1 / 23

Suppose G = (V, E) is a directed network. Each edge ( i , j ) in E has an associated

Shortest Paths. Suppose G = (V, E) is a directed network. Each edge ( i , j ) in E has an associated ‘length’ c ij (cost, time, distance, …). Determine a path of shortest length between two specified nodes s and t . The length

lmcleod
Download Presentation

Suppose G = (V, E) is a directed network. Each edge ( i , j ) in E has an associated

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Shortest Paths Suppose G = (V, E) is a directed network. Each edge (i,j) in E has an associated ‘length’ cij (cost, time, distance, …). Determine a path of shortest length between two specified nodes s and t. The length c(P) of a path P is the sum of its constituent arc lengths: c(P) = Σcij Applications: transportation planning (shortest route) telecommunications (most reliable connection) scheduling (critical path) pattern recognition, computer graphics subproblem to a larger optimization problem

  2. Different Types of SPP 1. Differentiated by the number of origin/destination(source/sink) vertices: • from s to t • from s to all other vertices • to t from all other vertices • from i to j 2. Differentiated by the properties of the edge length: • Nonnegative edge lengths • Negative edge lengths

  3. Assumptions & Algorithms • Integer data • Directed network • There is a path from s to all other vertices • There is no negative length directed cycle Two algorithms to be studied: Label-setting – no negative length edges Label-correcting – negative length edges allowed KEY IDEA: Distance labels – D(i) = how far away vertex i is from s

  4. Optimality Conditions Let dj correspond to the distance of the shortest path from vertex s to vertex j. Certainly ds=0. Moreover, the condition dj≤ di + cij (***) says that if we have a shortest path its length cannot be improved by means of an edge (i, j) not currently on the path. This condition (***) is used in both algorithms to efficiently solve the SPP. FOCUS: single source problems, first with nonnegative costs, then allow negatives.

  5. Shortest Path Trees c t Key observation: any subpath of a shortest path is also a shortest path When we find a shortest path from s to t, we also find a shortest path from s to any intermediate vertices. This leads to a way of representing all shortest paths from vertex s. Find a shortest path from s to some t; call it P1. Repeat with another vertex t not on path P1. Repeat until all vertices are used. This results in a shortest path tree for G. Namely, a directed tree in G whose paths s to j are all shortest paths from s to j. a s b

  6. i 1 2 3 4 5 6 p(i) - d(i) 0 Example 2 5 4 2 4 3 2 4 2 2 1 -1 6 1 0 6 8 1 • Use two arrays to represent the shortest path tree. • d(i) records the shortest distance to vertex i. • p(i) records the predecessor of vertex i in the shortest path 5 3 5 3 5 3 5 4 3 4

  7. Label Setting Algorithms Nonnegative edge lengths: At each step in the algorithm, temporarily labeled vertices have a distance label of D(i), which is an upper bound on d(i); permanently labeled vertices are guaranteed to have D(i) = d(i). Two sets to keep track of: S is the set of permanently labeled vertices; Ŝ = V – S is the set of temporarily labeled vertices Vertices are transferred from Ŝ to S, one at a time. Upon termination, all vertices are permanently labeled. The algorithm processes vertices in increasing distance from vertex s. At the kth iteration, the kth closest vertex to s will be permanently labeled. a s c t b ^ S S

  8. Topological(Acyclic) or Not Topological If network has a topological order, examine vertices from 1 to n (reaching). If no topological order exists, examine vertices in order of increasing distance from s (Dijkstra’s Algorithm). DIJKSTRA’S start with all vertices in Ŝ; give vertex s a distance D(s) = 0; give all other vertices the distance D(j) = ∞; while set S is not full pick a vertex, i, in Ŝ with the smallest distance; move that vertex from Ŝ to S; update distances for vertices reachable from i; for each (i,j) in E(i) if D(j) > D(i) + cij then D(j) = D(i) + cij and pred(j) = i;

  9. Example 4 2 4 2 2 1 3 6 1 2 4 1 3 5 3 move to S 2 4 2 2 0 1 3 6 1 2 4 1 3 5 3 move to S

  10. Example 2 4 2 4 2 2 0 1 3 6 1 2 4 1 3 5 3 move to S 2 4 2 4 2 2 0 1 3 6 1 2 4 1 3 5 3 3 move to S

  11. Example 6 2 4 2 4 2 2 0 1 3 6 1 2 4 1 3 5 3 3 move to S 6 2 4 2 4 2 2 0 1 3 6 1 2 4 1 3 5 6 3 3 move to S

  12. i 1 2 3 4 5 6 p(i) 0 1 2 2 3 5 D(i) 0 2 3 6 6 7 Example 2 6 4 2 4 2 2 7 0 1 3 6 1 2 4 1 3 5 3 6 3

