GRAPHS
OBJECTIVES
- Explain what a graph is
- Compare and contrast different types of graphs and their use cases in the real world
- Implement a graph using an adjacency list
- Traverse through a graph using BFS and DFS
- Compare and contrast graph traversal algorithms
WHAT ARE GRAPHS
A graph data structure consists of a finite (and possibly mutable) set of vertices or nodes or points, together with a set of unordered pairs of these vertices for an undirected graph or a set of ordered pairs for a directed graph.
WHAT
Nodes
+ Connections
A
B
E
C
F
D
A
B
C
D
E
F
NODES + CONNECTIONS
A
B
E
C
F
D
A
B
C
D
E
F
NODES + CONNECTIONS
A
B
E
C
F
D
A
B
C
D
E
F
NODES + CONNECTIONS
A
B
E
C
F
D
A
B
C
D
E
F
NODES + CONNECTIONS
USES FOR GRAPHS
- Social Networks
- Location / Mapping
- Routing Algorithms
- Visual Hierarchy
- File System Optimizations
- EVERYWHERE!
https://www.flickr.com/photos/kencf0618/6602925613/
Recommendations
- "People also watched"
- "You might also like..."
- "People you might know"
- "Frequently bought with"
Borderlands
Halo
Things in Common
Space
Future
Sci-Fi
Guns
TYPES OF GRAPHS
ESSENTIAL GRAPH TERMS
- Vertex - a node
- Edge - connection between nodes
- Weighted/Unweighted - values assigned to distances between vertices
- Directed/Undirected - directions assigned to distanced between vertices
A
B
E
C
F
D
A
B
C
D
E
F
TERMINOLOGY
Edge
Vertex
A
B
E
C
F
D
A
B
C
D
E
F
UNDIRECTED GRAPH
Facebook Friends Graph
Maria
Armie
Tim
Nan
Joan
A
B
E
C
F
D
A
B
C
D
E
F
DIRECTED GRAPH
Instagram Followers Graph
Maria
Justin
Bieber
Tim
Kanye
Joan
A
B
E
C
F
D
A
B
C
D
E
F
WEIGHTED GRAPH
10
17
20
8
15
7
90km
120km
65km
19km
3km
20km
40km
13km
REPRESENTING
A GRAPH
HOW DO WE STORE THIS???
ADJACENCY MATRIX
A
F
B
C
E
D
| - | A | B | C | D | E | F |
|---|---|---|---|---|---|---|
| A | 0 | 1 | 0 | 0 | 0 | 1 |
| B | 1 | 0 | 1 | 0 | 0 | 0 |
| C | 0 | 1 | 0 | 1 | 0 | 0 |
| D | 0 | 0 | 1 | 0 | 1 | 0 |
| E | 0 | 0 | 0 | 1 | 0 | 1 |
| F | 1 | 0 | 0 | 0 | 1 | 0 |
ADJACENCY LIST
[
[1,5],
[0,2],
[1,3],
[2,4],
[3,5],
[4,0]
]0
5
1
2
4
3
0
1
2
3
4
5
ADJACENCY LIST
{
A: ["B", "F"],
B: ["A", "C"],
C: ["B", "D"],
D: ["C", "E"],
E: ["D", "F"],
F: ["E", "A"]
}A
F
B
C
E
D
DIFFERENCES & BIG O
| OPERATION | ADJACENCY LIST | ADJACENCY MATRIX |
|---|---|---|
| Add Vertex | O(1) | O(|V^2|) |
| Add Edge | O(1) | O(1) |
| Remove Vertex | O(|V| + |E|) | O(|V^2|) |
| Remove Edge | O(|E|) | O(1) |
| Query | O(|V| + |E|) | O(1) |
| Storage | O(|V| + |E|) | O(|V^2|) |
|V| - number of vertices
|E| - number of edges
Adjacency
Matrix
Adjacency
List
- Takes up more space (in sparse graphs)
- Slower to iterate over all edges
- Faster to lookup specific edge
- Can take up less space (in sparse graphs)
- Faster to iterate over all edges
- Can be slower to lookup specific edge
VS.
What will we use?
