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--- fp:lab05 [2021/04/08 18:08]
pdmatei
+++ fp:lab05 [2022/03/31 08:53] (current)
pdmatei
@@ Line 1: / Line 1: @@
-===== 5. Processing using higher-order functions =====
+===== 5. Functional data representation =====
-The following is an input test. You can add more examples to it:
+==== 5.1. Nats ===
-<code haskell>
-l = ["matei@gmail.com", "mihai@gmail.com", "tEst@mail.com", "email@email.com", "short@ax.ro"]
-</code>
-Use ''map'', ''foldr''/''foldl'', instead of recursive functions. Wherever possible, use functional composition and closures.
+Consider the following toy implementation of the type ''Nat'' which encodes natural numbers.
-.1. Replace uppercases with lowercases from emails. (Do **not** use recursion). To be able to use character functions from the library, add ''import Data.Char'' at the beginning of the program. Use the Internet to find the appropriate character function.
+<code scala>
+trait Nat {}
-<code haskell>
+case object Zero extends Nat {}
-rem_upper l =
+case class Succ(n: Nat) extends Nat {}
 </code>
-.2. Write a function which removes emails longer than a given size.
+For instance, ''3'' will be encoded as the value: ''Succ(Succ(Succ(Zero)))''.
- Use anonymous functions in your implementation, then think about how you can replace them by a functional composition of more basic functions. **Hint:** Write your code in steps. Start with the basic idea, then think about how you can write it better and cleaner.
-<code haskell>
+**5.1.1.** Write a function which implements addition over Nats:
-longer :: Int -> [String] -> [String]
+<code scala>
-longer x =
+def add(n: Nat, m: Nat): Nat = ???
 </code>
-.3. Count the number of emails longer than 12 characters. Use a fold, anonymous functions and functional composition.
+**5.1.2.** Write a function which converts a ''Nat'' to an ''Int'':
-<code haskell>
+<code scala>
-howmany l =
+def toInt(n: Nat): Int = ???
 </code>
-.4. Split the list between first names and email domains. What ingredients (auxiliary functions) are necessary? Use either a fold or a tail-recursive function in your implementation. Example: ''names_emails ["matei@email.com"] = [ ["matei", "email.com"] ]''.
+**5.1.3.** Write a function which converts an ''Int'' to a ''Nat''.
-<code haskell>
+<code scala>
-names_emails :: [String] -> [[String]]
+def fromInt(i: Int): Nat
-names_emails l =
 </code>
-.5. Identify the list of the employed domain names (e.g. ''gmail.com''). Remove duplicates. Use no recursion and no additional prelude function apart from ''head'' and ''tail''. **Hint** think about the sequence of basic operations you want to perform and assemble them using functional composition.
+==== 5.2. Binary Search Trees ===
-<code haskell>
-domains :: [String] -> [String]
-domains =
-</code>
-(!) 5.6. In some previous exercise you have, most likely, implemented a split function using ''foldr''. Implement one with ''foldl''. **Hint:** use an example together with the ''foldl'' implementation to figure out what the accumulator should do.
+In a [[https://en.wikipedia.org/wiki/Binary_search_tree| binary search tree (BST)]], the key of the current node, is always:
+  * **smaller** or equal than **all** keys in the **right** sub-tree.
+  * **larger** or equal than **all** keys in the **left** sub-tree.
-<code haskell>
+Consider a binary search tree with keys as integers, encoded as follows:
-splitl :: String -> [String]
+<code scala>
-splitl =
+trait ITree {}
+case object Empty extends ITree
+case class INode(key: Int, left: ITree, right: ITree) extends ITree
 </code>
-.7. Write a function which extracts the domains from emails, without the dot part. (e.g. ''gmail''). Generalise the previous function ''splitl'' to ''splitBy:: Char -> String -> [String]'', and use it each time necessary, in your implementation. **Hint**: Wherever you want to mix pattern matching with guards, start with the patterns first.
+**5.2.1.** Create the tree shown below:
+<code scala>
+val tree = ???
+/*
+
+      /   \
+     7
+    / \     \
+  3      9
+*/
+</code>
+**5.2.2.** Implement the method ''size'' which determines the number of non-empty nodes from the BST.
+**5.2.3.** Define the method ''contains'', which checks if a given integer is a member of the BST.
+**5.2.4.** Implement the method ''ins'' which inserts a new integer in the BST. **Note:** the insertion must return a new BST (the //binary search tree// property mentioned above must hold after insertion).
+**5.2.5.** Implement a method ''flatten'' which converts a BST into a list of integers. You must carefully choose the flattening method in such a way as to obtain **a sorted list** from the BST. Hint: you may use the list concatenation operator '':::'' (triple colons; example usage: ''List(1,2,3):::List(4,5)''.
+**5.2.6.** Implement a method ''depth'' which returns the maximal depth of a BST. Hint: use the method: ''_.max(_)''.
+**(!) 5.2.8.** Implement a method ''minimum'' which returns the smallest integer from a BST. (If the tree is empty, we return -1). Hint: use the example above, to guide your implementation.
+**5.2.9.** Implement a similar method ''maximum''.
-<code haskell>
+**(!) 5.2.10.** Implement a method ''successor(k)'' which returns **the smallest** integer from the BST, which is **larger** than ''k''. Use the following examples for your implementation:
-domain :: [String] -> [String]
+<code>
-domain =
+             t.successor(2) = 5
+   / \            t.successor(5) = 6
+   7           t.successor(7) = 8
+     / \
+   8
 </code>
+** (!!) 5.2.11.** Implement a method ''remove(k)'' which removes element ''k'' from the BST.