1. © DEEDS 2008Graph Algorithms1 Remarks on Dijkstra  Determining the minimum marginal element (code line: v = the vertex in M 2 with minimum v.L; ) accounts for the substantial cost of the algorithm:  An implementation that uses lists requires to traverse the interlinking for each vertex once: time complexity O(|V| 2 )  An implementation that uses a priority queue for the margin instead of a list has a time complexity of O(|E| log |V|)  Implementations with Fibonacci-heaps give O(|V|log|V|+|E|)  |V|  max eE c(e) is allowed instead of   If the vertex weights are all 1, the Moore‘ Algorithm performs better (it is linear)

2. © DEEDS 2008Graph Algorithms2 Variants  Dijkstra‘s Algorithm can be enhanced:  In addition to the length of the shortest path the algorithm could save the path itself Carry the predecessor vertex along the shortest path Collecting these edges results in a spanning tree  Find the shortest path to a particular vertex Abort the algorithm, as soon as this vertex was moved to M 1  Dijkstra‘ algorithm does not work in a setting where negative vertex weights are allowed. Then use the Bellmann-Ford Algorithm p. 588

3. © DEEDS 2008Graph Algorithms3 Negative Edges  Why negative edges?  Energy consumption: uphill vs. downhill  Currency exchange: find cycles that “produce” money Search Google for “arbitrage” and see for yourselves… Instead of minimal sum, we have maximal product

4. © DEEDS 2008Graph Algorithms4 Negative Weight Edges  Graph G=(V, E), for all vertices vV reachable from source s, the distance is  δ(a,b) = 5  δ(a,h) = + ∞  δ(a,c) = ?  δ(a,f) = ? δ(a,c) = min {,,,…} δ(a,f) = min{,,,…} a b d f e h i j 5 7 2 c 2 g 3 -6

5. © DEEDS 2008Graph Algorithms5 c b a e d 0 8 2 7 5 10 8 76 21 2 3 -8 Dijkstra ?!  Why doesn’t Dijkstra’s algorithm work for negative weights? c b a e d f 0 8 2 7 5 9 10 8 76 21 2 3 9 f 9

6. © DEEDS 2008Graph Algorithms6 Bellmann-Ford  Solves the shortest path problem even with negative edges  The graph must not contain negative cycles!  Only works for directed graphs  However: negative cycles seldom appear in real-life problems

7. © DEEDS 2008Graph Algorithms7 Bellman-Ford Algorithm 0 ∞ ∞ ∞ ∞ 6 7 2 -2 -3 -4 7 9 s t x z y 6 7 4 2 2 -2 5 8 Round 1Round 2Round 3Round 4  Given: graph G = (V, E) with s, v  V and w : E  R  Problem: Solve single-source shortest-paths problem in the general case, returning FALSE if negative cycles exists  Strategy: Similar with Dijkstra’s algorithm, but relax all edges at each step, not just the neigbors… Algorithm BellmanFord(G, s) for all v  V if (v  s) d[v] = 0else d[v] =  for i  1 to |V|-1 do do for each e = (v,u)  E if (d[u] > d[v]+w(v,u)) d[u]  d[v]+w(v,u) for each e = (v,u)  E do if (d[u] > d[v]  w(u,v)) return FALSE return TRUE Relaxation order: (t,x),(t,y),(t,z),(y,x),(y,z),(z,x),(z,s),(x,t),(s,t),(s,y) p. 588

8. © DEEDS 2008Graph Algorithms8 BF Algorithm  Works even with negative-weight edges  Must assume directed edges (otherwise we would have negative-weight cycles)  Each iteration i finds all shortest paths of length i (lines 3-6)  If “improvements” can be made, negative cycles exist (line 7 to 10)  Proof in Cormen p 590  Running time:  Line 3: O(|V|)  Line 4: O(|E|)  Total: O(|V||E|)  Dijkstra: at least O(|E|log|V|) Algorithm BellmanFord(G, s) 1. 1.for all v  V 2.if (v  s) d[v] = 0else d[v] =  3. for i  1 to |V|-1 do 4. do for each e = (u,v)  E 5. if (d[v] > d[u]+w(u,v)) 6. d[v]  d[u]+w(u,v) 7. for each e = (u,v)  E 8. do if (d[v] > d[u]  weight(u,v)) 9. return FALSE 10. return TRUE

9. © DEEDS 2008Graph Algorithms9 Dijkstra Variants  Is Dijkstra the most suitable algorithm for finding the shortest path in geographical applications, such as route planning or computer games?

10. © DEEDS 2008Graph Algorithms10 Competition: A*  A* - (pronounced A-star) Similar to Dijkstra but each node also contains information about the minimum distance to the destination. For geographical applications this typically is the beeline (there is no shorter path!)  The algorithm first searches the nodes which sum of computed distance (from start) and minimal (approximated) distance to source

11. © DEEDS 2008Graph Algorithms11 A-Star Algorithm (simplified) /* Nodes labeledds(v)=(computed) distance from start, dd(v)=(approximated) distance to destination, M 3 implicitly defined as V \ ( M 1  M 2 ) */ for (all v  V) { /*Initialization*/ dd(v) = beelines; if ( v  s ) ds(v) =  ; } M 1 = { s }; M 2 = { }; ds(s) = 0; for (all v  s.neighbors( ) ) {/* fill M 2 initially */ M 2 = M 2  { v }; ds(v) = c( (s, v) ); } while ( M 2  { } && destination  M 1 ) { v = The node in M 2 with minimal (ds(v) + dd(v)); M 2 = M 2 \ { v }; M 1 = M 2  { v }; /* move v */ for (all w  v.neighbours( ) ) { if ( w  M 3 ) { /* move neighbors and adapt ds(w) */ M 2 = M 2  { w }; ds(w) = ds(v) + c( (v, w) ); } else if ( ds(w) > ds(v) + c( (v, w) ) ) ds(w) = ds(v) + c( (v,w) ); }

12. © DEEDS 2008Graph Algorithms12 Example  Destination: E dd(v) marked in green=beeline

13. © DEEDS 2008Graph Algorithms13 red: ds(v) in brackets: ds(v)+dd(v) Example

14. © DEEDS 2008Graph Algorithms14 Example

15. © DEEDS 2008Graph Algorithms15 Example

16. © DEEDS 2008Graph Algorithms16 Destination reached!  No other path is shorter, since the minimal distance (in brackets) is already shorter than the actual distance to the destination! Example A B C D E F G H I K L M O P N X Y Z 0 112 79 110 90 39 37 56 87 30 72 109 94 50 55 100 43 66 0 (87) 43 (152) 40 (150) 34 (113) 65 (121) 55 (110) 80 (110) 109 (148) 146 (196) 100 (143) 138 (175) 111 (111)

17. © DEEDS 2008Graph Algorithms17 A-Star  The actual algorithm is optimized by using a priority queue structure  Complexity: the same as Dijkstra, but better average case execution time  Why not always use A-Star?  Needs a known destination  Needs an approximated minimal distance