Most data in the real-world tends to lend itself to sparser and/or larger graphs
Why?
An Adjacency List
An Adjacency List
An Adjacency List
OUR GRAPH CLASS
class Graph {
constructor(){
this.adjacencyList = {}
}
}WE ARE BUILDING AN UNDIRECTED GRAPH
ADDING A VERTEX
- Write a method called addVertex, which accepts a name of a vertex
- It should add a key to the adjacency list with the name of the vertex and set its value to be an empty array
g.addVertex("Tokyo"){
"Tokyo": []
}YOUR
TURN
ADDING AN EDGE
- This function should accept two vertices, we can call them vertex1 and vertex2
- The function should find in the adjacency list the key of vertex1 and push vertex2 to the array
- The function should find in the adjacency list the key of vertex2 and push vertex1 to the array
- Don't worry about handling errors/invalid vertices
{
"Tokyo": [],
"Dallas": [],
"Aspen": []
}g.addEdge("Tokyo", "Dallas"){
"Tokyo": ["Dallas"],
"Dallas": ["Tokyo"],
"Aspen": []
}g.addEdge("Dallas", "Aspen"){
"Tokyo": ["Dallas"],
"Dallas": ["Tokyo", "Aspen"],
"Aspen": ["Dallas"]
}YOUR
TURN
REMOVING AN EDGE
- This function should accept two vertices, we'll call them vertex1 and vertex2
- The function should reassign the key of vertex1 to be an array that does not contain vertex2
- The function should reassign the key of vertex2 to be an array that does not contain vertex1
- Don't worry about handling errors/invalid vertices
g.removeEdge("Tokyo", "Dallas"){
"Tokyo": ["Dallas"],
"Dallas": ["Tokyo", "Aspen"],
"Aspen": ["Dallas"]
}{
"Tokyo": [],
"Dallas": ["Aspen"],
"Aspen": ["Dallas"]
}REMOVING AN EDGE
YOUR
TURN
- The function should accept a vertex to remove
- The function should loop as long as there are any other vertices in the adjacency list for that vertex
- Inside of the loop, call our removeEdge function with the vertex we are removing and any values in the adjacency list for that vertex
- delete the key in the adjacency list for that vertex
REMOVING A VERTEX
g.removeVertex("Hong Kong"){
"Tokyo": ["Dallas", "Hong Kong"],
"Dallas": ["Tokyo", "Aspen", "Hong Kong", "Los Angeles"],
"Aspen": ["Dallas"],
"Hong Kong": ["Tokyo", "Dallas", "Los Angeles"],
"Los Angeles": ["Hong Kong", "Dallas"]
}{
"Tokyo": ["Dallas"],
"Dallas": ["Tokyo", "Aspen","Los Angeles"],
"Aspen": ["Dallas"],
"Los Angeles": ["Dallas"]
}REMOVING A VERTEX
YOUR
TURN
GRAPH TRAVERSAL
Visiting/Updating/Checking
each vertex in a graph
GRAPH TRAVERSAL USES
- Peer to peer networking
- Web crawlers
- Finding "closest" matches/recommendations
-
Shortest path problems
- GPS Navigation
- Solving mazes
- AI (shortest path to win the game)
Borderlands
Halo
Things in Common
Space
Future
Sci-Fi
Guns
Facebook Friends Graph
Maria
Armie
Tim
Nan
Joan
DEPTH FIRST
Explore as far as possible down one branch before "backtracking"
A
B
E
C
F
D
A
B
C
D
E
F
DEPTH FIRST TRAVERSAL
(STARTING FROM "A")
A
B
E
C
F
D
A
B
C
D
E
F
DEPTH FIRST TRAVERSAL
(STARTING FROM "A")
A
B
E
C
F
D
A
B
C
D
E
F
{
"A":["B", "C"],
"B":["A", "D"],
"C":["A", "E"],
"D":["B", "E", "F"],
"E":["C", "D", "F"],
"F":["D", "E"]
}X
X
X
X
X
X
X
X
X
X
X
X
X
X
A
B
E
C
F
D
A
B
C
D
E
F
{
"A":["B", "C"],
"B":["A", "D"],
"C":["A", "E"],
"D":["B", "E", "F"],
"E":["C", "D", "F"],
"F":["D", "E"]
}{
"A": true,
"B": true,
"D": true,
"E": true,
"C": true,
"F": true
}{
"A": true,
"B": true,
"D": true,
"E": true,
"C": true,
}{
"A": true,
"B": true,
"D": true,
"E": true,
}{
"A": true,
"B": true,
"D": true,
}{
"A": true,
"B": true,
}{
"A": true,
}{
}g.addVertex("A")
g.addVertex("B")
g.addVertex("C")
g.addVertex("D")
g.addVertex("E")
g.addVertex("F")
