#### Contents

- Types of Graphs
- Types of Shortest Path Algorithms
- Algorithms
- Comparison of Algorithms
- Improvements

There are many variants of graphs. The first property is the directionality of its edges. Edges can either be unidirectional or bidirectional. If they are unidirectional, the graph is called a **directed** graph. If they are bidirectional (meaning they go both ways), the graph is called a **undirected** graph. In the case where some edges are directed and others are not, the bidirectional edges should be swapped out for 2 directed edges that fulfill the same functionality. That graph is now fully directed.

The same graph as above, but directed

The second property of a graph has to do with the weights of the edges. Edges can have no weight, and in that case the graph is called **unweighted**. If edges do have weights, the graph is said to be **weighted**. There is an extra caveat here: graphs can be allowed to have negative weight edges. The inclusion of negative weight edges prohibits the use of some shortest path algorithms.

The third property of graphs that affects what algorithms can be used is the existence of cycles. A cycle is defined as any path $p$p through a graph, $G$G, that visits that same vertex, $v$v, more than once. So, if a graph has any path that has a cycle in it, that graph is said to be **cyclic**. **Acyclic** graphs, graphs that have no cycles, allow more freedom in the use of algorithms.

Cyclic graph with cyclic path A -> E -> D -> B -> A

There are two main types of shortest path algorithms, single-source and all-pairs. Both types have algorithms that perform best in their own way. All-pairs algorithms take longer to run because of the added complexity. All shortest path algorithms return values that can be used to find the shortest path, even if those return values vary in type or form from algorithm to algorithm.

**Single-source**

Single-source shortest path algorithms operate under the following principle:

Given a graph $G$G, with vertices $V$V, edges $E$E with weight function $w(u, v) = w_{u, v}$w(u,v)=wu,v, and a single source vertex, $s$s, return the shortest paths from $s$s to all other vertices in $V$V.

If the goal of the algorithm is to find the shortest path between only two given vertices, $s$s and $t$t, then the algorithm can simply be stopped when that shortest path is found. Because there is no way to decide which vertices to "finish" first, all algorithms that solve for the shortest path between two given vertices have the same worst-case asymptotic complexity as single-source shortest path algorithms.

This paradigm also works for the *single-destination shortest path* problem. By reversing all of the edges in a graph, the single-destination problem can be reduced to the single-source problem. So, given a destination vertex, $t$t, this algorithm will find the shortest paths *starting* at all other vertices and ending at $t$t.

**All-pairs**

All-pairs shortest path algorithms follow this definition:

Given a graph $G$G, with vertices $V$V, edges $E$E with weight function $w(u, v) = w_{u, v}$w(u,v)=wu,v return the shortest path from $u$u to $v$v for all $(u, v)$(u,v) in $V$V.

The most common algorithm for the all-pairs problem is the floyd-warshall algorithm. This algorithm returns a matrix of values $M$M, where each cell $M_{i, j}$Mi,j is the distance of the shortest path from vertex $i$i to vertex $j$j. Path reconstruction is possible to find the actual path taken to achieve that shortest path, but it is not part of the fundamental algorithm.

**Bellman-Ford algorithm**

The Bellman-Ford algorithm solves the single-source problem in the general case, where edges can have negative weights and the graph is directed. If the graph is undirected, it will have to modified by including two edges in each direction to make it directed.

Bellman-Ford has the property that it can detect negative weight cycles reachable from the source, which would mean that no shortest path exists. If a negative weight cycle existed, a path could run infinitely on that cycle, decreasing the path cost to $- \infty$−∞.

If there is no negative weight cycle, then Bellman-Ford returns the weight of the shortest path along with the path itself.

**Dijkstra's algorithm**

Dijkstra's algorithm makes use of breadth-first search (which is not a single source shortest path algorithm) to solve the single-source problem. It does place one constraint on the graph: there can be no negative weight edges. However, for this one constraint, Dijkstra greatly improves on the runtime of Bellman-Ford.

