6.4 Parallel Complexity Theory
5 min read•july 30, 2024
Parallel complexity theory dives into the of algorithms running on multiple processors. It's all about measuring how fast and how much work these algorithms do, compared to their single-processor counterparts. This stuff is crucial for designing speedy parallel programs.
In this part of the chapter, we're looking at key concepts like parallel time and work complexity, theoretical models, and performance metrics. We'll also explore how problems are classified based on their parallel solvability and learn techniques for proving where a problem fits in the parallel complexity world.
Parallel Time and Work Complexity
Fundamental Concepts
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- Parallel measures execution time of parallel algorithms as function of input size, assuming unlimited processors
- Parallel work complexity represents total computational work performed by all processors in parallel algorithm
- Work-time principle defines relationship between parallel time and work complexity
- Product of time and number of processors used should not exceed work of best sequential algorithm
- in parallel computing calculated as ratio of sequential to parallel execution time
- Ideal linear speedup rarely achieved due to communication overhead and load balancing issues
- quantifies maximum achievable speedup through parallelization
- Considers fraction of program that can be parallelized
- Scalability in parallel algorithms refers to efficient utilization of increasing processor numbers
- Often measured by isoefficiency functions
Theoretical Frameworks
- Work-depth model analyzes parallel algorithms
- Focuses on depth (critical path length) and total work of computation
- PRAM (Parallel Random-Access Machine) model serves as theoretical foundation
- Used for analyzing parallel algorithms and defining complexity classes
- Circuit models provide alternative framework for parallel complexity analysis
- Circuit depth and size correspond to time and work in
Performance Metrics and Analysis
- Efficiency measures how well processors are utilized in parallel computation
- Calculated as speedup divided by number of processors
- Cost optimality achieved when parallel algorithm's work complexity matches best sequential algorithm
- relates work and depth to execution time on p processors
- Time ≤ (Work/p) + Depth
- provides alternative to Amdahl's Law for scaling problems
- Focuses on fixed time rather than fixed problem size
Problem Classification by Parallel Complexity
Complexity Classes
- (Nick's Class) contains problems solvable in polylogarithmic time using polynomial processors on PRAM
- P-complete problems considered inherently sequential
- Unlikely to benefit significantly from parallelization
- Form crucial boundary in parallel complexity theory
- (Randomized Nick's Class) probabilistic counterpart to NC
- Contains problems solvable by randomized parallel algorithms in polylogarithmic time with polynomial processors
- (AC^0, AC^1, ...) classifies problems based on circuit depth
- Each level corresponds to specific parallel complexity class
Hierarchies and Relationships
- classes form hierarchy within NC
- NC^k contains problems solvable in O(log^k n) time with polynomial processors
- Key relationships between complexity classes
- NC ⊆ P (open question whether P = NC)
- AC^0 ⊂ AC^1 ⊂ ... ⊂ NC
- problems represent upper bound for parallel computation
- Cannot be solved efficiently in parallel unless PSPACE = NC
Examples and Applications
- Matrix multiplication () in NC^2
- Sorting networks () in NC^1
- Maximum flow problem P-complete
- Primality testing in RNC ()
- Boolean formula evaluation in NC^1 (parallel prefix computation)
Proving Problem Membership in Parallel Classes
Proof Techniques
- Circuit depth and size key parameters for proving AC and NC hierarchy membership
- Reduction techniques (NC-reductions) essential for relating problems to known class members
- Uniformity in circuit families crucial for efficient circuit construction in proofs
- Proving NC membership involves designing parallel algorithms with polylogarithmic time complexity
- Analyze work-depth characteristics
- Randomized class (RNC) proofs consider probability of correctness and required random bits
- Simulation arguments establish relationships between complexity classes
- Show higher class problem simulated by lower class
- Adapted classical complexity theory techniques for parallel complexity proofs
- Diagonalization and padding establish class separation or prove lower bounds
Specific Proof Strategies
- Divide-and-conquer paradigm often used to design NC algorithms
- Prove logarithmic depth recursion with polynomial total work
