Graphs and Trees

Part 3 and 4 of

Practical Data Structures and

Algorithms for Interviews

slides.com/bgando/intro-graph-trees

EXERCISES

Common Interview DS

DS Common Uses FEM Course
Arrays & Strings ordered data, words Part 2 
Hash Tables optimization Part 2
Linked Lists data insert/delete Part 2
Stacks/Queues creative aux DS Part 2 
Graphs complex relationships Part 3 (this one)
Trees hierarchical data Part 4 (this one)

What will we cover today?

Trees

Graphs

Binary Search Trees

How to be effective

Rule #1 don't aim to memorize, this will not help!

Rule #2 find themes, but don't jump to conclusions

Rule #3 practice with a timer, speed matters

Rule #3.5 actually practice, reading doesn't count

Rule #4 communicate and be nice

Linear Data Structures

Array

Linked List

Stack

Queue

Linear Data Structures

Array

Insert at end:

Insert at beginning:

Insert (generally):

Remove from end:

Remove from beginning:

Remove (generally):

Find value:

Access value:

Linear Data Structures

Array

Insert at end: O(1)

Insert at beginning: O(n)

Insert (generally): O(n)

Remove from end: O(1)

Remove from beginning: O(n)

Remove (generally): O(n)

Find value: O(n)

Access value: O(1)

Linear Data Structures

Linked List

Insert at end:

Insert at beginning:

Insert (generally):

Remove from end:

Remove from beginning:

Remove (generally):

Find value:

Access value:

Linear Data Structures

Linked List

Insert at end: O(1)

Insert at beginning: O(1)

Insert (generally): O(1)

Remove from end: O(1)

Remove from beginning: O(1)

Remove (generally): O(1)

Find value: O(n)

Access value: O(n)

Linear Data Structures

Stack

Insert at end:

Insert at beginning:

Insert (generally):

Remove from end:

Remove from beginning:

Remove (generally):

Find value:

Access value:

Linear Data Structures

Stack

Insert at end: O(1)

Insert at beginning: N/A

Insert (generally): O(1)

Remove from end: O(1)

Remove from beginning: N/A

Remove (generally): O(1)

Find value: O(n)

Access value: O(n)

Linear Data Structures

Queue

Insert at end:

Insert at beginning:

Insert (generally):

Remove from end:

Remove from beginning:

Remove (generally):

Find value:

Access value:

Linear Data Structures

Queue

Insert at end: N/A

Insert at beginning: O(n) or O(1)

Insert (generally): O(1) with linked list

Remove from end: O(1)

Remove from beginning: N/A

Remove (generally): O(1)

Find value: O(n)

Access value: O(n)

Linear Data Structures

Array

Linked List

Stack

Queue

LINEAR DATA STRUCTURE NON-LINEAR DATA STRUCTURE
Overview arranged in an orderly manner where the elements are attached adjacently arranged with multiple relationships and properties to describe them
Traversing can be accessed in one time (single run) traversing is best handled recursively
Ease of implementation simpler complex
Levels involved single level multiple level
Examples array, queue, stack, linked list, etc. tree and graph

Anyone have a morning routine?

Expectation

Expectation

Reality

Create a data structure to model a chat-bot that helps you decide what to eat for breakfast by asking yes or no questions. 

{ 
  question: 'Do you feel like cooking?', 
  yes: <question>, 
  no: <question> 
}

1. create a data structure

2. model a chat-bot that

3. helps you decide what to eat for breakfast

4. by asking yes or no questions.

?

Yes

No

Recommendation

{ 
  question: 'Do you feel like cooking?', 
  yes: <question>, 
  no: <question> 
}

{ 
  question: 'Do you have milk?',
  yes: <question>,
  no: <question> 
}

1. create a data structure

2. model a chat-bot that

3. helps you decide what to eat for breakfast

4. by asking yes or no questions.

Yes

No

Recommendation

{ 
  question: 'Do you feel like cooking?', 
  yes: <question>, 
  no: <question> 
}

{ 
  question: 'Do you have milk?',
  yes: <question>,
  no: <question> 
}

{ 
  question: 'Do you have toast?', 
  yes: <question>, 
  no: <question>,
  recommendation: 'You should make toast for breakfast!' 
}

1. create a data structure

2. model a chat-bot that

3. helps you decide what to eat for breakfast

4. by asking yes or no questions.