Dijkstra's algorithm is also sometimes used to solve the all-pairs shortest path problem by simply running it on all vertices in $V$V. Again, this requires all edge weights to be positive.

**Topological Sort**

For graphs that are directed acyclic graphs (DAGs), a very useful tool emerges for finding shortest paths. By performing a topological sort on the vertices in the graph, the shortest path problem becomes solvable in linear time.

A topological sort is an ordering all of the vertices such that for each edge $(u, v)$(u,v) in $E$E, $u$u comes before $v$v in the ordering. In a DAG, shortest paths are always well defined because even if there are negative weight edges, there can be no negative weight cycles.

**Floyd-Warshall algorithm**

The Floyd-Warshall algorithm solves the all-pairs shortest path problem. It uses a dynamic programming approach to do so. Negative edge weight may be present for Floyd-Warshall.

Floyd-Warshall takes advantage of the following observation: the shortest path from A to C is either the shortest path from A to B plus the shortest path from B to C *or* it's the shortest path from A to C that's already been found. This may seem trivial, but it's what allows Floyd-Warshall to build shortest paths from smaller shortest paths, in the classic dynamic programming way.

**Johnson's Algorithm**

While Floyd-Warshall works well for dense graphs (meaning many edges), Johnson's algorithm works best for sparse graphs (meaning few edges). In sparse graphs, Johnson's algorithm has a lower asymptotic running time compared to Floyd-Warshall.

Johnson's algorithm takes advantage of the concept of reweighting, and it uses Dijkstra's algorithm on many vertices to find the shortest path once it has finished reweighting the edges.

Dijkstra Bellman-Ford Johnson Floyd-Warshall

Shortest-path algorithms are useful for certain types of graphs. For the graph below, which algorithm should be used to solve the single-source shortest path problem?

Graph

The runtimes of the shortest path algorithms are listed below.

See also: big O notation.

Algorithm | Runtime |

Bellman-Ford | $O(|V| \cdot |E|)$O(∣V∣⋅∣E∣) |

Dijkstra's (with list) | $O(|V|^2)$O(∣V∣2) |

Topological Sort | $O(|V| + |E|)$O(∣V∣+∣E∣) |

Floyd-Warshall | $O(|V|^3)$O(∣V∣3) |

Johnson's | *$O(|E| \cdot |V| + |V|^2 \cdot \log_2(|V|))$O(∣E∣⋅∣V∣+∣V∣2⋅log2(∣V∣)) |

**This runtime assumes that the implementation uses fibonacci heaps.*

Oftentimes, the question of which algorithm to use is not left up to the individual; it is merely a function of what graph is being operated upon and which shortest path problem is being solved.

For graphs with negative weight edges, the single source shortest path problem needs Bellman-Ford to succeed. For dense graphs and the all-pairs problem, Floyd-Warshall should be used.

However, there are some subtle differences. For sparse graphs and the all-pairs problem, it might be obvious to use Johnson's algorithm. However, if there are no negative edge weights, then it is actually better to use Dijkstra's algorithm with binary heaps in the implementation. Running Dijsktra's from each vertex will yield a better result.

From a space complexity perspective, many of these algorithms are the same. In their most fundemental form, for example, Bellman-Ford and Dijkstra are the exact same because they use the same representation of a graph. However, when these algorithms are sped up using advanced data structures like fibonacci or binary heaps, the space required to perform the algorithm increases. As is common with algorithms, space is often traded for speed.

These algorithms have been improved upon over time. Dijkstra's algorithm, for example, was initally implemented using a list, and had a runtime of $O(|V|^2)$O(∣V∣2). However, when a binary heap is used, a runtime of $O((|E|+|V|) \cdot \log_2(|V|))$O((∣E∣+∣V∣)⋅log2(∣V∣)) has been achieved. When a fibonacci heap is used, one implementation can achieve $O(|E| + |V| \cdot \log_2(|V|))$O(∣E∣+∣V∣⋅log2(∣V∣)) while another can do $O(|E| \cdot \log_2(\log_2(|C|)))$O(∣E∣⋅log2(log2(∣C∣))) where $|C|$∣C∣ is a bounded constant for edge weight.

