Big Tree

The "Big Three" List Operations Overview:

In Scheme (and Scamper), there are three key higher-order list operations: map, apply, and reduce (also called fold). These operations are powerful tools for manipulating lists and allow for both repetition and parallelism.

1. map Function:

• Purpose: The map function applies a given procedure to each element of a list and returns a new list with the results.
• Key Idea: Instead of using loops (like in other programming languages), map allows for a different way of thinking about repetition by applying the same function to each element in parallel.

Example of map:

(map (lambda (x) (* x 2)) (list 1 2 3 4)) ; produces (2 4 6 8)

• Benefits:
  • Parallel Processing: The function can be applied to multiple elements simultaneously, which is crucial in modern computing for speed (e.g., in Google and Amazon's systems).
  • Encourages decomposition: You don’t need to think about the whole list, just the operation on a single element.

2. apply Function:

• Purpose: apply takes a procedure and applies it to a list of arguments, combining the elements of the list into a single value.
• Key Idea: apply is useful for applying functions that can take an arbitrary number of arguments (like + or *) to a list.

Example of apply:

(apply + (list 1 2 3 4 5)) ; produces 15

• apply is great for reducing lists into a single value when you have a function that can take multiple inputs.

3. reduce (or fold) Function:

• Purpose: reduce combines neighboring elements of a list, using a function to repeatedly combine pairs until a single value remains.
• Key Idea: It’s useful when you need to accumulate results over a list but don’t have a function that works with multiple inputs (like string-append).

Example of reduce:

(reduce + (list 1 2 3 4)) ; produces 10

• In string manipulations:

  (reduce string-append (list "a" "b" "c")) ; produces "abc"
  

Combining Lists with Spaces:

• Consider reversing the words in a string and then combining them back with spaces:

  (reduce (section string-append _ " " _)
          (map (o list->string reverse string->list)
               (string-split "this is a test")))
  ; produces "siht si a tset"
  

Associativity and Efficiency:

• Associativity: Operations like addition, multiplication, and string-append are associative, meaning it doesn’t matter in what order the operations are performed.

  • This allows reduce to operate efficiently in parallel.

• Order of Operations:

  • Not all operations are associative. For example, subtraction produces different results depending on the order of operations. To support different orders, Scamper provides both reduce (left-to-right) and reduce-right (right-to-left).

4. filter Function:

• Purpose: filter selects elements from a list that satisfy a predicate function (i.e., a function that returns true or false).

Example of filter:

(filter odd? (list 1 2 3 4 5)) ; produces (1 3 5)

• Combining Functions: You can combine filter, map, and reduce for more complex operations. For example, filtering digits from a string, mapping them to numbers, and summing them:

  (reduce + (map char->integer (filter char-numeric? (string->list "123abc"))))
  

Using map with Multiple Lists:

• Key Idea: map can also apply a function to corresponding elements from multiple lists, combining them into a single list.

Example:

(map + (list 1 2 3) (list 4 5 6)) ; produces (5 7 9)

Self-Check Questions:

1. Check 1: Inconsistent Subtraction:

   • Find different results for the expression (4 - 1 - 6 - 3 - 2 - 10 - 8) using different orders of operation.

2. Check 2: Pipelining (‡):

   • Extend an expression using map, filter, and reduce so that:
     • The list only contains names in "last, first" format with at least 13 characters.
     • The sum of the lengths of all these names is computed.