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問題解決：木とグラフの探索

問題解決：木とグラフの探索. 白井 良明 立命館大学情報理工学部 知能情報学科 shirai@ci.ritsumei.ac.jp. ON (B, C) ON (C, A). ON (B, C) ON (C, A). 問題の記述：初期状態と目標状態. Ａ. Ｂ. Ｃ. Ｂ. Ａ. Ｃ. ONTABLE (A) ONTABLE (C). x. y. 問題の記述： オペレータ.

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問題解決：木とグラフの探索

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  1. 問題解決：木とグラフの探索 白井 良明 立命館大学情報理工学部 知能情報学科 shirai@ci.ritsumei.ac.jp

  2. ON (B, C) ON (C, A) ON (B, C) ON (C, A) 問題の記述：初期状態と目標状態 Ａ Ｂ Ｃ Ｂ Ａ Ｃ ONTABLE (A) ONTABLE (C)

  3. x y 問題の記述：オペレータ PICKUP（x）；テーブルの上のxをつかむ　　　　　　　　　　　　　　　　 前提条件： ONTABLE（x）,CLEAR（x）,EMPTY　　　　　　　　　　　削除リスト：ONTABLE（x）,CLEAR（x）,EMPTY　　　　　　　　　　　追加リスト：HOLD（x）；手はxをつかんでいる 　　　　　PUTDOWN（x）；xをテーブルの上に置く　　　　　　　　　　　　　　　　　前提条件： HOLD（x）　　　　　　　　　　　　　　　　　　　　　　　　　　　削除リスト：HOLD（x）　　　　　　　　　　　　　　　　　　　　　　　　　　　追加リスト： ONTABLE（x）,CLEAR（x）,EMPTYTAKEOFF（x,y）；yの上にあるxをつかむ　　　　　　　　　　　　　　　　前提条件：ON（x,y）,CLEAR（x）,EMPTY　　　　　　　　　　　　　　　削除リスト： ON（x,y）,CLEAR（x）,EMPTY　　　　　　　　　　　　　　　追加リスト：CLEAR（y）,HOLD（x）　　　　　　　　　　　　　　　　PUTON（x,y）；xをyの上へ置く　　　　　　　　　　　　　　　　　　　　　　　前提条件： HOLD（x）,CLEAR（y）　　　　　　　　　　　　　　　　　　　　削除リスト：HOLD（x）,CLEAR（y）　　　　　　　　　　　　　　　　　　　追加リスト：　 ON（x,y）,CLEAR（x）,EMPTY x x

  4. 木の探索 Ｓ Ａ Ｂ Ｃ Ｄ Ｅ Ｆ Ｈ Ｇ

  5. width-first procedure depth-first tree search • add (s, open); • LOOP: if open = null then exit(fail); • n:= first(open); • If goal(n) then exit (success); • Remove(n, open); • Expand(n); • Put all child nodes into the top of open and attach pointers to n; • Go to LOOP. last

  6. グラフの探索 Ｄ Ｂ Ｅ Ｇ Ｈ Ａ Ｃ Ｆ Ｓ Ｉ

  7. procedure depth-first tree search • add (s, open); • LOOP: if open = null then exit(fail); • n:= first(open); • If goal(n) then exit (success); • Remove(n, open); • Expand(n); • Put all child nodes into the top of open and attach pointers to n; • Go to LOOP. add(n, closed)

  8. procedure depth-first tree search • add (s, open); • LOOP: if open = null then exit(fail); • n:= first(open); • If goal(n) then exit (success); • Remove(n, open); add(n, closed) • Expand(n); • Put all child nodes into the top of open and attach pointers to n; • Go to LOOP. which are not in open nor closed

  9. 4 1 6 5 4 3 5 7 6 7 最適経路の探索 Ｓ ４ １ Ｂ Ａ ２ ３ １ ２ Ｅ Ｃ Ｆ Ｄ ４ ３ ２ １ Ｈ Ｇ１ Ｉ Ｇ２

  10. procedure optimal tree search • add (s, open); g^(s):= 0; • LOOP: if open = null then exit(fail); • n:= first(open); • If goal(n) then exit (success); • Remove(n, open); • Expand(n); • Put all child nodes into open and attach pointers to n; • Sort nodes in open in increasing order of g^; • Go to LOOP.

