bellman ford algorithm bellman ford algorithm

This means that, given a weighted graph, this algorithm will output the shortest distance from a selected node to all other nodes. v | Single source shortest path with negative weight edges. If there is such a cycle, the algorithm indicates that no solution exists. bellman_ford length, nodes, negative_cycle = bellman_ford (G, source, target, weight = 'weight') Compute shortest path and shortest path lengths between a source node and target node in weighted graphs using the Bellman-Ford algorithm. Let us now prove the following assertion: After the execution of $i_{th}$ phase, the Bellman-Ford algorithm correctly finds all shortest paths whose number of edges does not exceed $i$. j Consider a scenario, in which each edge has a negative edge weight, we can apply the Bellman-Ford algorithm. [6] Bannister, M. J.; Eppstein, D. Randomized speedup of the Bellman-Ford algorithm. {\displaystyle O(|V||E|)} Create an array dist [] of size |V| with all values as infinite except dist [s]. Moving on to understanding this algorithm more. Meyer and Sanders [ 48] show that a value of = (1/ d . Hence we will get the vertex $y$, namely the vertex in the cycle earliest reachable from source. ta cn chy n bc th n (ngha l i qua ti a n+1 nh). If a shorter path is still found, this means that there is a negative weight cycle in the graph. j In the loop, for each edge, we take the value of the vertex from where the edge is starting (D[U]) and add it to the edge cost. The graph can contain negative-weight edges, but it should not contain a negative-weight cycle that is reachable from the source vertex. So, the Bellman-Ford algorithm does not work for graphs that contains a negative weight cycle. Finally, it checks for negative cycles. | When expanded it provides a list of search options that will switch the search inputs to match the current selection. All the vertices are numbered $0$ to $n - 1$. For more on this topic see separate article, Finding a negative cycle in the graph. The distance to vertex A is updated to -5 units. Mt bin th phn tn ca thut ton Bellman-Ford c dng trong cc giao thc nh tuyn vector khong cch, chng hn giao thc RIP (Routing Information Protocol). Analytics Vidhya is a community of Analytics and Data Science professionals. Im sure Richard Bellman and Lester Ford Jr would be proud of you, just sleeping and smiling in their graves. The limitation of the algorithm is that there should not be negative cycles (a cycle whose sum of edges produces a negative value) in the graph. Begin create a status list to hold the current status of the selected node for all . An algorithm for finding shortest routes from all source nodes to a given destination in general networks. We define a. The distance to B is 9, so the distance to vertex F is 9 + (-5) = 4. package Combinatorica` . We are building the next-gen data science ecosystem https://www.analyticsvidhya.com. As we have already reached an optimized value already, so if we can relax an edge again that means we have encountered a negative cycle. Consider the edge (D, F). We run the same loop again, taking edges and relaxing them. The shortest path problem is about finding a path between $$2$$ vertices in a graph such that the total sum of the edges weights is minimum. Though discovering the algorithm after Ford he is referred to in the Bellman-Ford algorithm, also sometimes referred to as the Label Correcting Algorithm, computes single-source shortest paths in a weighted digraph where some of the edge weights may be negative. 1. O 1 A negative weight is just like a positive weight, a value on the top of an edge. Edge H-D can be relaxed since we know the distance to vertex H is -1. The `Graph` struct is defined to represent a connected, directed graph. 1 The algorithm often used for detecting negative cycles in a directed graph. The next edge is (1, 2). | {\displaystyle |V|-1} Although each edge is relaxed, the only edges that matter are the edges from S and from A since the distance to those vertices is already known. the penultimate vertex in the shortest path leading to it. This is done by relaxing all the edges in the graph for n-1 times, where n is the number of vertices in the graph. The above graph contains 6 vertices so we will go on relaxing till the 5 vertices. Ta s i tm ng i ngn nht t node 1 n cc node cn li . 