Programming in Scala”

Chapter notes 10
data structures, both classic and new, from the perspective of functional programming. We highly
recommend picking up this book if you’re interested in data structures. The dissertation that the
book is based on is also available from Carnegie Mellon University’s website³⁶.
Rose trees
The tree structure that we introduce at the end of the chapter is called a Rose Tree³⁷. It is a nonempty
multi-way tree that contains data at the nodes rather than at the leaves.
The algebra of data types
The “algebraic” in algebraic data types means something specific. This is a reference to the fact that
such data types are composed of sums and products of other types. More specifically, data types
form a seminearring³⁸.
See the following links:
• The Algebra of Algebraic Data Types³⁹ by Chris Taylor.
• Species and Functors and Types, Oh My!⁴⁰ by Brent Yorgey
• Clowns to the left of me, jokers to the right⁴¹ by Conor McBride
Zippers
Since an algebraic data type is a type-level function involving sums and products, we can take the
derivative of such a function, yielding a data structure called a zipper⁴². The zipper for a given
structure is like the original structure, but with a movable “focus” or pointer into the structure. This
can be used to insert, remove, and modify elements under the focus.
For example, a list zipper⁴³ consists of one element under focus together with two lists: one
enumerating all the elements to the left of the focus, and another enumerating all the elements to
the right of the focus. The focus can me moved left or right (like a zipper on a piece of clothing) and
elements will move through the focus. The element under focus can then be removed or modified,
and we can insert a new element by consing onto the lists to its left or right.
³⁶http://www.cs.cmu.edu/~rwh/theses/okasaki.pdf
³⁷http://en.wikipedia.org/wiki/Rose_tree
³⁸http://en.wikipedia.org/wiki/Near-semiring
³⁹http://chris-taylor.github.io/blog/2013/02/10/the-algebra-of-algebraic-data-types/
⁴⁰http://www.cis.upenn.edu/~byorgey/papers/species-pearl.pdf
⁴¹http://personal.cis.strath.ac.uk/~conor/Dissect.pdf
⁴²http://en.wikipedia.org/wiki/Zipper_%28data_structure%29
⁴³http://eed3si9n.com/learning-scalaz/Zipper.html