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Search Methodologies

Search Methodologies. Problem Solving. Missionaries and Cannibals Hanoi Towers Maze Puzzles (8-Puzzle, 15-Puzzle) Brix ( 消除方塊 ) Freecell ( 新接龍 ) Magic Cube (Rubik's Cube) Web Spidering Xmas Gift, etc. Problem Space: (1) Condition (restricts) (2) Actions (operators) - and costs

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Search Methodologies

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  1. Search Methodologies

  2. Problem Solving • Missionaries and Cannibals • Hanoi Towers • Maze • Puzzles (8-Puzzle, 15-Puzzle) • Brix (消除方塊) • Freecell (新接龍) • Magic Cube (Rubik's Cube) • Web Spidering • Xmas Gift, etc. Problem Space: (1) Condition (restricts) (2) Actions (operators) - and costs (3) Stats - initial/start state - goal state(s) - dead state(s) 最古老談論 State Search 是春秋時代的 ( )

  3. Example (Missionaries and Cannibals) • Start (M, C, Canoe) =(0,0,0) • operators • Move one cannibal across the river. • Move two cannibals across the river. • Move one missionary across the river. • Move two missionaries across the river. • Move one missionary and one cannibal.

  4. (A,B,C)()() (B,C)(A)() (B,C)()(A) (C)(A)(B) (C)(B)(A) (A,C)()(B) (C)()(A,B) (A,C)(B)() (C)(A,B)() (C)()(A,B) ()(C)(A,B) (C)(A,B)() ()(A,B)(C) Example (Hanoi Tower)

  5. Example (8-Puzzle)

  6. Example (Maze)

  7. Example (Maze, dead states) Infinite Cycle Pitfall

  8. Example (Brix) (Dead) https://www.youtube.com/watch?v=kPi8p42pzKI#t=1m43s

  9. Example (Brix, example 2)

  10. Example (Freecell) Possibly Dead

  11. Contents • Brute force search • Depth-first search • Breadth-first search • Depth first iterative deepening • Properties of search methods • Implementations • Heuristics • Hill climbing • Best first search • Beam search • Optimal paths • A* algorithms • Uniform cost search • Greedy search

  12. Outline Problem Solving Data-Driven (forward-chaining) Goal-Driven (back-chaining) e.g. Brute-Force Search (exhaustive Search) Beam Search Maze, Proof Blind Search (Generate & Test) Heuristic Search (Informed Method) DFS BFS Iterative Deepening Search Hill Climbing Best-first S. A* family A* Alg. British Museum Procedure (identifying the optimal path) Uniform Cost Search (Branch and Bound, Dijkstra) Not always complete (infinite loop) Evaluation function Greedy Search

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  14. Brute Force Search • Search methods that examine every node in the search tree – also called exhaustive. • Generate and test is the simplest brute force search method: • Generate possible solutions to the problem. • Test each one in turn to see if it is a valid solution. • Stop when a valid solution is found. • The method used to generate possible solutions must be carefully chosen.

  15. Depth-First Search • An exhaustive search method. • Follows each path to its deepest node, before backtracking to try the next path.

  16. Breadth-First Search • An exhaustive search method. • Follows each path to a given depth before moving on to the next depth.

  17. Comparison of Depth-First and Breadth-First Search • In some situations, DFS will never find a solution, even though one exists. (即使重複的node已先排除, e.g. 永遠選最右邊的叉路) • DFS – resource (memory) saving BFS – near optimal solution e.g. 解的數量及分佈 DFID (next page)

  18. Depth-First Iterative Deepening • Based on both depth-first and breadth-first search. • Carries out DFS to depth of 1, 2, 3 … • Not as inefficient in time as it might appear, particularly for very large trees, in which it only needs to examine the largest row (the last one) once. • As good as DFS  使用 Revocation 只記action，節省記憶體 • As good as BFS  避免 infinite loop，而且找到的近似最佳解 DFID, IDS (Iterative Deepening Search): Case 1. Resource 有限 (memory/旅遊時間) Case 2. Game Tree, Min-Max 變化大，限時

