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CS 312: Algorithm Analysis

This work is licensed under a Creative Commons Attribution-Share Alike 3.0 Unported License . CS 312: Algorithm Analysis. Lecture #35: Branch and Bound Design Options - State Spaces. Slides by: Eric Ringger, with contributions from Mike Jones, Eric Mercer, and Sean Warnick. Announcements.

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CS 312: Algorithm Analysis

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  1. This work is licensed under a Creative Commons Attribution-Share Alike 3.0 Unported License. CS 312: Algorithm Analysis Lecture #35: Branch and Bound Design Options - State Spaces Slides by: Eric Ringger, with contributions from Mike Jones, Eric Mercer, and Sean Warnick

  2. Announcements • Homework #25 due now • Homework #26 optional • Project #7: TSP • We continue discussing ideas today • Early: Wednesday • Due: next Friday

  3. Objectives • Consider an alternate way to structure the state space. • i.e., an alternate way to generate successor states • Provide more options to complete the project.

  4. Review the Ingredients of B&B

  5. Review: Branch and Bound functionBandB(v) BSSF  quick-solution(v) // BSSF.cost holds cost Agenda.clear() v.b  bound(v) Agenda.add(v, v.b) while !Agenda.empty() and time remains and BSSF.cost != v.bdo u  Agenda.first() Agenda.remove_first() children = generate_children_ascending(u) for each w in childrendo • if ! time remains then break w.b  bound(w) if(w.b < BSSF.cost) then if criterion(w) then BSSF  w • Agenda.prune(BSSF.cost) else ifpartial_criterion(w) then Agenda.add(w, w.b) return BSSF Compute bound when constructing staterather than as separate step.

  6. Review: Branch and Bound functionBandB(v) BSSF  quick-solution(v) // BSSF.cost holds cost Agenda.clear() Agenda.add(v, v.b) while !Agenda.empty() and time remains and BSSF.cost != v.bdo u  Agenda.first() Agenda.remove_first() children = generate_children_ascending(u) for each w in childrendo • if ! time remains then break if(w.b < BSSF.cost) then if criterion(w) then BSSF  w • Agenda.prune(BSSF.cost) else ifpartial_criterion(w) then Agenda.add(w, w.b) return BSSF Compute bound when constructing state rather than as separate step.

  7. Review: Branch and Bound functionBandB(v) BSSF  quick-solution(v) // BSSF.cost holds cost Agenda.clear() Agenda.add(v, v.b) while !Agenda.empty() and time remains and BSSF.cost != v.bdo u  Agenda.first() Agenda.remove_first() children = generate_children_ascending(u) for each w in childrendo • if ! time remains then break if(w.b < BSSF.cost) then if criterion(w) then BSSF  w • Agenda.prune(BSSF.cost) else ifpartial_criterion(w) then Agenda.add(w, w.b) return BSSF Apply partial_criterion() when generating children.

  8. Simplify: Branch and Bound functionBandB(v) BSSF  quick-solution(v) // BSSF.cost holds cost Agenda.clear() Agenda.add(v, v.b) while !Agenda.empty() and time remains and BSSF.cost != v.bdo u  Agenda.first() Agenda.remove_first() children = generate_children_ascending(u) for each w in childrendo • if ! time remains thenbreak if(w.bound< BSSF.cost) then if criterion(w) then BSSF  w • Agenda.prune(BSSF.cost) else Agenda.add(w, w.b) return BSSF Apply partial_criterion() when generating children.

  9. Simplify: Branch and Bound functionBandB(v) BSSF  quick-solution(v) // BSSF.cost holds cost Agenda.clear() Agenda.add(v, v.b) while !Agenda.empty() and time remains and BSSF.cost != v.bdo u  Agenda.first() Agenda.remove_first() children = generate_children_ascending(u) for each w in childrendo • if ! time remains thenbreak if(w.bound< BSSF.cost) then if criterion(w) then BSSF  w • Agenda.prune(BSSF.cost) else Agenda.add(w, w.b) return BSSF Use init_state() function.Compute bound as partof initial state.

  10. Simplify: Branch and Bound functionBandB() s  init_state() BSSF  quick-solution(s) // BSSF.cost holds cost Agenda.clear() Agenda.add(s, s.bound) while !Agenda.empty() and time remains and BSSF.cost != s.bounddo u  Agenda.first() Agenda.remove_first() children = generate_children_ascending(u) for each w in childrendo • if ! time remains thenbreak if(w.bound< BSSF.cost) then if criterion(w) then BSSF  w • Agenda.prune(BSSF.cost) else Agenda.add(w, w.b) return BSSF Use init_state() function.Compute bound as partof initial state. Requirement for Proj. #7:Use B&B pseudo-code.

