OpenMP Reduction Technique

Reduction in OpenMP

• Reduction is a parallel computing technique

• Multiple threads compute partial results independently

• Partial results are combined into a single final result

• Used for operations like sum, max, min, product, and logical AND/OR

• The reduction(operator: variable) clause lets each thread maintain a private copy of the variable

• Private copies are merged using the specified operator at the end of the parallel region

Syntax of Reduction

• #pragma omp parallel for reduction(operator:variable) is the basic structure

• operator specifies how values should be combined (e.g., +, *, max, min)

• variable stores the final result

Common Reduction Operators in OpenMP

• Sum (+) uses reduction(+:sum)
• Product (*) uses reduction(*:prod)
• Subtraction (-) uses reduction(-:diff) (not recommended)
• Bitwise AND (&) uses reduction(&:bit_and_result)
• Bitwise OR (|)
• Bitwise XOR (^) uses reduction(^:bit_xor_result)
• Logical AND (&&) uses reduction(&&:logical_and_result)
• Maximum value (max) uses reduction(max:max_value)
• Minimum value (min) uses reduction(min:min_value)

Maximum Reduction

• Each thread finds a partial maximum
• The reduction(max:max_value) clause ensures the highest value is retained

Minimum Reduction

• The reduction(min:min_value) clause ensures the smallest value is kept

Important Notes

• The reduction variable should be initialized before the parallel region

• Reduction variables must be scalar (e.g., int, float, double); arrays cannot be directly reduced

• Reduction avoids race conditions

• OpenMP automatically optimizes reduction, which reduces synchronization overhead

Binary Tree-Based Approach

• A tree-based approach is a parallel technique that improves performance

Time Complexity of Tree-Based Reduction

• For an input array of size n:
  • The number of levels in the reduction tree is O(log n)
  • Computations are performed in parallel at each level
  • Each step reduces the problem size by half, leading to O(log n) complexity
• O(n) is the time complexity of a sequential algorithm

Broadcast in OpenMP

• Broadcasting distributes a value from a single thread (typically the master thread) to all other threads in a parallel region

• Broadcasting ensures all threads share a single consistent value efficiently, unlike reduction where multiple values ​are aggregated

• OpenMP doesn't have a direct broadcast function like MPI

• Broadcasting can be achieved using:

  • #pragma omp single ensures only one thread initializes the value
  • #pragma omp barrier ensures all threads wait before proceeding
  • #pragma omp master ensures only the master thread executes the code

Broadcasting Using #pragma omp single

• Only one thread (not necessarily the master) executes a specific section of code using this directive

• #pragma omp single ensures that only one thread initializes the variable

• #pragma omp barrier ensures that all threads wait until the broadcast value is ready before using it

Broadcasting Using #pragma omp master

• The master directive ensures that only the master thread (thread 0) executes the code

Key Differences Between single and master

• With single, any one thread executes the code and an implicit barrier (unless nowait is used) exists
• With master, only the master thread (thread 0) executes the code, and there is no implicit barrier, therefore a manual #pragma omp barrier is needed

Broadcasting an Array (Multi-Value Broadcast)

• Use OpenMP's single directive along with a barrier to broadcast an entire array

• #pragma omp single: One thread initializes the array

• #pragma omp barrier: Ensures all threads see the updated values before using them

Optimize Performance Using nowait

• An implicit barrier is not required after single in some cases

• Adding nowait after single allows threads to continue execution without waiting

• If nowait is used, some threads may read an uninitialized value if they execute before the broadcast completes

• Use #pragma omp barrier if needed to ensure correctness

Parallel Prefix Sum (Cumulative Sum)

• The cumulative sum (prefix sum) of an array is an operation where each element at index i is replaced with the sum of all previous elements, including itself

• Given an array A = [1, 2, 3, 4, 5], the prefix sum S = [1, 3, 6, 10, 15] where:

  • S[0] = A[0]
  • S[1] = A[0] + A[1]
  • S[2] = A[0] + A[1] + A[2], and so on

Sequential (Single-Threaded) Approach

• The sequential approach iterates over the array and computes the cumulative sum

• Runs in O(n) time without leveraging parallelism

Complexity Analysis

Alternative OpenMP Implementation Using Scan

• Using OpenMP scan to parallelize prefix sum is a simpler approach

Applications of Parallel Prefix Sum Computation

• Parallel prefix sum (cumulative sum) is a fundamental operation in parallel computing and GPU-based optimizations

• Heavily used in parallel computing frameworks like OpenMP, CUDA, and MPI for efficient computation

• Example: CUDA-based parallel scan operations for large datasets

• Radix sort uses prefix sums to efficiently distribute elements into buckets during counting sort

• The prefix sum helps compute the final positions of elements after each digit-wise pass

