Friday, November 13, 2009

OPL: Branch-and-Bound & Graphical Models

Branch-and-Bound computational patterns generally involve finding an optimal solution to a problem by iteratively searching through a large space of all possible solutions. Typically, to perform such a search, no known polynomial time algorithm is known. The Branch-and-Bound pattern comprises of four steps: Branching, Evaluation, Bounding, Pruning.

1. Examples: The authors discuss 3-SAT and Traveling Salesman Problem as examples. Yet, a more fundamental example to grasp is the knapsack problem. That said, what might be some other simpler examples of branch-and-bound that you are familiar with?

+One example that I will steal from Wikipedia is a chess program. The incredibly large search space of possible moves 1 to n moves ahead explodes at an exponential rate, and being able to prune the search space efficiently is key for the system to be able to identify the optimal or almost optimal move.

2. Trade-offs: How does the nature of branch-and-bound pattern make it especially hard to deal with incurring low communication overhead while maximizing concurrent computation?

+When branch finds an optimal solution or constraints to apply to boundaries, that must be communicated to the other branches in order to aid their pruning. This could potentially mean that every branch needs to communicate with every other branch, which in turn would lead to a significant amount of chatter. There is a direct relationship between the amount of communication and the pruning efficiency.

3. Solutions and Parallelization:
a. The authors discuss three possible techniques for parallelization: operation-based, structure-based, and construction-based. What makes structure-based parallelization preferable? Are there any scenarios where you would use the operation-based or construction-based technique instead?

+I think this question should be answered by the authors of the paper. I do not feel that they are giving the topic a thorough analysis if they simply skip operation- and construction-based patterns in favor of structure-based ones without much justification.

b. For the structure-based technique, the authors describe four different methods for implementation: Synchronous Single Pool, Asynchronous Single Pool, Synchronous Multiple Pool, and Asynchronous Multiple Pool. Which method do you think would be hardest to debug for correctness? Which method do you think would be hardest to optimize for performance and scalability?

+Correctness: AMP. This complicates the scenario both by distributing data and by having updates happen with no definite period.

+Optimize: ASP. This may be hard to optimize for efficiency because the shared task queue needs synchronized access. It is also complicated by the fact that each processor can request data at any time. If all processors were updated at once, synchronization on the queue would likely be easier because just one process could handle the global lock & coordination.

c. The authors discuss two categories of task assignment: static assignment and dynamic assignment. In real-world applications, how important do you think the choice of the task assignment method is to performance?

+I think it depends on the problem domain. Static assignment can likely be more efficient if tasks size can be known with a reasonable degree of accuracy a priori. That way, the overhead of load balancing does not need to be incurred. It thus follows that dynamic assignment is beneficial in the opposite case. Given that, I think that choosing static assignment in the wrong situations can severely hurt performance by causing many processors to sit idle, but I do not imagine that choosing dynamic assignment in the wrong cases would lead to much of a loss in efficiency.

4. Pattern Comparison: How might fork/join or task queue pattern relate or fit with the branch-and-bound pattern?

+Task queue is an easy one since it is explicitly called out in the paper as being useful as the central data store in the SSP & ASP styles of structure-based parallelism. As for fork-join, each branch can be a fork until you get down to a low threshold, at which point it is best to compute the result sequentially.

Graphical Models are often times used in probability theory to represent uncertainty of a collection of (possibly independent) events. Given the probability of occurrence of some events in the network, one can use the model to infer probabilities of other events within the network.

1. Examples: Did the authors' examples (e.g. hidden Markov Models, Bayesian Netoworks) help illuminate what a graphical model is? Can you think of other simple example problems or scenarios that you could model as a network/chain of probabilistic events?

+The examples made sense only because of my prior exposure to HMMs and Bayesian networks in CS 512. One example of how they could be used is determining how likely you are to see certain sequences of genes or proteins. If the authors wish to talk about HMMs and Bayesian networks, they should really give examples. Although many readers will be familiar with basic graph theory, these concepts go beyond that.

2. Solutions and Parallelization:
a. The authors make a distinction between exact and approximate estimation of probability. In what example problems do you think approximate estimation would work best? In what examples exact estimation?

+It depends on the definition of "work best." Exact algorithms would always be ideal, but a faster approximate algorithm can typically be found & be beneficial from a computational resources perspective. The example above of finding or predicting certain patterns in a genetic sequence is a good candidate for using an approximate algorithm over an exact one due to the immense number of nucleotides that need to be processed.

b. The authors mention the Junction Tree Algorithm, indicating that searching for the correct factorizations can be done in parallel. What other characteristics of graphical models do you think could benefit from parallelism?

+Not sure. The paper is a bit too short for me to really wrap my head around parallelization opportunities, and the mention of the JTA does not help since I am unfamiliar with the pattern.

3. Pattern Comparison: What might this pattern have in common with the Branch-and-Bound pattern? Do you think it has similarity to the Graph Algorithms patterns?

Not sure.

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