- Parallel prefix computation technique proves many problems in NC^1
- (List ranking, expression evaluation)
- Algebraic techniques used for problems like matrix inversion and determinant computation
- Prove membership in NC^2
- Randomized algorithms often easier to parallelize
- Used to prove membership in RNC (maximal independent set problem)
Case Studies and Examples
- Proving planarity testing in NC using Euler tour technique and tree contraction
- Demonstrating graph connectivity in RNC using random walks
- Establishing P-completeness of Circuit Value Problem through logspace reductions
- Proving membership of perfect matching in RNC using
- Showing Fast Fourier Transform (FFT) in NC^1 using butterfly network
Parallel vs Sequential Complexity
Fundamental Relationships
- P^NC = P establishes parallel algorithms with polylogarithmic time and polynomial processors simulated by polynomial-time sequential algorithms
- Time-processor tradeoffs in parallel algorithms illustrate diminishing returns of increasing processors
- Communication overhead limits effectiveness
- Efficient parallel algorithms achieve polylogarithmic time while keeping total work within polynomial factor of best sequential algorithm
- Key inclusions relate parallel complexity to sequential space and time classes
- NC^1 ⊆ L ⊆ NL ⊆ NC^2 ⊆ P
Theoretical Implications
- P-completeness theory identifies likely inherently sequential problems
- Resistant to significant parallelization
- Parallel decision trees compared to sequential decision trees offer insights into potential speedups
- (Binary search tree vs. parallel binary search)
- Parallel approximation algorithms sometimes provide faster solutions to NP-hard problems
- Trade-off between speed and solution quality (parallel approximation for traveling salesman problem)
Practical Considerations
- Amdahl's Law implications for real-world parallel speedup
- Even small sequential portions significantly limit overall speedup
- Memory hierarchy effects on parallel vs. sequential performance
- Cache coherence and false sharing in parallel systems
- Communication complexity often dominates parallel algorithm performance
- May lead to sublinear speedups in practice
- Load balancing challenges in parallel implementations
- Static vs. dynamic scheduling strategies (parallel quicksort with work stealing)
Key Terms to Review (29)
AC Hierarchy: The AC hierarchy is a framework in parallel complexity theory that categorizes decision problems based on their computability and the resources required to solve them in parallel computation. It separates problems into classes defined by their parallel time complexity and allows for a better understanding of the power of parallel algorithms and the limitations of different computational models.
AKS Sorting Network: An AKS sorting network is a parallel sorting algorithm that achieves a time complexity of O(log n) for sorting n elements using a specific network of comparators. It is a significant development in the realm of parallel complexity theory, showcasing how sorting can be performed efficiently in parallel architectures by utilizing a structured arrangement of comparison operations. This network is notable for its deterministic approach and optimality in both time and size.
Amdahl's Law: Amdahl's Law is a formula that helps to find the maximum improvement of a system's performance when only part of the system is improved. This concept is crucial in parallel computing, as it illustrates the diminishing returns of adding more processors or resources when a portion of a task remains sequential. Understanding Amdahl's Law allows for better insights into the limits of parallelism and guides the optimization of both software and hardware systems.
Brent's Theorem: Brent's Theorem is a foundational principle in parallel complexity theory that provides a method for analyzing the time complexity of certain algorithms, particularly those related to parallel computing. This theorem establishes a relationship between the work done by an algorithm and its depth, which helps in determining the efficiency of algorithms when executed on multiple processors. It highlights how minimizing depth while maximizing work can lead to better performance in parallel systems.
Cook's Theorem: Cook's Theorem states that the Boolean satisfiability problem (SAT) is NP-complete, which means it is at least as hard as the hardest problems in NP. This theorem laid the foundation for understanding computational complexity by establishing the concept of NP-completeness, helping to identify problems that are computationally challenging. As a result, it connects various aspects of complexity theory and highlights the significance of polynomial-time reductions between problems.