g.addEdge("A","B")
g.addEdge("A","C")
g.addEdge("B","D")
g.addEdge("C","E")
g.addEdge("D","E")
g.addEdge("D","F")
g.addEdge("E","F")A
B
E
C
F
D
A
B
C
D
E
F
DEPTH FIRST TRAVERSAL
(STARTING FROM "A")
DFS(vertex):
if vertex is empty
return (this is base case)
add vertex to results list
mark vertex as visited
for each neighbor in vertex's neighbors:
if neighbor is not visited:
recursively call DFS on neighbor
DFS PSEUDOCODE
Recursive
{
"A": true,
"B": true,
"D": true
}VISITING THINGS
DEPTH FIRST TRAVERSAL
-
The function should accept a starting node
-
Create a list to store the end result, to be returned at the very end
-
Create an object to store visited vertices
-
Create a helper function which accepts a vertex
-
The helper function should return early if the vertex is empty
-
The helper function should place the vertex it accepts into the visited object and push that vertex into the result array.
-
Loop over all of the values in the adjacencyList for that vertex
-
If any of those values have not been visited, recursively invoke the helper function with that vertex
-
-
Invoke the helper function with the starting vertex
-
Return the result array
Recursive
DFS-iterative(start):
let S be a stack
S.push(start)
while S is not empty
vertex = S.pop()
if vertex is not labeled as discovered:
visit vertex (add to result list)
label vertex as discovered
for each of vertex's neighbors, N do
S.push(N)DFS PSEUDOCODE
Iterative
DEPTH FIRST TRAVERSAL
-
The function should accept a starting node
-
Create a stack to help use keep track of vertices (use a list/array)
-
Create a list to store the end result, to be returned at the very end
-
Create an object to store visited vertices
-
Add the starting vertex to the stack, and mark it visited
-
While the stack has something in it:
-
Pop the next vertex from the stack
-
If that vertex hasn't been visited yet:
-
Mark it as visited
-
Add it to the result list
-
Push all of its neighbors into the stack
-
-
-
Return the result array
Iterative
DEPTH FIRST TRAVERSAL
- The function should accept a starting node
- Create an object to store visited nodes and an array to store the result
- Create a helper function which accepts a vertex
- The helper function should place the vertex it accepts into the visited object and push that vertex into the results
- Loop over all of the values in the adjacencyList for that vertex
- If any of those values have not been visited, invoke the helper function with that vertex
- return the array of results
BREADTH FIRST
Visit neighbors at current depth first!
A
B
E
C
F
D
A
B
C
D
E
F
BREADTH FIRST TRAVERSAL
(STARTING FROM "A")
BREADTH FIRST
- This function should accept a starting vertex
- Create a queue (you can use an array) and place the starting vertex in it
- Create an array to store the nodes visited
- Create an object to store nodes visited
- Mark the starting vertex as visited
- Loop as long as there is anything in the queue
- Remove the first vertex from the queue and push it into the array that stores nodes visited
- Loop over each vertex in the adjacency list for the vertex you are visiting.
- If it is not inside the object that stores nodes visited, mark it as visited and enqueue that vertex
- Once you have finished looping, return the array of visited nodes
A
B
E
C
F
D
A
B
C
D
E
F
YOUR
TURN
YOUR
TURN
Shortest Path Algorithms
When working with weighted and directed/undirected graphs, we very commonly want to know how to get from one vertex to another! Better yet, how to do it quickly.
What's the fastest way to get from point A to point B?