• Example: Counting sort step in parallel radix sort

• Used in contrast enhancement techniques like histogram equalization

• Prefix sum helps compute the cumulative histogram, used for pixel intensity transformation

• Example: Enhancing image contrast in OpenCV

• Helps compute node ranks, edge weights, and paths in large graphs

• Example: Parallel BFS using prefix sum to track frontier expansion

• Used in dynamic programming (DP) problems like optimal parenthesization of matrices and sequence alignment

• DP problems rely on cumulative sums for state transitions

• Example: Parallel computation of DP table values

• Used in numerical integration, particle simulations, and molecular dynamics

• Useful in cumulative force calculations, time series processing, and physics simulations

• Example: Computing cumulative forces in N-body simulations

• Used for fast data compression

• Prefix sums are used to compute symbol frequencies and encoding tables efficiently

• Example: Parallel construction of Huffman trees

• Used in parallel SQL aggregation operations like SUM, COUNT, and AVG

• Speeds up window functions and range-based queries

• Example: Parallel execution of OLAP queries

• Used in Computing GC content, pattern matching, and sequence alignment

• Helps in fast DNA sequence processing and genome alignment

• Example: Fast GC content calculation in large genomes

• Used in cumulative profit/loss calculations, moving averages, and risk assessments

• Allows for efficient stock trend analysis and financial modeling

• Example: Parallel calculation of moving averages in stock data

Matrix Multiplication

• Root process initializes matrices A and B

• B is broadcasted to all processes

• A is scattered among processes and each process computes its part

• Computed results are gathered back to form the final matrix C

OpenMP Matrix Multiplication

• Program uses OpenMP to parallelize the nested loops of matrix multiplication
• #pragma omp parallel for collapse(2) parallelizes both i and j loops
• Threads run independently on different rows/columns
• Shared memory allows quick access to matrices without communication overhead

Sorting

• Sorting is a fundamental operation in computer science
• Parallel computing improves sorting performance by dividing the workload among multiple processing elements

Odd-Even Transposition Sort

• A parallel variant of Bubble Sort
• Consists of two alternating phases: Odd Phase and Even Phase
  • Odd Phase: Compare and swap adjacent odd-indexed pairs
  • Even Phase: Compare and swap adjacent even-indexed pairs
• Two phases continue until the array is fully sorted
• The algorithm benefits from parallel execution, where multiple swaps occur simultaneously

Algorithm

1. Distribute array elements across processors
2. Repeat k for 1 to n (number of elements):
   • Perform Odd Phase: odd-indexed processor sends element & receives from the next processor (if it exists)
   • Perform Even Phase: even-indexed processor sends element & receives from the next processor (if it exists)
3. Collect all elements after k iterations

Parallelization with OpenMP

• #pragma omp parallel for distributes loop iterations across threads

MPI Parallelization

• The parallel version improves performance as multiple elements are processed at once

Time Complexity

Parallelism in OpenMP

• Multiple threads handle swaps in the odd and even phases concurrently
• Synchronization is required between odd and even phases to prevent conflicts

Bitonic Merge Sort

• A parallel sorting algorithm that efficiently works on parallel architectures like SIMD (Single Instruction, Multiple Data)
• Divides the array into two parts:
  • Bitonic Sequence Generation: Converts an unsorted array into a bitonic sequence
  • Bitonic Merge: Merges bitonic sequences into sorted order

Steps of Bitonic Sort

1. Divide the array into two halves
2. Sort the first half in ascending order and the second half in descending order to form a bitonic sequence
3. Perform Bitonic Merge
   • Compare and swap elements at distance d
   • Reduce d until the array is fully sorted

Parallelism in CUDA

• CUDA threads process different pairs simultaneously
• Sorting is performed using bitonic merging, which reduces inter-thread communication overhead

Time Complexity

Parallel Quick Sort

• A divide-and-conquer algorithm that:
  • Chooses a pivot
  • Partitions the array into two halves:
    • Elements less than pivot go to the left
    • Elements greater than pivot go to the right
  • Recursively sorts both halves
• The parallel version splits tasks into separate threads

Stages in Algorithm

1. Parallel Partitioning
   • Select a pivot element and partition the array into two subarrays: elements less than pivot & greater than pivot
   • Distribute the two subarrays across available processors/threads
2. Concurrent Sorting
   • Each processor sorts its assigned subarray using a sequential Quick Sort algorithm (or another suitable sorting algorithm)
3. Merge Sorted Subarrays (If Necessary)
   • If needed, merge the sorted subarrays to obtain the final sorted array

Parallelism in OpenMP

• The two recursive calls after partitioning run in separate threads
• OpenMP sections divide work across multiple CPU cores

Time Complexity