Yes

No

Recommendation

const chatBot = { 
  question: 'Do you feel like cooking?', 
  yes: { 
    question: 'Do you have milk?',
    yes: <question>,
    no: <question> 
  }, 
  no: { 
    question: 'Do you have toast?', 
    yes: <question>, 
    no: <question> 
  } 
}

1. create a data structure

2. model a chat-bot that

3. helps you decide what to eat for breakfast

4. by asking yes or no questions.

?

Yes

No

Recommendation

Trees - Terminology

eat toast!

eat toast!

The DOM is a tree.

Parsers use trees to model the structure of code

Autocomplete features use a special kind of tree called a trie.

Trees - Terminology

Non-Linear to Linear? 

const chatBot = { 
  question: 'Do you feel like cooking?', 
  yes: { 
    question: 'Do you have milk?',
    yes: <question>,
    no: <question> 
  }, 
  no: { 
    question: 'Do you have toast?', 
    yes: <question>, 
    no: <question> 
  } 
}




const mother = { name: 'Ashleigh' };
const son = { name: 'Sammy' };
const daughter = { name: 'Alex' };

Family Trees 

const mother = { name: 'Ashleigh' };
const son = { name: 'Sammy' };
const daughter = { name: 'Alex' };


const mother = {
  name: 'Ashleigh',
  children: [son, daughter]
};

Family Trees 

/** Class representing a Tree. */

class Tree {

  constructor(value) {


  }
  /*
  * Adds a new value as a child of the tree
  * @param {*} value the value to add
  */
  insertChild(value) {


  }

  /*
  * Removes a value from the tree
  * @param {*} value the value to remove
  */
  removeChild(value) {


  }
}

const myTree = new BinaryTree();

console.log(myTree)

EXERCISE

click me

Insert: O(1)

Remove: ??

Traverse: ??

class Tree {
    constructor(value) {
        this.value = value;
        this.children = [];
    }

    insertChild(value) {
        const newTree = new Tree(value);
        this.children.push(newTree);
        return newTree;
    }

    removeChild(value) {
        // ??
    }
}

 

Count all possible

chat-bot recommendations.

 

const chatBot = { 
  question: 'Do you feel like cooking?', 
  yes: { 
    question: 'Do you have milk?',
    yes: <question>,
    no: <question> 
  }, 
  no: { 
    question: 'Do you have toast?', 
    yes: <question>, 
    no: <question> 
  } 
}

Q1

Q2

Q3

const chatBot = { 
  question: 'Do you feel like cooking?', 
  yes: { 
    question: 'Do you have milk?',
    yes: <question>,
    no: <question> 
  }, 
  no: { 
    question: 'Do you have toast?', 
    yes: <question>, 
    no: <question> 
  } 
}

Q1

Q2

Q3

Leaf nodes are the recommendations!

Binary Trees - Traversals

Preorder: Node, then children (starting left) 

 

 

Q1

Q2

Q3

Q1

Q2

Q3

function traverse(tree) {

...

}
const mother = { name: 'Ashleigh' };
const son = { name: 'Sammy' };
const daughter = { name: 'Alex' };


const mother = {
  name: 'Ashleigh',
  children: [son, daughter]
};
function traverse(tree) {

...

}
const family = {
 name: 'Ashleigh',
 children: [{
     name: 'Sammy',
     children: [{
       name: 'Bowser',
       children: [{ name: 'P' }],
     }],
   },
   { name: 'Alex' }]
};

/** Class representing a Binary Tree. */

class BinaryTree {

  constructor(value) {
  }
  /*
  * Explores all the nodes in the tree
  */
  traverse() {
  }

  /*
  * Returns true if value is found
  */
  contains(value) {
  }
}

const myTree = new BinaryTree();

console.log(myTree)
                                Fixed Operations 
contains(val), size(tree), traverse(tree)


                                Dynamic Operations
insert(x, y), remove(node), 
reOrder(x, y)

