ellman-Ford is in its basic structure very similar to Dijkstra's algorithm, but instead of greedily selecting the minimum-weight node not yet processed to relax, it simply relaxes all the edges, and does this |V| − 1 times, where |V| is the number of vertices in the graph. The repetitions allow minimum distances to accurately propagate throughout the graph, since, in the absence of negative cycles, the shortest path can only visit each node at most once. Unlike the greedy approach, which depends on certain structural assumptions derived from positive weights, this straightforward approach extends to the general case.
Bellman-Ford runs in O(|V|·|E|) time, where |V| and |E| are the number of vertices and edges respectively. procedure BellmanFord(list vertices, list edges, vertex source) // This implementation takes in a graph, represented as lists of vertices // and edges, and modifies the vertices so that their distance and // predecessor attributes store the shortest paths. // Step 1: Initialize graph for eachvertex v in vertices: if v is source then v.distance := 0 else v.distance := infinity v.predecessor := null // Step 2: relax edges repeatedly for i from 1 tosize(vertices)-1: for each edge uv in edges: // uv is the edge from u to v u := uv.source v := uv.destination if u.distance + uv.weight < v.distance: v.distance := u.distance + uv.weight v.predecessor := u // Step 3: check for negative-weight cycles for each edge uv in edges: u := uv.source v := uv.destination if u.distance + uv.weight < v.distance: error "Graph contains a negative-weight cycle"
Proof of correctnessThe correctness of the algorithm can be shown by induction. The precise statement shown by induction is:Lemma. After i repetitions of forcycle:
Proof. For the base case of induction, consider i=0 and the moment before for cycle is executed for the first time. Then, for the source vertex, source.distance = 0, which is correct. For other vertices u, u.distance = infinity, which is also correct because there is no path from source to u with 0 edges.
For the inductive case, we first prove the first part. Consider a moment when a vertex's distance is updated by v.distance := u.distance + uv.weight. By inductive assumption, u.distance is the length of some path from source to u. Then u.distance + uv.weight is the length of the path from source to v that follows the path from source to u and then goes to v.
For the second part, consider the shortest path from source to u with at most i edges. Let vbe the last vertex before u on this path. Then, the part of the path from source to v is the shortest path from source to v with at most i-1 edges. By inductive assumption, v.distanceafter i-1 cycles is at most the length of this path. Therefore, uv.weight + v.distance is at most the length of the path from s to u. In the ith cycle, u.distance gets compared with uv.weight + v.distance, and is set equal to it if uv.weight + v.distance was smaller. Therefore, after i cycles, u.distance is at most the length of the shortest path from source to u that uses at most i edges.
If there are no negative-weight cycles, then every shortest path visits each vertex at most once, so at step 3 no further improvements can be made. Conversely, suppose no improvement can be made. Then for any cycle with vertices v[0],..,v[k-1],
v[i].distance <= v[i-1 (mod k)].distance + v[i-1 (mod k)]v[i].weight
Summing around the cycle, the v[i].distance terms and the v[i-1 (mod k)] distance terms cancel, leaving
0 <= sum from 1 to k of v[i-1 (mod k)]v[i].weight
I.e., every cycle has nonnegative weight.
Applications in routingA distributed variant of the Bellman-Ford algorithm is used in distance-vector routing protocols, for example the Routing Information Protocol (RIP). The algorithm is distributed because it involves a number of nodes (routers) within an Autonomous system, a collection of IP networks typically owned by an ISP. It consists of the following steps:The main disadvantages of the Bellman-Ford algorithm in this setting are
The Bellman-Ford algorithm computes single-source shortest paths in a weighted digraph.For graphs with only non-negative edge weights, the faster Dijkstra's algorithm also solves the problem. Thus, Bellman-Ford is used primarily for graphs with negative edge weights. The algorithm is named after its developers, Richard Bellman and Lester Ford, Jr.
An algorithm.
The program itself is the solution. All programs are a solution to a given problem; that's the entire point of writing a program, to solve a problem. The program's algorithm specifies how the problem is solved and it's the programmer's job to convert that algorithm into working code.
i put true
P is the class of problems for which there is a deterministic polynomial time algorithm which computes a solution to the problem. NP is the class of problems where there is a nondeterministic algorithm which computes a solution to the problem, but no known deterministic polynomial time solution
algorithm
No, not every problem has a clear-cut solution. Some problems are complex, with multiple factors and unknown variables, making it difficult to find a definitive answer. In such cases, solutions may require compromise, trade-offs, or continued efforts to address the issue effectively.
Algorithm
Strange as it may seem, we don't actually use algorithms to solve problems; an algorithm is the end-product of problem-solving. In short, every problem that has a solution already has an algorithm. Moreover, every problem that is known to have no solution has a proof to demonstrate that fact. But problems that have yet to be solved have no known algorithm or proof -- and that's precisely why they remain unsolved (for now).
An intractable problem is one for which there is an algorithm that produces a solution - but the algorithm does not produce results in a reasonable amount of time. Intractable problems have a large time complexity. The Travelling Salesman Problem is an example of an intractable problem.
I think you mean Algorithm....which is a logical arithmetical procedure, that if correctly applied, ensures the solution of a problem.
1 Define the problem 2 Analyze the problem 3 Develop an algorithm/method of solution 4 Write a computer program corresponding to the algorithm 5 Test and debug the program 6 Document the program (how it works and how to use it)