Shortest path in undirected weighted graph Paths and connectivity are defined as in the case of unweighted graphs. The graph can either be directed or undirected. This is the first algorithm to break the O(m + n log n) time bound for real-weighted sparse graphs by Dijkstra’s algorithm with Fibonacci But Dijkstra and Prim/Kruskal calulcate two inherently different things. If the weight of an edge corresponds to length, then the length of a path is the sum of the lengths of the edges on the path the distance between two nodes is the length of a shortest path between them Sep 17, 2024 · Directed Uniform Weighted Graph 01 BFS 01 BFS is a slightly different variation of BFS algorithm, which is used to calculate shortest path between vertices in graph in with binary weighted edges ARandomizedAlgorithmforSingle-SourceShortestPathon UndirectedReal-WeightedGraphs A Randomized Algorithm for Single-Source Shortest Path on Undirected Real-Weighted Graphs We present a new scheme for computing shortest paths on real-weighted undirected graphs in the fundamental comparison-addition model. Maybe you need to find the shortest path between point A and B, but maybe you need to shortest path between point A and all other points in the graph. If there are 2 different Oct 16, 2023 · Here is a weighted graph showing the connections between a set of vertices: In a weighted graph, each connection (or edge) between two vertices has a weight associated with it. Dijkstra’s Algorithm: To handle, we use Dijkstra Aug 13, 2025 · Let's discover the shortest path in an undirected graph. My initial thought was that each edge is weighted with weight 1. Types of Weighted Graphs Weighted graphs can be classified based on the nature of the graph and the weights assigned to the edges. For a directed graph you'll be looking to find a minimum cost aborescence, which can't be Level up your coding skills and quickly land a job. Aug 12, 2024 · Why isn't the shortest path always the fastest? Dive into the world of graphs—directed, undirected, and weighted—and discover how they solve real-world problems like optimizing routes and analyzing networks. Instead of directly minimizing the product, we transform the problem by taking log of each edge weight. But what if edges have different ‘costs’? In a weighted directed graph, a path that goes from node ni to nj and for which the sum of the weights of the edges included in this path is the minimum over all possible paths from ni to nj is called the shortest path from ni to nj. Learn efficient algorithms for finding optimal routes in unweighted graphs. Dijkstra’s algorithm. So, given this information, I would want to find the shortest path from a starting node to an ending node. You are given an array graph where graph[i] is a list of all the nodes connected with node i by an edge. For example, the single-source shoretest path problem requires finding the shortest Jul 9, 2023 · In undirected graphs with real non-negative weights, we give a new randomized algorithm for the single-source shortest path (SSSP) problem with running time O(m log n ⋅ log log n− −−−−−−−−−−−√) in the comparison-addition model. Dijkstra's Jun 6, 2018 · One way to think of this question is to improve the running time of using Dijkstra's algorithm to find the shortest path between two vertices in the undirected weighted graph. One … Shortest Path on Weighted Graphs BFS finds the shortest paths from a source node s to every vertex v in the graph. The total weight of a path is the sum of the weights of its edges. Jul 11, 2025 · Given an unweighted, undirected graph of V nodes and E edges, a source node S, and a destination node D, we need to find the shortest path from node S to node D in the graph. Approach: Mark all vertices unvisited. Shortest path (A, C, E, D, F), blue, between vertices A and F in the weighted directed graph In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized. We present a new scheme for computing shortest paths on real-weighted undirected graphs in the fundamental comparison-addition model. ) therefore having a single algorithm that Sep 28, 2020 · Dijkstra's Algorithm finds the shortest path between a given node (which is called the "source node") and all other nodes in a graph. May 23, 2015 · I was revising single source shortest path algorithms and in the video, the teacher mentions that BFS/DFS can't be used directly for finding shortest paths in a weighted graph (I guess everyone knows May 28, 2025 · Whereas BFS can find the shortest path from one vertex to another in an unweighted graph (we saw that on Wednesday), Dijkstra's algorithm finds shortest paths in weighted graphs. 