Bellmanford
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The document discusses hierarchical routing in computer networks. It describes how routing is divided into intra-autonomous system (intra-AS) routing within an AS and inter-autonomous system (inter-AS) routing between ASes. Special routers called gateways perform both intra-AS routing with other routers in their AS and inter-AS routing with gateways in other ASes. This hierarchical structure allows scaling to large networks with many destinations by aggregating routers into regions.
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This document discusses algorithms for finding shortest paths in weighted graphs:
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The Bellman-Ford routing algorithm computes the shortest paths from a single source vertex to all other vertices in a weighted digraph. It is also known as the distance vector algorithm. The algorithm uses the principle of relaxation to iteratively update the cost of the shortest paths by relaxing edges until it either finds the shortest paths or detects a negative cost cycle. It runs in O(|V||E|) time in the worst case. Each node maintains a distance vector with the estimated cost to all other nodes and shares this information with neighbors to iteratively update the costs until reaching an optimal solution.
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The document discusses and compares different routing algorithms:
1. It classifies routing protocols as either link state or distance vector, and describes how each type works. Link state algorithms use flooding to share link information, while distance vector algorithms iteratively calculate the best paths.
2. It explains that distance vector algorithms can have issues with counting to infinity and routing loops. Techniques like poison reverse are used to address this.
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- Bellman-Ford algorithm can handle graphs with negative edge weights by relaxing all edges V-1 times to detect negative cycles. It runs in O(VE) time.
- Examples are provided to illustrate the relaxation process and execution of both algorithms on sample graphs. Key properties like handling of negative weights and cycles are also explained.
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The document discusses and compares different routing algorithms:
1. It classifies routing protocols as either link state or distance vector, and describes how each type works. Link state algorithms use flooding to share link information, while distance vector algorithms iteratively calculate the best paths.
2. It explains that distance vector algorithms can have issues with counting to infinity and routing loops. Techniques like poison reverse are used to address this.
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Dijkstra's algorithm is a link-state routing algorithm that computes the least-cost paths from one node to all other nodes in a network. It works by iteratively building a set N' of nodes whose least-cost paths are known, starting from a source node. In each iteration, it finds the node w not in N' with the minimum path cost and updates the cost estimates to nodes adjacent to w. This process repeats until all nodes are in N', providing the shortest path forwarding table for the source node. The algorithm runs in O(n^2) time for n nodes but more efficient implementations exist. Oscillations can potentially occur if link costs change based on traffic loads.
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Bellman Ford's algorithm finds the shortest paths from a source vertex to all other vertices in a weighted graph, even if edge weights are negative. It works by repeatedly relaxing all edges to update the distance and previous vertex for each vertex. After iterating through all edges |V|-1 times, if an edge can still be relaxed, then a negative cycle exists in the graph.
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Dijkstra's algorithm is used to find the shortest path between a source node and all other nodes in a graph. It works by maintaining two sets - S of nodes whose shortest paths have been determined, and V-S of remaining nodes. It initializes distances and predecessors, then iteratively relaxes edges of the closest node in V-S until V-S is empty, determining the shortest path to each node. The algorithm runs in O(|E| + |V|)log|V|) time.
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Prim's Algorithm on minimum spanning tree
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The document discusses several shortest path algorithms for graphs, including Dijkstra's algorithm, Bellman-Ford algorithm, and Floyd-Warshall algorithm. Dijkstra's algorithm finds the shortest path from a single source node to all other nodes in a graph with non-negative edge weights. Bellman-Ford can handle graphs with negative edge weights but is slower. Floyd-Warshall can find shortest paths in a graph between all pairs of nodes.
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The Floyd-Warshall algorithm is used to find the shortest paths between all pairs of vertices in a weighted graph. It creates distance and sequence tables with rows and columns for each vertex. The distance table stores the weights of edges between vertices. The sequence table stores the path vertices to reach between each pair. It iterates through the tables, updating distances and paths at each iteration based on whether a new shortest path is found through a third vertex. When complete, the final distance table shows the shortest distance between each pair of vertices, and the sequence table shows the shortest path.