Bellman-Ford has been implemented in $O(|V|^2 \cdot \log_2(|V|))$O(∣V∣2⋅log2(∣V∣)). This implementation can be efficient if used on the right kind of graph (sparse).

## FAQs

### Which algorithm is best for finding shortest path? ›

**What Is the Best Shortest Path Algorithm?**

- Dijkstra's Algorithm. Dijkstra's Algorithm stands out from the rest due to its ability to find the shortest path from one node to every other node within the same graph data structure. ...
- Bellman-Ford Algorithm. ...
- Floyd-Warshall Algorithm. ...
- Johnson's Algorithm. ...
- Final Note.

**Does Google Maps use Dijkstra? ›**

**Google M a**

**aA* is an informed search algorithm, or a best-first search, meaning that it is formulated in terms of weighted graphs: starting from a specific starting node of a graph, it aims to find a path to the given goal node having the smallest cost (least distance travelled, shortest time, etc.).https://en.wikipedia.org › wiki**

**ps essentially uses two Graph algorithms – Dijkstra's algorithm and A* algorithm**, to calculate the shortest distance from point A ( Source) to point B ( destination).

**What is shortest path algorithm explain with example? ›**

The target of shortest path algorithms is **to find a route between any pair of vertices along the edges, so the sum of weights of edges is minimum**. If the edges are of equal weights, the shortest path algorithm aims to find a route having minimum number of hops.

**How do you solve the shortest path? ›**

Basically you take a single pair of vertices. Or points. And you will find the shortest path between

**Which is the best algorithm? ›**

**Top Machine Learning Algorithms You Should Know**

- Linear Regression.
- Logistic Regression.
- Linear Discriminant Analysis.
- Classification and Regression Trees.
- Naive Bayes.
- K-Nearest Neighbors (KNN)
- Learning Vector Quantization (LVQ)
- Support Vector Machines (SVM)

**Why is the shortest path important? ›**

Abstract. Finding the shortest path (SP) in a large-scale network analysis between any two nodes is a tough but very significant task. **The SP can help us to analyze the information spreading performance and research the latent relationship in the weighted social network**, and so on.

**Is Dijkstra algorithm still used? ›**

**Yes, Dijkstra's algorithm is used in modern maps systems**.

**Does Apple Maps use Dijkstra? ›**

Dijkstra's work on the shortest path algorithm that eventually was named after him – the Dijkstra's algorithm that made Navigation possible. **The core of this algorithm is what powers the navigate functionality at Google Maps, Apple Maps, Here, OpenStreetMap and any other digital map that you probably use**.

**Why Dijkstra algorithm is used? ›**

Dijkstra's algorithm **solves the shortest-path problem for any weighted, directed graph with non-negative weights**. It can handle graphs consisting of cycles, but negative weights will cause this algorithm to produce incorrect results.

**How do you write an algorithm example? ›**

...

**An Algorithm Development Process**

- Step 1: Obtain a description of the problem. This step is much more difficult than it appears. ...
- Step 2: Analyze the problem. ...
- Step 3: Develop a high-level algorithm. ...
- Step 4: Refine the algorithm by adding more detail. ...
- Step 5: Review the algorithm.

### Does a * find the shortest path? ›

**it can be used to find a shortest path**.

**What is the shortest path in data structure? ›**

In d

**a path that has the least cost as compared to all other existing paths**.

**How does Dijkstras algorithm work? ›**

Dijkstra's Algorithm finds the shortest path between a given node (which is called the "source node") and all other nodes in a graph. This algorithm **uses the weights of the edges to find the path that minimizes the total distance (weight) between the source node and all other nodes**.

**Is Dijkstra A greedy algorithm? ›**

Abstract.

