A Survey of Suboptimal Search Algorithms

Transcription

1 A Survey of Search Algorithms Jordan T. Thayer and Wheeler Ruml jtd7, ruml at cs.unh.edu slides at: jtd7/papers/ Jordan Thayer and Wheeler Ruml (UNH) Search 1 / 28

2 This is Search Search duplicates structs Search Types Search Tips Outline Not Discussed Bounded initial state expand generates all successor states, implicitly computes g(n) goal test h(n) estimates cost of reaching a goal admissibility: non-overestimating consistency: obeys triangle inequality state: problem data node: state, g, parent pointer Jordan Thayer and Wheeler Ruml (UNH) Search 2 / 28

3 What is a duplicate? Search duplicates structs Search Types Search Tips Outline Not Discussed 1. while open has nodes 2. remove n from open with minimum h(n) 3. if n is a goal then return n 4. otherwise expand n, inserting its children into open 5. return failure Bounded same state, different path Jordan Thayer and Wheeler Ruml (UNH) Search 3 / 28

4 Common Data Structures Search duplicates structs Search Types Search Tips Outline Not Discussed 1. while open has nodes 2. remove n from open with minimum h(n) 3. if n is a goal then return n 4. otherwise expand n, inserting its children into open 5. return failure Bounded openlist: heap: handles real costs, large ranges of values, etc bucket list: more efficient, requires integer values closed list: hash table: in practice, also includes open nodes Jordan Thayer and Wheeler Ruml (UNH) Search 4 / 28

5 Different Kinds of Search Search duplicates structs Search Types Search Tips Outline Not Discussed Grid Four-way 35% A* greedy Bounded total nodes generated width 800 suboptimal search scales better than optimal search Jordan Thayer and Wheeler Ruml (UNH) Search 5 / 28

6 Different Kinds of Search Search duplicates structs Search Types Search Tips Outline Not Discussed Bounded total nodes generated Grid Four-way 35% A* greedy final sol cost relative to A* greedy A* Grid Four-way 35% width width 800 it does so at the cost of solution quality Jordan Thayer and Wheeler Ruml (UNH) Search 5 / 28

7 Different Kinds of Search Search duplicates structs Search Types Search Tips Outline Not Discussed Bounded total nodes generated Grid Four-way 35% A* wa*, w = 1.5 greedy final sol cost relative to A* Grid Four-way 35% greedy wa*, w = 1.5 A* Size Size 800 bounded suboptimal search presents a middle ground Jordan Thayer and Wheeler Ruml (UNH) Search 5 / 28

8 Simple Tips for Efficient Search Search duplicates structs Search Types Search Tips Outline Not Discussed Bounded store open nodes in closed list as well prevents multiple copies of a state being open at once duplicate checking and delaying or dropping correct tie breaking varies by search, generally prefer high g goal test on generation then prune for bounded suboptimal search recursive expansions (to reduce heap operations) Jordan Thayer and Wheeler Ruml (UNH) Search 6 / 28

9 About the Tutorial Search duplicates structs Search Types Search Tips Outline Not Discussed Bounded Jordan and I will alternate bibliography at the end the pseudo code not presented during talk included for later review not comprehensive due to time constraints each section covers greatest hits our personal experience and biases Jordan Thayer and Wheeler Ruml (UNH) Search 7 / 28

10 Things We Will Discuss Search duplicates structs Search Types Search Tips Outline Not Discussed suboptimal search minimize solving time greedy best-first search beam search LSS-learning real-time Search (LSS-LRTA*) Bounded bounded suboptimal search weighted A* optimistic search anytime search anytime repairing A* anytime weighted A* restarting weighted A* balance time and cost unknown deadlines Jordan Thayer and Wheeler Ruml (UNH) Search 8 / 28

11 Things We Won t Discuss Search duplicates structs Search Types Search Tips Outline Not Discussed Bounded optimal search strategies bounded-depth tree search local search strategies constructing heuristics using disk parallel algorithms anything relying on distance estimates (come back next session) Jordan Thayer and Wheeler Ruml (UNH) Search 9 / 28

