Binary Search Tree Assignment

We will implement a binary search tree data structure as well as Higher Order Function Hall of Fame inductees: find and fold.

Background

student record example

val bst = BinarySearchTree.create_empty(String.compare, (fn(s:student) => (#wustl_key s)))
val (bst, _) = BinarySearchTree.insert(bst, {first_name="Bruce", last_name="Wayne", wustl_key="wayne.b", bear_bucks=999999.99, courses=[("Business", 101)]})
val (bst, _) = BinarySearchTree.insert(bst, {first_name="Peter", last_name="Parker", wustl_key="webslinger", bear_bucks=12.34, courses=[("Biology", 101)]})
val (bst, _) = BinarySearchTree.insert(bst, {first_name="Diana", last_name="Prince", wustl_key="amazon_diana", bear_bucks=234.56, courses=[("Anthropology", 101)]})
val (bst, _) = BinarySearchTree.insert(bst, {first_name="Clark", last_name="Kent", wustl_key="i.m.superman", bear_bucks=34.56, courses=[("Journalism", 101)]})
val (bst, _) = BinarySearchTree.insert(bst, {first_name="Bruce", last_name="Banner", wustl_key="gamma.ray", bear_bucks=456.78, courses=[("Physics", 101)]})

student record example with int as key

datatype department = COMPUTER_SCIENCE | ENGLISH | ARCHITECTURE | BIOCHEMISTRY
type student = {id: int, name: string, major: department}

val tennessee_williams = {id=401936, name="Tennessee Williams", major=ENGLISH}
val charles_eams = {id=401927, name="Charles Eams", major=ARCHITECTURE}
val rochelle_walensky = {id=401991, name="Rochelle Walensky", major=BIOCHEMISTRY}

val bst = BinarySearchTree.create_empty(Int.compare, fn(s:student) => #id s)
val (bst,_) = BinarySearchTree.insert(bst, tennessee_williams)
val (bst,_) = BinarySearchTree.insert(bst, charles_eams)
val (bst,_) = BinarySearchTree.insert(bst, rochelle_walensky)

order

SML's General structure's order

datatype order = LESS | EQUAL | GREATER

Values of type order are used when comparing elements of a type that has a linear ordering.

Functions which take (('k * 'k) -> order) functions behave as Int.compare does:

compare (i, j)

returns LESS, EQUAL, or GREATER when i is less than, equal to, or greater than j, respectively.

list

We will implement some of the higher ordered functions list provides on our binary tree.

traversal order

• In-order LNR

• Reverse in-order RNL

Code to Implement

file:	src/main/sml/binary_tree/binary_tree.sml
functions:	create_empty find insert remove fold_lnr fold_rnl debug_message to_graphviz_dot

signature BINARY_SEARCH_TREE = sig
    type 'k compare_function = (('k * 'k) -> order)
    type ('e,'k) to_key_function = 'e -> 'k

    type ('e,'k) tree;

    val create_empty : ('k compare_function * ('e,'k) to_key_function) -> ('e,'k) tree

    val find : (('e,'k) tree * 'k) -> 'e option
    val insert : (('e,'k) tree * 'e) -> (('e,'k) tree * 'e option)
    val remove : (('e,'k) tree * 'k) -> (('e,'k) tree * 'e option)

    val fold_lnr : ((('e * 'b) -> 'b) * ('b) * (('e,'k) tree)) -> 'b 
    val fold_rnl : ((('e * 'b) -> 'b) * ('b) * (('e,'k) tree)) -> 'b 
    
    val debug_message : (('e -> string) * (('e,'k) tree)) -> string
    val to_graphviz_dot : (('e -> string) * ('k -> string) * (('e,'k) tree)) -> string
end

type

provided compare_function and to_key_function type synonyms

 type 'k compare_function = (('k * 'k) -> order)
type ('e,'k) to_key_function = 'e -> 'k 

required (at minimum) tree

(* TODO: replace unit with the datatype(s) and/or type synonym(s) you decide upon *)
type ('a,'k) tree = unit

create_empty

fun create_empty(cmp : 'k compare_function, to_key : ('e,'k) to_key_function) : ('e,'k) tree =
	raise Fail("NotYetImplemented")

find

reference: List.find

fun find(t : ('e,'k) tree, key : 'k) : 'e option = 
	raise Fail("NotYetImplemented")

insert

fun insert(t : ('e,'k) tree, element : 'e) : (('e,'k) tree * 'e option) =
	raise Fail("NotYetImplemented")

NOTE: if the key for the specified item matches a key already in the tree, the previous item is replaced.

insert returns a pair containing the new tree and the (optional) replaced value.

remove

fun remove(t : ('e,'k) tree, key : 'k) : (('e,'k) tree * 'e option) =
	raise Fail("NotYetImplemented")

remove the item whose key matches item_key, if it is found.

returns a pair of the modified tree and the (optional) removed item.