EXERCISE

click me 

function traverse(tree) {
    console.log(tree.name);
    tree.children.forEach((child)=> {
        traverse(child);
    });
}
// if node is null, return 0.
// elseIf yes and no child nodes are null, return 1
// else recursively calculate 
    // Leaf count of left subtree + 
    // Leaf count of right subtree

function countReccos(tree) {
    console.log(tree.name);
    tree.children.forEach((child)=> {
        traverse(child);
    });
}

Insert:

Remove:

Traverse:

countReccos:

Preorder: Node, then children (starting left) 

 

 

Inorder: Left child recursively, Node, Right child 

 

 


Postorder: Children recursively, then Node
 +

  

Q1

Q2

Q3

Q1

Q2

Q3

Q3

Q2

Q1

Q3

Q2

Q1

Binary Trees - Traversals 


// left, root, right
function inOrder(node, func = console.log) {
    if (node.left) {
        node.left.inOrder(func);
    }

    func(node);

    if (node.right) {
        node.right.inOrder(func);
    }
}

// root, left, right
function preOrder(node, func = console.log) {
    func(node);

    if (node.left) {
        node.left.preOrder(func);
    }

    if (node.right) {
        node.right.preOrder(func);
    }
}
// left, right, root
function postOrder(node, func = console.log) {
    if (node.left) {
        node.left.postOrder(func);
    }

    if (node.right) {
        node.right.postOrder(func);
    }

    func(node);
}

No

Do you want toast?

No!

Am I being helpful?

...?

Let's create a data structure to model a better breakfast recommendation engine!

Draw it!

Consider the breakfast foods of myself and my friends. How could you represent this as a graph?

 

 

 

A   −   B,

 

A   −  C,

 

A   −   E,

 

B    −   D,

 

C   −    D,

 

C .   −   E.

 

 

 

A   −   B,

 

A   −  C,

 

A   −   E,

 

B    −   D,

 

C   −    D,

 

C .   −   E.

 

 

 

A   −   B,

 

A   −  C,

 

A   −   E,

 

B    −   D,

 

C   −    D,

 

C .   −   E.

 

 

 

A   −   B,

 

A   −  C,

 

A   −   E,

 

B    −   D,

 

C   −    D,

 

C .   −   E.

 

 

 

A   −   B,

 

A   −  C,

 

A   −   E,

 

B    −   D,

 

C   −    D,

 

C .   −   E.

 

 

 

A   −   B,

 

A   −  C,

 

A   −   E,

 

B    −   D,

 

C   −    D,

 

C .   −   E.

 

 

 

A   −   B,

 

A   −  C,

 

A   −   E,

 

B    −   D,

 

C   −    D,

 

C .   −   E.

Graph

Graph Implementation

Let's create a data structure to model a better breakfast recommendation engine!

Aren't you so excited?

First Step

What data needs to be stored?

What data needs to be stored?

1

4

3

2

5

First Step

Adjacency Matrix

Adjacency Matrix

Adjacency Matrix

Vertices

Vertices

Edges

Adjacency Matrix

Vertices

Vertices

Edges

 [[0,1,0,0,1],
  [1,0,1,1,1],
  [0,1,0,1,0],
  [0,1,1,0,1],
  [1,1,0,1,0]];





addEdge(v1,v2);

Adjacency Matrix

Vertices

Vertices

Edges

 [[0,1,0,0,1],
  [1,0,1,1,1],
  [0,1,0,1,0],
  [0,1,1,0,1],
  [1,1,0,1,0]];
function addEdge(v1, v2) {
  adjMatrix[v1][v2] = 1;
  adjMatrix[v2][v1] = 1;
}

addEdge(v1,v2);

Adjacency Matrix

Vertices

Vertices

Edges

 [[0,1,0,0,1],
  [1,0,1,1,1],
  [0,1,0,1,0],
  [0,1,1,0,1],
  [1,1,0,1,0]];
function addEdge(v1, v2) {
  adjMatrix[v1][v2] = 1;
  adjMatrix[v2][v1] = 1;
}

addEdge(v1,v2);







removeEdge(v1,v2)

