CS 584

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CS 584
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Discrete Optimization Problems

• A discrete optimization problem can be expressed
  as (S, f)
• S is the set of all feasible solutions
• f is the cost function
• Goal Find a feasible solution xopt such that
• f(xopt) lt f(x) for all x in S

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Discrete Optimization Problems

• Examples
• VLSI layout
• Robot motion planning
• Test pattern generation
• In most problems, S is very large
• S can be converted to a state-space graph and
  then can be reformulated to a search problem.

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Discrete Optimization Problems

• NP-hard
• Why parallelize?
• Consider real-time problems
• robot motion planning
• speech understanding
• task scheduling
• Faster search through bigger search spaces.

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Search Algorithms

• Depth First Search
• Breadth First Search
• Best First Search
• Branch and Bound
• Use cost to determine expansion
• Iterative Deepening A
• Use cost heuristic value to determine expansion

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Parallel Depth First Search

• Critical issue is distribution of search space.

Static partitioning of unstructured trees leads
to poor load balancing.
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Dynamic Load Balancing

• Consider sequential DFS

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Parallel DFS

• Each processor performs DFS on a disjoint section
  of the tree. (Static load assignment)
• After the processor finishes, it requests
  unsearched portions of the tree from other
  processors.
• Unexplored sections are stored in the stack
• Pop off a section from the stack and give it to
  somebody else.

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Parallel DFS Problems

• Splitting up the work
• How much work should you give to another
  processor?
• Determining a donor processor
• Who do you request more work from?

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Work Splitting Strategies

• When splitting up a stack, consider
• Sending too little or too much increases work
  requests
• Ideally, rather than splitting the stack, you
  would split the search space.
• HARD
• Nodes high in tree --gt big subtrees, vice-versa

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Work Splitting Strategies

• To avoid sending small amounts of work, nodes
  beyond a specified stack depth are not sent.
• Cut-off depth
• Strategies
• Send only nodes near bottom of stack
• Send nodes near cut-off depth
• Send 1/2 of nodes between bottom and cut-off

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Load Balancing Schemes(Who do I request work
from?)

• Asynchronous Round Robin
• each processor maintains target
• Ask from target then increment target
• Global Round Robin
• target is maintained by master node
• Random Polling
• randomly select a donor
• each processor has equal probability

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Speedups of DFS
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Best-First Search

• Heuristic is used to direct the search
• Maintains 2 lists
• Open
• Nodes unsearched
• Sorted by heuristic value
• Closed
• Expanded nodes
• Memory requirement is linear in the size of the
  search space explored.

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Parallel Best-First Search

• Concurrent processors pick the most promising
  node from the open list
• Newly generated nodes are placed back on the open
  list
• Centralized Strategy

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Global list maintained
at designated processor
Put expanded
nodes
Get
current
best node
Lock the list
Lock the list
Place generated
Place generated
Lock the list
nodes in the list
nodes in the list
Place generated
Pick the best node
Pick the best node
from the list
nodes in the list
from the list
Pick the best node
Unlock the list
Unlock the list
from the list
Expand the node to
Expand the node to
Unlock the list
generate successors
generate successors
Expand the node to
generate successors
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Centralized Best-First Search

• Termination condition
• A processor may find a solution but not the best
  solution.
• Modify the termination criteria (how?)
• Centralization leads to congestion
• Open list must be locked when accessed
• Extra work

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Decentralizing Best-First Search

• Let each processor maintain its own open list
• Issues
• Load balancing
• Termination (make sure it is the best)

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Communication Strategies

• Random
• Periodically send some of the best nodes to a
  random processor
• Ring
• Periodically exchange best nodes with neighbors
• Blackboard
• Select best node from open list
• If l-value is OK then expand
• If l-value is BAD then get some from blackboard
• If l-value is GREAT then give some to blackboard

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Ring Communication
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Blackboard
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What about searching a graph?

• Problem node replication
• Possible solution
• Assign each node to a processor
• Use hash function
• Whenever a node is generated, check to see if it
  already has been searched
• Costly

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Speedup Anomalies

• Due to nature of the problem, speedup can vary
  greatly from one execution to the next.
• Two anomaly types
• Acceleration
• Deceleration

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Termination Detection

• Dijkstra's Token Termination Detection
• When idle, send idle token to next processor
• When idle token is received again, all done
• Tree-Based Termination Detection
• Associate a weight of 1 with initial work load
• Assign portions of the weight
• When finished give the weight portion back
• When processor 0 has weight of 1 --gt all done.