Dijkstra
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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 having each node broadcast its link states so all nodes have the same network topology information. The algorithm is iterative - after k iterations it knows the least-cost path to k destinations. It works by initializing a set of nodes with known least-cost paths (N') and then finding the node with the lowest cost path not in N' to add to N' and update the costs of neighboring nodes on each iteration until all nodes are in N'.
Dijkstra
- 1. A Link-State Routing Algorithm Dijkstra’s algorithm Notation: ❒ net topology, link costs ❒ c(x,y): link cost from node known to all nodes x to y; = ∞ if not direct ❍ accomplished via “link neighbors state broadcast” ❒ D(v): current value of cost ❍ all nodes have same info of path from source to ❒ computes least cost paths dest. v from one node (‘source”) to ❒ p(v): predecessor node all other nodes along path from source to v ❍ gives forwarding table for that node ❒ N': set of nodes whose least cost path definitively ❒ iterative: after k known iterations, know least cost path to k dest.’s Network Layer 4-75
- 2. Dijsktra’s Algorithm 1 Initialization: 2 N' = {u} 3 for all nodes v 4 if v adjacent to u 5 then D(v) = c(u,v) 6 else D(v) = ∞ 7 8 Loop 9 find w not in N' such that D(w) is a minimum 10 add w to N' 11 update D(v) for all v adjacent to w and not in N' : 12 D(v) = min( D(v), D(w) + c(w,v) ) 13 /* new cost to v is either old cost to v or known 14 shortest path cost to w plus cost from w to v */ 15 until all nodes in N' Network Layer 4-76
- 3. Dijkstra’s algorithm: example Step N' D(v),p(v) D(w),p(w) D(x),p(x) D(y),p(y) D(z),p(z) 0 u 2,u 5,u 1,u ∞ ∞ 1 ux 2,u 4,x 2,x ∞ 2 uxy 2,u 3,y 4,y 3 uxyv 3,y 4,y 4 uxyvw 4,y 5 uxyvwz 5 v 3 w 2 5 u 2 1 z 3 1 2 x 1 y Network Layer 4-77
- 4. Dijkstra’s algorithm: example (2) Resulting shortest-path tree from u: v w u z x y Resulting forwarding table in u: destination link v (u,v) x (u,x) y (u,x) w (u,x) z (u,x) Network Layer 4-78
- 5. Dijkstra’s algorithm, discussion Algorithm complexity: n nodes ❒ each iteration: need to check all nodes, w, not in N ❒ n(n+1)/2 comparisons: O(n2) ❒ more efficient implementations possible: O(nlogn) Oscillations possible: ❒ e.g., link cost = amount of carried traffic 1 A A A A 1+e 2+e 0 0 2+e 2+e 0 D 0 0 B D B D B D 1+e 1 B 1+e 1 0 0 0 e 0 0 1 1+e 0 e 1 C C C C 1 e … recompute … recompute … recompute initially routing Network Layer 4-79