This article will explore 20 of the most common graph algorithms and various ways to use them in real-life scenarios. In recent years, graphs have become a valuable tool for real-world data modeling. They’re used in various fields, including economics, mathematics, physics, aeronautics, biology (for DNA analysis), etc. They have also found use in social media networks, websites and web links, and routes and locations in GPS. Let’s start with a little bit of theory.

## What is graph theory?

Graph theory is the study of graphs and their properties. A graph consists of vertices (or nodes) and the edges (or relationships) connecting them. Graphs can be used to model real-world situations, such as social networks, transportation networks, or electrical networks.

## What are graph algorithms?

Graph algorithms have a non-linear data structure of edges and nodes. The edges are arcs or lines that connect any two nodes in a graph. The following are some of the most common graph algorithms:

### 1. Breadth-first search

The breadth-first search algorithm finds the shortest path between two nodes in a graph. A graph traversal algorithm begins at the root node and works its way down through the adjacent nodes. Breadth-first search explores all of the previously unknown neighboring nodes for each new node discovered. When traversing using BFS, any node in the graph can be considered the root node. BFS is a repetitive algorithm when searching vertices of a graph data structure. It categorizes every vertex of the graph into two categories: visited and unvisited. It selects one node in a graph and visits all nodes adjacent to the one it selected.

#### Applications of the BFS algorithm

- BFS may be used to discover the locations nearest to a specific origin point.
- The BFS algorithm is used in peer-to-peer networks as a search technique to discover all neighboring nodes. uTorrent, BitTorrent, and other similar torrent clients use this method to look for “peers” and “seeds” in the network.
- BFS is used in web crawlers to build web page indexes. It starts at the source page and works its way through all of the links connected with it. Every web page is treated as a node in the network graph.

### 2. Depth-first search

The depth-first search (DFS) algorithm is a graph-traversing algorithm that works its way down the graph data structure beginning from the root node through to the adjacent nodes. The DFS algorithm is a recursive algorithm that uses the idea of backtracking. It involves exhausting searches of all nodes by moving ahead if possible until there are no more nodes to explore in the current path, at which time it begins backtracking. The algorithm will follow the current route until all of the unvisited nodes have been visited, at which point a new path will be chosen.

#### Applications of DFS algorithm

- Depth-first search is employed in all of these situations: in scheduling problems, cycle detection in graphs, topological sorting, and finding solutions for puzzles that have only one solution, e.g., sudoku and mazes.
- The depth-first search method is used in network analysis, for example, to test if a graph is bipartite. It is frequently employed as a subroutine in the Ford-Fulkerson algorithm and other network flow algorithms.
- The depth-first search is frequently employed as a subroutine in more complicated algorithms. For instance, Hopcroft–Karp incorporates DFS to assist in discovering a matching in a graph.
- DFS algorithm is used in scheduling, finding spanning trees, and mapping routes.

### 3. Dijkstra’s algorithm

Dijkstra’s graph search algorithm finds the shortest path between two nodes in a graph. It is an iterative algorithm that starts with the source node and works its way to the destination node. For each new node discovered, Dijkstra’s algorithm calculates the shortest path to the destination node using the currently known distances. When traversing using Dijkstra’s algorithm, any node in the graph can be considered the root node.

#### Applications of Dijkstra’s algorithm

- Dijkstra’s algorithm is used in network routing protocols, such as RIP, OSPF, and BGP, to calculate the best route between two nodes.
- It is used in algorithms for solving the shortest path problem, such as the A* algorithm.
- The Bellman-Ford algorithm uses Dijkstra’s algorithm to find the shortest path from a source node to all other nodes in a graph.
- Dijkstra’s algorithm is used in many artificial intelligence applications, such as game playing and search engines.

### 4. A* algorithm

The A* algorithm finds the shortest path between two nodes in a graph. It is similar to Dijkstra’s algorithm, but it is a more sophisticated algorithm that considers the cost of each edge in the graph. The cost of an edge is usually determined by its length or by some other measure of distance, such as time or money.

#### Applications of A* algorithm

- It is commonly used in web-based maps and games to find the shortest path at the highest possible efficiency.
- A* is used in many artificial intelligence applications, such as search engines.
- It is used in other algorithms such as the Bellman-Ford algorithm to solve the shortest path problem.
- The A* algorithm is used in network routing protocols, such as RIP, OSPF, and BGP, to calculate the best route between two nodes.

### 5. Shortest path algorithm

The shortest path algorithm is a graph search algorithm that calculates the shortest path between two nodes in a graph. It is similar to the A* algorithm, but it is a simpler algorithm that does not take into account the cost of each edge in the graph.

