Algorithm Assessment and Simulating Large Machines

Tom Kelliher, CS 315

Jan. 29, 1999

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Assignment

From Last Time

Sorting on a linear array, algorithm assessment.

Outline

1. Algorithm assessment.

2. Simulating larger machines.

3. Examples.

Coming Up

Bit models, lower bounds.

Algorithm Assessment

How can we measure a parallel algorithm? Apply following to sorting on the linear array.

1. Run time: T.

2. Number of processors: N.

3. Speedup: Let be the running time of the fastest sequential algorithm. Then:

   

   What is linear speedup?

   Say you have P processors. Speedup is at most P. Why? (Hint: a parallel algorithm that runs in T steps on P processors can be simulated on a sequential machine in how many steps?)

4. Work: W = TP.

   Measures algorithm inefficiencies due to idle processors.

5. Efficiency of processor utilization:

   

6. Is it always possible to minimize T and E?

   Example: consider two algorithms for solving a problem of size M:

   1. M steps on an M processor machine.

   2. steps on an processor machine.

   If we have an processor machine and need to run the algorithm X times, which algorithm should we use? Assume the processor machine may be split in M M processor machines.

   (Recall: .)

Simulating a Large Array of Processors with a Small Array of Processors

Let's say we want to solve a problem of size N with P processors.

1. We generally assume that P = N. Is this valid? --- More later.

2. Suppose, we have fine-grain processors in a linear array. Can we sort N numbers?

   Fine-grain vs. coarse-grain processing.

3. Assuming appropriate granularity, here's how we simulate 12 processors on 3 processors:

   

   What about simulating processors on processors, where ?

   What slowdown do we encounter?

4. Is the scaled-down algorithm less efficient?

5. So, why do we assume, in general, that P = N?

   Implications:

   1. Given an algorithm efficient for large P, we can construct one efficient for small P.

   2. What about the converse?

Problems

1.2

Consider two algorithms for solving a a problem of size M, one that runs in M steps on an M-processor machine and one that runs in steps on an -processor machine. Which algorithm is more efficient?

Assume that the cost of running a P-processor machine for T steps is , where and are constants and P = M or . For what values of and is the first algorithm cheaper to run? For what values of and is the second algorithm cheaper to run?

1.3

Consider two algorithms for solving a problem of size M, one that runs in M steps on an M-processor machine and one that runs in steps on an -processor machine. Which algorithm will run faster on an N-processor machine? (Hint: Your answer depends on the relative sizes of M and N, and you will need to use the fact that an N-processor machine can simulate a P-processor machine with slowdown .)