2 Dijkstra's Correctness In the previous lecture, we introduced Dijkstra's algorithm, which, given a positive-weighted graph G = In fact, the shortest paths algorithms like Dijkstra's algorithm or Bellman-Ford algorithm give us a relaxing order. Do leave some feedback, I am really looking forward to it. Hence in the code, we adopted additional measures against the integer overflow as follows: The above implementation looks for a negative cycle reachable from some starting vertex $v$; however, the algorithm can be modified to just looking for any negative cycle in the graph. However be careful, because this algorithm is deterministic and it is easy to create counterexamples that make the algorithm run in $O(n m)$. Dijkstras cant work on this problem then. Now the first iteration is completed. If the loop is iterated more than 5 times then also the answer will be the same, i.e., there would be no change in the distance between the vertices. We have to go from this vertex, through the predecessors, until we get back to the same vertex $y$ (and it will happen, because relaxation in a negative weight cycle occur in a circular manner). The Bellman-Ford algorithm|V-1| times relaxes every edge of the graph, hence the time complexity of the algorithm is O (VE). Do , trng_s(v, u) + khong_cch(v) c gi tr khng vt qu di ca ng i t s ti u. Trong ln lp th i, khong_cch(u) c ly gi tr nh nht ca khong_cch(v) + trng_s(v, u) vi mi v c th. To avoid this, it is possible to create a counter that stores how many times a vertex has been relaxed and stop the algorithm as soon as some vertex got relaxed for the $n$-th time. If the graph contains negative -weight cycle . Bellman-Ford algorithm is a single source shortest path algorithm that finds the shortest path from the source vertex to all other vertices in a given weighted graph. Bellman-Ford algorithm finds all shortest path lengths from a source s V to all v V or determines that a negative weight cycle exists. The Bellman-Ford algorithm is an extension of Dijkstra's algorithm which calculates the briefest separation from the source highlight the entirety of the vertices. {\displaystyle |V|-1} ) In the above graph, we consider vertex 1 as the source vertex and provides 0 value to it. | Your task is to complete the function bellman_ford( ) which takes a number of vertices V and an E-sized list of lists of three integers where the three integers are u,v, and w; denoting there's an edge from u to v, which has a weight of w and source node S as input parameters and returns a list of integers where the ith integer denotes the . Author of An Illustrative Introduction to Algorithms. Since (9 - 15) equals to -6 which is less than -5 so update: Since the graph contains 4 vertices, so according to the bellman ford algorithm, there would be only 3 iterations. The predecessor of C is A. ] ( He has over a decade of software engineering experience. Divide & Conquer Method vs Dynamic Programming, How to solve a dynamic programming problem, Dynamic Programming vs Divide and Conquer, Traveling Salesperson problem using branch and bound, Single Source Shortest Path in a directed Acyclic Graphs. The Bellman-Ford algorithm is a graph search algorithm that finds the shortest path between a given source vertex and all other vertices in the graph. Since (0 + 5) equals to 5 which is greater than -4 so there would be no updation in the vertex 3. If we examine another iteration, there should be no changes. : In this section, we will understand the Bellman-Ford algorithm with example and also implement the Bellman ford algorithm in a Java program. Accordingly, Dijkstra's algorithm has more applications, since charts with negative loads are typically viewed as an uncommon case. While Dijkstra's algorithm simply works for edges with positive distances, Bellman Ford's algorithm works for negative distances also. Bellman This Applet demonstrates the Bellman-Ford Algorithm. The only input graph that Bellman-Ford algorithm has issue is the input graph with negative weight cycle reachable from the source vertex s. However, Bellman-Ford can be used to detect if the input graph contains at least one negative weight cycle reachable from the source vertex s by using the corollary of Theorem 2: . Table 1 shows Bellman -Ford algorithm [2] [3], whose input is a given graph G = (V, E), the edge weight setting cost, number of nodes n and the single source node v. The dist [u] to store the . A web tool to build, edit and analyze graphs. We then relax the edges numVertices 1 times. [ Do , cu trc d liu lu cng cn lu khi khai bo. tree algorithms graph data-structures topological-sort dag dijkstra-algorithm strongly-connected-components eulerian-path adjacency-matrix bellman-ford-algorithm graphtheory adjacency-list bridges articulation-point. The router is used to find the optimal . To get the vertices that are guaranteed to lie in a negative cycle, starting from the vertex $x$, pass through to the predecessors $n$ times. But if optimal time is not the highest priority then no doubt Bellman Ford is a better shortest path algorithm. There might be a negative-weight