  19. Properties of Search Methods • Complexity (say, #(states)  Problem/Representation) • How much time and memory the method uses. • Completeness (infinite loop?  Algorithm) • A complete method is one that is guaranteed to find a goal if one exists. • Optimality & Admissibility ( Algorithm) • A method is optimal (or admissible) if it is guaranteed to find the best path to the goal. • Irrevocability ( Implementation) • Example: DFS to solve 15-puzzle • Counter example: DFS to solve Brix Representation: e.g.砝碼秤重問題 N個砝碼有一個瘕疪品 O(N), O(log2N), O(log3N)? 有些實作的方法 可以只記錄 action (而非 state) 再使用 revocation 搜尋

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  21. Outline Problem Solving Data-Driven (forward-chaining) Goal-Driven (back-chaining) e.g. Brute-Force Search (exhaustive Search) Beam Search Maze, Proof Blind Search (Generate & Test) Heuristic Search (Informed Method) DFS BFS Iterative Deepening Search Hill Climbing Best-first S. A* family A* Alg. British Museum Procedure (identifying the optimal path) Uniform Cost Search (Branch and Bound, Dijkstra) Not always complete Evaluation function Greedy Search

  22. Implementations • Depth-first search can be implemented using a queue to store states • But a stack makes more sense • And enables us to create a recursive depth-first search function • Breadth-first search implementations are almost identical to depth-first • Except that they place states on the back of the queue instead of on the front.

  23. Heuristics • Heuristic: a rule or other piece of information that is used to make methods such as search more efficient or effective. • In search, often use a heuristic evaluation function, f(n): • f(n) tells you the approximate distance of a node, n, from a goal node. • f(n) may not be 100% accurate, but it should give better results than pure guesswork.

  24. Heuristics – how informed? h(n)  h*(n) 與「保證最佳解」有關，稍後A* 完整討論 • The more informed a heuristic is, the better it will perform. • Heuristic h is more informed than j, if: • j(n) h(n)h*(n) for all nodes n. • A search method using h will search more efficiently than one using j. • A heuristic should reduce the number of nodes that need to be examined. 8-puzzles: h1(),h2(),h3(); Maze: straight-line distance 如何找 h(n)? Relaxing Problems

  25. Hill Climbing DFS  Hill Climbing  Best First  A* • An informed, irrevocable search method. • Easiest to understand when considered as a method for finding the highest point in a three dimensional search space: • Check the height one foot away from your current location in each direction; North, South, East and West. • As soon as you find a position whose height is higher than your current position, move to that location, and restart the algorithm. 展開過程中遇到更好的點，就直接走過去，走一步算一步； 樹形沒記錄，向前無法回頭，不是實用的解題法。另有一說， Hill Climbing是展開的點先排好，再Stack到OPEN list去，便可回溯。

  26. Best-First Search DFS  Hill Climbing  Best First  A* • Works rather like hill-climbing: • Picks the most likely path (based on heuristic value) from the partially expanded tree at each stage. • Tends to find a shorter path than depth-first or breadth-first search, but does notguarantee to find the best path. 讓最有希望的解，在 Queue 上插隊到前面去 不保證最佳解的原因是?  題示： f = g + h

  27. Beam Search • Breadth-first method. • Only expands the best few paths at each level. • Thus has the memory advantages of depth-first search. • Not exhaustive, and so may not find the best solution. • May not find a solution at all.

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  29. Outline Problem Solving Data-Driven (forward-chaining) Goal-Driven (back-chaining) e.g. Brute-Force Search (exhaustive Search) Beam Search Maze, Proof Blind Search (Generate & Test) Heuristic Search (Informed Method) DFS BFS Iterative Deepening Search Hill Climbing Best-first S. A* family A* Alg. British Museum Procedure (identifying the optimal path) Uniform Cost Search (Branch and Bound, Dijkstra) Not always complete Evaluation function Greedy Search

  30. Optimal Paths • An optimal path through a tree is one that is the shortest possible path from root to goal. • In other words, an optimal path has the lowest cost (not necessarily at the shallowest depth, if edges have costs associated with them). • The British Museum procedure finds an optimal path by examining every possible path, and selecting the one with the least cost. • There are more efficient ways. BM: 所有State 都要能回溯其 path/actions, 找出所有解再選最佳解

  31. A* Algorithms OPEN List & CLOSED List • Uses the cost function: • f(node) = g(node) + h(node). • g(node) is the cost of the path so far leading to the node. • h(node) is an underestimate of how far node is from a goal state. • f is a path-based evaluation function. • An A* algorithm expands paths from nodes that have the lowest f value.

