Big O Notation

• Big O Notation is used to describe the upper bound of a particular algorithm. Big O is used to describe worst case scenarios

Little O Notation

• Little O Notation is also used to describe an upper bound of a particular algorithm; however, Little O provides a bound that is not asymptotically tight

Big Ω Omega Notation

• Big Omega Notation is used to provide an asymptotic lower bound on a particular algorithm

Little ω Omega Notation

• Little Omega Notation is used to provide a lower bound on a particular algorithm that is not asymptotically tight

Theta Θ Notation

• Theta Notation is used to provide a bound on a particular algorithm such that it can be "sandwiched" between two constants (one for an upper limit and one for a lower limit) for sufficiently large values

Linked List

• A Linked List is a linear collection of data elements, called nodes, each pointing to the next node by means of a pointer. It is a data structure consisting of a group of nodes which together represent a sequence.
• Singly-linked list: linked list in which each node points to the next node and the last node points to null
• Doubly-linked list: linked list in which each node has two pointers, p and n, such that p points to the previous node and n points to the next node; the last node's n pointer points to null
• Circular-linked list: linked list in which each node points to the next node and the last node points back to the first node
• Time Complexity:
  • Access: O(n)
  • Search: O(n)
  • Insert: O(1)
  • Remove: O(1)

Stack

• A Stack is a collection of elements, with two principle operations: push, which adds to the collection, and pop, which removes the most recently added element
• Last in, first out data structure (LIFO): the most recently added object is the first to be removed
• Time Complexity:
  • Access: O(n)
  • Search: O(n)
  • Insert: O(1)
  • Remove: O(1)

Queue

• A Queue is a collection of elements, supporting two principle operations: enqueue, which inserts an element into the queue, and dequeue, which removes an element from the queue
• First in, first out data structure (FIFO): the oldest added object is the first to be removed
• Time Complexity:
  • Access: O(n)
  • Search: O(n)
  • Insert: O(1)
  • Remove: O(1)

Tree

• A Tree is an undirected, connected, acyclic graph

Binary Tree

• A Binary Tree is a tree data structure in which each node has at most two children, which are referred to as the left child and right child
• Full Tree: a tree in which every node has either 0 or 2 children
• Perfect Binary Tree: a binary tree in which all interior nodes have two children and all leave have the same depth
• Complete Tree: a binary tree in which every level except possibly the last is full and all nodes in the last level are as far left as possible

Binary Search Tree

• A binary search tree, sometimes called BST, is a type of binary tree which maintains the property that the value in each node must be greater than or equal to any value stored in the left sub-tree, and less than or equal to any value stored in the right sub-tree
• Time Complexity:
  • Access: O(log(n))
  • Search: O(log(n))
  • Insert: O(log(n))
  • Remove: O(log(n))

<img src="/img/data_structure/BST.png?raw=true" alt="Binary Search Tree" width="400" height="500">

Trie

• A trie, sometimes called a radix or prefix tree, is a kind of search tree that is used to store a dynamic set or associative array where the keys are usually Strings. No node in the tree stores the key associated with that node; instead, its position in the tree defines the key with which it is associated. All the descendants of a node have a common prefix of the String associated with that node, and the root is associated with the empty String.

Fenwick Tree

• A Fenwick tree, sometimes called a binary indexed tree, is a tree in concept, but in practice is implemented as an implicit data structure using an array. Given an index in the array representing a vertex, the index of a vertex's parent or child is calculated through bitwise operations on the binary representation of its index. Each element of the array contains the pre-calculated sum of a range of values, and by combining that sum with additional ranges encountered during an upward traversal to the root, the prefix sum is calculated
• Time Complexity:
  • Range Sum: O(log(n))
  • Update: O(log(n))

Segment Tree

• A Segment tree, is a tree data structure for storing intervals, or segments. It allows querying which of the stored segments contain a given point
• Time Complexity:
  • Range Query: O(log(n))
  • Update: O(log(n))

Heap

• A Heap is a specialized tree based structure data structure that satisfies the heap property: if A is a parent node of B, then the key (the value) of node A is ordered with respect to the key of node B with the same ordering applying across the entire heap. A heap can be classified further as either a "max heap" or a "min heap". In a max heap, the keys of parent nodes are always greater than or equal to those of the children and the highest key is in the root node. In a min heap, the keys of parent nodes are less than or equal to those of the children and the lowest key is in the root node
• Time Complexity:
  • Access Max / Min: O(1)
  • Insert: O(log(n))
  • Remove Max / Min: O(log(n))

<img src="/img/data_structure/heap.png?raw=true" alt="Max Heap" width="400" height="500">