12 Greedy Beam LSS-LRTA* Bounded Unboundedly Search Jordan Thayer and Wheeler Ruml (UNH) Search 10 / 28

13 Search: Outline Greedy Beam LSS-LRTA* Bounded greedy best-first search variant of best-first search beam search: best-first, breadth-first irrevocable pruning real-time search: LSS-LRTA* interleaves planning and acting next session: speedy search, A* on length, beam search on d. Jordan Thayer and Wheeler Ruml (UNH) Search 11 / 28

14 Greedy Best-first Search Doran and Michie 1966 Greedy Beam LSS-LRTA* Bounded 1. Best-first Search On Cost-to-go Estimate, h(n). A* root with admissible h, f rises resulting in vacillation in optimal search, h prunes, in greedy it guides Jordan Thayer and Wheeler Ruml (UNH) Search 12 / 28

15 Greedy Best-first Search Doran and Michie 1966 Greedy Beam LSS-LRTA* Bounded 1. while open has nodes 2. remove n from open with minimum h(n) 3. if n is a goal then return n 4. otherwise expand n, inserting its children into open 5. return failure Jordan Thayer and Wheeler Ruml (UNH) Search 12 / 28

16 Greedy Best-first Search Doran and Michie 1966 Greedy Beam LSS-LRTA* Bounded 1. while open has nodes 2. remove n from open with minimum h(n) 2a. break ties on h in favor of low g(n) 3. if n is a goal then return n 4. otherwise expand n, inserting its children into open 5. return failure Jordan Thayer and Wheeler Ruml (UNH) Search 12 / 28

17 Greedy Best-first Search Doran and Michie 1966 Greedy Beam LSS-LRTA* Bounded 1. while open has nodes 2. remove n from open with minimum h(n) 2a. break ties on h in favor of low g(n) 3. if n is a goal then return n 4. otherwise for each child c of n 4a. if c was ever expanded, discard it 4b. if c is in open, keep c with smallest g 4c. otherwise insert c into open 5. return failure Jordan Thayer and Wheeler Ruml (UNH) Search 12 / 28

18 Greedy Best-first Search Doran and Michie 1966 Greedy Beam LSS-LRTA* Bounded total nodes generated Grid Four-way 35% expand duplicates drop duplicates final sol cost Grid Four-way 35% drop duplicates expand duplicates 400 width Size 800 dropping duplicates improves time, harms quality Jordan Thayer and Wheeler Ruml (UNH) Search 12 / 28

19 Greedy Best-first Search Doran and Michie 1966 Greedy Beam LSS-LRTA* Bounded greedy best-first search: h(n) A*: f(n) = g(n)+h(n) Jordan Thayer and Wheeler Ruml (UNH) Search 12 / 28

20 Greedy Best-first Search Doran and Michie 1966 Greedy Beam LSS-LRTA* Bounded greedy best-first search: h(n) A*: f(n) = g(n)+h(n) basic brittle Jordan Thayer and Wheeler Ruml (UNH) Search 12 / 28

21 Beam Search Rich and Knight 1991, Bisani 1992 Greedy Beam LSS-LRTA* Bounded Best-First Beam Search 1. run A* with a fixed sized open list 2. filter out nodes with high f(n) fixed memory (sometimes) conflates propulsion with pruning doesn t work well in practice Wilt et al SoCS-10 Jordan Thayer and Wheeler Ruml (UNH) Search 13 / 28

22 Beam Search Rich and Knight 1991, Bisani 1992 Greedy Beam LSS-LRTA* Bounded Best-First Beam Search 1. while open has nodes 2. remove n from open with minimum f(n) 3. if n is a goal then return n 4. otherwise for each child c of n 5. insert c into open 6. if open is larger than width 7. discard worst node on open 8. return failure Jordan Thayer and Wheeler Ruml (UNH) Search 13 / 28

23 Beam Search Rich and Knight 1991, Bisani 1992 Greedy Beam LSS-LRTA* Bounded Breadth-First Beam Search 1. run breadth-first search with a fixed sized open list 2. filter out nodes with high f(n) Jordan Thayer and Wheeler Ruml (UNH) Search 13 / 28

24 Beam Search Rich and Knight 1991, Bisani 1992 Greedy Beam LSS-LRTA* Bounded Breadth-First Beam Search 1. while open has nodes 2. for each n open 3. if n is a goal, return n 4. otherwise expand n, adding to children 5. open becomes best width nodes in children 5a. best according to f(n) = g(n)+h(n) 6. return failure Jordan Thayer and Wheeler Ruml (UNH) Search 13 / 28