  11. 4 1 6 5 4 3 5 7 6 7 最適経路の探索 Ｓ ４ １ Ｂ Ａ ２ ３ １ ２ Ｅ Ｃ Ｆ Ｄ ４ ３ ２ １ Ｈ Ｇ１ Ｉ Ｇ２

  12. グラフ最適経路の探索 Ｄ Ｂ Ｅ Ｇ ６ ５ ４ ５ ３ ６ Ｈ Ａ ４ ５ Ｃ Ｆ ６ Ｓ Ｉ

  13. procedure optimal tree search • add (s, open); g^(s):= 0; • LOOP: if open = null then exit(fail); • n:= first(open); • If goal(n) then exit (success); • Remove(n, open); • Expand(n); • Put all child nodes into open and attach pointers to n; • Sort nodes in open in increasing order of g^; • Go to LOOP.

  14. procedure optimal tree search • add (s, open); g^(s):= 0; • LOOP: if open = null then exit(fail); • n:= first(open); • If goal(n) then exit (success); • Remove(n, open); • Expand(n); • Put all child nodes into open, and attach pointers to n; • Sort nodes in open in increasing order of g^; • Go to LOOP. add(n, closed)

  15. procedure optimal tree search • add (s, open); g^(s):= 0; • LOOP: if open = null then exit(fail); • n:= first(open); • If goal(n) then exit (success); • Remove(n, open); add(n, closed) • Expand(n); • Put all child nodes into open, and attach pointers to n; • Sort nodes in open in increasing order of g^; • Go to LOOP. which are not in open nor closed. If child node j was in open and g^(n, j) < g^(j), then attach pointers to n;

  16. グラフ最適経路の探索 Ｄ Ｂ Ｅ Ｇ ６ ５ ４ ５ ３ ６ Ｈ Ａ ４ ５ Ｃ Ｆ ６ Ｓ Ｉ

  17. グラフ最適経路の探索 ４ Ｂ Ｇ ３ Ｃ Ａ ２ Ｈ Ｄ Ｅ ４ Ｓ Ｆ

  18. グラフ最適経路の探索 ４ Ｂ Ｇ ３ Ｃ Ａ ２ 7 Ｈ Ｄ Ｅ ４ Ｓ Ｆ

  19. グラフ最適経路の探索 ４ Ｂ Ｇ ３ Ｃ Ａ ２ / ５ 7 Ｈ Ｄ Ｅ ４ Ｓ Ｆ

  20. グラフ最適経路の探索 ４ Ｂ Ｇ ８ ３ Ｃ Ａ ２ ６ ５ Ｈ Ｄ Ｅ ４ ６ Ｓ Ｆ

  21. ２ １ ３ ０ Ｄ Ｂ Ｅ Ｇ ２ ３ Ａ Ｈ ４ ３ Ｃ Ｆ ４ ６ Ｓ Ｉ グラフの最良優先探索

  22. ２ ２ １ ３ ０ Ｄ Ｂ Ｅ Ｇ ２ ３ Ａ Ｈ ４ ４ ３ Ｃ Ｆ ４ ６ Ｓ Ｉ グラフの最良優先探索 ３ ６

  23. ２ １ ２ １ ３ ３ ０ Ｄ Ｂ Ｅ Ｇ ２ ３ Ａ Ｈ ４ ４ ３ Ｃ Ｆ ４ ６ Ｓ Ｉ グラフの最良優先探索 ３ ６

  24. ２ １ ２ １ ３ ０ ３ ０ Ｄ Ｂ Ｅ Ｇ ２ ２ ３ Ａ Ｈ ４ ４ ３ Ｃ Ｆ ４ ６ Ｓ Ｉ グラフの最良優先探索 ３ ６

  25. procedure best-first search • add (s, open); • LOOP: if open = null then exit(fail); • n:= first(open); • If goal(n) then exit (success); • Remove(n, open); add(n, closed) • Expand(n); • Put all child nodes except in open or closed into open, Sort nodes in open in increasing order of h^ ; • Go to LOOP;