Adjacency Matrix

Vertices

Vertices

Edges

 [[0,1,0,0,1],
  [1,0,1,1,1],
  [0,1,0,1,0],
  [0,1,1,0,1],
  [1,1,0,1,0]];
function addEdge(v1, v2) {
  adjMatrix[v1][v2] = 1;
  adjMatrix[v2][v1] = 1;
}

addEdge(v1,v2);


function removeEdge(v1, v2) {
  adjMatrix[v1][v2] = 0;
  adjMatrix[v2][v1] = 0;
}

removeEdge(v1,v2)

Adjacency Matrix

Vertices

Vertices

Edges

 [[0,1,0,0,1],
  [1,0,1,1,1],
  [0,1,0,1,0],
  [0,1,1,0,1],
  [1,1,0,1,0]];
function addEdge(v1, v2) {
  adjMatrix[v1][v2] = 1;
  adjMatrix[v2][v1] = 1;
}

addEdge(v1,v2);


function removeEdge(v1, v2) {
  adjMatrix[v1][v2] = 0;
  adjMatrix[v2][v1] = 0;
}

removeEdge(v1,v2)

Time Complexity?

Vertices

Vertices

Edges

 [[0,1,0,0,1],
  [1,0,1,1,1],
  [0,1,0,1,0],
  [0,1,1,0,1],
  [1,1,0,1,0]];

Vertices

Edges

Adjacency List 






addEdge(v1,v2);

const adjList = {
    1:[2,5],
    2:[1,5,3,4],
    3:[2,4],
    4:[2,5,3],
    5:[4,1,2]
}

Vertices

Edges

Adjacency List 

function addEdge(v1, v2) {
  adjList[v1].push(v2);
  adjList[v2].push(v1);
}

addEdge(v1,v2);

const adjList = {
    1:[2,5],
    2:[1,5,3,4],
    3:[2,4],
    4:[2,5,3],
    5:[4,1,2]
}

Vertices

Adjacency List

Edges

function addEdge(v1, v2) {
  adjList[v1].push(v2);
  adjList[v2].push(v1);
}

addEdge(v1,v2);









removeEdge(v1,v2)
const adjList = {
    1:[2,5],
    2:[1,5,3,4],
    3:[2,4],
    4:[2,5,3],
    5:[4,1,2]
}

Vertices

Adjacency List

Edges

function addEdge(v1, v2) {
  adjList[v1].push(v2);
  adjList[v2].push(v1);
}

addEdge(v1,v2);


function removeEdge(v1, v2) {
  const v2Index = adjList[v1].indexOf(v2);
  const v1Index = adjList[v2].indexOf(v1);
  adjList[v1].splice(v2Index, 1);
  adjList[v2].splice(v1Index, 1);
}

removeEdge(v1,v2)
const adjList = {
    1:[2,5],
    2:[1,5,3,4],
    3:[2,4],
    4:[2,5,3],
    5:[4,1,2]
}

Vertices

Adjacency List

Edges

function addEdge(v1, v2) {
  adjList[v1].push(v2);
  adjList[v2].push(v1);
}

addEdge(v1,v2);


function removeEdge(v1, v2) {
  const v2Index = adjList[v1].indexOf(v2);
  const v1Index = adjList[v2].indexOf(v1);
  adjList[v1].splice(v2Index, 1);
  adjList[v2].splice(v1Index, 1);
}

removeEdge(v1,v2)
const adjList = {
    1:[2,5],
    2:[1,5,3,4],
    3:[2,4],
    4:[2,5,3],
    5:[4,1,2]
}

Vertices

Time Complexity?