25 Beam Search Rich and Knight 1991, Bisani 1992 Greedy Beam LSS-LRTA* Bounded Breadth-First Beam Search 1. while open has nodes 2. for each n open 3. for each child c of n 4. if c is a goal, return it 5. otherwise add c to children 6. open becomes best width nodes in children 6a. best according to f(n) = g(n)+h(n) 6b. break ties in favor of high g(n) 7. return failure Jordan Thayer and Wheeler Ruml (UNH) Search 13 / 28

26 Beam Search Rich and Knight 1991, Bisani 1992 for the uninitiated, the 15 puzzle Greedy Beam LSS-LRTA* Bounded Jordan Thayer and Wheeler Ruml (UNH) Search 13 / 28

27 Beam Search Rich and Knight 1991, Bisani 1992 Greedy Beam LSS-LRTA* Bounded Solution Quality Korf's Puzzles Solution Quality Life Four-way Grids 35% Obstacles A* wa* greedy breadth IDA* breadth wa* greedy 0-3 log10 total raw cpu time 0 0 log10 total raw 1 cpu time 2 beam search might be best approach, or it might be awful Jordan Thayer and Wheeler Ruml (UNH) Search 13 / 28

28 Beam Search Rich and Knight 1991, Bisani 1992 Greedy Beam LSS-LRTA* Bounded no goals dead ends (left) and many duplicates (right) cause problems Jordan Thayer and Wheeler Ruml (UNH) Search 13 / 28

29 LSS-LRTA* Koenig And Sun, 2008 Greedy Beam LSS-LRTA* Bounded real-time search: interleave planning and acting time bound on planning per action 1. run A* for a fixed number of expansions 2. commit to best f 3. back-up heuristic values using djikstra variant 4. act and repeat Jordan Thayer and Wheeler Ruml (UNH) Search 14 / 28

30 LSS-LRTA* Koenig And Sun, 2008 Greedy Beam LSS-LRTA* Bounded real-time search: interleave planning and acting time bound on planning per action 1. run A* for a fixed number of expansions 2. commit to best f 3. back-up heuristic values using djikstra variant 4. act and repeat Jordan Thayer and Wheeler Ruml (UNH) Search 14 / 28

31 LSS-LRTA* Koenig And Sun, 2008 Greedy Beam LSS-LRTA* Bounded real-time search: interleave planning and acting time bound on planning per action 1. run A* for a fixed number of expansions 2. commit to best f 3. back-up heuristic values using djikstra variant 4. act and repeat Jordan Thayer and Wheeler Ruml (UNH) Search 14 / 28

32 LSS-LRTA* Koenig And Sun, 2008 Greedy Beam LSS-LRTA* Bounded real-time search: interleave planning and acting time bound on planning per action 1. run A* for a fixed number of expansions 2. commit to best f 3. back-up heuristic values using djikstra variant 4. act and repeat Jordan Thayer and Wheeler Ruml (UNH) Search 14 / 28

33 LSS-LRTA* Koenig And Sun, 2008 Greedy Beam LSS-LRTA* Bounded real-time search: interleave planning and acting time bound on planning per action 1. run A* for a fixed number of expansions 2. commit to best f 3. back-up heuristic values using djikstra variant 4. act and repeat Jordan Thayer and Wheeler Ruml (UNH) Search 14 / 28

34 LSS-LRTA* Koenig And Sun, 2008 Greedy Beam LSS-LRTA* Bounded Solution Quality Korf's Puzzles Solution Quality Life Four-way Grids 35% Obstacles greedy breadth LSS-LRTA* breadth LSS-LRTA* greedy log10 total raw cpu time log10 total raw cpu time don t commit if you don t have to! Jordan Thayer and Wheeler Ruml (UNH) Search 14 / 28

35 Section Greedy Beam LSS-LRTA* Bounded duplicate policy affects solution cost and solving time beam searches breadth-first generally better than best-first need closed lists, see Wilt et al SoCS-10 topology dictates algorithm choice greedy if problems are small beam search if problems too big for greedy many dead necessitates complete searches many duplicates necessitates duplicate dropping real-time only if you have real-time constraints Jordan Thayer and Wheeler Ruml (UNH) Search 15 / 28