  26. ２ ０ Ｂ Ｇ ４ ３ Ａ Ｃ ３ ４ ２ Ｈ Ｄ Ｅ ６ ４ Ｓ Ｆ グラフの最良優先探索

  27. ２ ２ ０ Ｂ Ｇ ４ ４ ３ ３ Ａ Ｃ ３ ４ ２ Ｈ Ｄ Ｅ ６ ６ ４ Ｓ Ｆ グラフの最良優先探索 ３

  28. ２ ２ ０ Ｂ Ｇ ４ ４ ３ ３ Ａ Ｃ ３ ４ ４ ２ ２ Ｈ Ｄ Ｅ ６ ６ ４ Ｓ Ｆ グラフの最良優先探索 ３

  29. ２ ２ ０ ０ Ｂ Ｇ ４ ４ ３ ３ Ａ Ｃ ３ ４ ４ ２ ２ Ｈ Ｄ Ｅ ６ ６ ４ ４ Ｓ Ｆ グラフの最良優先探索 ３

  30. A*アルゴリズム f^(n):= g^(n) + h^(n) ２ １ ３ ０ Ｄ Ｂ Ｅ Ｇ ６ ３ ２ Ａ Ｈ ４ ３ Ｃ Ｆ ４ ６ ６ Ｓ Ｉ

  31. ６ ８ A*アルゴリズム ２ １ ３ ０ Ｄ Ｂ Ｅ Ｇ ６ ３ ２ Ａ Ｈ ４ ３ Ｃ Ｆ ４ ６ ６ Ｓ Ｉ

  32. ６ ６ ８ ８ A*アルゴリズム ２ １ ３ ０ Ｄ Ｂ Ｅ Ｇ ６ ３ ２ Ａ Ｈ ４ ３ Ｃ Ｆ ４ ６ ６ Ｓ Ｉ

  33. ６ ６ ８ ６ ８ ８ A*アルゴリズム ２ １ ３ ０ Ｄ Ｂ Ｅ Ｇ ６ ３ ２ Ａ Ｈ ４ ３ Ｃ Ｆ ４ ６ ６ Ｓ Ｉ

  34. procedure optimal tree search • add (s, open); g^(s):= 0; • LOOP: if open = null then exit(fail); • n:= first(open); • If goal(n) then exit (success); • Remove(n, open); add(n, closed) • Expand(n); • Put all child nodes into open, and attach pointers to n; • Sort nodes in open in increasing order of g^; • Go to LOOP. which are not in open nor closed. If child node j was in open,select min{g^(j), g^(n, j)},

  35. procedure optimal tree search • add (s, open); g^(s):= 0; • LOOP: if open = null then exit(fail); • n:= first(open); • If goal(n) then exit (success); • Remove(n, open); add(n, closed) • Expand(n); • Put all child nodes into open, and attach pointers to n; • Sort nodes in open in increasing order of g^; • Go to LOOP. which are not in open nor closed. If child node j was in open,select min{g^(j), g^(n, j)},

  36. procedure optimal search • add (s, open); g^(s):= 0; • LOOP: if open = null then exit(fail); • n:= first(open); • If goal(n) then exit (success); • Remove(n, open); add(n, closed) • Expand(n); • Put all child nodes except in open or closed into open, compute g, and attach pointers to n; and attach pointers to n; • Sort nodes in open in increasing order of g^; • Go to LOOP. If child node j was in open,select min{g^(j), g^(n, j)},