  32. 即使goal, i.e., Z,已放在OPEN 而且經評估h=0, 唯尚未確定其為lowest之前都不算找到最佳解 A ～ ～ ～ ～ X Y ～ ～ Check if it is a goal when we popup it. Z replacement or insertion For any X in the opt. path, 如果 A…X...Z 才是 Opt. Solution 則 ZY將被 永久放在 OPEN List f(X)≤f*(X)<f*(ZY)=f(ZY) Add START to OPEN list while OPEN not empty get node n from OPEN that has the lowest f(n) if n is GOAL then return path move n to CLOSED for each n’ = CanMove(n , direction) g(n’) = g(n) + cost(n,n’) calculate f(n’)=g(n’)+h(n’) if n’ in OPEN list and new n’ is not better , continue if n’ in CLOSED list and new n’ is not better , continue remove any n’ from OPEN and CLOSED add n as n’s parent add n’ to OPEN …從OPEN開始 end for end while if we get here , then there is No Solution

  33. A* Alg (admissible heuristic) • A* algorithms are optimal: • They are guaranteed to find the shortest path to a goal node. • Provided h(node) is always an underestimate. (h is an admissible heuristic). • A* methods are also optimally efficient – they expand the fewest possible paths to find the right one. • If h is not admissible, the method is called A, rather than A*.

  34. A 8 2 4 4 B 5 C 9 5 3 1 2 3 5 E6 D 3 F 4 3 4 1 2 2 8 K 2 1 G 2 H 3 4 2 7 5 L 1 1 M 0 A* Alg (example) Note: (1) Why OPEN list (rather than a tree…?) 不必用 DFS 或 BFS 把每條路徑都找出來比較 (2) Why “underestimate?” 試想如果 E and C 高估, Step 6 可能已得到非最佳解 (M/12) 還有「取代既有 CLOSED」未在此例 若有此情形，則復活者重新加入OPEN

  35. Uniform Cost Search • Also known as branch and bound. • Like A*, but uses: • f(node) = g(node); h( )=0 • g(node) is the cost of the path leading up to node. • Note: All Cost >0 or Dijkstra Recall: 此即以前看過的 Dijkstra Algorithm shortest path searching 從未要求一開始就開圖， 遇到的時候可以查知即可

  36. Greedy Search Greedy Search: f(n)=h(n) 不變， 無可能替換親源節點，所以效果 和 Best-first Search相同，差別在Gr-有CLOSED list，剔除重複節點 Be-重複的節點可能再次展開 • Like A*, but uses: • f(node) = h(node); g( )=0 • Hence, always expands the node that appears to be closest to a goal, regardless of what has gone before. • Not optimal, and not guaranteed to find a solution at all. • Can easily be fooled into taking poor paths.

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  38. Extended Research • Real time A* • Iterative deepening A* • Parallel search • Bidirectional search • Nondeterministic search • Nonchronological backtracking

  39. Recall: Outline Problem Solving Data-Driven (forward-chaining) Goal-Driven (back-chaining) e.g. Brute-Force Search (exhaustive Search) Beam Search Maze, Proof Blind Search (Generate & Test) Heuristic Search (Informed Method) DFS BFS Iterative Deepening Search Hill Climbing Best-first S. A* family A* Alg. British Museum Procedure (identifying the optimal path) Uniform Cost Search (Branch and Bound, Dijkstra) Not always complete Evaluation function Greedy Search