0 means there is no edge): Jul 23, 2025 · Given a weighted undirected graph G and an integer S, the task is to print the distances of the shortest paths and the count of the number of the shortest paths for each node from a given vertex, S. In undirected graphs with real non-negative weights, we give a new randomized algorithm for the single-source shortest path (SSSP) problem with running time O(m log n ⋅ log log n− −−−−−−−−−−−√) in the comparison-addition model. But when edge weights are different, BFS becomes inefficient - it explores nodes in the order they appear and may process the same nodes multiple times to update distances, which is slow and insufficient. 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. shortest_path # shortest_path(csgraph, method='auto', directed=True, return_predecessors=False, unweighted=False, overwrite=False, indices=None) # Perform a shortest-path graph search on a positive directed or undirected graph. This weight can be used to represent various things. The graph consists of m edges represented by a 2D array edges, where edges [i] = [ai, bi, wi] indicates that there is an edge between nodes ai and bi with weight wi. You may start and stop at any node, you may revisit nodes multiple times Jul 23, 2025 · What are the Shortest Path Algorithms? The shortest path algorithms are the ones that focuses on calculating the minimum travelling cost from source node to destination node of a graph in optimal time and space complexities. Sep 11, 2021 · To find shortest path in undirected weighted graph I was comparing BFS and dijkstra's algo to understand why we need priority queue. An undirected, weighted graph There are also different types of shortest path algorithms. One obvious application is in finding the shortest route from one address to another, however shortest paths have many other application. The traveling salesman problem asks for the circuit of minimum total weight in a weighted, com-plete, undirected graph that visits each vertex exactly once and returns to its starting point. Jul 15, 2025 · Given a directed graph and a source vertex in the graph, the task is to find the shortest distance and path from source to target vertex in the given graph where edges are weighted (non-negative) and directed from parent vertex to source vertices. Dijkstra's algorithm is used for solving single-source shortest path problems for directed or undirected paths. Can you solve this real interview question? Shortest Path Visiting All Nodes - You have an undirected, connected graph of n nodes labeled from 0 to n - 1. We will look at a version of the problem assuming we are finding distances from a single source to all Jul 13, 2024 · You are given a weighted undirected graph having n vertices numbered from 1 to n and m edges along with their weights. A heap is a complete binary tree with the heap property that every parent node is smaller (greater) than its children nodes in the tree in a min heap (a Shortest Paths # Compute the shortest paths and path lengths between nodes in the graph. Nov 15, 2024 · Our algorithm exploits the strong connection between Shortest Odd Path and the problem of finding two internally vertex-disjoint paths between two terminals in an undirected edge-weighted graph. Undirected Weighted Graph In an undirected weighted graph, the edges have no direction, and the weights represent the cost or distance between the two connected vertices, irrespective of direction. We talked about it with the help of real world example and it will help May 20, 2025 · What is Dijkstra‘s Algorithm? Dijkstra‘s algorithm solves the single-source shortest path problem for weighted graphs with non-negative edge weights. The weight of a path is the sum of the weights of the edges on the path. Building a Graph using Dictionaries Approach: The idea is to store the adjacency list into the dictionaries, which helps to store the graph in any format Nov 15, 2025 · How to find the minimum-product path from a source node to a destination node in undirected weighted graph? Yes, this can be solved using Dijkstra’s algorithm. If a vertex is unreachable from the source node, then Modify Graph Edge Weights - You are given an undirected weighted connected graph containing n nodes labeled from 0 to n - 1, and an integer array edges where edges [i] = [ai, bi, wi] indicates that there is an edge between nodes ai and bi with weight wi. Problem Statement Given a Undirected Graph of N vertices form 0 to N-1 and M edges and a 2D integer array edges, where there is a edge from vertex edge [i] [0] to vertex edge [i] [1] of unit weight. Feb 17, 2020 · Below are implementations for finding shortest paths in weighted & unweighted graphs. Find the shortest path between the vertex 1 and the vertex n, if there exists a path, and returna list of integers whose first element is the weight of the path, and the rest consist of the nodes on that path. Given for digraphs but easily modified to work