Edges

Adjacency List 

function addEdge(v1, v2) {
  adjList[v1].push(v2);
  adjList[v2].push(v1);
}

addEdge(v1,v2);


function removeEdge(v1, v2) {
  const v2Index = adjList[v1].indexOf(v2);
  const v1Index = adjList[v2].indexOf(v1);
  adjList[v1].splice(v2Index, 1);
  adjList[v2].splice(v1Index, 1);
}

removeEdge(v1,v2)
function addEdge(v1, v2) {
  adjMatrix[v1][v2] = 1;
  adjMatrix[v2][v1] = 1;
}

addEdge(v1,v2);


function removeEdge(v1, v2) {
  adjMatrix[v1][v2] = 0;
  adjMatrix[v2][v1] = 0;
}

removeEdge(v1,v2)

Adjacency Matrix 

/** Class representing a Graph. */

class Graph {
  constructor(value) {
  }
  
                                    
  * Adds a new value
  * @param {*} value the value to add
  */
  addNode(value) {
  }
  
                                    
  * Adds an edge between nodeA and nodeB
  * @param {*} nodes to connect
  */
  addEdge(nodeA, nodeB) {
  }
  
                                  
  * Removes a node
  * @param {*} node to remove
  */
  removeNode(node) {
  }
}

const myGraph = new Graph();

console.log(myGraph)

EXERCISE

click me

 

Let's make some recommendations!

 

A

C

6

3

4

B

2

8

const adjList = {
    A:[2,3,4],
    2:[A],
    3:[A,B],
    4:[A,B,C],
    B:[4,3,6]
    ...
}

DFS vs BFS

4

7

DFS vs BFS

DFS vs BFS

4

7

 

Depth First!

DFS

4

7

DFS

1

DFS

1

2

DFS

1

2

3

DFS

1

2

3

4

DFS

1

2

5

3

4

DFS

1

2

5

3

6

4

DFS

1

2

5

3

6

4

7

DFS

1

2

5

3

6

8

4

7

DFS

1

2

5

9

3

6

8

4

7

DFS

1

2

5

9

3

6

8

4

7

10

Graph Exploration 

const adjList = {
    1:[2,3,4],
    2:[1],
    3:[1,5],
    4:[1,5,7],
    5:[4,3,6]
    ...
}

1. Add unvisited vertex to stack

2. Mark vertex as visited

3. If vertex has unvisited children

    - repeat 1-2 with child

4. If vertex has no unvisited children

    - pop from stack

5. repeat until stack is empty

DFS

1. Add unvisited vertex to stack

2. Mark vertex as visited

3. If vertex has unvisited children

    - repeat 1-2 with child

4. If vertex has no unvisited children

    - pop from stack

5. repeat until stack is empty

1

2

5

9

3

6

8

4

7

10

DFS 

{ 1: [2, 3, 4] }
const stack = [1          ]

visited = {




}

1. Add unvisited vertex to stack

DFS 

{ 1: [2, 3, 4] }
const stack = [1          ]

visited = {
  1: true,



}

2. Mark vertex as visited

DFS 

{ 1: [2, 3, 4] }
{ 2: [5] }
const stack = [1, 2      ]

visited = {
  1: true,
  2: true,


}

3. If vertex has unvisited children

    - repeat 1-2 with child

DFS 

{ 1: [2, 3, 4] }
{ 2: [5] }
{ 5: [9] }
const stack = [1, 2, 5   ]

visited = {
  1: true,
  2: true,
  5: true,

}

DFS 

{ 1: [2, 3, 4] }
{ 2: [5] }
{ 5: [9] }
{ 9: [] }
const stack = [1, 2, 5, 9]

visited = {
  1: true,
  2: true,
  5: true,
  9: true
}

DFS 

{ 1: [2, 3, 4] }
{ 2: [5] }
{ 5: [9] }
{ 9: [] }
const stack = [1, 2, 5, 9]

visited = {
  1: true,
  2: true,
  5: true,
  9: true
}

DFS 

{ 1: [2, 3, 4] }
{ 2: [5] }
{ 5: [9] }
const stack = [1, 2, 5,  ]

visited = {
  1: true,
  2: true,
  5: true,
  9: true
}

DFS 

{ 1: [2, 3, 4] }
{ 2: [5] }
{ 5: [9] }
const stack = [1, 2, 5,  ]

visited = {
  1: true,
  2: true,
  5: true,
  9: true
}

DFS 

{ 1: [2, 3, 4] }
{ 2: [5] }
const stack = [1, 2,     ]

visited = {
  1: true,
  2: true,
  5: true,
  9: true
}

DFS 

{ 1: [2, 3, 4] }
const stack = [1,        ]

