1. PFDS 10.1.2
Binary Random-Access
List Revisited
@rf0444

2. a Seq = NIL | CONS of a * (a * a) Seq
NIL
CONS 1 NIL
CONS 1 (CONS (2, 3) NIL)
CONS 1 (CONS (2,3) (CONS ((4,5),(6,7)) NIL))

3. a Seq = NIL | CONS of a * (a * a) Seq
NIL 0個
CONS 1 NIL 1個
CONS 1 (CONS (2, 3) NIL) 3個
CONS 1 (CONS (2,3) (CONS ((4,5),(6,7)) NIL))
7個

4. a Seq = NIL | CONS of a * (a * a) Seq
CONS 1 (CONS (2,3) (CONS ((4,5),(6,7)) NIL))
1個 2個 4個
a a*a (a * a) * (a * a)

5. 9.2.1 Binary Random-Access Lists
a Tree = LEAF of a | NODE of a Tree * a Tree
a Digit = ZERO | ONE of a Tree
a RList = a Digit list
ONE ZERO ONE
1 2 3 4 5

6. 9.2.1 Binary Random-Access Lists
型だけ見ると下の構成もできるように見える
ONE ZERO ONE
サイズが 5
1 2
おかしい 3 4
完全じゃない

7. 10.1.2 Binary Random-Access
Lists Revisited
a Seq = NIL | ZERO of (a * a) Seq
| ONE of a * (a * a) Seq
((a * a) * (a * a)) * ((a * a) * (a * a)) Seq
ONE ZERO ONE
1 ((2, 3), (4,5))
a
a*a (a * a) * (a * a)

8. head, tail
大体 9.2.1 と同じ

9. lookup
lookup 5
ONE ONE ZERO ONE
1 (2, 3) (((4, 5), (6, 7)), ((8, 9), (10, 11)))

10. lookup
lookup 5
ONE ONE ZERO ONE
1 (2, 3) (((4, 5), (6, 7)), ((8, 9), (10, 11)))
lookup 4
ZERO ONE ZERO ONE
(2, 3) (((4, 5), (6, 7)), ((8, 9), (10, 11)))

11. lookup
lookup 4
ZERO ONE ZERO ONE
(2, 3) (((4, 5), (6, 7)), ((8, 9), (10, 11)))
lookup 2 の左
ONE ZERO ONE
(2, 3) (((4, 5), (6, 7)), ((8, 9), (10, 11)))

12. lookup
lookup 2 の左
ONE ZERO ONE
(2, 3) (((4, 5), (6, 7)), ((8, 9), (10, 11)))
lookup 1 の左
ZERO ZERO ONE
(((4, 5), (6, 7)), ((8, 9), (10, 11)))

13. lookup
lookup 1 の左
ZERO ZERO ONE
(((4, 5), (6, 7)), ((8, 9), (10, 11)))
lookup 0 の右 の左
ZERO ONE
(((4, 5), (6, 7)), ((8, 9), (10, 11)))

14. lookup
lookup 0 の右 の左
ZERO ONE
(((4, 5), (6, 7)), ((8, 9), (10, 11)))
lookup 0 の左 の右 の左
ONE
(((4, 5), (6, 7)), ((8, 9), (10, 11)))

15. lookup
lookup 0 の左 の右 の左
ONE
(((4, 5), (6, 7)), ((8, 9), (10, 11)))
(((4, 5), (6, 7)), ((8, 9), (10, 11))) の左 の右 の左
((4, 5), (6, 7)) の右 の左
(6, 7) の左
6

16. update
update 5 20
ONE ONE ZERO ONE
1 (2, 3) (((4, 5), (6, 7)), ((8, 9), (10, 11)))

17. update
update 5 20
ONE ONE ZERO ONE
1 (2, 3) (((4, 5), (6, 7)), ((8, 9), (10, 11)))
cons 1 . update 4 20
ZERO ONE ZERO ONE
(2, 3) (((4, 5), (6, 7)), ((8, 9), (10, 11)))

18. update
cons 1 . update 4 20
ZERO ONE ZERO ONE
(2, 3) (((4, 5), (6, 7)), ((8, 9), (10, 11)))
cons 1 . ZERO . update 2 (20, 7)
ONE ZERO ONE
(2, 3) (((4, 5), (6, 7)), ((8, 9), (10, 11)))

19. update
cons 1 . ZERO . update 2 (20, 7)
ONE ZERO ONE
(2, 3) (((4, 5), (6, 7)), ((8, 9), (10, 11)))
cons 1 . ZERO . cons (2, 3) . update 1 (20, 7)
ZERO ZERO ONE
(((4, 5), (6, 7)), ((8, 9), (10, 11)))

