a_4, a_5, a_6, a_7] -> [a_0, a_1, a_2, a_3, a_4, a_5, a_6, a_7] ]
Level 1: Combine results from Level 0:
[ a_3 = a_1 + a_3, a_7 = a_5 + a_7 ]
Result: [ [a_0, a_1, a_2, a_3, a_4, a_5, a_6, a_7] -> [a_0, a_1, a_2, a_3, a_4, a_5, a_6, a_7] ]
Level 2 (Root): Combine results from Level 1:
[ a_7 = a_3 + a_7 ]
Final Up-Sweep Tree (Partial Sums at Each Node): [ [a_0, a_1, a_2, a_3, a_4, a_5, a_6, a_7] ]
At this point, the root (rightmost element, (a_7)) contains the total sum of the array.
Step 2: Down-Sweep
Now, we propagate the partial sums downward to calculate the prefix sums.
Initialize the root with a neutral value for the operator (0 for addition).
Traverse the tree in reverse (from root to leaves), distributing the partial results and computing the final prefix sums.
Detailed Steps:
Level 2 (Root):
The root (a_7) is initialized to (0).
Propagate values to the left and right children:
Left child (a_3 = 0 + a_3).
Right child (a_7 = a_3 + a_7).
Level 1:
Propagate results to children:
(a_1 = a_0 + a_1), (a_5 = a_4 + a_5).
Level 0:
Distribute to the leaves:
Compute (a_0, a_2, a_4, a_6).
Final Prefix Sum Results
After the down-sweep completes, we have: [ P = [a_0, a_0 + a_1, a_0 + a_1 + a_2, a_0 + a_1 + a_2 + a_3, a_0 + a_1 + a_2 + a_3 + a_4, ...] ]
Example with Numbers
Let (A = [1, 2, 3, 4, 5, 6, 7, 8]).
Up-Sweep (Partial Sums):
Level 0: [ [1, 2] -> 3, [3, 4] -> 7, [5, 6] -> 11, [7, 8] -> 15 ] Array: ([1, 3, 3, 7, 5, 11, 7, 15])
Level 1: [ [3, 7] -> 10, [11, 15] -> 26 ] Array: ([1, 3, 3, 10, 5, 11, 7, 26])
Level 2 (Root): [ [10, 26] -> 36 ] Array: ([1, 3, 3, 10, 5, 11, 7, 36])
Down-Sweep:
Initialize Root: Set (a_7 = 0).
Level 2: Propagate values: [ a_3 = 0 + 10, a_7 = 10 + 26 ] Array: ([1, 3, 3, 10, 5, 11, 7, 36])
Level 1: Propagate: [ a_1 = 0 + 3, a_5 = 10 + 11 ]
Level 0: Compute final sums: [ a_0 = 1, a_2 = a_0 + a_2 ]
Reference
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