visited = {
  1: true,
  2: true,
  5: true,
  9: true
}

5. repeat until stack is empty

DFS 

{ 1: [2, 3, 4] }
const stack = [1,3     ]

visited = {
  1: true,
  2: true,
  5: true,
  9: true,
  3: true,


}
{ 3: [6, 7] }

DFS 

{ 1: [2, 3, 4] }
const stack = [1,3,6   ]

visited = {
  1: true,
  2: true,
  5: true,
  9: true,
  3: true,
  6: true,

}
{ 3: [6, 7] }
{ 6: [10] }

DFS 

{ 1: [2, 3, 4] }
const stack = [1,3,6,10]

visited = {
  1: true,
  2: true,
  5: true,
  9: true,
  3: true,
  6: true,
  10: true
}
{ 3: [6, 7] }
{ 6: [10] }
{ 10: [] }

DFS 

{ 1: [2, 3, 4] }
const stack = [1,3,6,10]

visited = {
  1: true,
  2: true,
  5: true,
  9: true,
  3: true,
  6: true,
  10: true
}
{ 3: [6, 7] }
{ 6: [10] }
{ 10: [] }

DFS 

{ 1: [2, 3, 4] }
const stack = [1,3,6,  ]

visited = {
  1: true,
  2: true,
  5: true,
  9: true,
  3: true,
  6: true,
  10: true
}
{ 3: [6, 7] }
{ 6: [10] }

DFS 

{ 1: [2, 3, 4] }
const stack = [1,3     ]

visited = {
  1: true,
  2: true,
  5: true,
  9: true,
  3: true,
  6: true,
  10: true,

}
{ 3: [6, 7] }

DFS 

{ 1: [2, 3, 4] }
const stack = [1,3,7   ]

visited = {
  1: true,
  2: true,
  5: true,
  9: true,
  3: true,
  6: true,
  10: true,
  7: true
}
{ 3: [6, 7] }
{ 7: [] }

DFS 

{ 1: [2, 3, 4] }
const stack = [1,3,7   ]

visited = {
  1: true,
  2: true,
  5: true,
  9: true,
  3: true,
  6: true,
  10: true,
  7: true
}
{ 3: [6, 7] }
{ 7: [] }

DFS 

{ 1: [2, 3, 4] }
const stack = [1,3     ]

visited = {
  1: true,
  2: true,
  5: true,
  9: true,
  3: true,
  6: true,
  10: true,
  7: true
}
{ 3: [6, 7] }

DFS 

{ 1: [2, 3, 4] }
const stack = [1       ]

visited = {
  1: true,
  2: true,
  5: true,
  9: true,
  3: true,
  6: true,
  10: true,
  7: true
}

DFS 

{ 1: [2, 3, 4] }
const stack = [1,4       ]

visited = {
  1: true,
  2: true,
  5: true,
  9: true,
  3: true,
  6: true,
  10: true,
  7: true,
  4: true

}
{ 4: [8] }

DFS 

{ 1: [2, 3, 4] }
const stack = [1,4,8      ]

visited = {
  1: true,
  2: true,
  5: true,
  9: true,
  3: true,
  6: true,
  10: true,
  7: true,
  4: true,
  8: true
}
{ 4: [8] }
{ 8: [] }

DFS 

{ 1: [2, 3, 4] }
const stack = [1,4        ]

visited = {
  1: true,
  2: true,
  5: true,
  9: true,
  3: true,
  6: true,
  10: true,
  7: true,
  4: true,
  8: true
}
{ 4: [8] }

DFS 

{ 1: [2, 3, 4] }
const stack = [1      ]

visited = {
  1: true,
  2: true,
  5: true,
  9: true,
  3: true,
  6: true,
  10: true,
  7: true,
  4: true,
  8: true
}

DFS 

const stack = [          ]

visited = {
  1: true,
  2: true,
  5: true,
  9: true,
  3: true,
  6: true,
  10: true,
  7: true,
  4: true,
  8: true
}
const DFS = graph => {
  
}

DFS

EXERCISE

click me

DFS - Time Complexity

 

Breadth First!