36 Bounded Weighted A* Optimistic Search Bounded Search Jordan Thayer and Wheeler Ruml (UNH) Search 16 / 28

37 Bounded Search: Outline Bounded Weighted A* Optimistic Search weighted A* larger w faster search optimistic search selecting an appropriate optimism handling duplicates effectively next tutorial: skeptical search, A ǫ, EES Jordan Thayer and Wheeler Ruml (UNH) Search 17 / 28

38 Weighted A* Pohl 1970 Bounded Weighted A* Optimistic Search 1. Best-first Search on f (n) = g(n)+w h(n) w 1 placing additional emphasis on h should encourage progress admissible h ensures bound. Jordan Thayer and Wheeler Ruml (UNH) Search 18 / 28

39 Weighted A* Pohl 1970 Bounded Weighted A* Optimistic Search 1. while open has nodes 2. remove n from open with minimum f (n) 3. if n is a goal then return n 4. otherwise expand n, inserting its children into open 5. return failure placing additional emphasis on h should encourage progress Jordan Thayer and Wheeler Ruml (UNH) Search 18 / 28

40 Weighted A* Pohl 1970 Bounded Weighted A* Optimistic Search 1. while open has nodes 2. remove n from open with minimum f (n) 2a f (n) = g(n)+w h(n) 2b. break ties on in favor of low f(n) 3. if n is a goal then return n 4. otherwise for each child c of n 4a. if c was ever expanded, discard it 4b. if c is in open, keep c with smallest g 4c. otherwise insert c into open 5. return failure discarding duplicates greatly improves performance at the cost of solution quality only applicable with consistent heuristics Jordan Thayer and Wheeler Ruml (UNH) Search 18 / 28

41 Weighted A* Pohl 1970 Bounded Weighted A* Optimistic Search total nodes generated relative to A* 4 2 Life Four-way Grid World expand duplicates drop_duplicates final sol cost 2.7e e+06 Life Four-way Grid World drop duplicates expand duplicates 2 ity 4 2 ity 4 discard duplicates when possible Jordan Thayer and Wheeler Ruml (UNH) Search 18 / 28

42 Weighted A* Pohl 1970 Dynamic Robot Motion Planning wa* Dock Robot wa* Bounded Weighted A* Optimistic Search total nodes generated 2e+07 1e+07 total nodes generated 8e+06 4e ity 3 40 ity 80 larger w does not always mean better performance Jordan Thayer and Wheeler Ruml (UNH) Search 18 / 28

43 Bound for Weighted A* Bounded Weighted A* Optimistic Search f(n) = g(n)+h(n) f (n) = g(n)+w h(n) p is the deepest node on an optimal path to opt. g(sol) f (sol) f (p) g(p)+w h(p) w (g(p)+h(p)) w f(p) w f(opt) w g(opt) 1. works for any f (p) w f(p) 2. g(p)+w h(p) w(g(p)+h(p))! Jordan Thayer and Wheeler Ruml (UNH) Search 19 / 28

44 Optimistic Search Thayer and Ruml, ICAPS-08 Bounded Weighted A* Optimistic Search run weighted A with a high weight. expand node with lowest f value after a solution is found. continue until w best f f(sol) this clean up guarantees solution quality. Jordan Thayer and Wheeler Ruml (UNH) Search 20 / 28

45 Optimistic Search Thayer and Ruml, ICAPS-08 Bounded Weighted A* Optimistic Search p is the deepest node on an optimal path to opt. best f isthenodewiththe smallest f value. f(p) f(opt) f(best f ) f(p) best f provides a lower bound on solution cost Determine best f via priority queue sorted on f Jordan Thayer and Wheeler Ruml (UNH) Search 20 / 28

46 Optimistic Search Thayer and Ruml, ICAPS-08 Bounded Weighted A* Optimistic Search 1. run weighted A with weight (bound 1) expand node with lowest f value after a solution is found. Continue until w best f f(sol) this clean up guarantees solution quality. Jordan Thayer and Wheeler Ruml (UNH) Search 20 / 28

47 Optimistic Search Thayer and Ruml, ICAPS-08 Bounded Weighted A* Optimistic Search 1. run weighted A with a high weight. 2. expand node with lowest f value after a solution is found. continue until w best f f(sol) this clean up guarantees solution quality. Jordan Thayer and Wheeler Ruml (UNH) Search 20 / 28