  13. i 1 2 3 4 5 6 D(i) 0 ∞ ∞ ∞ ∞ ∞ D(i) 0 D(i) 0 D(i) 0 D(i) 0 D(i) 0 Example – Topological 7 5 2 4 3 3 6 8 4 1 6 2 6 4 3 3 5 6

  14. i 1 2 3 4 5 6 D(i) 0 ∞ ∞ ∞ ∞ ∞ D(i) 0 D(i) 0 D(i) 0 D(i) 0 D(i) 0 Example – Dijkstra’s 7 5 2 4 4 3 1 8 4 1 6 2 6 4 3 3 5 6

  15. Example 6 3 5 1 6 12 1 1 2 4 6 6 4 2 3 10 5 6 0 3 3 1 1 1 4 s 2 7 5 11 t 2 1 2 2 3 6 6 1 9 5 4 3 1 3 5 3 8 13 4 7 5

  16. Label Correcting Suppose G = (V, E) is a directed network. Each edge (i,j) in E has an associated ‘length’ cij= positive, zero, negative d(j) = minimum length of a path from s to j D(j) = distance label, current length of some path from s to j D(j) = d(j) for all j in V iff D(j) ≤ D(i) + cij for all (i,j) in E(i) and D(s) = 0 In a network with negative edge lengths, maintain a LIST of vertices with changed labels (Bellman-Ford Algorithm). Search edge list and update until no distance is changed. Maintain LIST as: • Queue – FIFO • Stack – LIFO • Dequeue – vertex selected from top;first time vertex is added, it is placed at the bottom; and anytime after that, a vertex re-enters at top of list

  17. List Structures leave • Queue • Stack • Dequeue (Pape) enter leave enter leave enter re-enter

  18. 3 2 4 queue i 1 2 3 4 5 6 3 2 D(i) 0 ∞ ∞ ∞ ∞ -2 -1 6 {1} ∞ 1 3 2 3 1 {2, 3} D(i) 0 3 2 3 ∞ ∞ ∞ 5 4 =List D(i) 0 D(i) 0 =List D(i) 0 =List D(i) 0 =List D(i) 0 =List 0 D(i) =List D(i) 0 =List D(i) 0 =List D(i) 0

  19. 3 2 4 i 1 2 3 4 5 6 dequeue 3 2 D(i) 0 ∞ ∞ ∞ ∞ -2 -1 6 {1} ∞ 1 3 2 3 1 {2, 3} D(i) 0 ∞ 3 2 3 ∞ ∞ 5 4 =List D(i) 0 D(i) 0 =List D(i) 0 =List D(i) 0 =List D(i) 0 =List D(i) 0

  20. Negative cycle detection A negative length cycle will cause some distance labels to decrease without limit. With label correcting algorithms, negative length cycles can be detected in a variety of ways. 1. If C is max of all |cij|, then any D(j) falling below -nC. 2. In the queue(FIFO) implementation, more than n-1 updates for a single vertex. 3. If the graph based on predecessors fails to be a tree. -3 2 4 3 2 -2 -1 1 6 3 2 3 1 3 5 4

  21. All Pairs Shortest Path 1. Apply Dijkstra’s n times, once for each vertex. Good for sparse network. 2. Floyd-Warshall Algorithm finds best in phases. Good for dense network. Assume network is strongly connected. Matrix entry D(i, j) = distance from i to j. Floyd-Warshall begin set D(i, j) = cij, pred(i, j) = i for each (i, j) in E; set D(i, i) = cii, pred(i, i) = i for each i; fork = 1 to n do for all i,j = 1 to n do if D(i, j) > D(i, k) + D(k, j)then D(i, j) = D(i, k) + D(k, j)and pred(i, j) = pred(k, j); end

  22. Example k = 0 k = 1 4 2 4 1 D = pred= 2 1 1 1 3 3 5 2 k = 2 D = pred=

  23. Example 4 2 4 1 2 1 1 1 3 3 5 2 k = 5 D = pred= What is shortest path from 1 to 4? What is shortest path from 5 to 3?

More Related