  40. Outline Problem Solving Data-Driven (forward-chaining) Goal-Driven (back-chaining) e.g. Brute-Force Search (exhaustive Search) Beam Search Maze, Proof Blind Search (Generate & Test) Heuristic Search (Informed Method) DFS BFS Iterative Deepening Search Hill Climbing Best-first S. A* family • Real time A*, Iterative deepening A* • Parallel search • Bidirectional search • Island-Driven Search • Nondeterministic search • Nonchronological backtracking A* Alg. Uniform Cost Search (Branch and Bound, Dijkstra) Greedy Search

  41. Iterative Deepening A* • A* is applied iteratively, with incrementally increasing limits on f(n). • Works well if there are only a few possible values for f(n). • The method is complete, and has a low memory requirement, like depth-first search. 結合A* 與 Beam Search Methods 補看更多 branches 看能否得解或得更好的解。 See also: Real-time A* -- 不再謀定而後動，設定時間參數，時限到，就走到目前OPENlist的首位節點，它代表目前最佳的中途點，改為起點，再重來。

  42. Parallel Search • Some search methods can be easily split into tasks which can be solved in parallel. • Important concepts to consider are: • Task distribution • Load balancing • Tree ordering 同時pop N個 nodes 出來展開 宣告 OpenTmp, 並暫記thisNode 比對的 Openlist  N+1 thisNode被取代 刪除這個暫記區 沒被取代的OpenTmp抄到Open 存取主要Open/Close 要lock

  43. Bidirectional Search • Also known as wave search. • Useful when the start and goal are both known. • Starts two parallel searches – one from the root node and the other from the goal node. • Paths are expanded in a breadth-first fashion from both points. • Where the paths first meet, a complete and optimal path has been formed. (if all costs are equal) 從頭(起點)從尾(終點)兩端互找， e.g. 1 – 智慧拼盤 OPEN(CLOSED) 共同點 e.g. 2 – 迷宮中留下如何到達此點的記錄 Note: 目前的作法不能保證為最佳解 See also: Island-Driven Search (c.f. And-or tree, i.e., goal tree)

  44. Nondeterministic Search • Useful when very little is known about the search space. • Combines the depth first and breadth first approaches randomly. • Avoids the problems of both, but does not necessarily have the advantages of either. • New paths are added to the queue in random positions, meaning the method will follow a random route through the tree until a solution is found. 亂跳式搜尋 (DFS x BFS ) 在OPEN list 隨機放前放後， 也可直接以樹形搜尋，在葉 節點上隨機展開。 早期的變化方法(無hueristics) 可避免DFS與BFS各自的風險

  45. Nonchronological backtracking • Depth first search uses chronological backtracking. • Does not use any additional information to make the backtracking more efficient. • Nonchronological backtracking involves going back to forks in the tree that are more likely to offer a successful solution, rather than simply going back to the next unexplored path. 回溯到“直系”親代任一點，該點提供最好的“旁系”路徑分支，此法可以不搭配Open list 板凳，直接以樹形回溯 早期的變化方法(有hueristics) Say,本來預估的成本怎麼變差了?

  46. Extended Study • Can you write a Navigation Program? • How do you ride to Taipei 101? • Heuristic function and Real-time A* (開圖的錯覺) • In case you need to insert a midway point? (注意附加功能) • What can Innovation Play in Problem Solving? • Creative idea upon states/actions/heuristics … • Edge computing (e.g.state=checklist, problem=TSP…) • e.g. leaving the kid’s scissors home when eating outdoors…

  47. Preview Advanced Search Constructive Methods (search tree for goals) • General Metaheuristics • Exchanging heuristics • Iterated local search • (Ant Colony Optimization) • Taboo Search • Simulated annealing • Genetic algorithms 由這兩個例子 來看這些 methods Search for Goals (8-Queens/conflict=0) Search for Optima (TSP/ lowest cost) Constraint Satisfaction forward checking (Dead Ends?) DFS (64 x 63 x…) DFS (n x (n-1) x…) A* Algorithm Heuristic (which branch first?) Most-Constrained Variables Local Search (metaheuristics) Heuristic Repair k-opt exchanging search space search tree search plane = untraceable branches Iterative Methods (search plane for optima)

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