cycle that is reachable from the source. You know the source and need to reach all the other vertices through the shortest path. Now use the relaxing formula: Therefore, the distance of vertex D is 5. " ()" is published by Yi-Ning. Where |V| is number of vertices. The distance to vertex B is 0 + 6 = 6. From vertex E, we can move to vertex D only. {\displaystyle \Pi (k,i)=\min(\{\Pi (k-1,i)\}\cup \{\Pi (k-1,j)+L[j][i]\})}. Order of edges: (B, E), (D, B), (B, D), (A, B), (A, C), (D, C), (B, C), (E, D). Mi nt gi bng thng tin ca mnh cho tt c cc nt ln cn. There are various other algorithms used to find the shortest path like Dijkstra algorithm, etc. Distance is represented by the variable d and the predecessor is represented by the variable . ] To begin, all the outbound edges are recorded in a table in alphabetical order. Following is an implementation of the Bellman-Ford with the retrieval of shortest path to a given node $t$: Here starting from the vertex $t$, we go through the predecessors till we reach starting vertex with no predecessor, and store all the vertices in the path in the list $\rm path$. Copyright 2011-2021 www.javatpoint.com. The last edge, S-A, yields a different result. Gii bi ton c th. * CSES - Cycle Finding, Bellman-Ford - finding shortest paths with negative weights, Euclidean algorithm for computing the greatest common divisor, Deleting from a data structure in O(T(n) log n), Dynamic Programming on Broken Profile. , (Cycle Cancellation Algorithms), - Similarly, from A to E, the cost is 2, however, since the distance to A is infinity, the value of E remains infinity. Coding, Tutorials, News, UX, UI and much more related to development. | Read every story from Dino Cajic (and thousands of other writers on Medium). pp. The main difference between this algorithm with Dijkstra's the algorithm is, in Dijkstra's algorithm we cannot handle the negative weight, but here we can handle it easily. The Bellman-Ford algorithm is an algorithm that computes shortest paths from a single source vertex to all of the other vertices in a weighted digraph. Repeat the following |V| - 1 times. Summary: In this tutorial, well learn what the Bellman-Ford algorithm is, how it works, and how to find the cost of the path from the source vertex to all other vertices in a given graph using the algorithm in C++, Java, and Python. Denote vertex '3' as 'u' and vertex '2' as 'v'. A Bellman-Ford-algoritmus egy algoritmus, amely kiszmtja a legrvidebb utat egyetlen forrstl (vertex) az sszes tbbi cscshoz egy slyozott digrfban. There are various other algorithms used to find the shortest path like Dijkstra algorithm, etc. Fill in the following table with the intermediate distance values of all the nodes at the end of . It is slower than Dijkstra's algorithm, but more versatile, as it is capable of handling graphs in which some of the edge weights are negative numbers. O Looking at edges B-F, C-B, C-H, F-G, G-B, and H-D, we can see that they all yield the same result, infinity. Suppose that we are given a weighted directed graph $G$ with $n$ vertices and $m$ edges, and some specified vertex $v$. In such a case the algorithm will be terminated. D. From vertex D, we can move to vertex B and C. Calculate the distance from vertex D to other vertices. He has a B.S. Now use the relaxing formula: Since (4 + 7) equals to 11 which is less than , so update. It is claimed that $n-1$ phases of the algorithm are sufficient to correctly calculate the lengths of all shortest paths in the graph (again, we believe that the cycles of negative weight do not exist). The check if (d[e[j].a] < INF) is needed only if the graph contains negative weight edges: no such verification would result in relaxation from the vertices to which paths have not yet found, and incorrect distance, of the type $\infty - 1$, $\infty - 2$ etc. V Vertex Bs predecessor is updated to vertex A. in Computer Science, a minor in Biology, and a passion for learning. If this graph had a negative cycle, after the iteration is repeated n-1 times, theoretically the Bellman-Ford algorithm should have found the shortest paths to all vertices. For solving such problems, there is no polynomial-time algorithm exists. 