#### Applications of shortest path algorithm

- Computer games - Finding the shortest/best route from one point to another is a common task in video games.
- Maps - Finding the shortest and/or most affordable route for a car from one city to another.
- Satellite navigation systems - to show drivers the fastest path they can drive from one point in a city to the other.

### 6. Minimum spanning tree algorithm

The minimum spanning tree algorithm entails connecting all vertices with the fewest total edge weights. In other words, it is a spanning tree that has the lowest possible sum of edge weights.

#### Applications of minimum spanning tree algorithm

- Network designs - cable and telephone networks use minimum spanning tree algorithms to create a list of possible connections with the lowest cost.
- Find approximate solutions to complex mathematical problems, e.g., the Traveling Salesman Problem.
- Real-time tracking and verification of human faces.
- Use to avoid network cycles in computer science protocols.

### 7. Maximum flow algorithm

The maximum flow algorithm is a graph theory algorithm used to find the maximum possible flow between two nodes in a graph. It is similar to the Ford–Fulkerson algorithm, but it is a more sophisticated algorithm that considers each edge’s capacity in the graph.

#### Applications of maximum flow algorithm

- Network routing - The maximum flow algorithm can be used to calculate the maximum possible traffic that can flow through a network.
- Computer science - Can be used to find a solution to the Maximum Flow Problem, also known as the Max Flow/Min Cut problem.
- Operations research - Is used to solve Network Flow Problems.

### 8. Connected components algorithm

The connected components algorithm is a graph theory algorithm used to find the connected components of a graph. It is a simple algorithm that takes into account the edges and vertices of a graph.

#### Applications of connected components algorithm

- Computer science - The connected components algorithm can be used to find a solution to the Connected Components Problem, also known as the Graph Isomorphism Problem.
- Networks - Can be used to find all the isolated nodes in a network.
- VLSI design - The algorithm can be used to find all the connections in a chip layout.
- Electronics - Can be used to find all the circuits in an electronic schematic.

### 9. Fleury’s Algorithm

Fleury’s algorithm displays the Euler circuit or Euler path from a graph. The algorithm starts at one edge and moves adjacent vertices by removing previous ones. The graph gets less complicated in each step towards finding the Euler or circuit path.

#### Applications of Fleury’s algorithm

- Computer science - Fleury’s algorithm can be used to find a solution to the Euler Circuit Problem, also known as the Euler Path Problem.
- Networks - Can be used to find all the circuits in a network.

### 10. Johnson’s algorithm

Johnson’s algorithm finds the shortest paths between every pair of vertices in an edge-weighted directed graph. Edge weights can have negative numbers, but negative-weight cycles are not allowed. Johnson’s algorithm aims to reweigh all edges, make them positive, then run Dijkstra’s algorithm on each vertex.

#### Applications of Johnson’s algorithm

- Computer science - Johnson’s algorithm can be used to find a solution to the Shortest Path Problem.
- Networks - Finding all the shortest paths between nodes in a network.
- Operations research - Used to solve Network Flow Problems.

### 11. Tarjan’s algorithm

Tarjan’s algorithm is a graph theory algorithm used to find the strongly connected components of a graph. A graph is considered strongly connected if each vertex can be reached from every other vertex.

#### Applications of Tarjan’s algorithm

- Computer science - Tarjan’s algorithm can be used to find a solution to the Strongly Connected Components Problem, also known as the Connected Components Problem.
- Used to obtain the Dulmage–Mendelsohn decomposition, a bipartite graph edges classification method.
- Used in social networks to discover groups of strongly connected people and make recommendations based on shared interests.

### 12. Kosaraju’s algorithm

Kosaraju’s algorithm is similar to Tarjan’s algorithm in that both are used to find strongly connected components of a graph. While Tarjan’s and Kosaraju’s algorithms have a linear time complexity, their SCC computing methodologies differ. Tarjan’s technique partitions the graph using nodes’ timestamps in a DFS, but Kosaraju’s method partitions the graph using two DFSs and is comparable to the method for discovering topological sorting.

#### Applications of Kosaraju’s algorithm

- Computer science - Kosaraju’s algorithm can be used to find a solution to the Strongly Connected Components Problem, also known as the Connected Components Problem.
- Used to obtain the Dulmage–Mendelsohn decomposition, a bipartite graph edges classification method.
- Used in social networks to discover groups of strongly connected people and make recommendations based on shared interests.