Remove contains the most challenging aspect of this studio. When instructed to remove a node from a tree, there are several cases:

not found

What will indicate that you reached the point where you know the node is not found in the tree?
note: this has a trivial solution.

no child in the left tree

How will you detect this pattern?
note: this has a trivial solution.

no child in the right tree

how will you detect this pattern?
note: this has a trivial solution.

both children are present

If you need to remove a node and it has both children, now you have a legit problem. You must maintain a correct binary search tree.

standard approach

A common approach is to choose one of the following:

• remove the right most descendant in the left child, and promote it to be the node at the current level, or
• remove the left most descendant in the right child, and promote it to be the node at the current level

The image below shows finding the left most child in the right subtree for promotion:

Building a helper function will likely be helpful.

Wikipedia BST Deletion

alternate approach?

Can you come up with a different approach that produces a clean solution while still providing O(lg(N)) expected performance?

fold

reference: List.foldl foldr

LNR would produce: ABCDEFGHI
RNL would produce: IHGFEDCBA

fold_lnr

(*
 * depth-first, in-order traversal
 * https://en.wikipedia.org/wiki/Tree_traversal#In-order_(LNR)
 *)
fun fold_lnr(f, init, t) = 
	raise Fail("NotYetImplemented")

fold_rnl

(*
 * depth-first, reverse in-order traversal
 * https://en.wikipedia.org/wiki/Tree_traversal#Reverse_in-order_(RNL)
 *)
fun fold_rnl(f, init, t) = 
	raise Fail("NotYetImplemented")

Debugging

debug_message

fun debug_message(element_to_string, t) =
	"(Optional) TODO: BinarySearchTree.debug_message"

debug_binary_search_tree

file: debug_binary_search_tree.sml

in folder: src/test/sml/run_binary_search_tree_testing

BinarySearchTree to_graphviz_dot

	fun to_graphviz_dot(element_to_string, key_to_string, t) =
		let
			
			(* TODO: bind root *)
			val root = raise Fail("NotYetImplemented")
			(* TODO: bind to_key *)
			val to_key = raise Fail("NotYetImplemented")
			

			fun node_to_dot(element) =
				"\t" ^ key_to_string(to_key(element)) ^ " [label= \"{ " ^ element_to_string(element) ^ " | { <child_left> | <child_right> } }\"]"

			fun edge_to_dot(parent_element_opt, tag, element) = 
				case parent_element_opt of
				  NONE => ""
				| SOME(parent_element) => "\t" ^ key_to_string(to_key(parent_element)) ^ tag ^ " -> " ^ key_to_string(to_key(element))

			fun nodes_to_dot(bst) =
					raise Fail("NotYetImplemented")

			fun edges_to_dot(bst, parent_element_opt, tag) =
					raise Fail("NotYetImplemented")

		in
			"digraph g {\n\n\tnode [\n\t\tshape = record\n\t]\n\n\tedge [\n\t\ttailclip=false,\n\t\tarrowhead=vee,\n\t\tarrowtail=dot,\n\t\tdir=both\n\t]\n\n" ^ nodes_to_dot(root) ^ edges_to_dot(root, NONE, "") ^ "\n}\n"
		end

graphviz dot vs code

search for "graphviz dot" in vs code extensions marketplace

Testing

Complete

source folder:	src/test/sml/binary_search_tree
how to run with CM.make verbosity off:	sml -Ccm.verbose=false run_binary_search_tree_testing.sml
how to run with CM.make verbosity on:	sml run_binary_search_tree_testing.sml

note: ensure that you have removed all printing to receive credit for any assignment.

SML Error Messages

Without Remove

sml -Ccm.verbose=false run_binary_search_tree_testing.sml --remove=false

sml run_binary_search_tree_testing.sml --remove=false

Pledge, Acknowledgments, Citations

file:

studio-binary-search-tree-pledge-acknowledgments-citations.txt

More info about the Honor Pledge