**Dutch computer scientist, programmer, software engineer, systems scientist, and science essayist**.

**It is a type of greedy algorithm**. It only works on weighted graphs with positive weights.

**What are the applications of shortest-path problem? ›**

Dijkstra's algorithm is one of the most popular algorithms for solving many single-source shortest path problems having non-negative edge weight in the graphs i.e., it is **to find the shortest distance between two vertices on a graph**. It was conceived by computer scientist Edsger W.

**What algorithms should I learn? ›**

Important search algorithms include **binary search and depth search**. Must-know sorting algorithms include heap sort, merge sort, quick sort, number of inversions, and insertion sort. Hashing is also an important skill to learn that combines algorithms and data structures.

**What are 3 examples of algorithms? ›**

Common examples include: **the recipe for baking a cake, the method we use to solve a long division problem, the process of doing laundry, and the functionality of a search**

**a searchA* is an informed search algorithm, or a best-first search, meaning that it is formulated in terms of weighted graphs: starting from a specific starting node of a graph, it aims to find a path to the given goal node having the smallest cost (least distance travelled, shortest time, etc.).https://en.wikipedia.org › wiki**

**engine**are all examples of an algorithm.

**What is a famous algorithm? ›**

**Top 25 Algorithms Every Programmer Should Know**

- Binary Search Algorithm.
- Breadth First Search (BFS) Algorithm.
- Depth First Search (DFS) Algorithm.
- Merge Sort Algorithm.
- Quicksort Algorithm.
- Kruskal's Algorithm.
- Floyd Warshall Algorithm.
- Dijkstra's Algorithm.

**Do people use shortest path? ›**

**In almost 90% of cases people choose routes that are less than 5 minutes longer than the shortest time routes**. Commute routes deviate from shortest time routes slightly more in percentage compared with non-commute routes.

**What is meant by shortest path? ›**

(classic problem) Definition: **The problem of finding the shortest path in a graph from one vertex to another**. "Shortest" may be least number of edges, least total weight, etc. Also known as single-pair shortest-path problem.

### What is shortest path analysis? ›

INTRODUCTION Shortest path analysis **finds the path with the minimum cumulative impedance between nodes on a network**. The path may connect just two nodes an origin and a destination or have specific stops between the nodes.

**What algorithm does Google use? ›**

**PageRank (PR)** is an algorithm used by Google Search to rank web pages in their search engine results.

**Which algorithm is used in Google Maps? ›**

Google Maps uses **Dijkstra's Algorithm** [63] of finding the shortest paths between nodes in a graph, which may represent, for example, road networks [64] .

**Which is better Dijkstra or A *? ›**

Even though Dijkstra's algorithm and the A* algorithm both find the same shortest paths, **the A* algorithm does it almost 60 times faster**!

**What is the Google Earth algorithm? ›**

Google maps is using **Dijkstra's Shortest Path Algorithm**. It calculates the connections between pairs of elements or so called nodes.

**What applications use Dijkstra's algorithm? ›**

- It is used in Google Maps.
- It is used in finding Shortest Path.
- It is used in geographical Maps.
- To find locations of Map which refers to vertices of graph.
- Distance between the location refers to edges.
- It is used in IP routing to find Open shortest Path First.
- It is used in the telephone network.

**What 3 algorithms are part of a navigation app? ›**

There are three major shortest path algorithms: **Bellman Ford's Algorithm, Dijkstra's Algorithm, and Floyd–Warshall's Algorithm**.

**Is Dijkstra a best first search? ›**

Conclusion: **Best-First Search should be prefered over dijkstra when you have some knowledge on the graph, and can estimate a distance from target**. If you don't - an uninformed Best-First Search that uses h(v) = 0 , and relays only on already explored vertices, decays into Dijkstra's algorithm.

**What are the limitations of Dijkstra's algorithm? ›**

The major disadvantage of the algorithm is the fact that **it does a blind search there by consuming a lot of time waste of necessary resources**. Another disadvantage is that it cannot handle negative edges. This leads to acyclic graphs and most often cannot obtain the right shortest path.