Distributed computing model: A distributed computing model refers to a computing paradigm where processing is spread across multiple interconnected computers that communicate and coordinate their actions by passing messages. This model allows for enhanced performance, resource sharing, and improved fault tolerance by utilizing the collective power of numerous machines, making it essential for solving complex problems in parallel complexity theory.
Efficiency: Efficiency in computing refers to the ability of a system to maximize its output while minimizing resource usage, such as time, memory, or energy. In parallel and distributed computing, achieving high efficiency is crucial for optimizing performance and resource utilization across various models and applications.
Gustafson's Law: Gustafson's Law is a principle in parallel computing that argues that the speedup of a program is not limited by the fraction of code that can be parallelized but rather by the overall problem size that can be scaled with more processors. This law highlights the potential for performance improvements when the problem size increases with added computational resources, emphasizing the advantages of parallel processing in real-world applications.
Leslie Valiant: Leslie Valiant is a prominent computer scientist known for his contributions to the field of computational complexity, particularly in the context of parallel computing and learning theory. He introduced the concept of Probably Approximately Correct (PAC) learning, which has profound implications for understanding the computational limits and capabilities of algorithms in parallel settings.
Logcfl: logcfl refers to the class of languages that can be decided by a logarithmic space-bounded, concurrent, and deterministic finite automaton. It is an important concept in parallel complexity theory as it explores how computation can be performed in parallel while using limited space resources, helping to classify problems based on their computational efficiency and resource requirements.
Lovász's Algorithm: Lovász's Algorithm is a combinatorial optimization technique used to find a maximum independent set in a graph. This algorithm is significant in the realm of parallel complexity theory as it provides an efficient way to approach problems related to graph theory, particularly in determining the largest set of vertices such that no two vertices in the set are adjacent.
Miller-Rabin Algorithm: The Miller-Rabin algorithm is a probabilistic method used to determine if a number is prime or composite. It works by using properties of modular arithmetic and is particularly useful in the field of cryptography, where determining the primality of large numbers efficiently is crucial for secure communications.
MPI: MPI, or Message Passing Interface, is a standardized and portable message-passing system designed for parallel programming, which allows processes to communicate with one another in a distributed computing environment. It provides a framework for developing parallel applications by enabling data exchange between processes, regardless of whether they are on the same machine or across different nodes in a cluster. Its design addresses challenges in synchronization, performance, and efficient communication that arise in high-performance computing.
Nc: In parallel complexity theory, 'nc' stands for 'nick's class', which is a complexity class that encompasses decision problems solvable by a parallel computer using a polynomial number of processors in polylogarithmic time. This class highlights the power of parallel computation, allowing for more efficient algorithms for problems that may be infeasible to solve in a reasonable time using sequential methods.
Nc^k: The term $n^{c^k}$ represents a class of functions that grow polynomially with respect to the size of input $n$, where $c$ is a constant and $k$ is a fixed positive integer. This notation is crucial in understanding the performance of parallel algorithms, as it helps categorize the efficiency and scalability of computations performed on multiple processors. The exponent $c^k$ indicates how the problem size affects resource requirements, shaping our understanding of algorithmic complexity in parallel systems.
Np-completeness: NP-completeness refers to a class of problems in computational complexity theory that are both in NP (nondeterministic polynomial time) and as hard as the hardest problems in NP. This means that if a polynomial-time algorithm can be found for any NP-complete problem, then all problems in NP can be solved in polynomial time. Understanding NP-completeness is crucial because it helps categorize problems based on their computational difficulty and has significant implications for parallel and distributed computing.
OpenMP: OpenMP is an API that supports multi-platform shared memory multiprocessing programming in C, C++, and Fortran. It provides a simple and flexible interface for developing parallel applications by enabling developers to specify parallel regions and work-sharing constructs, making it easier to utilize the capabilities of modern multicore processors.
P-class: In parallel complexity theory, a p-class is a class of decision problems that can be solved in polynomial time using a parallel computing model. This concept helps in understanding how efficiently problems can be computed when multiple processors work together, highlighting the differences between sequential and parallel processing. The p-class is essential for classifying problems based on their computational complexity and efficiency in parallel settings.