  11. 9 1 1 8 2 2 5 10 6 3 4 12 7 3 4 4 Review: A Bound on TSP Tour

  12. 1 0 1 0 0 2 5 9 0 0 2 6 1 3 4 1 Reduce Rows

  13. 1 0 1 0 0 2 5 9 0 0 2 6 0 3 4 1 2. Reduce Columns

  14. Review: Pruning • Under what conditions can a search state be pruned? • Lower bound for state  BSSF (least upper bound) • Could be true when the state is created • Could happen when the BSSF is updated and the agenda is pruned • The state is not feasible • Where feasibility may include any test you might think of: • Already occupied • Cycle among sub-set of cities • No options remaining • …

  15. Focus on State Space functionBandB() s  init_state() BSSF  quick-solution(s) // BSSF.cost holds cost Agenda.clear() Agenda.add(s, s.bound) while !Agenda.empty() and time remains and BSSF.cost != s.bounddo u  Agenda.first() Agenda.remove_first() children = generate_children_ascending(u) for each w in childrendo • if ! time remains thenbreak if(w.bound< BSSF.cost) then if criterion(w) then BSSF  w • Agenda.prune(BSSF.cost) else Agenda.add(w, w.b) return BSSF

  16. Structure of State Space • Recall the structure of the state space from last time?

  17. Review: State Space #1 • Generate all children one step away • each child state is generated by including one of the edges leaving city j, where j is the destination of the last edge in the parent state • Pro: Does not include premature cycles. Why not? • Pro: Simple algorithm: pick all neighbors that remain from vertex j, (leave the starting vertex for the end) • Con: On many instances, while searching with B&B, will result in excessively deep agendas. Why? • Can B&B in this state space be optimal? (i,j) Include edge (j,1) … (j,N) (j,2) (j,3)

  18. Structure of State Space • Could we structure the space another way? • How else could we generate children states?

  19. State Space #2 Use the same method for representing states computing bounds. New method for generating children: Include edge (i,j) Exclude edge (i,j) Why can branch and bound in this state space be optimal?

  20. 1 0 1 0 0 2 5 9 0 2 0 6 0 3 4 1 Excluding an Edge What happens to a state’s cost matrix when an edge is excluded? Exclude 4  2

  21. State Space: Strategy #2

  22. Which edge to select? • What strategies for edge selection can you think of? • Idea: try to reduce the size of the part of the state-space you actually need to explore!

  23. 1 1 2 2 5 5 3 3 4 4 Try #1: Consider edge (3,2) Parent: bound = 21 include (3,2) exclude (3,2) bound = 21 bound = 22 0 1 0 0 0 0 0 9 9 1 0 0 1 0 6 0 Both children similarlycompetetive. 6 0 1

  24. 1 1 2 2 5 5 3 3 4 4 Try #2: Consider edge (2,3) Parent: bound = 21 include (2,3) exclude (2,3) bound = 28 bound = 21 1 0 1 0 0 0 0 9 0 9 0 1 0 Left child more likely tobe pruned. 1 0 6 0 6 1 1

  25. 1 1 2 2 5 5 3 3 4 4 Try #3: Consider edge (5,1) Parent: bound = 21 include (5,1) exclude (5,1) bound = 21 bound = 30 1 0 1 0 0 0 0 0 Right child more likelyto be pruned. 9 0 0 1 0 1 0 6 6 0 0 1 1

  26. State Space so far bound = 21 include (5,1) exclude (5,1) bound = 21 bound = 30

  27. 1 1 2 2 5 5 3 3 4 4 Consider edge (2,5) Parent: bound = 21 include (2,5) exclude (2,5) bound = 21 bound = 28 1 1 0 0 0 0 Again, right child morelikely to be pruned. 0 0 1 0 0 6 0 0 1 1

  28. Search Space so far 21 include (5,1) exclude (5,1) 21 30 include (2,5) exclude (2,5) 21 28

  29. 1 1 2 2 5 5 3 3 4 4 Consider edge (3,2) Parent: bound = 21 include (3,2) exclude (3,2) bound = 21 bound = 22 1 0 0 0 0 0 0 0 0

  30. Search Space so far 21 include (5,1) exclude (5,1) 21 30 include (2,5) exclude (2,5) 21 28 include (3,2) exclude (3,2) 21 22

  31. 1 2 5 3 4 Search Space so far 21 include (5,1) exclude (5,1) 21 30 include (2,5) exclude (2,5) 21 28 include (3,2) exclude (3,2) 0 21 22 include (4,3),(1,4) ... 0 21

  32. Summarize: State Space #2 • Pick an edge e = (i,j) and generate two children: • Include edge e • Exclude edge e • One might pick an edge by: • maximizing the difference between the included edge child and the excluded edge child • maximizing the bound on the excluded edge child • minimizing the bound on the included edge child • One must be careful not to create a cycle prematurely • How to prevent? Include edge (i,j) Exclude edge (i,j)

  33. Conclusions • You have options! • Having some control without sacrificing optimality is important. • You may be able to prune more aggressively without sacrificing optimality.

  34. Assignment • HW #26: Optional • Include-exclude state-space generation • Kill two birds with one stone: • Read the project instructions and lecture notes • Do HW #26 on whiteboard with a partner • Have your “whiteboard experience” for Project #7 • Consider your options • Design a B&B algorithm • Work and talk through an example • Read the assigned selection from Russell & Norvig • Use your homework key credentials • Happy General Conference weekend!

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