20. update
cons 1 . ZERO . cons (2, 3) . update 1 (20, 7)
ZERO ZERO ONE
(((4, 5), (6, 7)), ((8, 9), (10, 11)))
cons 1 . ZERO . cons (2, 3) . ZERO .
update 0 ((4, 5), (20, 7))
ZERO ONE
(((4, 5), (6, 7)), ((8, 9), (10, 11)))

21. update
cons 1 . ZERO . cons (2, 3) . ZERO .
update 0 ((4, 5), (20, 7))
ZERO ONE
(((4, 5), (6, 7)), ((8, 9), (10, 11)))
cons 1 . ZERO . cons (2, 3) . ZERO . ZERO .
update 0 (((4, 5), (20, 7)), ((8, 9), (10, 11)))
ONE
(((4, 5), (6, 7)), ((8, 9), (10, 11)))

22. update
cons 1 . ZERO . cons (2, 3) . ZERO . ZERO .
update 0 (((4, 5), (20, 7)), ((8, 9), (10, 11)))
ONE
(((4, 5), (6, 7)), ((8, 9), (10, 11)))
cons 1 . ZERO . cons (2, 3) . ZERO . ZERO
ONE
(((4, 5), (20, 7)), ((8, 9), (10, 11)))

23. update
cons 1 . ZERO . cons (2, 3) . ZERO . ZERO
ONE
(((4, 5), (20, 7)), ((8, 9), (10, 11)))
ONE ONE ZERO ONE
1 (2, 3) (((4, 5), (20, 7)), ((8, 9), (10, 11)))

24. update
先頭要素が ZERO の時に毎回 lookup が走る
結果、update は O(log^2 n)
変更する値ではなく、変更する関数を渡すよ
うにする

25. update
update 5 20
ONE ONE ZERO ONE
1 (2, 3) (((4, 5), (6, 7)), ((8, 9), (10, 11)))
fupdate (fn _ -> 20) 5
ONE ONE ZERO ONE
1 (2, 3) (((4, 5), (6, 7)), ((8, 9), (10, 11)))

26. update
fupdate (fn _ -> 20) 5
ONE ONE ZERO ONE
1 (2, 3) (((4, 5), (6, 7)), ((8, 9), (10, 11)))
cons 1 . fupdate (fn _ -> 20) 4
ZERO ONE ZERO ONE
(2, 3) (((4, 5), (6, 7)), ((8, 9), (10, 11)))

27. update
cons 1 . fupdate (fn x -> 20) 4
ZERO ONE ZERO ONE
(2, 3) (((4, 5), (6, 7)), ((8, 9), (10, 11)))
cons 1 . ZERO . fupdate (fn (x, y) -> (20, y)) 2
ONE ZERO ONE
(2, 3) (((4, 5), (6, 7)), ((8, 9), (10, 11)))

28. update
cons 1 . ZERO . fupdate (fn (x, y) -> (20, y)) 2
ONE ZERO ONE
(2, 3) (((4, 5), (6, 7)), ((8, 9), (10, 11)))
cons 1 . ZERO . cons (2, 3) .
fupdate (fn (x, y) -> (20, y)) 1
ZERO ZERO ONE
(((4, 5), (6, 7)), ((8, 9), (10, 11)))

29. update
cons 1 . ZERO . cons (2, 3) .
fupdate (fn (x, y) -> (20, y)) 1
ZERO ZERO ONE
(((4, 5), (6, 7)), ((8, 9), (10, 11)))
cons 1 . ZERO . cons (2, 3) . ZERO .
fupdate (fn (w, (x, y)) -> (w, (20, y))) 0
ZERO ONE
(((4, 5), (6, 7)), ((8, 9), (10, 11)))

30. update
cons 1 . ZERO . cons (2, 3) . ZERO .
fupdate (fn (w, (x, y)) -> (w, (20, y))) 0
ZERO ONE
(((4, 5), (6, 7)), ((8, 9), (10, 11)))
cons 1 . ZERO . cons (2, 3) . ZERO . ZERO .
fupdate (fn ((w, (x, y)), z) -> ((w, (20, y)), z)) 0
ONE
(((4, 5), (6, 7)), ((8, 9), (10, 11)))

31. update
cons 1 . ZERO . cons (2, 3) . ZERO . ZERO .
fupdate (fn ((w, (x, y)), z) -> ((w, (20, y)), z)) 0
ONE
(((4, 5), (6, 7)), ((8, 9), (10, 11)))
cons 1 . ZERO . cons (2, 3) . ZERO . ZERO
ONE
(((4, 5), (20, 7)), ((8, 9), (10, 11)))

32. update
cons 1 . ZERO . cons (2, 3) . ZERO . ZERO
ONE
(((4, 5), (20, 7)), ((8, 9), (10, 11)))
ONE ONE ZERO ONE
1 (2, 3) (((4, 5), (20, 7)), ((8, 9), (10, 11)))