### 13. Labeled graph

A labeled graph is a graph with a set of labels or tags associated with each vertex and edge in the graph. The purpose of labeling a graph is to make it easier to identify the vertices and edges in the graph.

#### Applications of labeled graphs

- Computer science - Labeled graphs can be used to represent data structures, such as arrays, lists, and trees.
- Networks - Can be used to represent networks, such as social networks and transportation networks.
- Mathematics - Are used to represent mathematical structures, such as groups and rings.

### 14. Topological Sort algorithm

The topological sort algorithm is a graph theory algorithm used to find the order in which the vertices of a graph should be visited. The algorithm works by starting with the first vertex in the graph and visiting each vertex in order until all the vertices have been visited.

#### Applications of topological sorting

- Computer science - Topological sorting can be used to solve the DAG problem, also known as the Directed Acyclic Graph Problem.
- Networks - Can be used to create a network flow diagram.
- Operations research - Is used to solve Network Flow Problems.

### 15. Floyd–Warshall algorithm

The Floyd–Warshall algorithm is a graph theory algorithm used to find the shortest path between all pairs of vertices in a graph. The algorithm works by constructing a table of shortest paths from each vertex to every other vertex in the graph. Like the Dijkstra’s and Bellman-Ford algorithms, it calculates the shortest path in a graph. However, unlike the two algorithms that use only one source to compute the shortest distance, the Floyd–Warshall algorithm calculates the shortest distances for all pairs of vertices in a graph.

#### Applications of Floyd–Warshall

- Computer science - Floyd–Warshall can be used to find the best path between two vertices in a graph.
- Networks - Is used to find the shortest path between two points in a network.
- Optimal routing - Finds the path with the maximum flow between two vertices.
- Pathfinder networks - Can be used for fast computations of Pathfinder networks.

### 16. Prim’s algorithm

Prim’s algorithm is a graph theory algorithm used to find the shortest path between a given source vertex and all other vertices in a graph. The algorithm works by constructing a table of distances from the source vertex to all other vertices in the graph. The shortest path from the source vertex to any other vertex is then found by looking at the minimum distance in the table.

#### Applications of Prim’s algorithm

- In scenarios where finding the shortest path would lower costs, e.g., landing cables, LAN networks, electric grid, natural gas pipes, drinking water pipe network, etc.
- In single-link clusters to find the pair of elements closest to each other.

### 17. Kruskal algorithm

The Kruskal algorithm is a graph theory algorithm used to find the minimum spanning tree for a given graph. The algorithm works by starting with all the vertices in the graph and connecting them to form a tree. The tree is then trimmed by removing the edges that are not part of the minimum spanning tree.

#### Applications of Kruskal

- Designing rail and road networks to connect several cities
- Placing microwave towers
- Designing irrigation channels
- Designing fiber-optic grids

### 18. Bellman-Ford algorithm

The Bellman-Ford procedure is a method for finding the shortest paths between a single source vertex and each of the other vertices in a weighted digraph. Give the algorithm a weighted graph, and it’ll return the shortest path between the selected node and all other nodes. It is similar to Dijkstra’s algorithm but slower. However, its versatility, which means it can handle both negative-weighted and positive-weighted edges, makes it a popular choice.

#### Applications of Bellman-Ford

- It is used in distance-vector routing protocols, e.g., in Routing Information Protocols (RIPs).
- It is used in the Maximum Weighted Matching problem, a problem in graph theory.
- It is used in the All-Pairs Shortest Paths problem.
- In chemical reactions computing the smallest possible heat loss/gain.
- Identifying the currency conversion method that’d be most efficient

### 19. Boyer-Moore algorithm

The Boyer-Moore algorithm is a string search algorithm that is used to find a substring in a larger string. The algorithm works by trying all possible matches for the substring and then checking whether each match is a prefix of the larger string. If the match is not a prefix, then the algorithm moves on to the next possible match.

#### Applications of Boyer-Moore

- String searching, such as finding occurrences of a word in a text document.
- Text editing, such as spell-checking and correcting typos.
- Pattern recognition, such as recognizing characters in an image.

### 20. Greedy algorithm

A greedy algorithm is an algorithm that always chooses the best possible option at each step without considering future steps. The algorithm typically starts with a small solution and then improves it by making local changes that do not affect the global optimum.

#### Applications of greedy algorithms

- Optimizing search results by ordering them according to some heuristic.
- Scheduling tasks for a limited number of machines so that the most important tasks are completed first.
- Choosing a subset of items from a larger set so that the resulting set has the largest possible value.
- Packing items into a container in a way that minimizes transportation costs.