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Discrete Optimization: Search Algorithms and Load Balancing Strategies

This article explores discrete optimization problems and their expression as sets of feasible solutions and cost functions. It discusses various search algorithms, including depth-first search (DFS), breadth-first search (BFS), best-first search, and branch and bound. It also covers load balancing strategies for parallel DFS, work splitting strategies, communication strategies, and termination detection techniques. Lastly, it addresses speedup anomalies and termination detection in graph searching.

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Discrete Optimization: Search Algorithms and Load Balancing Strategies

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Presentation Transcript

  1. CS 584

  2. 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) <= f(x) for all x in S

  3. 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.

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

  5. 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

  6. Parallel Depth First Search • Critical issue is distribution of search space. Static partitioning of unstructured trees leads to poor load balancing.

  7. Dynamic Load Balancing • Consider sequential DFS

  8. 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.

  9. 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?

  10. 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 --> big subtrees, & vice-versa

  11. 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

  12. 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

  13. Speedups of DFS

  14. 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.

  15. 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

  16. 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

  17. Decentralizing Best-First Search • Let each processor maintain its own open list • Issues: • Load balancing • Termination (make sure it is the best)

  18. 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

  19. Ring Communication

  20. Blackboard

  21. 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

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

  23. 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 --> all done.

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