Dijkstra's Algorithm
OBJECTIVES
- Understand the importance of Dijkstra's
- Implement a Weighted Graph
- Walk through the steps of Dijkstra's
- Implement Dijkstra's using a naive priority queue
- Implement Dijkstra's using a binary heap priority queue
WHAT IS IT
One of the most famous and widely used algorithms around!
Finds the shortest path between two vertices on a graph
"What's the fastest way to get from point A to point B?"
WHO WAS HE?
Edsger Dijkstra was a Dutch programmer, physicist, essayist, and all around smarty-pants
He helped to advance the field of computer science from an "art" to an academic discipline
Many of his discoveries and algorithms are still commonly used to this day.
HE
DID
A
LOT...
What is the shortest way to travel from Rotterdam to Groningen, in general: from given city to given city. It is the algorithm for the shortest path, which I designed in about twenty minutes. One morning I was shopping in Amsterdam with my young fiancée, and tired, we sat down on the café terrace to drink a cup of coffee and I was just thinking about whether I could do this, and I then designed the algorithm for the shortest path. As I said, it was a twenty-minute invention. Eventually that algorithm became, to my great amazement, one of the cornerstones of my fame.
— Edsger Dijkstra, in an interview with Philip L. Frana, Communications of the ACM, 2001[2]
A QUOTE
WHY IS IT USEFUL?
- GPS - finding fastest route
- Network Routing - finds open shortest path for data
- Biology - used to model the spread of viruses among humans
- Airline tickets - finding cheapest route to your destination
- Many other uses!
90km
120km
65km
19km
3km
20km
40km
13km
HOW DOES GPS WORK?
Find the shortest path from Santiago to Antigua
A
B
90km
120km
65km
19km
3km
20km
40km
13km
OUR GRAPH CAN'T DO THIS YET!
Let's Write A Weighted Graph
Now on to Dikstra's
Djikstra's
Dijkstra's
A
B
C
D
E
F
2
4
4
1
1
3
3
2
Find the shortest path from A to E
- Every time we look to visit a new node, we pick the node with the smallest known distance to visit first.
- Once we’ve moved to the node we’re going to visit, we look at each of its neighbors
- For each neighboring node, we calculate the distance by summing the total edges that lead to the node we’re checking from the starting node.
- If the new total distance to a node is less than the previous total, we store the new shorter distance for that node.
THE APPROACH
A
B
C
D
E
F
2
4
4
1
1
3
3
2
FIND THE SHORTEST PATH
FROM A TO E
| Vertex | Shortest Dist From A |
|---|---|
| A | |
| B | |
| C | |
| D | |
| E | |
| F |
[]{
A: null,
B: null,
C: null,
D: null,
E: null,
F: null
}Previous:
Visited:
A
B
C
D
E
F
2
4
4
1
1
3
3
2
FIND THE SHORTEST PATH
FROM A TO E
| Vertex | Shortest Dist From A |
|---|---|
| A | 0 |
| B | Infinity |
| C | Infinity |
| D | Infinity |
| E | Infinity |
| F | Infinity |
[]{
A: null,
B: null,
C: null,
D: null,
E: null,
F: null
}Previous:
Visited:
A
B
C
D
E
F
2
4
4
1
1
3
3
2
FIND THE SHORTEST PATH
FROM A TO E
| Vertex | Shortest Dist From A |
|---|---|
| A | 0 |
| B | Infinity |
| C | Infinity |
| D | Infinity |
| E | Infinity |
| F | Infinity |
[A]{
A: null,
B: null,
C: null,
D: null,
E: null,
F: null
}Previous:
Visited:
Pick The Smallest...A
A
B
C
D
E
F
2
4
4
1
1
3
3
2
