Video: Graphs and Graph Algorithms (Bulgarian)

November 03, 2020
Nikolay Kostov (Nikolay.IT)

Topics covered:

Graph Definitions and Terminology
Representing Graphs
Graph Traversal Algorithms
Connectivity
Dijkstra’s Algorithm
Topological sorting
Prim and Kruskal

Video (in Bulgarian)

Presentation Content

Graph Data Structure

Set of nodes with many-to-many relationship between them is called graph
- Each node has multiple predecessors
- Each node has multiple successors

Graph Definitions

Node(vertex)
- Element of graph
- Can have name or value
- Keeps a list of adjacent nodes
Edge
- Connection between two nodes
- Can be directed / undirected
- Can be weighted / unweighted
- Can have name / value
Weighted graph
- Weight (cost) is associated with each edge
Path (in undirected graph)
- Sequence of nodes n₁, n₂, … n_k
- Edge exists between each pair of nodes n_i, n_i+1
- Examples:
  - A,B,C is a path
  - H,K,C is not a path
Path (in directed graph)
- Sequence of nodes n₁, n₂, … n_k
- Directed edge exists between each pair of nodes n_i, n_i+1
- Examples:
  - A,B,C is a path
  - A,G,K is not a path
Cycle
- Path that ends back at the starting node
- Example:
  - A,B,C,G,A
Simple path
- No cycles in path
Acyclic graph
- Graph with no cycles
- Acyclic undirected graphs are trees
Two nodes are reachable if
- Path exists between them
Connected graph
- Every node is reachable from any other node

Graphs and Their Applications

Graphs have many real-world applications
- Modeling a computer network like Internet
  - Routes are simple paths in the network
- Modeling a city map
  - Streets are edges, crossings are vertices
- Social networks
  - People are nodes and their connections are edges
- State machines
  - States are nodes, transitions are edges

Simple C# Representation

public class Graph
{
  List<int>[] childNodes;
  public Graph(List<int>[] nodes)
  {
    this.childNodes = nodes;
  }
}

Graph g = new Graph(new List<int>[] {
  new List<int> {3, 6}, // successors of vertice 0
  new List<int> {2, 3, 4, 5, 6},// successors of vertice 1
  new List<int> {1, 4, 5}, // successors of vertice 2
  new List<int> {0, 1, 5}, // successors of vertice 3
  new List<int> {1, 2, 6}, // successors of vertice 4
  new List<int> {1, 2, 3}, // successors of vertice 5
  new List<int> {0, 1, 4}  // successors of vertice 6
});

Advanced C# Representation

Using OOP:
- Class Node
- Class Connection (Edge)
- Class Graph
- Optional classes
Using external library:
- QuickGraph- http://quickgraph.codeplex.com/

Graph Traversal Algorithms

Depth-First Search (DFS) and Breadth-First Search (BFS) can traverse graphs
- Each vertex should be visited at most once

BFS(node)
{
  queue ← node
  visited[node] = true
  while queue not empty
    v ← queue
    print v
    for each child c of v
      if not visited[c]
        queue ← c
        visited[c] = true
}

DFS(node)
{
  stack ← node
  visited[node] = true
  while stack not empty
    v ← stack
    print v
    for each child c of v
      if not visited[c]
        stack ← c
        visited[c] = true
}

Recursive DFS Graph Traversal

void TraverseDFSRecursive(node)
{
  if (not visited[node])
  {
    visited[node] = true;
    print node;
    foreach child node c of node
    {
       TraverseDFSRecursive(c);
    }
  }
}

void Main()
{
  TraverseDFS(firstNode);
}

Connectivity

Connected component of undirected graph
- A sub-graph in which any two nodes are connected to each other by paths
A simple way to find number of connected components
- A loop through all nodes and start a DFS or BFS traversing from any unvisited node
Each time you start a new traversing
- You find a new connected component!
Algorithm:

foreach node from graph G
{
   if node is unvisited
   {
      DFS(node);
      countOfComponents++;
   }
}

Note: Do not forget to mark each node in the DFS as visited!
Connected graph
- A graph with only one connected component
In every connected graph a path exists between any two nodes
Checking whether a graph is connected
- If DFS / BFS passes through all vertices → graph is connected!

Dijkstra’s Algorithm

Find the shortest path from vertex A to all other vertices
- The path is a directed path between them such that no other path has a lower weight.
Assumptions
- Edges can be directed or not
- Weight does not have to be distance
- Weights are positive or zero
- Shortest path is not necessary unique
- Not all edges need to be reachable
In non-weighted graphs or edges with same weight finding shortest path can be done with BFS
- Note: Path from A to B does not matter – triangle inequality
In weighted graphs – simple solution can be done with breaking the edges in sub-vertexes
- Note:Too much memory usage even for smaller graphs!
Solution to this problem:
- Priority queue instead of queue
- Keeping information about the shortest distance so far
Steps:
- Enqueue all distances from S
- Get the lowest in priority - B
- If edge (B, A) exists, check (S, B) + (B, A) and save the lower one
- Overcome the triangle inequality miss
Dijkstra’s algorithm:
1. Set the distance to every node to Infinity except the source node – must be zero
2. Mark every node as unprocessed
3. Choose the first unprocessed node with smallest non-infinity distance as current. If such does not exist, the algorithm has finished
4. At first we set the current node our Source
5. Calculate the distance for all unprocessed neighbors by adding the current distance to the already calculated one
6. If the new distance is smaller than the previous one – set the new value
7. Mark the current node as processed
8. Repeat step 3.