Parallel graph algorithms: Parallel graph algorithms are computational methods designed to solve graph problems using multiple processors or computing units simultaneously. They are essential in the realm of parallel complexity theory as they allow for faster processing and more efficient handling of large-scale graphs by distributing workloads across available resources, thus significantly reducing execution time compared to sequential algorithms.
Parallel sorting algorithm: A parallel sorting algorithm is a method used to sort a collection of data elements by dividing the task among multiple processors or threads, allowing for concurrent execution. This approach enhances the sorting process by reducing the time complexity compared to traditional serial sorting methods, leveraging the capabilities of parallel computing environments. The efficiency of these algorithms often depends on how well the data can be partitioned and the coordination among the processing units.
PRAM Model: The PRAM (Parallel Random Access Machine) model is an abstract computational model that facilitates the study of parallel algorithms by providing a simplified framework for understanding parallel computation. It assumes multiple processors that can access a shared memory simultaneously, making it easier to analyze and design efficient algorithms for parallel systems. This model is crucial for understanding how complexity and performance can be evaluated in parallel computing.
Pspace-complete: Pspace-complete refers to a class of decision problems that are the hardest in the complexity class PSpace, meaning they can be solved using a polynomial amount of space. If any pspace-complete problem can be solved in polynomial time, then all problems in PSpace can also be solved in polynomial time. This concept is crucial when examining computational resources, particularly how space is utilized in parallel and distributed computing environments.
Rnc: rnc stands for 'randomized nondeterministic complexity', which is a class of decision problems where a randomized algorithm is allowed to make guesses and accept or reject based on those guesses. This complexity class highlights the power of randomness in computation and is particularly useful for analyzing problems that are hard for deterministic algorithms. It represents a way to understand how randomness can lead to more efficient solutions for certain computational tasks.
Savitch's Theorem: Savitch's Theorem states that any problem that can be solved by a non-deterministic Turing machine using space $$S(n)$$ can also be solved by a deterministic Turing machine using space $$S(n)^2$$. This theorem highlights the relationship between non-deterministic and deterministic complexity classes, particularly in the realm of space complexity. It provides a critical insight into how problems classified as NP can be addressed with deterministic algorithms, although often at a higher resource cost.
Space complexity: Space complexity refers to the amount of memory space required by an algorithm to execute as a function of the length of the input. This measure includes both the space needed for the inputs themselves and any additional space used for variables, data structures, and function calls. Understanding space complexity is crucial in parallel complexity theory, where algorithms often require efficient memory usage to achieve better performance on multi-core or distributed systems.
Speedup: Speedup is a performance metric that measures the improvement in execution time of a parallel algorithm compared to its sequential counterpart. It provides insights into how effectively a parallel system utilizes resources to reduce processing time, highlighting the advantages of using multiple processors or cores in computation.
Stephen Cook: Stephen Cook is a renowned computer scientist best known for his groundbreaking work in computational complexity theory, particularly the concept of NP-completeness. His introduction of the Cook's theorem in 1971 established that the Boolean satisfiability problem (SAT) is NP-complete, creating a foundation for understanding the relationships between different computational problems and their complexities in parallel computing contexts.
Strassen's Algorithm: Strassen's Algorithm is a divide-and-conquer method for matrix multiplication that significantly reduces the computational complexity compared to the conventional method. By breaking down large matrices into smaller submatrices, it uses clever mathematical techniques to minimize the number of necessary multiplications, leading to a time complexity of approximately $$O(n^{2.81})$$ instead of the traditional $$O(n^3)$$. This efficiency makes it especially relevant in the study of parallel complexity theory.
Time Complexity: Time complexity is a computational concept that describes the amount of time an algorithm takes to complete as a function of the length of the input. It helps in evaluating the efficiency of algorithms, especially in parallel computing, by providing a way to analyze how the execution time grows as the input size increases. Understanding time complexity is crucial for designing efficient algorithms that can handle large data sets effectively, especially when considering how tasks can be divided and executed simultaneously in parallel environments.