FIND THE SHORTEST PATH
FROM A TO E
| Vertex | Shortest Dist From A |
|---|---|
| A | 0 |
| B | Infinity |
| C | Infinity |
| D | Infinity |
| E | Infinity |
| F | Infinity |
[A]{
A: null,
B: null,
C: null,
D: null,
E: null,
F: null
}Previous:
Visited:
Pick The Smallest...A
A
B
C
D
E
F
2
4
4
1
1
3
3
2
FIND THE SHORTEST PATH
FROM A TO E
| Vertex | Shortest Dist From A |
|---|---|
| A | 0 |
| B |
|
| C | Infinity |
| D | Infinity |
| E | Infinity |
| F | Infinity |
[A]{
A: null,
B: null,
C: null,
D: null,
E: null,
F: null
}Previous:
Visited:
Pick The Smallest...A
A
B
C
D
E
F
2
4
4
1
1
3
3
2
FIND THE SHORTEST PATH
FROM A TO E
| Vertex | Shortest Dist From A |
|---|---|
| A | 0 |
| B |
|
| C | Infinity |
| D | Infinity |
| E | Infinity |
| F | Infinity |
[A]{
A: null,
B: A,
C: null,
D: null,
E: null,
F: null
}Previous:
Visited:
Pick The Smallest...A
A
B
C
D
E
F
2
4
4
1
1
3
3
2
FIND THE SHORTEST PATH
FROM A TO E
| Vertex | Shortest Dist From A |
|---|---|
| A | 0 |
| B |
|
| C | Infinity |
| D | Infinity |
| E | Infinity |
| F | Infinity |
[A]{
A: null,
B: A,
C: null,
D: null,
E: null,
F: null
}Previous:
Visited:
Pick The Smallest...A
A
B
C
D
E
F
2
4
4
1
1
3
3
2
FIND THE SHORTEST PATH
FROM A TO E
| Vertex | Shortest Dist From A |
|---|---|
| A | 0 |
| B |
|
| C |
|
| D | Infinity |
| E | Infinity |
| F | Infinity |
[A]{
A: null,
B: A,
C: null,
D: null,
E: null,
F: null
}Previous:
Visited:
Pick The Smallest...A
A
B
C
D
E
F
2
4
4
1
1
3
3
2
FIND THE SHORTEST PATH
FROM A TO E
| Vertex | Shortest Dist From A |
|---|---|
| A | 0 |
| B |
|
| C |
|
| D | Infinity |
| E | Infinity |
| F | Infinity |
[A]{
A: null,
B: A,
C: A,
D: null,
E: null,
F: null
}Previous:
Visited:
Pick The Smallest...A
A
B
C
D
E
F
2
4
4
1
1
3
3
2
FIND THE SHORTEST PATH
FROM A TO E
| Vertex | Shortest Dist From A |
|---|---|
| 0 | |
| B |
|
| C |
|
| D | Infinity |
| E | Infinity |
| F | Infinity |
[A]{
A: null,
B: A,
C: A,
D: null,
E: null,
F: null
}Previous:
Visited:
Pick The Smallest...C
A
B
C
D
E
F
2
4
4
1
1
3
3
2
FIND THE SHORTEST PATH
FROM A TO E
| Vertex | Shortest Dist From A |
|---|---|
| 0 | |
| B |
|
| C |
|
| D | Infinity |
| E | Infinity |
| F | Infinity |
[A,C]{
A: null,
B: A,
C: A,
D: null,
E: null,
F: null
}Previous:
Visited:
Pick The Smallest...C
A
B
C
D
E
F
2
4
4
1
1
3
3
2
FIND THE SHORTEST PATH
FROM A TO E
| Vertex | Shortest Dist From A |
|---|---|
| 0 | |
| B |
|
| C |
|
| D |
|
| E | Infinity |
| F | Infinity |
[A,C]{
A: null,
B: A,
C: A,
D: null,
E: null,
F: null
}Previous:
Visited:
Pick The Smallest...C
A
B
C
D
E
F
2
4
4
1
1
3
3
2
FIND THE SHORTEST PATH
FROM A TO E
| Vertex | Shortest Dist From A |
|---|---|
| 0 | |
| B |
|
| C |
|
| D |
|
| E | Infinity |
| F | Infinity |
[A,C]{
A: null,
B: A,
C: A,
D: C,
E: null,
F: null
}Previous:
Visited:
Pick The Smallest...C
A
B
C
D
E
F
2
4
4
1
1
3
3
2
FIND THE SHORTEST PATH
FROM A TO E
| Vertex | Shortest Dist From A |
|---|---|
| 0 | |
| B |
|
| C |
|
| D |
|
| E | Infinity |
| F | Infinity |
[A,C]{
A: null,
B: A,
C: A,
D: C,
E: null,
F: null
}Previous:
Visited:
Pick The Smallest...C
A
B
C
D
E
F
2
4
4
1
1
3
3
2
FIND THE SHORTEST PATH
FROM A TO E
| Vertex | Shortest Dist From A |
|---|---|
| 0 | |
| B |
|
| C |
|
| D |
|
| E | Infinity |
| F |
|
[A,C]{
A: null,
B: A,
C: A,
D: C,
