Gain insights into array and linked list-based implementations, delve into advanced structures like skiplists and trees, and explore template-based collections for efficient data optimization and retrieval.

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Data structures and algorithms are essential in computer science since they play a crucial role in efficient information retrieval and processing, dealing with files, storing contacts on phones, social networks and web searches.

In this course, you’ll learn about the array-based implementation of various linear data structures, stack, and queues. You’ll also learn about linked list-based implementation. Next, you’ll explore advanced data structures like skiplists and hashing. You’ll learn how to implement a variety of trees and graphs, and data structures related to bits of an integer. Toward the end of the course, you’ll learn the implementation of structures based on external storage.

After completing this course, you’ll be able to create reusable programs with template-based collections that can efficiently analyze how to optimize the storage and retrieval of very large amounts of data. Overall, this course will enhance your productivity and performance as a software developer.

Data Structures with Generic Types in C++

# Addition

To implement `add(x)` in a `RedBlackTree`, we perform a standard `BinarySearchTree` insertion to add a new leaf, `u`, with `u.x = x` and set `u.colour =
red`. Note that this does not change the black height of any node, so it
does not violate the black-height property. It may, however, violate the
left-leaning property (if `u` is the right child of its parent), and it may
violate the no-red-edge property (if `u`’s parent is red). To restore these
properties, we call the `addFixup(u)` method.

 

boolean add(T x) {
  Node < T > u = newNode(x);
  u.colour = red;
  boolean added = add(u);
  if (added)
    addFixup(u);
  return added;
}

A single round in the process of fixing Property 2 after an insertion is illustrated in figure below:

The `addFixup(u)` method takes as input a
node `u` whose colour is red and which may violate the no-red-edge property and/or the left-leaning property. The following discussion is probably impossible to follow without referring to Figure 9.8 or recreating it on
a piece of paper. Indeed, the reader may wish to study this figure before
continuing.

If `u` is the root of the tree, then we can colour `u` black to restore both
properties. If u’s sibling is also red, then `u`’s parent must be black, so both
the left-leaning and no-red-edge properties already hold.

Otherwise, we first determine if `u`’s parent, `w`, violates the left-leaning
property and, if so, perform a `flipLeft(w)` operation and set `u = w`. This leaves us in a well-defined state: `u` is the left child of its parent, `w`, so `w` now satisfies the left-leaning property. All that remains is to ensure the no-red-edge property at `u`. We only have to worry about the case in which
`w` is red, since `u` already satisfies the no-red-edge property.


We are not done yet, `u` is red and `w` is red. The no-red-edge property (which is only violated by `u` and not by `w`) implies that `u`’s grandparent `g` exists and is black. If `g`’s right child is red, then the left-leaning
property ensures that both `g`’s children are red, and a call to `pushBlack(g)`
makes `g` red and `w` black. This restores the no-red-edge property at `u`, but may cause it to be violated at g, so the whole process starts over with
`u = g`.

If `g’`s right child is black, then a call to `flipRight(g)` makes `w` the
(black) parent of `g` and gives `w` two red children, `u` and `g`. This ensures
that `u` satisfies the no-red-edge property and `g` satisfies the left-leaning property. In this case, we can stop.

void addFixup(Node < T > u) {
  while (u.colour == red) {
    if (u == r) { // u is the root - done
      u.colour = black;
      return;
    }
    Node < T > w = u.parent;
    if (w.left.colour == black) { // ensure left-leaning
      flipLeft(w);
      u = w;
      w = u.parent;
    }
    if (w.colour == black)
      return; // no red-red edge = done
    Node < T > g = w.parent; // grandparent of u
    if (g.right.colour == black) {
      flipRight(g);
      return;
    } else {
      pushBlack(g);
      u = g;
    }
  }
}

The `insertFixup(u)` method takes constant time per iteration and each iteration either finishes or moves `u` closer to the root. Therefore,
the `insertFixup(u)` method finishes after $O(\log n)$ iterations in $O(\log n)$ time.


# Removal
The `remove(x)` operation in a `RedBlackTree` is the most complicated to
implement, and this is true of all known red-black tree variants. Just
like the `remove(x)` operation in a `BinarySearchTree`, this operation boils
down to finding a node `w` with only one child, `u`, and splicing `w` out of the
tree by having `w.parent` adopt `u`.

The problem with this is that, if `w` is black, then the black-height
property will now be violated at `w.parent`. We may avoid this problem, temporarily, by adding `w.colour` to `u.colour`. Of course, this introduces two other problems: (1) if `u` and `w` both started out black, then
`u.colour` + `w.colour` = 2 (double black), which is an invalid colour. If
`w` was red, then it is replaced by a black node `u`, which may violate the
left-leaning property at `u.parent`. Both of these problems can be resolved
with a call to the `removeFixup(u)` method.

Learn to add and remove an element in red-black trees.

Overview

Array-Based Lists

Linked Lists

Skiplists

Hash Tables

Binary Trees

Random Binary Search Trees

Scapegoat Trees

Red-Black Trees

Heaps

Sorting Algorithms

Graphs

Data Structures for Integers

External Memory Searching

Wrap Up

Red-Black Tree Operations