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Dijkstra's algorithm finds the shortest paths from a source node to all other nodes in a graph. It works by maintaining two sets - one for nodes whose shortest paths are determined, and one for remaining nodes. It iteratively selects the node with the smallest distance from the source, calculates the distances to its neighbors, and moves them to the determined set until all nodes are processed. Some applications include finding fastest routes in transportation networks like road maps or flight schedules.
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The document describes Dijkstra's algorithm for finding the shortest paths between nodes in a graph. It begins with an overview of how the algorithm works by solving subproblems to find the shortest path from the source to increasingly more nodes. It then provides pseudocode for the algorithm and analyzes its time complexity of O(E+V log V) when using a Fibonacci heap. Some applications of the algorithm include traffic information systems and routing protocols like OSPF.
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Dijkstra's algorithm is a link-state routing algorithm that computes the least-cost paths from one node to all other nodes in a network. It works by iteratively building a set N' of nodes whose least-cost paths are known, starting from a source node. In each iteration, it finds the node w not in N' with the minimum path cost and updates the cost estimates to nodes adjacent to w. This process repeats until all nodes are in N', providing the shortest path forwarding table for the source node. The algorithm runs in O(n^2) time for n nodes but more efficient implementations exist. Oscillations can potentially occur if link costs change based on traffic loads.
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Bellman Ford's algorithm finds the shortest paths from a source vertex to all other vertices in a weighted graph, even if edge weights are negative. It works by repeatedly relaxing all edges to update the distance and previous vertex for each vertex. After iterating through all edges |V|-1 times, if an edge can still be relaxed, then a negative cycle exists in the graph.
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This document compares the parallel implementations of Dijkstra's and Bellman Ford algorithms for solving the single source shortest path problem in graphs. It discusses distributing graph nodes across processors, using parallel prefix to find the shortest path, and broadcasting results. Performance analyses show Dijkstra's and Bellman Ford have near-linear speedups and better scaling to larger graphs in parallel compared to sequential implementations.
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This document describes Floyd's algorithm for solving the all-pairs shortest path problem in graphs. It begins with an introduction and problem statement. It then describes Dijkstra's algorithm as a greedy method for finding single-source shortest paths. It discusses graph representations and traversal methods. Finally, it provides pseudocode and analysis for Floyd's dynamic programming algorithm, which finds shortest paths between all pairs of vertices in O(n3) time.
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Dijkstra's algorithm is used to find the shortest path between a source node and all other nodes in a graph. It works by maintaining two sets - S of nodes whose shortest paths have been determined, and V-S of remaining nodes. It initializes distances and predecessors, then iteratively relaxes edges of the closest node in V-S until V-S is empty, determining the shortest path to each node. The algorithm runs in O(|E| + |V|)log|V|) time.
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This document describes the Bellman-Ford algorithm, a single-source shortest path algorithm that can detect negative edge cycles. It provides pseudocode for the algorithm, runs through an example, discusses the relaxation equation and time/space complexity. It also lists some applications of Bellman-Ford in routing protocols and mentions some references for more information.
Dijkstra s algorithm
Dijkstra's algorithm is a solution to the single-source shortest path problem in graph theory. It finds the shortest paths from a source vertex to all other vertices in a weighted graph where all edge weights are non-negative. The algorithm uses a greedy approach, maintaining a set of vertices whose final shortest path from the source vertex has already been determined.
Prim's Algorithm on minimum spanning tree
Prim's algorithm is used to find the minimum spanning tree of a connected, undirected graph. It works by continuously adding edges to a growing tree that connects vertices. The algorithm maintains two lists - a closed list of vertices already included in the minimum spanning tree, and a priority queue of open vertices. It starts with a single vertex in the closed list. Then it selects the lowest cost edge that connects an open vertex to a closed one, adds it to the tree and updates the lists. This process repeats until all vertices are in the closed list and connected by edges in the minimum spanning tree. The algorithm runs in O(E log V) time when using a binary heap priority queue.
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1) Each router maintains a routing table containing the outgoing link and distance to reach each destination node. 2) Routers periodically share their routing tables with neighbors so each can update its own table. 3) This allows routers to continuously determine the shortest paths to all destinations as network conditions change.
Computer Communication Networks-Routing protocols 1
This document provides an overview of routing protocols in computer networks. It discusses:
1) The goal of routing is to deliver data packets from source to destination(s) using forwarding tables updated by routing protocols. Routing can be unicast (one-to-one) or multicast (one-to-many).
