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Parallel algorithm lecture 5 : PRAM Models

Parallel processing and parallel algorithms theory and compuation. A deterministic model produces a unique result for a given input. A non-deterministic model produces one of two or more possible results for a given input. Any non-deterministic model contains a deterministic model as a subset.


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The complexity classes for decision problems are listed in the table below. The prefix N stands for non-deterministic.

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The Complexity Classes for Decision Problems. Complexity class P is the set of decision problems that are solvable by a deterministic model within polynomial P time.


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  • This is the class of problems that we can solve feasibly, efficiently or quickly. We say that we can solve these quickly. Consider the addition of two n by n matrices. A typical algorithm will require n 2 steps to solve the problem. A typical CPU executes 10 9 operations per second. So the solution should be available in 0. Complexity class NP is the set of decision problems that are solvable by a non-deterministic N model within polynomial P time.

    This is the class of problems that we can verify in polynomial time but not necessarily solve efficiently. We say that we can verify these quickly. An example is the subset sum problem. We can verify this in 3 additions. There is no known algorithm to solve this problemin polynomial time. This problem is in NP quickly verifiable but not necessarily in P quickly solvable. Complexity class EXP is the set of decision problems that are solvable deterministically within exponential time.

    A typical CPU executes 10 9 operations per seconds. So the problem will finish in 1. This is longer than the known age of our universe. For the subset sum problem described above, there is an algorithm to find such a subset in exponential time, but there is no known algorithm in polynomial time. An interesting and important question arises if we focus on problems in NP that are hard to solve.

    Parallel algorithm - Wikipedia

    A problem is hard to solve if every problem in its class can be reduced to that hard problem. If a problem is hard for its class, we say that the problem is complete. It is the hardest problem for its own class. We call the set of problems that are hard for NP the set of NP-hard problems. The class of NP-complete problems contains the most difficult problems in NP-hard.

    The unsolved question that arises here is whether the most difficult problems in NP - those that can be verified in polynomial time but do not have a solution - can be solved in polynomial time; that is, is there any point in trying to find an algorithm. If P is not NP-Complete, then there are problems that are harder to solve than they are to verify and those problems cannot be solved in polynomial time.

    Parallel processing – systolic arrays

    This is one of the major unsolved problems of computer science. Any solution to a computing problem that contains several independent sets of instructions is a candidate for concurrent computing. For large problems, a concurrent programming solution should take significantly less time than a serial solution.

    Serial computing uses a single processor to execute all of the instructions in a program. The processor executes the instructions one after another. Each vertical bar in the figure below represents a single instruction waiting to enter the processor. Serial Computing. Concurrent computing uses multiple processors to execute the instructions in a program.

    The set of processors executes the sets of instructions concurrently. Each processor executes its own set of instructions sequentially. Parallel Computing. Concurrent program design starts with identifying the sets of instructions within a program that are independent of one another. Concurrent program design includes coordinating the results produced by multiple processors so as to integrate the results for the program as a whole.

    We use six fundamental components in the design of concurrent programs and group them into three categories:. The symbols used in these notes are illustrated below. We specify order within a program by arranging these symbols with respect to one another and connecting them using dependency symbols. Fundamental Components of an Algorithm. The relation of one individual unit to a task. There are two types of relation:. Michael Flynn categorized hardware architectures in terms of two independent streams:. Flynn's Taxonomy. Scalability is the ability to increase performance by adding resources.

    Two distinct types:. Elapsed time is the first and foremost measure of performance. Process time may also important in optimizations. Tracking the process time on each computational unit helps us identify bottlenecks within an application. Reducing the process time of a critical task results in a decrease in the application's elapsed time. Ideally, only those algorithms that lead to reductions in process time require our attention.

    Process time is not the same as elapsed time. Since CPUs and GPUs may execute their processes concurrently, individual process times are not necessarily crucial in the final result.


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    • Once we have identified the principal bootlenecks, we analyze the algorithms that are implemented at those bottlenecks. Analysis estimates the growth rate in solution time with problem size and helps up determine the resources needed to solve the computational problem for inputs of variable length.

      We analyze each critical algorithm in both time and space. In temporal analysis, we compare the computational steps within the algorithm with respect to problem size.