On 2010-05-14 08:17, Yoriyuki Yamagata wrote:
> When I read the balancing function of stdlib's Set/Map several years
> ago, I thought I have understand how it works.  But now, I read it again
> and I'm less confident now.  Could someone answer my questions?  Here is
> the snippet of the code.
> 
>  let bal l v r =
>       let hl = match l with Empty -> 0 | Node(_,_,_,h) -> h in
>       let hr = match r with Empty -> 0 | Node(_,_,_,h) -> h in
>       if hl > hr + 2 then begin
>         match l with
> 
>           Empty -> invalid_arg "Set.bal"
>         | Node(ll, lv, lr, _) ->
>             if height ll >= height lr then
>               create ll lv (create lr v r)
>             else begin
>               match lr with
> 
>                 Empty -> invalid_arg "Set.bal"
>               | Node(lrl, lrv, lrr, _)->
>                   create (create ll lv lrl) lrv (create lrr v r)
>             end
>       end else if hr > hl + 2 then begin
> 
>         match r with
>           Empty -> invalid_arg "Set.bal"
>         | Node(rl, rv, rr, _) ->
>             if height rr >= height rl then
>               create (create l v rl) rv rr
>             else begin
> 
>               match rl with
>                 Empty -> invalid_arg "Set.bal"
>               | Node(rll, rlv, rlr, _) ->
>                   create (create l v rll) rlv (create rlr rv rr)
>             end
> 
>       end else
>         Node(l, v, r, (if hl >= hr then hl + 1 else hr + 1))
[...]
> Another question is that why OCaml implementation allows 
> a balancing factor up to *2*, which is usually allowed only up to 1?

I guess the balancing factor of -2/+2 can only occur temporarily
during insert/delete operations (and such). See for instance (set.ml):

let rec add x = function
    Empty -> Node(Empty, x, Empty, 1)
  | Node(l, v, r, _) as t ->
      let c = Ord.compare x v in
      if c = 0 then t else
      if c < 0 then bal (add x l) v r else bal l v (add x r)

let rec remove x = function
    Empty -> Empty
  | Node(l, v, r, _) ->
      let c = Ord.compare x v in
      if c = 0 then merge l r else
      if c < 0 then bal (remove x l) v r else bal l v (remove x r)

This is what bal is for: to fix the balance in the tree root.

I guess this: http://en.wikipedia.org/wiki/AVL_tree#Operations is
more or less correct. :-)

STF

http://eisenbits.homelinux.net/~stf/
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