on undirected graphs. May 20, 2025 · You’re given an undirected, unweighted graph represented as an adjacency list and a source vertex src. This is the first algorithm to break the O(m + n log n) time bound for real-weighted sparse graphs by Dijkstra's algorithm with Fibonacci Aug 26, 2016 · We progress through the four most important types of graph models: undirected graphs (with simple connections), digraphs graphs (where the direction of each connection is significant), edge-weighted graphs (where each connection has an software associated weight), and edge-weighted digraphs (where each connection has both a direction and a weight). The shortest path from s to t in a weighted graph is a path P from s to t (or a s - t path) with minimum weight w(P) = ∑ w(P) . Learn the right graph for your needs with practical examples! Dec 8, 2024 · You are given an Undirected Graph having unit weight of the edges, find the shortest path from src to all the vertex and if it is unreachable to reach any vertex, then return -1 for that vertex. In an efficient preprocessing phase our algorithm creates a linear-size structure that facilitates single-source shortest path computations in O(m log $\\alpha$) time, where $\\alpha$ = $\\alpha$(m,n) is the very slowly growing inverse-Ackermann function, m the Abstract. Shortest Paths (SSAD) 2 Given a weighted graph, and a designated node S, we would like to find a path of least total weight from S to each of the other vertices in the graph. Jan 10, 2025 · Given a weighted line-graph (undirected connected graph, all vertices of degree 2, except two endpoints which have degree 1), devise an algorithm that preprocesses the graph in linear time and can return the distance of the shortest path between any two vertices in constant time. Create a set of all unvisited vertices. In an efficient preprocessing phase our al-gorithm creates a linear-size structure that facilitates single-source shortest path computations in O(mlog α) time, where α = α(m,n) is the very slowly growing inverse-Ackermann function, m the number of Dijkstra's algorithm is often considered to be the most straightforward algorithm for solving the shortest path problem. Shortest path algorithms have many applications. Find the shortest path from the source to all other nodes in this graph. This is represented by a 2D array edges of length n - 1, where edges [i] = [ui, vi, wi] indicates an undirected edge from node ui to vi with weight wi. The task is to find the shortest path (in terms of edge count) from the source to every 5 days ago · In a weighted graph, if all edges had the same weight (like 1), BFS could easily find the shortest path since each edge adds the same cost. This is the best place to expand your knowledge and get prepared for your next interview. For example, if the vertices represent towns, the weight could represent the distance between the towns, or it might represent the cost of a train ticket between the Jul 15, 2025 · Given an undirected and unweighted graph and two nodes as source and destination, the task is to print all the paths of the shortest length between the given source and destination. [1] The problem of finding the shortest path between two intersections on a road map may be modeled as a special Find Edges in Shortest Paths - You are given an undirected weighted graph of n nodes numbered from 0 to n - 1. Single-source means that one vertex is chosen to be the start, and the algorithm will find the shortest path from that vertex to all other vertices. This converts multiplication into addition, allowing Dijkstra to always pick the path with the minimum total log-sum, which The algorithm creates a tree of shortest paths from the starting vertex, the source, to all other points in the graph. Outline of this Lecture Recalling the BFS solution of the shortest path problem for unweighted (di)graphs. You may start and stop at any node, you may revisit nodes multiple times Feb 18, 2020 · Tada! All nodes have been visited so the algorithm is finished! The shortest path to any node from Node 0 can be found by following the path in teal. Types of Shortest Path Algorithms: As we know there are various types of graphs (weighted, unweighted, negative, cyclic, etc. May 13, 2025 · Given an undirected, weighted graph with V vertices numbered from 0 to V-1, and E edges represented as a 2D array edges[][], where each element edges[i] = [u, v, w] denotes an edge between nodes u and v with weight w, and all edge weights are positive integers, your task is to find the minimum weight cycle in the graph. Note: A cycle in a graph is a path that starts and ends at the same vertex Nov 6, 2023 · Matching Algorithms to Specific Use Cases Shortest Path Problems: Use Dijkstra’s Algorithm for finding the shortest path in weighted graphs without