2) Common routing algorithms include distance-vector routing (Bellman-Ford algorithm) and link-state routing (Dijkstra's algorithm). Distance-vector routing uses distance vectors exchanged between neighbors to calculate least-cost paths.
3) Issues with distance-vector routing include slow convergence and counting to infinity when link costs increase.
Scalable Online Betweenness Centrality in Evolving Graphs
Scalable Online Betweenness Centrality in Evolving Graphs.
This framework maintains both vertex & edge betweenness up-to-date for same computational cost handles both additions and removals of vertices and edges in a unified approach has reduced space overhead and is truly scalable and amenable to real-world deployment can be parallelized and deployed on top of modern DSPEs.
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Dynamic routing protocols calculate routes dynamically based on information exchanged between routers. Routing Information Protocol (RIP) is an example of a distance vector routing protocol. Distance vector protocols work by having each router periodically send its full routing table to neighbors. This can allow routes to increase without bound in the presence of failures, known as the count-to-infinity problem. Solutions include advertising the full path a route takes, not just the next hop.
Introducing ATM Technology unicast.ppt
This chapter discusses unicast routing and routing algorithms. It introduces the concepts of least-cost routing and least-cost trees to model routing between nodes in a network. It then describes three common routing algorithms: distance-vector routing, link-state routing, and path-vector routing. Distance-vector routing uses the Bellman-Ford algorithm and routing vectors to exchange routing information. Link-state routing uses flooding of link-state advertisements and the Dijkstra algorithm to build a global link-state database and compute least-cost trees. Path-vector routing additionally propagates path vectors to avoid routing loops.
Dijkstra
This document provides an overview of representing graphs and Dijkstra's algorithm in Prolog. It discusses different ways to represent graphs in Prolog, including using edge clauses, a graph term, and an adjacency list. It then explains Dijkstra's algorithm for finding the shortest path between nodes in a graph and provides pseudocode for implementing it in Prolog using rules for operations like finding the minimum value and merging lists.
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Bellmanford
- 3. Distance Table: example 7 8 1 2 1 2 loop! loop! A E D C B D () A B C D A 1 7 6 4 B 14 8 9 11 D 5 5 4 2 E cost to destination via destination D (C,D) E c(E,D) + min {D (C,w)} D w = = 2+2 = 4 D (A,D) E c(E,D) + min {D (A,w)} D w = = 2+3 = 5 D (A,B) E c(E,B) + min {D (A,w)} B w = = 8+6 = 14
- 4. Distance table gives routing table A B C D A,1 D,5 D,4 D,4 Outgoing link to use, cost destination Distance table Routing table D () A B C D A 1 7 6 4 B 14 8 9 11 D 5 5 4 2 E cost to destination via destination
- 6. Distance Vector Algorithm: 1 Initialization: 2 for all adjacent nodes v: 3 D (*,v) = infinity /* the * operator means "for all rows" */ 4 D (v,v) = c(X,v) 5 for all destinations, y 6 send min D (y,w) to each neighbor /* w over all X's neighbors */ X X X w At all nodes, X:
- 7. Distance Vector Algorithm (cont.): 8 loop 9 wait (until I see a link cost change to neighbor V 10 or until I receive update from neighbor V) 11 12 if (c(X,V) changes by d) 13 /* change cost to all dest's via neighbor v by d */ 14 /* note: d could be positive or negative */ 15 for all destinations y: D (y,V) = D (y,V) + d 16 17 else if (update received from V wrt destination Y) 18 /* shortest path from V to some Y has changed */ 19 /* V has sent a new value for its min DV(Y,w) */ 20 /* call this received new value is "newval" */ 21 for the single destination y: D (Y,V) = c(X,V) + newval 22 23 if we have a new min D (Y,w)for any destination Y 24 send new value of min D (Y,w) to all neighbors 25 26 forever w X X X X X w w
- 8. Distance Vector Algorithm: example X Z 1 2 7 Y
- 9. Distance Vector Algorithm: example X Z 1 2 7 Y D (Y,Z) X c(X,Z) + min {D (Y,w)} w = = 7+1 = 8 Z D (Z,Y) X c(X,Y) + min {D (Z,w)} w = = 2+1 = 3 Y