BFS

1

BFS

1

2

BFS

1

2

3

BFS

1

2

3

4

BFS

1

2

3

4

5

BFS

1

2

3

4

5

6

BFS

1

2

3

4

5

6

7

BFS

1

2

3

4

5

6

7

8

BFS

1

2

3

4

5

6

7

9

8

BFS

1

2

3

4

5

6

7

9

10

8

BFS 

const BFS = graph => {
    set start vertex to visited

    load it into queue
    
    while queue not empty
    
       for each edge incident to vertex
    
            if its not visited
    
                load into queue
    
                mark vertex
}

BFS

Order of visitation Queue contents after processing node
S
A
C
D
E
B 		[ S ]
[ A C D E ]
[ C D E B ]
[ D E B ]
[ E B ]
[ ]

BFS 

const BFS = graph => {
  initialize unexplored list

  for the first vertex
    add to the unexplored list
    optional: store pointer to parent vertex to null
    mark as visited
  while unexplored list is not empty
    remove next vertex from unexplored list
    optional: process vertex
    for each edge
        if un-visited
            optional: process edge
            add all un-visited/explored vertices to list
            optional: store pointer to parent vertex
            mark as visited
    mark vertex as explored
}

BFS 

const BFS = graph => {
  
}

BFS

EXERCISE

click me

BFS - Time Complexity

 

What should we use BFS or DFS?

 

Logging or validating the contents of each edge and/or vertex.

Copying a graph, or converting between adjacency matrix or list.

Counting the number of edges and/or vertices.

Identifying the connected components.

Finding paths or cycles between two vertices.

Common tasks for BFS/DFS 

const adjList = {
    1:[2,5],
    2:[1,5,3,4],
    3:[2,4],
    4:[2,5,3],
    5:[4,1,2]
};

Vertices

Edges

Vertices

Edges

const adjMatrix = [
  [0,1,0,0,1],
  [1,0,1,1,1],
  [0,1,0,1,0],
  [0,1,1,0,1],
  [1,1,0,1,0]
];

Directed, Undirected

Weighted, Unweighted

Self loops

Sparse, Dense

Cyclic, Acyclic

 

Let's find something specific

Apple

Bread

Cereal

Donut

Eggs

Fudge

Linear Search

Apple

Bread

Cereal

Donut

Eggs

Fudge

index = 0
length = 6

Linear Search

Apple

Bread

Cereal

Donut

Eggs

Fudge

index = 1
length = 6

Linear Search

Apple

Bread

Cereal

Donut

Eggs

Fudge

index = 2
length = 6

Linear Search

Apple

Bread

Cereal

Donut

Eggs

Fudge

index = 3
length = 6

Linear Search

Apple

Bread

Cereal

Donut

Eggs

Fudge

index = 4
length = 6

Linear Search

Apple

Bread

Cereal

Donut

Eggs

Fudge

index = 4
length = 6

Linear Search

Search/Find: O(n)

Apple

Bread

Cereal

Donut

Eggs

Fudge

firstIndex = 0
lastIndex = 5
mid = floor((lastIndex - firstIndex) / 2)

0

5

Binary Search

Apple

Bread

Cereal

Donut

Eggs

Fudge

firstIndex = 0
lastIndex = 5
mid = 2

0

5

Binary Search

Apple

Bread

Cereal

Donut

Eggs

Fudge

firstIndex = 0
lastIndex = 5
mid = 2

check mid - 1

0

5

Binary Search

Apple

Bread

Cereal

Donut

Eggs

Fudge

firstIndex = 0
lastIndex = 5
mid = 2

check mid + 1

0

5

Binary Search

Apple

Bread

Cereal

Donut

Eggs

Fudge

firstIndex = 0
lastIndex = 5
mid = 2

firstIndex = mid + 1

0

5

Binary Search

Apple

Bread

Cereal

Donut

Eggs

Fudge

firstIndex = 3
lastIndex = 5
mid = 
  (lastIndex - firstIndex / 2) 
  + firstIndex