  37. A-algorithm procedure optimal search f^(s):=h^(s); • add (s, open); g^(s):= 0; • LOOP: if open = null then exit(fail); • n:= first(open); • If goal(n) then exit (success); • Remove(n, open); add(n, closed) • Expand(n); • Put all child nodes except in open or closed into open, compute g, and attach pointers to n; • Sort nodes in open in increasing order of g^; • Go to LOOP. f^(n, j):= g^(n, j) + h^(n); If child node j was in open,select min{g^(j), g^(n, j)},

  38. procedure A-algorithm • add (s, open);f^(s):=h^(s); • LOOP: if open = null then exit(fail); • n:= first(open); • If goal(n) then exit (success); • Remove(n, open); add(n, closed) • Expand(n); • Put all child nodes except in open or closed into open, compute g; f^(n, j):= g^(n, j) + h^(n); and attach pointers to n; • Sort nodes in open in increasing order of g^; • Go to LOOP. If child node j was in open,select min{g^(j), g^(n, j)}, min{f^(j), f^(n, j)}; f^;

  39. procedure optimal search procedure A-algorithm search procedure A-algorithm search • add (s, open); f^(s):= h^(s); • LOOP: if open = null then exit(fail); • n:= first(open); • If goal(n) then exit (success); • Remove(n, open); add(n, closed) • Expand(n); • Put all child nodes except in open or closed into open, compute g; f^(n, j):= g^(n, j) + h^(n); if node j is in open or closed, select min{f^(j), f^(n, j)} and put into open; Sort nodes in open in increasing order of f^; • Go to LOOP;

  40. A*アルゴリズム ２ ０ ６ Ｂ Ｇ ４ ３ ６ ６ Ａ Ｃ ３ ４ ２ Ｄ Ｅ Ｈ ６ ６ ４ Ｓ Ｆ

  41. A*アルゴリズム ２ ０ ６ Ｂ Ｇ ４ ３ ６ ６ Ａ Ｃ ３ ４ ２ ８ 10 Ｄ Ｅ Ｈ ６ ６ ４ Ｓ Ｆ

  42. A*アルゴリズム ２ ０ ６ Ｂ Ｇ ４ ３ ６ ６ Ａ Ｃ ８ ３ ４ ２ ８ Ｄ Ｅ Ｈ ６ ６ １０ ４ Ｓ Ｆ

  43. A*アルゴリズム ２ ８ ０ ６ Ｂ Ｇ ４ ３ ６ ６ Ａ Ｃ ８ ３ ４ ２ ８ 10 ８ Ｄ Ｅ Ｈ ６ ６ １０ ４ Ｓ Ｆ

  44. CLOSED, OPEN を用いる例 １０ Ｓ ３ １ ９ ９ ６ １０ Ａ Ｂ １ ６ ４ ５ ８ ７ １４ ７ １５ Ｅ Ｄ Ｃ

  45. CLOSED, OPEN を用いる例 １０ Ｓ ３ １ ８ ９ ６ １０ Ａ Ｂ １ ６ ４ ５ ８ ７ １４ １５ ７ １５ Ｅ Ｄ Ｃ

  46. CLOSED, OPEN を用いる例 １０ Ｓ ３ １ ８ ９ ６ １０ Ａ Ｂ １ ６ ４ ５ ８ ７ １４ １５ １３ ７ １４ １５ Ｅ Ｄ Ｃ

  47. A*アルゴリズムが有効な例 Ｓ （７） ４ ５ ６ ９ ９ ８ （５） （４） （２） Ａ Ｂ Ｃ ３ ３ ３ ２ ２ ２ １ １ １ １０ ８ ９ （２） （３） （３） （３） （３） （３） （１） Ｄ Ｅ Ｋ Ｆ Ｈ Ｉ Ｊ ４ ４ ４ ４ ４ ４ ４ ２ １ ３ ３ ４ ４ ４ ９ ８ Ｇ２ Ｇ１ Ｇ３ Ｇ４ Ｇ５ Ｇ６ Ｇ７ Ｇ８

  48. A*アルゴリズムとならない例 Ｓ ５ ２ ３ ５ ３ ４ ７ Ａ Ｂ ４ ２ Ｇ ６

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