Data Structures

In computer science, a data structure is a particular way of organizing data in a computer so that it can be used efficiently.

Common Data Structures

  • Arrays
  • Linked Lists
  • Trees

and more, including

  • Hashes

A couple of those look familiar from JavaScript and Ruby. Arrays and Hashes are present in both languages. (In JavaScript, hashes are called Objects). Most of your programming needs can be solved using one of those popular structures. However, to call yourself a good Computer Scientist you should be knowledable about a few others.

**For the most part, in your day-to-day programming, stick to the data structures provided by your language. Don't start your first day on some job by implementing your own linked list.**


You probably have a good understanding of Arrays from using them in your day-to-day programming. They are one of the simplest and most common data structures.

foods = ["kale", "broccoli", "carrots"]
foods[0] # accesses the first element (i.e., "kale")
foods[1] # accesses the second element (i.e., "broccoli")


Remember the quote at the top of the page. Not only do we want to organize data, but we want to do it efficiently. Let's think about how good or bad Arrays might be for:

  • retrieving an element
    • from the beginning
    • from the middle
    • from the end
  • changing an element
  • removing an element
  • adding an element

To do this, we may want to pause and think about how Arrays are stored in memory. What is an Array, really?


Your computer's memory can be thought of as a place to store stuff. Maybe we can think of them as mailboxes. Depending on how much memory you have, you might have X mailboxes, each with their own address.

An Array requests access to mailboxes that are at sequential addresses. If I want to have an Array with 10 elements in it, I reserve 10 mailboxes that are all right next to each other.

This is nice because it's very fast to find the nth element in an Array. You just take the starting address and move over n spots.

This is bad because if you want to add an element to the array, you need to acquire an additional mailbox, but one that is next to your existing mailboxes might not be available. In that case you'll need to request 11 sequential addresses and copy your whole array into the new space.

**This is an oversimplification but it is close enough of a model to illustrate the tradeoffs being made with a theoretical Array. In practice, Arrays in JavaScript or Ruby will be smarter than this and do a better job of managing memory.**

Linked List

A Linked List (or Singly Linked List) is another way of organizing a list of elements.

Instead of requesting n contiguous memory locations, you just create n nodes (or vertices) and link them yourself. Each node can be stored anywhere in memory, and you also store where to find the next element in the list in each node.

There is also a version called a Doubly Linked List that has links going in both directions. So, instead of each node having a only a next property, they also have a previous property.


We already thought about the performance characteristics of the following for Arrays. How do they compare for Linked Lists?

  • retrieving an element
    • from the beginning
    • from the middle
    • from the end
  • changing an element
  • removing an element
  • adding an element


Implement a Linked List

  1. Need to create a representation of a Node (or Vertex).
  2. Write a method add_to_tail that appends a new value to the end.
  3. Write a method length that returns the length of a list.
  4. Overide the to_s method to nicely print the list.
  5. Bonus: Write a function remove to remove a node from the list.


# (1) Ruby is an Object-Oriented language, so creating a class to
# represent a Node seems like a reasonable idea.
class Node
  attr_accessor :val, :next

  def initialize(value)
    @val = value

  # (2) In order for a method `add_to_tail` to append a value to the
  # end, it needs to locate the end, then add a new Node, linking it
  # to the previous tail.
  def add_to_tail(value)
    if @next == nil
      @next =

  # (3) In order to calculate the length of the list, we can use a
  # recursive call. We're starting to see some of the shortcomings of
  # our design decisions :)
  def length
    if @next == nil
      @val ? 1 : 0
      1 + @next.length

  # (4) We can also use a recursive call to nicely print out the list.
  def to_s
    if @next == nil
      "#{@val} #{@next}"

  # (5) I chose a limited representation by only have Nodes be linked
  # to other Nodes. So the `remove` method should return the new head
  # of the list in case the old head was the Node removed. Perhaps
  # this indicates I should have used a more robust setup, such as a
  # LinkedList class that encapsulates the Nodes, but I liked this for
  # simplicity and illustrative purposes.
  def remove(value)
    if @val == value
      if @next
        return @next
    elsif @next and @next.val == value
      @next =
    return self

# You can test our all the code we just wrote with something like:
list =

puts "list.length is #{list.length}"
puts "list is (#{list})"

list = list.remove(21)
list = list.remove(14)

puts "list.length is #{list.length}"
puts "list is (#{list})"


In a way, a Tree is like a more generalized version of a Linked List where each Node can have more than one child.

  • Tree nodes have a parent-child relationship.
  • Trees cannot contain cycles (i.e., branches can't intertwine)
  • There is a starting node called a root node.
  • A leaf node is one that has no children.


Searching a Tree is more complicated than searching a linear data structure such as an Array or Linked List.

The two main approaches are:

DFS is the easiest to implement is the one you should tackle first in your programming careers.


Implement a Tree

  1. Need to create a representation of a Node (or Vertex).
  2. Write a method add_child to add a new child with the given value.
  3. Write a method find to search a Tree for a value. (Hint: DFS)

Specific Types of Trees

  • Binary Trees
    • Each node has zero, one, or two children. This assertion makes many tree operations simple and efficient.
  • Binary Search Trees
    • A binary tree where any left child node has a value less than its parent node and any right child node has a value greater than or equal to that of its parent node.
  • Heaps
    • See wikibooks for proper definition.
    • Simplified version: a (usually binary) tree where the biggest element is always at the top.
  • Tries
    • Useful for autocompletion.

Big O Notation

In computer science, big O notation is used to classify algorithms by how they respond (e.g., in their processing time or working space requirements) to changes in input size.

Since a big issue when discussing data structures is their efficiency in the face of certain tasks, we want to use a common language to discuss such (in)efficiencies. Normally, we are interested in how much memory or processing time is needed to complete the task depending on the size of the input. We often let n represent the input size.

So, an algorithm that is O(1), is said to be "Big O of 1" or "constant", and does not vary depending on the size of the input. This is good. This is fast even for very large n.

An algorithm that is O(n), is said to be "Big O of n" or "linear", and this indicates that the resources required grow proportionally to the size of the input. This is reasonable performance.

Another that is O(n2), is said to be "Big O of n squared" and it means the resources grow in proportion to the square of the input. This is slow. Think of really big numbers and then think of them squared.

See the Cheat Sheet for some other common time (processing time) and space (memory) complexities and their notations.