5

3

Binary Search

Apple

Bread

Cereal

Donut

Eggs

Fudge

firstIndex = 3
lastIndex = 5
mid = 4

5

3

Binary Search

Apple

Bread

Cereal

Donut

Eggs

Fudge

firstIndex = 3
lastIndex = 5
mid = 4

5

3

Binary Search

Apple

Bread

Cereal

Donut

Eggs

Fudge

firstIndex = 3
lastIndex = 5
mid = 4

5

3

Binary Search

Apple

Bread

Cereal

Donut

Eggs

Fudge

firstIndex = 3
lastIndex = 5
mid = 4

5

3

Binary Search

Search/Find: O(log n)

Apple

Bread

Cereal

Donut

Eggs

Fudge

5

3

Binary vs Linear Search

O(log n)

 

Apple

Bread

Cereal

Donut

Eggs

Fudge

O(n)

 

Apple

Bread

Cereal

Donut

Eggs

Fudge

5

3

Binary vs Linear Search

O(log n)

Needs to be sorted

Apple

Bread

Cereal

Donut

Eggs

Fudge

O(n)

Unsorted

Apple

Bread

Cereal

Donut

Eggs

Fudge

5

3

Binary vs Linear Search

O(log n)

Needs to be sorted

Apple

Bread

Cereal

Donut

Eggs

Fudge

O(n)

Unsorted

What is the TC of sorting?

O(nlog(n)) *

Binary Search Trees 

const BST = {
    root: {
        value: 19,
        left: {
            value: 11,
            left: {7},
            right: {16},
        },
        right: {
            value: 35,
            left: {23},
            right: null,
        },
    },
};

_

_

_

_

_

_

_

_

_

Values: 1, 2, 3, 4, 5, 6, 7, 8

_

5

7

4

8

1

6

2

3

Values: 1, 2, 3, 4, 5, 6, 7, 8

Insertion

20

?

20

20

?

20

Insertion

20

20

Insertion

20

20

Insertion

20

20

Insertion

20

20

Insertion

20

Insertion 

/** Class representing a Binary Search Tree. */

class BinarySearchTree {

  constructor() {
  }
  /*
  * Adds a new value
  * @param {*} value the value to add
  */
  insert(value) {
  }
}

const myBST = new BinarySearchTree();

console.log(myBST)

EXERCISE

click me

Find/Search Operation 

/** Class representing a Binary Search Tree. */

class BinarySearchTree {

  constructor() {
  }
  /*
  * Searches for a value
  * @param {*} value the value to find
  * @returns Node
  */
  contains(value) {
    ...
  }
}

const myBST = new BinarySearchTree();

console.log(myBST)

EXERCISE

click me

_

5

7

4

8

1

6

2

3

Apple

Bread

Cereal

Donut

Eggs

Fudge

Apple

Bread

Cereal

Donut

Eggs

Fudge

5

3

Binary Search on Array: O(log n)

BST Search: O(h)

Linear Search on Array: O(n)

_

5

7

4

8

1

6

2

3

Apple

Bread

Cereal

Donut

Eggs

Fudge

Apple

Bread

Cereal

Donut

Eggs

Fudge

5

3

BST Insert + Delete:  O(h)

Array Insert + Delete: O(n)

Compare

Don't forget about hash tables!

Same values, different heights

Balanced if...

1) The subtrees height differ by no more than 1

2) The left subtree is balanced
3) The right subtree of is balanced

Same values, different heights

Insert Order

Insert order: [4, 2, 1, 3, 6, 5, 7, 8]

Insert order: [1, 2, 3, 4, 5, 6, 7, 8]

Further Resources

On Frontend Masters:

Brian Holt's CS in 5hrs

 

Geeks for Geeks

https://github.com/jwasham/coding-interview-university

Cracking the Coding Interview

Hacker Rank / Leetcode

Interviewing.io & Pramp

A special shout out to Christopher Arriola, Jamil Lawrence, Angela Huang for all their help!  

 

If you enjoy the comics and illustrations, follow the artists on instagram: 

https://www.instagram.com/illustrationsbyem/

https://www.instagram.com/yeob_mi/

Introduction to Trees and Graphs

By Bianca Gandolfo

Introduction to Trees and Graphs

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