negative weights. Directed Weighted An example of a graph is shown below. Given a starting vertex (source), it calculates the shortest path from that vertex to all other vertices in the graph. Jul 23, 2025 · Given a directed graph where every edge has weight as either 1 or 2, find the shortest path from a given source vertex ‘s’ to a given destination vertex ‘t’. Moreover, Dijkstra finds the lowest-cost path from some source vertex to every other vertex in a graph! Shortest Paths Given a graph where edges are labaled with weights (or distances) and a source vertex, what is the shortest path between the source and some other vertex? Problems requiring us to an-swer such queries are broadly known as shortest-paths problems. The shortest path problem for weighted digraphs. Dijkstra calculates the shortest path tree from one node whereas Prim/Kruskal calculates the minimum spanning tree between all nodes. Therefore, I decided to do it using an altered form of a depth first search. These algorithms work with undirected and directed graphs. You are given a weighted undirected graph with n vertices numbered from 1 to n and m edges along with their weights. However, the graph is undirected, so Djikstras would not be an ideal fit. This can be used for any graph with weighted Jul 15, 2025 · Prerequisites: BFS for a Graph Dictionaries in Python In this article, we will be looking at how to build an undirected graph and then find the shortest path between two nodes/vertex of that graph easily using dictionaries in Python Language. Weighted Paths The weight w(π) of a path π in a weighted graph is the sum of weights of edges in the path The (weighted) shortest path from s ∈ V to t ∈ V is path of minimum weight from s to t δ(s, t) = inf{w(π) | path π from s to t} is the shortest-path weight from s to t Shortest Paths In this chapter we will cover problems involving finding the shortest path between vertices in a graph with weights (lengths) on the edges. Return the length of the shortest path that visits every node. Some edges have a weight of -1 (wi = -1), while others have a positive weight (wi > 0). The concept of an MST is not defined for directed graphs - the connections have to be undirected. Feb 15, 2020 · I have a complete undirected graph of locations (nodes), where each edge represents the distance between its connected nodes, and I want to find the shortest path starting from a start node without. Can you solve this real interview question? Shortest Path in a Weighted Tree - You are given an integer n and an undirected, weighted tree rooted at node 1 with n nodes numbered from 1 to n. So in this case, you can use a binary heap as the data structure. Find the shortest path between vertex 1 and vertex n. This is useful for problems where you must find the least expensive or shortest path. Find Edges in Shortest Paths - You are given an undirected weighted graph of n nodes numbered from 0 to n - 1. Dijkstra’s algorithm, published in 1959 and named after its creator Dutch computer scientist Edsger Dijkstra, can be applied on a weighted graph. Jul 26, 2005 · Abstract. In an efficient preprocessing phase our al-gorithm creates a linear-size structure that facilitates single-source shortest path computations in O(mlog α) time, where α = α(m,n) is the very slowly growing inverse-Ackermann function, m the number of Jul 15, 2025 · Given an undirected and unweighted graph and two nodes as source and destination, the task is to print all the paths of the shortest length between the given source and destination. Here, the length of a path is simply the number of edges on the path. I wrote some code modifying BFS to find the shortest path to all nodes in a given graph. Consider all the shortest paths from node 0 to node n - 1 in the graph. In an efficient preprocessing phase our al-gorithm creates a linear-size structure that facilitates single-source shortest path computations in O(mlog α) time, where α = α(m,n) is the very slowly growing inverse-Ackermann function, m the number of I have to find an algorithm that finds the SSSP (single-source shortest path - shortest paths from one source vertex to all other vertices) on a weighted undirected graph. I really hope you liked my article and found it helpful. There are implementations for both adjacency list & adjacency matrix graph representations (note that for adjacency matrix, instead of using a boolean matrix we use an integer matrix. In this problem statement, we have assumed the source vertex to be 0. Nov 25, 2024 · Weighted graphs: Each edge has a weight or cost. Shortest-paths problem come in several flavors. Jun 4, 2024 · Graph Data Structure | Graph Theory | Graph in DSA #dsa #graph #datastructure What is Graphs in DSA and why do we need it. Anything non 0 represents the weight of the edge.