1 E Since vertex B can be reached with a shorter distance by going through edge C-B, the table remains the same. Using vertex. [1][], Initialize the distance from the source to all vertices as infinite. The program starts by including the necessary libraries for the program to function. i This problem could be solved easily using (BFS) if all edge weights were ($$1$$), but here weights can take any value. The worst case of this algorithm is equal to the $O(n m)$ of the Bellman-Ford, but in practice it works much faster and some people claim that it works even in $O(m)$ on average. Hence we obtain the criterion for presence of a cycle of negative weights reachable for source vertex $v$: after $(n-1)_{th}$ phase, if we run algorithm for one more phase, and it performs at least one more relaxation, then the graph contains a negative weight cycle that is reachable from $v$; otherwise, such a cycle does not exist. The minimum time it takes for all nodes to receive the signal is 2. A free video tutorial from Loony Corn. This algorithm was named after its inventors. Edge B-F can now be relaxed. The Bellman-Ford algorithm will iterate through each of the edges. | Denote vertex 'C' as 'u' and vertex 'E' as 'v'. It repetitively loops over all the edges and updates the distances at the start node, the same as in Dijkstra's algorithm. n The distance to C is 5 + (-10) = -5. Now another point of optimization to notice carefully. During the first phase, the edge $(p_0,p_1)$ has been checked by the algorithm, and therefore, the distance to the vertex $p_1$ was correctly calculated after the first phase. In the beginning we fill it as follows: $d[v] = 0$, and all other elements $d[ ]$ equal to infinity $\infty$. {\displaystyle D:{\texttt {Dist}}[v],P:{\texttt {Pred}}[v]}, https://zh.wikipedia.org/w/index.php?title=-&oldid=71758509. We iterate through all the edges and update the distances if a shorter path is found. Therefore, the distance of vertex 4 is 11. The next edge is (1, 2). Denote vertex '2' as 'u' and vertex '4' as 'v'. In simpler terms, let V be the number of vertices, E be the number of edges, S be the starting node, and D be an array which tracks the best distance between the source node and rest vertices. And then it starts relaxing the estimates by discovering the new paths which are shorter than the previous ones. , trong V l s nh v E l s cung ca th. This is something that even the Bellman ford algorithm cant defeat. The runtime complexity of the algorithm is O(v*e) and space complexity is O(v). Since (1 - 1) equals to 0 which is less than 5 so update: The next edge is (C, E). The bellman ford algorithm does not produce a correct answer if the sum of the edges of a cycle is negative. If the distance varies, it means that the bellman ford algorithm is not providing the correct answer. In this tutorial, we learned what the Bellman-Ford algorithm is, how it works, and how to implement Bellman-Ford algorithm in C++, Java, and Python to find the cost of the path. The first point to know about the algorithm would be that is doesnt work on a greedy algorithm like Dijkstra. Consider the edge (A, C). So that is how the step of relaxation works. So it's necessary to identify these cycles. The next edge is (3, 2). {\displaystyle |V|-1} After that, we will traverse towards each vertex from the source node. Therefore, if you do not limit the number of phases to $n - 1$, the algorithm will run indefinitely, constantly improving the distance from these vertices. By varying in the range , we get a spectrum of algorithms with varying degrees of processing time and parallelism. Consider the edge (A, D). Now, why does our algorithm fail in front of negative cycles? Output The shortest paths from start to all other vertices. But then what about the gloomy part? The next edge is (1, 2). Since (10 - 15) equals to -5 which is less than -4 so update: Now again we will check all the edges. Share. Modify it so that it reports minimum distances even if there is a negative weight cycle. -, - From vertex C we cannot move to any vertex, so we will visit the next vertex i.e. Bellman-Ford algorithm can also work with a non-negative undirected graph, but it can only handle negative edges in a directed graph. Can we use Dijkstra's algorithm for shortest paths for graphs with negative weights - one idea can be, to calculate the minimum weight value, add . Follow. b) Integer. As soon as that happens, the IF condition becomes true and the return statement is executed, ending the function else the array D is printed. JavaTpoint offers too many high quality services. Trang ny c sa ln cui vo ngy 6 thng 4 nm 2022, 15:57. It deals with the negative edge weights. i For n vertices, we relax the edges for n-1 times where n is the number of edges. Java. Do , khong_cch(u) + trng_s(u, v) l di ca ng i t ngun ti u ri ti v. Chng minh cu 2: Xt ng i ngn nht t ngun ti u qua ti a i cung. For this, it is sufficient to remember the last vertex $x$ for which there was a relaxation in $n_{th}$ phase. 