```The discussion for this quiz was very interesting.  It's probably worth your
time to go back and read those messages, if you haven't already.  Some points I
got out of the discussion were:

* It's tricky to get ropes right.  Clever implementations may loose key
rope qualities, like the O(1) concatenation time.
* The functional approach, building immutable ropes, is probably
superior considering how ropes are intended to be used.
* Autorebalancing didn't hurt much, at least in the quiz test case.
* Copy-On-Write is helpful, when it works.

Many of these points came out due to Eric Mahurin's work in benchmarking the
submitted solutions.  Eric also submitted several variations of rope classes, a
few of which I want to take a look at below.  Let's begin with the class Eric
uses to build the rope trees:

module Mahurin
class Rope
include Enumerable
# form a binary tree from two ropes (possibly sub-trees)
def initialize(left,right)
@left = left
@right = right
@llength = @left.length
@length = @llength+@right.length
@depth = [left.depth, right.depth].max+1
end
# number of elements in this rope
def length
@length
end
# depth of the tree (to help keep the tree balanced)
def depth
@depth
end
# left rope (not needed when depth==0)
def left
@left
end
# right rope (not needed when depth==0)
def right
@right
end

# ...

Here we see the setup and relevant attributes of Rope objects.  First we have
the fact that they are binary trees, with left and right subtrees.  Next we see
that Eric is going to track two lengths for Rope objects, both the total length
and the length of just the left subtree.  The reasoning for that will become
apparent when we examine indexing.  Finally, Eric tracks a depth, for use in
rebalancing.

There are really two major operations that are key to a Rope implementation:
concatenation and indexing.  Here's the concatenation side of the puzzle:

# ...

# appended rope (non-modifying)
def +(other)
# balance as an AVL tree
balance = other.depth-@depth
if balance>+1
left = other.left
right = other.right
if left.depth>right.depth
# rotate other to right before rotating self+other to left
(self + left.left) + (left.right + right)
else
# rotate self+other to left
(self + left) + right
end
elsif balance<-1
if @right.depth>@left.depth
# rotate self to left before rotating self+other to right
(@left + @right.left) + (@right.right + other)
else
# rotate self+other to right
@left + (@right + other)
end
else
self.class.new(self, other)
end
end
alias_method(:<<, :+)

# ...

This method is only this long because it automatically rebalances the tree as
needed.  In fact, if you glance down to the final else clause, you will see the
trivial implementation, which is just to construct a new Rope from the current
Rope and the concatenated element.

The rebalancing done here is, as the comment suggests, a textbook AVL
implementation.  With an AVL tree, you subtract the left depth from the right
depth to get a tree's balance factor.  Anything in the range of -1 to 1 is a
balanced tree.  If the factor is outside of that range, one or two rotations are
required to rebalance the tree.

I'm not going to go into the specific rotations.  If you would like to read up
on them, I recommend:

http://fortheloot.com/public/AVLTreeTutorial.rtf

Let's move on to the indexing methods used to pull information back out of a
Rope:

# ...

# slice of the rope
def slice(start, len)
return self if start.zero? and len==@length
rstart = start-@llength
return @right.slice(rstart, len) if rstart>=0
llen = @llength-start
rlen = len-llen
if rlen>0
@left.slice(start, llen) + @right.slice(0, rlen)
else
@left.slice(start, len)
end
end
# element at a certain position in the rope
def at(index)
rindex = index-@llength
if rindex<0
@left.at(index)
else
@right.at(rindex)
end
end

# ...

Have a look at the at() method first, because it's easier to digest and it shows
the concept of indexing this tree structure.  Essentially, to index in the tree
you check a position against the left length.  If it's less than, index the left
side.  If it's greater than, index the right.  This search tactic is the
hallmark attribute of binary trees.

The slice() method works the same way.  It's just more complicated because it
has to work with two indices instead of one.  Finding the start index is the
same strategy we saw in at().  If that index is in the right subtree, the end
will be as well and the code makes the recursive hand-off without further
checks.  When it's in the left, the end must be located.  If that end point
turns out to also be in the left, the hand-off is made to the left side.  When
it is in the right, a partial slice() is made from both halves and combined.
This covers all the cases.

Eric added a couple more methods to the Rope class that cover iteration and
converting to a String:

# ...

# iterate through the elements in the rope
def each(&block)
@left.each(&block)
@right.each(&block)
end
# flatten the rope into a string (optionally starting with a prefix)
def to_s(s="")
@right.to_s(@left.to_s(s))
end
end

# ...

That covers the tree implementation.  What we haven't seen yet though, are the
leaf nodes.  We need two types for the implementation I want to examine,
EmptyRope and StringRope.  Here's the first of those:

# ...

EmptyRope = Object.new
class << EmptyRope
include Enumerable
def length
0
end
def depth
0
end
def +(other)
other
end
alias_method(:<<, :+)
def slice(start, len)
self
end
def each
end
def to_s
""
end
end

# ...

This implementation is kind of a poor man's singleton instance (the design
pattern, not the Ruby concept, though we see both here).  There shouldn't be any
surprises in this code.  The attributes are zeroed, concatenation results in
whatever the concatenated element is, and slice()ing just returns self which
doubles as an empty String.

That leaves just one last leaf, StringRope:

# ...

class StringRope
include Enumerable
def self.new(*args)
if args.empty?
EmptyRope
else
super
end
end
def initialize(data)
@data = data
end
def length
@data.length
end
def depth
0
end

# ...

This class just wraps a String to support the Rope API we've been examining.
About the only interesting trick here is the is that the default new() for this
class is overridden to allow returning an EmptyRope as needed.  Anytime the
argument is provided though, this method does hand-off to the default new()
implementation.

Here's the concatenation method:

# ...

def +(other)
balance = other.depth
if balance>1
left = other.left
right = other.right
if left.depth>right.depth
# rotate other to right before rotating self+other to left
(self + left.left) + (left.right + right)
else
# rotate self+other to left
(self + left) + right
end
else
Rope.new(self, other)
end
end
alias_method(:<<, :+)

# ...

We see some more balance work here, but the algorithm is simplified since only
the right side can be a subtree.  Beyond that, we've seen this code before.

Here are the final few methods:

# ..

def slice(start, len)
return self if start.zero? and len==@data.length
# depend on ruby's COW mechanism to just reference the slice data
self.class.new(@data.slice(start, len))
end
def at(index)
@data[index]
end
def each(&block)
@data.each_char(&block)
end
def to_s(s="")
s.concat(@data.to_s)
end
end
end

Note here that slice() was overridden to return a StringRope instead of a
String.  As the comment says, Ruby internally uses some Copy On Write semantics
to reference sliced Strings.  This should keep it from wildly duplicating the
data, but it was found that a couple of solutions had problems with this for
unknown reasons.

That covers a basic Rope implementation.  We won't bother to go into the
destructive methods as you are probably better off working without them.

My thanks to all who explored this newly popular data structure with us.  It
looks like there will be a talk on ropes at this year's Rubyconf, so hopefully
we gave the speaker some extra material to work with.

This week's Ruby Quiz will start a day early, to adjust for my Lone Star
Rubyconf schedule this weekend.  The no-spoiler period is still 48 hours and I
will still summarize it at the same time, you just get an extra day to work on
it.  Stay tuned for that problem in just a few minutes...

```