**What are the 4 types of algorithm? ›**

Introduction To Types of Algorithms

**Brute Force algorithm**. Greedy algorithm. Recursive algorithm. Backtracking algorithm.

### How do you solve algorithms? ›

**Be Strategic, Think First**

- Analyze the problem.
- Restate the problem.
- Write out examples of input and output.
- Break the problem into its component parts.
- Outline a solution in psuedo-code.
- Step through your example data with your psuedo-code.

**WHY A * algorithm is best? ›**

**A* achieves better performance by using heuristics to guide its search**. Compared to Dijkstra's algorithm, the A* algorithm only finds the shortest path from a specified source to a specified goal, and not the shortest-path tree from a specified source to all possible goals.

**WHY A * algorithm is used? ›**

Wh

**to find the shortest path between an initial and a final point**. It is a handy algorithm that is often used for map traversal to find the shortest path to be taken.

**IS A * algorithm Complete? ›**

A* is complete and will always find a solution if one exists. Have a look at the wikipedia article. If further the heuristics is admissible and monotonic the algorithm will also be admissible(i.e. optimal).

**How many types of shortest path algorithms are there? ›**

There are **two main types** of shortest path algorithms, single-source and all-pairs.

**Can the shortest path be negative? ›**

A negative cycle is a directed cycle whose total weight (sum of the weights of its edges) is negative. **The concept of a shortest path is meaningless if there is a negative cycle**. Accordingly, we consider edge-weighted digraphs with no negative cycles. Bellman-Ford algorithm.

**Which of the following is true about shortest path algorithm? ›**

Shortest Paths MCQ Question 4

The correct answer is “option 4”. CONCEPT: **Dijkstra's algorithm is an algorithm used for finding the shortest paths between nodes or vertices in a graph**. This algorithm is basically used to find the shortest path from a starting node to a target node in a weighted graph.

**What type of algorithm is Dijkstra? ›**

Dijkstra's algorithm is the **iterative algorithmic process** to provide us with the shortest path from one specific starting node to all other nodes of a graph. It is different from the minimum spanning tree as the shortest distance among two vertices might not involve all the vertices of the graph.

**What does Dijkstra's algorithm return? ›**

You can return **distance between two nodes, distances between a node and all other nodes, distance and a path, distance and a previous node** (which is enough to construct a path). So in the case of wikipedia article - it returns you distances to all vertices and what is the previous vertex in the path to get your path.

**Does Dijkstra visit every node? ›**

Dijkstra's algorithm in default form computes the shortest distance from a starting node to all connected nodes. Even in this form **it does not visit all nodes**: only the vertices of a connected component have to be checked.

### Is Dijkstra BFS or DFS? ›

According to this page, Dijkstra's algorithm is just **BFS** with a priority queue.

**Is Kruskal algorithm greedy? ›**

**It is a greedy algorithm in graph theory** as in each step it adds the next lowest-weight edge that will not form a cycle to the

**a subset of the edges of a connected, edge-weighted undirected graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight**.

**Which algorithm is used to find shortest distances in a graph? ›**

**Bellman-Ford algorithm** is used to find all pairs shortest distances in a graph and it is dynamic programming technique.

**Which shortest path algorithm is best? ›**

**What Is the Best Shortest Path Algorithm?**

- Dijkstra's Algorithm. Dijkstra's Algorithm stands out from the rest due to its ability to find the shortest path from one node to every other node within the same graph data structure. ...
- Bellman-Ford Algorithm. ...
- Floyd-Warshall Algorithm. ...
- Johnson's Algorithm. ...
- Final Note.

**What is single source shortest path algorithm? ›**

The Single-Source Shortest Path (SSSP) problem consists of **finding the shortest paths between a given vertex v and all other vertices in the graph**. Algorithms such as Breadth-First-Search (BFS) for unweighted graphs or Dijkstra [1] solve this problem.

**Is Dijkstra's algorithm the best? ›**

As we can see, **Dijkstra's algorithm is better when it comes to reducing the time complexity**. However, when we have negative weights, we have to go with the Bellman-Ford algorithm. Also, if we want to know whether the graph contains negative cycles or not, the Bellman-Ford algorithm can help us with that.