20 is a reduced value from the earlier 25. Another difference is that the Dijkstra algorithm looks only to the immediate neighbors of a vertex, Bellman-Ford goes through each edge in every iteration. In Step 3, we check for negative-weight cycles by iterating through all the edges again and seeing if we can still find a shorter path. Now use the relaxing formula: Since (4 + 3) is greater than 5, so there will be no updation. d) Double. Bellman-Ford algorithm is used to find minimum distance from the source vertex to any other vertex. It will always keep finding a more optimized, that is, a more negative value than before. Xt thi im khi khong cch ti mt nh c cp nht bi cng thc AFAICS from the data I've seen during testing, those "inefficiencies" come from the fact that exchange rates are more volatile over course of minutes than the Bid-Ask spread. The graph may contain negative weight edges. {\displaystyle n} The Bellman-Ford algorithm helps us find the shortest path from a vertex to all other vertices of a weighted graph. Edge B-C can be reached in 6 + 2 = 8. Dist This algorithm can be used on both weighted and unweighted graphs. Enjoy! {\displaystyle |V|-1} Conclusion. You want to find the length of shortest paths from vertex $v$ to every other vertex. Let us assume that the graph contains no negative weight cycle. Since (0 + 4) equals to 4 so there would be no updation in the vertex 2. The algorithm is implemented as BellmanFord[g, v] in the Wolfram Language package Combinatorica` . Bellman-Ford algorithm can also work with a non-negative undirected graph, but it can only handle negative edges in a directed graph. From the "Maximum Number of Iterations" section, we already know that the algorithm runs through n-1 iterations, where n is the number of nodes. would appear. Tnh ng n ca thut ton c th c chng minh bng quy np. ) Make way for negative cycles. Continuing in the loop, the edge 4 9 makes the value of 9 as 200. From vertex B, we can move to vertex C, D and E. Calculate the distance from B to other vertices, we get. How Bellman Ford Algorithm works? Though it is slower than Dijkstra's algorithm, Bellman . Its because Bellman ford Relaxes all the edges. It is slower than Dijkstra's algorithm for the same problem, but more versatile, as it is capable of handling graphs in which some of the edge weights are negative numbers. The limitation of the algorithm is that it cannot be applied if the graph has negative edge weights. -, -, | A list of tasks that can be solved using the Bellman-Ford algorithm: See also the problem list in the article Finding the negative cycle in a graph. During the third iteration, the Bellman-Ford algorithm examines all the edges again. Mathematics is a way of dealing with tasks that require e#xact and precise solutions. The algorithm bears the name of two American scientists: Richard Bellman and Lester Ford. {\displaystyle n} Lets look at a quick example. The distance to S is 0, so the distance to A is 0 + 3 = 3. Edges S-A and S-B yield nothing better, so the second iteration is complete. It is simple to understand and easy to implement. V JavaTpoint offers college campus training on Core Java, Advance Java, .Net, Android, Hadoop, PHP, Web Technology and Python. Thut ton c th c pht biu chnh xc theo kiu quy np nh sau: Trng hp c bn: Xt i = 0 v thi im trc khi vng for c chy ln u tin. The Bellmann Ford algorithm returns _______ value. , | Since (5 - 2) equals to 3 so there would be no updation in the vertex C. The next edge is (D, F). In other words, for any vertex $a$ let us denote the $k$ number of edges in the shortest path to it (if there are several such paths, you can take any). If we examine the graph closely, we can see that A-B-C yields a negative value: 5 + 2 + (-10) = -3. The distance to A is 3, so the distance to vertex B is 3 + 5 = 8. The algorithm sees that there are no changes, so the algorithm ends on the fourth iteration. Since (0 +5) equals to 5 which is greater than -6 so there would be no change in the vertex 3. The `BellmanFord` function implements the Bellman-Ford algorithm to find the shortest path from source to all other vertices in the graph. It is a single-source shortest path (minimum weight) algorithm very similar to Dijkstra's algorithm. The Bellman-Ford algorithm will iterate through each of the edges. in Computer Science and a minor in Biology. {\displaystyle |E|} E The working of the Bellman-Ford algorithm is the same as Dijkstra's algorithm. The Bellman-Ford Algorithm works by repeatedly relaxing each edge in the graph, updating the estimated shortest path between the source vertex and all other vertices. The predecessor of G is F. Edge G-B can now be relaxed. Thut ton Dijkstra gii cng bi ton ny tuy nhin Dijkstra c thi gian chy nhanh hn, n gin l i hi trng s ca cc cung phi c gi tr khng m. - By doing this repeatedly for all vertices, we can guarantee that the . [ The next edge is (3, 2). This process is repeated at most (V-1) times, where V is the number of vertices in the