48 Optimistic Search Thayer and Ruml, ICAPS-08 Bounded Weighted A* Optimistic Search Jordan Thayer and Wheeler Ruml (UNH) Search 20 / 28

49 Optimistic Search Thayer and Ruml, ICAPS-08 Bounded Weighted A* Optimistic Search 1. w = (bound 1) optimism 2. while open and clean contain nodes 3. if no incumbent has been found 4. remove n from open with minimum f (n) 5. remove n from clean 5. if n is a goal, set incumbent to n 6. otherwise expand n, inserting its children into open and c 7. otherwise remove n from clean with minimum f(n) 8. if bound f(n) f(incumbent), return incumbent 9. otherwise expand n, inserting its children into open 10. return failure In practice, clean isn t constructed until an incumbent is found. Jordan Thayer and Wheeler Ruml (UNH) Search 20 / 28

50 Optimistic Search Thayer and Ruml, ICAPS-08 Bounded Weighted A* Optimistic Search log10 total nodes generated Dynamic Robot Motion Planinng wa* Optimism 2 Optimism 5 Optimism ity proper optimism depends on h accuracy Jordan Thayer and Wheeler Ruml (UNH) Search 20 / 28

51 Section Bounded Weighted A* Optimistic Search use weighted A* as a first approach to a problem use optimistic search if you know weighted A* works well duplicate handling can be important wa* can drop, requires consistent heuristic other algorithms can delay solving problems and showing bounds can be separate steps Jordan Thayer and Wheeler Ruml (UNH) Search 21 / 28

52 Bounded wa* Variants Jordan Thayer and Wheeler Ruml (UNH) Search 22 / 28

53 Algorithms: Outline Bounded wa* Variants anytime repairing A* anytime weighted A* restarting weighted A* come back next session for d-fenestration, size-cost search Jordan Thayer and Wheeler Ruml (UNH) Search 23 / 28

54 Anytime Algorithms Based on Weighted A* Bounded wa* Variants anytime weighted A*, Hansen and Zhou run weighted A* 2. if you find a goal, keep going. anytime repairing A*, Likhachev et al, run weighted A* 2. if you find a duplicate, don t look at it just yet. 3. if you find a goal 4. dump duplicates into open, reduce w, keep going. restarting weighted A*, Richter et al run weighted A* 2. if you find a goal, start over with a lower weight. Jordan Thayer and Wheeler Ruml (UNH) Search 24 / 28

55 Anytime Algorithms Based on Weighted A* Bounded wa* Variants anytime weighted A*, Hansen and Zhou run weighted A* 2. if you find a goal, keep going. anytime repairing A*, Likhachev et al, run weighted A* 2. if you find a duplicate, don t look at it just yet. 3. if you find a goal 4. dump duplicates into open, reduce w, keep going. restarting weighted A*, Richter et al run weighted A* 2. if you find a goal, start over with a lower weight. Jordan Thayer and Wheeler Ruml (UNH) Search 24 / 28

56 Anytime Algorithms Based on Weighted A* Bounded wa* Variants continued search, Hansen and Zhou run some complete suboptimal search 2. if you find a goal, keep going. repairing search, Likhachev et al, run some complete parameterized search 2. if you find a duplicate, don t look at it just yet. 3. if you find a goal 4. dump duplicates into open, change parameter, keep going. restarting search, Richter et al run some complete parameterized search 2. if you find a goal, start over with a different parameter. Jordan Thayer and Wheeler Ruml (UNH) Search 24 / 28

57 Anytime Algorithms Based on Weighted A* anytime weighted A*, Hansen and Zhou 1997 Bounded wa* Variants 1. while open has nodes 2. remove n from open with minimum f (n) 3. if n is a goal set n as incumbent 4. otherwise for each child c of n 5. if f(c) < f(incumbent) insert c into open 6. return incumbent anytime repairing A*, Likhachev et al, 2003 restarting weighted A*, Richter et al 2010 Jordan Thayer and Wheeler Ruml (UNH) Search 24 / 28

58 Anytime Algorithms Based on Weighted A* anytime weighted A*, Hansen and Zhou 1997 anytime repairing A*, Likhachev et al, 2003 Bounded wa* Variants 1. while open has nodes 2. remove n from open with minimum f (n) 3. if n is a goal 4. set n as incumbent 5. empty delay into open 6. reduce w 7. otherwise for each child c of n 8. if c was ever expanded, add it to delay 9. otherwise insert c into open 10. return incumbent restarting weighted A*, Richter et al 2010 Jordan Thayer and Wheeler Ruml (UNH) Search 24 / 28