E: null,
F: null
}Previous:
Visited:
Pick The Smallest...C
A
B
C
D
E
F
2
4
4
1
1
3
3
2
FIND THE SHORTEST PATH
FROM A TO E
| Vertex | Shortest Dist From A |
|---|---|
| 0 | |
| B |
|
| C |
|
| D |
|
| E | Infinity |
| F |
|
[A,C]{
A: null,
B: A,
C: A,
D: C,
E: null,
F: C
}Previous:
Visited:
Pick The Smallest...C
A
B
C
D
E
F
2
4
4
1
1
3
3
2
FIND THE SHORTEST PATH
FROM A TO E
| Vertex | Shortest Dist From A |
|---|---|
| 0 | |
| B |
|
|
|
|
| D |
|
| E | Infinity |
| F |
|
[A,C]{
A: null,
B: A,
C: A,
D: C,
E: null,
F: C
}Previous:
Visited:
Pick The Smallest...B
A
B
C
D
E
F
2
4
4
1
1
3
3
2
FIND THE SHORTEST PATH
FROM A TO E
| Vertex | Shortest Dist From A |
|---|---|
| 0 | |
| B |
|
|
|
|
| D |
|
| E | Infinity |
| F |
|
[A,C]{
A: null,
B: A,
C: A,
D: C,
E: null,
F: C
}Previous:
Visited:
Pick The Smallest...B
A
B
C
D
E
F
2
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2
FIND THE SHORTEST PATH
FROM A TO E
| Vertex | Shortest Dist From A |
|---|---|
| 0 | |
| B |
|
|
|
|
| D |
|
| E |
|
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WE ARE DONE!!!
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A simple PQ
Notice we are sorting which is O(N * log(N))
class PriorityQueue {
constructor(){
this.values = [];
}
enqueue(val, priority) {
this.values.push({val, priority});
this.sort();
};
dequeue() {
return this.values.shift();
};
sort() {
this.values.sort((a, b) => a.priority - b.priority);
};
}Can we do better?
We sure can! Using a min binary heap
Right now, let's focus on Dijkstra's
Dijkstra's Pseudocode
- This function should accept a starting and ending vertex
- Create an object (we'll call it distances) and set each key to be every vertex in the adjacency list with a value of infinity, except for the starting vertex which should have a value of 0.
- After setting a value in the distances object, add each vertex with a priority of Infinity to the priority queue, except the starting vertex, which should have a priority of 0 because that's where we begin.
- Create another object called previous and set each key to be every vertex in the adjacency list with a value of null
- Start looping as long as there is anything in the priority queue
- dequeue a vertex from the priority queue
- If that vertex is the same as the ending vertex - we are done!
- Otherwise loop through each value in the adjacency list at that vertex
- Calculate the distance to that vertex from the starting vertex
- if the distance is less than what is currently stored in our distances object
- update the distances object with new lower distance
- update the previous object to contain that vertex
- enqueue the vertex with the total distance from the start node
YOUR
TURN
Improving Dijkstra's
Dijkstra's algorithm is greedy! That can cause problems!
We can improve this algorithm by adding a heuristics (a best guess)
Recap
- Graphs are collections of vertices connected by edges
- Graphs can be represented using adjacency lists, adjacency matrices and quite a few other forms.
- Graphs can contain weights and directions as well as cycles
- Just like trees, graphs can be traversed using BFS and DFS
- Shortest path algorithms like Dijkstra can be altered using a heuristic to achieve better results like those with A*
Graphs
By colt_steele