graph. The distance to all other vertices is infinity. Now, infinite levels are too high for us, stress is building up. It is used in situations where a source vertex is selected and the shortest paths to every other vertex in the graph need to be determined. V V According to this statement, the algorithm guarantees that after $k_{th}$ phase the shortest path for vertex $a$ will be found. Get Solution. The Bellman-Ford Algorithm has Bellman-Ford algorithm finds shortest path from the source vertex to all vertices in the graph. between two given vertices. - Bc 0: Ta nh du nh xut pht = 0, cc inh cn li bng v cc. The Bellman-Ford Algorithm has many applications in computer science and beyond. In this image, the vertices B, C, and D form a cycle where the starting node is B which is also the ending node. At this time, all shortest paths should have been found. i If any edge can be relaxed, then it means the given graph has a negative cycle. The next edge is (4, 3). IT Leader with a B.S. Bellman ford algorithm is a single-source shortest path algorithm. Now coming to your original question, yes Bellman Ford algorithm can relax the edges in any arbitrary order as nicely answered by @ead above. The Bellman-Ford algorithm is an algorithm similar to Dijkstra that is it finds the shortest path in a graph from a single source vertex to all other vertices in a weighted graph but it works even . Use the convention that edges (u,v) are relaxed in lexicographic order, sorting first by u then by v . (Bellman Ford Algorithm) Bangla tutorial , Single source shortest path, For that, let's create another array $p[0 \ldots n-1]$, where for each vertex we store its "predecessor", i.e. Other algorithms that can be used for this purpose include The first edge is (1, 3). ( It first calculates the shortest distances which have at-most one edge in the path. Edges A-C and A-E yield the same results. , Consider the below graph. A cycle is a path where the first and the last vertex is the same, that is, it is a closed path. Okay? It can be used to detect negative cycles in a graph. D all the vertices of the graph), and any simple path with a V number of vertices cannot have more than V-1 edges. The distance to vertex F is 4, so the distance to vertex G is 4 + 2 = 6. {\displaystyle O(|V|\cdot |E|)} Improve this answer. V The Bellman-Ford Algorithm is a single-source shortest-path algorithm that can find the shortest path between a source vertex and all other vertices in a weighted graph. In the second iteration, we again check all the edges. ] To change consent settings at any time please visit our privacy policy using the link below.. Nu tn ti chu trnh m m t nh ngun c th i n c th s khng tn ti ng i nh nht (v mi ln i quanh chu trnh m l mt ln gim trng s ca ng). Other algorithms that can be used for this purpose include Dijkstra's algorithm and reaching algorithm. The problem with Dijkstra's Algorithm is, if . How Bellman Ford's algorithm works. Proof. Edge G-B cannot be relaxed. I hope you guys liked this blog. Ti nh A c nh B i vo c chi ph hin ti (2) < chi ph trc () => cp nht li chi ph nh A, Ti nh C c nh B i vo c chi ph hin ti (6) < chi ph trc () => cp nht li chi ph nh C, Ti nh C c nh A i vo c chi ph hin ti (5) < chi ph trc (6) => cp nht li chi ph nh C, Ti nh D c nh C i vo c chi ph hin ti (8) < chi ph trc () => cp nht li chi ph nh D, Ti nh D c nh A i vo c chi ph hin ti (7) < chi ph trc (8) => cp nht li chi ph nh D, C ng i ngn nht t B->D: B->A->C->D, Nu bc 4 khng ging bc 3 => kt lun khng c ng i ngn nht t B->D. {\displaystyle |V|} The first edge is (1, 3). Bellman-Ford algorithm is a well-known solution to "the single-source shortest path (SSSP)" problem. Here are some examples: Feel Free to Ask Queries via LinkedIn and to Buy me Coffee : ), Security Researcher | Bug Hunter | Web Pentester | CTF Player | TryHackme Top 1% | AI Researcher | Blockchain Developer | Writeups https://0dayinventions.tech. Let us now consider how to modify the algorithm so that it not only finds the length of shortest paths, but also allows to reconstruct the shortest paths. V The current distance to S is 0, so the distance from S to A is 0 + 5 = 5. | Thut ton BellmanFord l mt thut ton tnh cc ng i ngn nht ngun n trong mt th c hng c trng s (trong mt s cung c th c trng s m). It is s. Starting the loop, the first edge we take is 0 1, after which 1 is assigned the value 5. Now use the relaxing formula: Therefore, the distance of vertex C is 4. The table with the distances and the predecessors is constructed. After that, it is guaranteed that no relaxation will improve the distance to some vertex. Deal with mathematic questions. It can be used to find the shortest path between two cities on a road network with variable traffic conditions.