59 Anytime Algorithms Based on Weighted A* Bounded wa* Variants anytime weighted A*, Hansen and Zhou 1997 anytime repairing A*, Likhachev et al, 2003 restarting weighted A*, Richter et al while open has nodes 2. remove n from open with minimum f (n) 3. if n is a goal 4. set n as incumbent 5. reduce w 6. if w < 1, return incumbent 7. otherwise restart the search 8. otherwise for each child c of n 9. if f(c) < f(incumbent) insert c into open 10. return incumbent Jordan Thayer and Wheeler Ruml (UNH) Search 24 / 28

60 Anytime Algorithms Based on Weighted A* Life Four-way Grids 35% Obstacles Korf's Puzzles Bounded wa* Variants Solution Quality 0.4 Solution Quality ARA* AwA* RWA* 0.8 RwA* ARA* AwA* 0 20 raw cpu time raw cpu time 300 no one best anytime algorithm (or even framework) generally, try repairing first. Jordan Thayer and Wheeler Ruml (UNH) Search 24 / 28

61 Anytime Algorithms Based on Weighted A* effort of anytime A* Bounded wa* Variants Jordan Thayer and Wheeler Ruml (UNH) Search 24 / 28

62 Anytime Algorithms Based on Weighted A* effort of anytime repairing A* Bounded wa* Variants Jordan Thayer and Wheeler Ruml (UNH) Search 24 / 28

63 Anytime Algorithms Based on Weighted A* effort of restarting A* Bounded wa* Variants Jordan Thayer and Wheeler Ruml (UNH) Search 24 / 28

64 Bounded wa* Variants ARA*, AwA*, RwA* describe general frameworks (wa* not important) continued few duplicates, tight lower bound repairing many duplicates, high initial bound restarting low h bias, cheap expansions great for unknown deadlines for known deadlines, deadline aware search (Dionne et al, SoCS-11) Jordan Thayer and Wheeler Ruml (UNH) Search 25 / 28

65 Bounded Bibliography Jordan Thayer and Wheeler Ruml (UNH) Search 26 / 28

66 Things We Discussed Bounded Bibliography suboptimal search no guarantees on solution quality so use inadmissible heuristics and drop duplicate states minimize solving time bounded suboptimal search balance time and cost bounds on solution quality drop duplicates when possible looser bounds always better performance anytime search unknown deadlines automatically trades time for quality deadline agnostic frameworks can be used with any complete algorithm Jordan Thayer and Wheeler Ruml (UNH) Search 27 / 28

67 Bibliography Bounded Bibliography J. E. Dorand and D. Michie, Experiments with the Graph Traverser Program, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Ira Pohl, Heuristic Search Viewed as Path Finding in a Graph, Artificial Intelligence volume 1, Elain Rich and Kevin Knight, Artificial Intelligence, Sven Koenig and Xiaoxun Sun, Comparing Real-Time and Incremental Heuristic Search for Real-Time Situated Agents, AAMAS Jordan T. Thayer and Wheeler Ruml, Faster Than Weigthed A*: An Optimistic Approach to Bounded Search, ICAPS Jordan Thayer and Wheeler Ruml (UNH) Search 28 / 28

68 Bibliography Bounded Bibliography Eric A. Hansen, Shlomo Zilberstein and Victor A. Danilchenko, Anytime Heuristic Search: First Results, University of Massachusetts, Amherst Technical Repart 97-50, Maxim Likhachev, Geoff Gordon and Sebastian Thrun ARA*: Anytime A* with Provable Bounds on Sub-Optimality, NIPS Eric A. Hansen and Rong Zhou, Anytime Heuristic Search, Journal of Artificial Intelligence Research volume 28, Silvia Richter, Jordan T. Thayer and Wheeler Ruml, The Joy of Forgetting: Faster via Restarting, ICAPS Chris Wilt, Jordan T. Thayer, and Wheeler Ruml, A Comparison of Greedy Search Algorithms, SoCS Jordan Thayer and Wheeler Ruml (UNH) Search 28 / 28