**Why Floyd warshall is better than Dijkstra? ›**

Unlike Dijkstra's algorithm, Floyd Warshall can be implemented in a distributed system, making it suitable for data structures such as Graph of Graphs (Used in Maps). Lastly **Floyd Warshall works for negative edge but no negative cycle**, whereas Dijkstra's algorithm don't work for negative edges.

**Which algorithm is best suitable for finding the shortest path between a single source to any destination with positive and negative? ›**

**Floyd\u2013Warshall's Algorithm** is used to find the shortest paths between between all pairs of vertices in a graph, where each edge in the graph has a weight which is positive or negative.

**What's the difference between Dijkstra and Bellman-Ford? ›**

The only difference between the two is that **Bellman-Ford is also capable of handling negative weights whereas Dijkstra Algorithm can only handle positives**.

**Which is better A * or Dijkstra? ›**

A* algorithm is just like Dijkstra's algorithm, and the only difference is that **A* tries to look for a better path by using a heuristic function, which gives priority to nodes that are supposed to be better than others while Dijkstra's just explore all possible ways**.

### Is Dijkstra algorithm still used? ›

**Yes, Dijkstra's algorithm is used in modern maps systems**.

**Which is better BFS or Dijkstra? ›**

**Dijkstra's algorithm is more general than BFS**,in deed it is a generalization of BFS where edges' weights no longer have to be equal - this is “THE” only significant difference. For efficiency reason,a FIFO queue in BFS generalizes to a priority queue in Dijkstra.

**Which is better Dijkstra or Floyd? ›**

The biggest difference is that **Floyd's algorithm finds the shortest path between all vertices and Dijkstra's algorithm finds the shortest path between a single vertex and all other vertices**. The space overhead for Dijkstra's algorithm is considerably more than that for Floyd's algorithm.

**Why do we need Floyd-Warshall algorithm? ›**

Floyd-Warshall Algorithm is an algorithm for **finding the shortest path between all the pairs of vertices in a weighted graph**. This algorithm works for both the directed and undirected weighted graphs. But, it does not work for the graphs with negative cycles (where the sum of the edges in a cycle is negative).

**Is Floyd algorithm greedy? ›**

The Floyd-Warshall algorithm takes into account all possible routes so that there are some routes are displayed while **the greedy algorithm checks every node that is passed to select the shortest route (Local Optimum) so that the time needed in searching is faster**.

**How many types of shortest path algorithms are there? ›**

There are **two main types** of shortest path algorithms, single-source and all-pairs.

**Why topological sort is needed for shortest path? ›**

Topological Sorting basically sorts the vertices of graph in increasing order of their dependencies(by dependencies of vertex we mean indegree of a edge) or their indegree and Since **shortest path between source vertex and a particulat vertex should involve minimum intermediate edges** hence finding topologcial sort first ...

**Which algorithm will you use to determine the path from the source to the destination? ›**

At the heart of any routing protocol is the algorithm (the "**routing algorithm**") that determines the path for a packet. The purpose of a routing algorithm is simple: given a set of routers, with links connecting the routers, a routing algorithm finds a "good" path from source to destination.

**Is Dijkstra better than DFS? ›**

**Most people prefer Dijkstra to DFS in pathfinding because Dijkstra is so accurate**. Well, Dijkstra finds the shortest path from the starting point. DFS does not guarantee shortest path, it would just generate a path that visits very nodes in the graph. Dijkstra finds the shortest path for weighted graphs.

**Is Dijkstra greedy or Dynamic Programming? ›**

**Dutch computer scientist, programmer, software engineer, systems scientist, and science essayist**.

**greedy algorithm**. It only works on weighted graphs with positive weights.

### Can Dijkstra detect cycle? ›

**It is certainly possible to modify Dijkstra's algorithm to detect negative cycles**, but there is no point in doing so, because you have a stronger restriction of having no negative edges.