Pseudo code

set all nodes DIST = INFINITY;
set current node the source and distance = 0;
Q -> all nodes from graph, ordered by distance;
while (Q is not empty)
    a = dequeue the smallest element (first in PriorityQueue);
    if (distance of a == INFINITY) break

    foreach neighbour v of a     
        potDistance = distance of a + distance of (a-v)      
        if (potDistance < distance of v)
            distance of v = potDistance;
            reorder Q;

Modifications
- Saving the route
- Having a target node
- Array implementation, Queue, Priority Queue
- A*
Complexity
- O((|V| + |E|).log(|V|))
Applications – GPS, Networks, Air travels, etc.

Topological Sorting

Topological ordering of a directed graph
- Linear ordering of its vertices
- For every directed edge (U, V), U comes before V in the ordering
Example:
- 7, 5, 3, 11, 8, 2, 9, 10
- 3, 5, 7, 8, 11, 2, 9, 10
- 5, 7, 3, 8, 11, 10, 9, 2
Rules
- Undirected graph cannot be sorted
- Directed graphs with cycles cannot be sorted
- Sorting is not unique
- Various sorting algorithms exists and they give different results
Source removal algorithm
1. Create an Empty List
2. Find a Node without incoming Edges
3. Add this Node to the end of the List
4. Remove the Edge from the Graph
5. Repeat until the Graph is empty
Pseudo code

L ← Empty list that will contain the sorted elements
S ← Set of all nodes with no incoming edges
while S is non-empty do
    remove a node n from S
    insert n into L
    for each node m with an edge e from n to m do
        remove edge e from the graph
        if m has no other incoming edges then
            insert m into S
if graph has edges then
    return error (graph has at least one cycle)
else
    return L (a topologically sorted order)

DFS algorithm
- Create an empty List
- Find a Node without Outgoing Edges
- Mark the Node as visited
- Add the node to the List
- Stop when reach visited node
- Reverse the List and get the TS of the Elements
Pseudo code

L ← Empty list that will contain the sorted nodes
while there are unmarked nodes do
    select an unmarked node n
    visit(n)
function visit(node n)
    if n has a temporary mark then stop (not a DAG)
    if n is not marked (i.e. has not been visited yet) then
        mark n temporarily
        for each node m with an edge from n to m do
            visit(m)
        mark n permanently
        add n to head of L

topological-sorting

Minimum Spanning Tree

Spanning Tree
- Subgraph (Tree)
- Connects all vertices together
All connected graphs have spanning tree
Minimum Spanning Tree
- weight <= weight(all other spanning trees)
First used in electrical network
- Minimal cost of wiring
Minimum Spanning Forest – set of all minimum spanning trees (when the graph is not connected)

Prim’s Algorithm

Create a tree T containing a single vertex (chosen randomly)
Create a set S from all the edges in the graph
Loop until every edge in the set connects two vertices in the tree
- Remove from the set an edge with minimum weight that connects a vertex in the tree with a vertex not in the tree
- Add that edge to the tree
Note: the graph must be connected
Note: at every step before adding an edge to the tree we check if it makes a cycle in the tree or if it is already in the queue
When we add a vertex we check if it is the last which is not visited
We build a tree with the single vertex A
Priority queue which contains all edges that connect A with the other nodes (AB, AC, AD)
The tree still contains the only vertex A
We dequeue the first edge from the priority queue (4) and we add the edge and the other vertex (B) form that edge to the tree
We push all edges that connect B with other nodes in the queue
- Note that the edges 5 and 9 are still in the queue
Now the tree contains vertices A and B and the edge between them
We dequeue the first edge from the priority queue (2) and we add the edge and the other vertex (D) from that edge to the tree
We push all edges that connect D with other nodes in the queue
Now the tree contains vertices A, B and D and the edges (4, 2) between them
We dequeue the first edge from the priority queue (5) and we add the edge and the other vertex © from that edge to the tree
We push all edges that connect C with other nodes in the queue
Now the tree contains vertices A, B, D and C and the edges (4, 2, 5) between them
We dequeue the first edge from the priority queue (7) and we add the edge and the other vertex (E) from that edge to the tree
We push all edges that connect C with other nodes in the queue
Now the tree contains vertices A, B, D, C and E and the edges (4, 2, 5, 7) between them
We dequeue the first edge from the priority queue (8)
This edge will cost a cycle
So we get the next one – 9
- This edge will also cost a cycle
So we get the next one – 12
- We add it to the tree
- We add the vertex F to the tree

Kruskal’s Algorithm

The graph may not be connected
- If the graph is not connected – minimum spanning forest
Create forest F (each tree is a vertex)
Set S – all edges in the graph
While S is not empty and F is not spanning
- Remove edge with min cost from S
- If that edge connects two different trees – add it to the forest (these two trees are now a single tree)
- Else discard the edge
The graph may not be connected
We build a forest containing all vertices from the graph
We sort all edges
Edges are – 2, 4, 5, 7, 8, 9, 12, 20
At every step we select the edge with the smallest weight and remove it from the list with edges
If it connects two different trees from the forest we add it and connect these trees
We select the edge 2
This edge connects the vertices B and D (they are in different trees)
We add it
We select the edge 4
This edge connects the vertices A and B (they are different trees)
We add it
We select the edge 5
This edge connects the vertices A and C (they are different trees)
We add it
We select the edge 7
This edge connects the vertices C and E (they are different trees)
We add it
We select the edge 8
This edge connects the vertices E and D (they are not different trees)
We don’t add it
We select the edge 9
This edge connects the vertices A and D (they are not different trees)
We don’t add it
We select the edge 12
This edge connects the vertices E and F (they are not different trees)
We add it
We can have a function that checks at every step if all vertices are connected and the tree that we build is spanning
If we have such function we stop
Otherwise we check for the other edges
- We just won’t add them to the tree