ECE250: Algorithms and Data Structures Single Source Shortest Paths Dijkstra s Algorithm

Transcription

1 ECE0: Algorithms and Data Strctres Single Sorce Shortest Paths Dijkstra s Algorithm Ladan Tahildari, PEng, SMIEEE Associate Professor Software Technologies Applied Research (STAR) Grop Dept. of Elect. & Comp. Eng. Uniersity of Waterloo Materials from CLRS: Chapter 4.3, pp

2 Acknowledgements The following resorces hae been sed to prepare materials for this corse: Ø MIT OpenCorseWare Ø Introdction To Algorithms (CLRS Book) Ø Data Strctres and Algorithm Analysis in C++ (M. Wiess) Ø Data Strctres and Algorithms in C++ (M. Goodrich) Thanks to many people for pointing ot mistakes, proiding sggestions, or helping to improe the qality of this corse oer the last ten years: Ø ECE0

3 Shortest Path Generalize distance to weighted setting Digraph G = (V,E) with weight fnction W: E R (assigning real ales to edges) Weight of path p = 1 k is k 1 wp ( ) = w ( i, i+ 1) i= 1 Shortest path = a path of the minimm weight Applications Ø static/dynamic network roting Ø robot motion planning Ø map/rote generation in traffic ECE0 3

4 Shortest-Path Problems Single-sorce (single-destination). Find a shortest path from a gien sorce (ertex s) to each of the ertices. The topic of this lectre. Single-pair. Gien two ertices, find a shortest path between them. Soltion to single-sorce problem soles this problem efficiently, too. All-pairs. Find shortest-paths for eery pair of ertices. Dynamic programming algorithm. Unweighted shortest-paths Ø BFS ECE0 4

5 Optimal Sbstrctre Theorem: Sbpaths of shortest paths are shortest paths Proof ( ct and paste ) Ø if some sbpath were not the shortest path, one cold sbstitte the shorter sbpath and create a shorter total path ECE0

6 Negatie Weights and Cycles? Negatie edges are OK, as long as there are no negatie weight cycles (otherwise paths with arbitrary small lengths wold be possible) Shortest-paths can hae no cycles (otherwise we cold improe them by remoing cycles) Ø Any shortest-path in graph G can be no longer than n 1 edges, where n is the nmber of ertices ECE0 6

7 Shortest Path Tree The reslt of the algorithms a shortest path tree. For each ertex, it Ø records a shortest path from the start ertex s to..parent() gies a predecessor of in this shortest path Ø gies a shortest path length from s to, which is recorded in.d(). Vertex ADT with operations: Ø adjacent():vertexset Ø keyd():int and setd(k:int) Ø parent():vertex and setparent(p:vertex) ECE0 7

8 Relaxation For each ertex in the graph, we maintain.d(), the estimate of the shortest path from s, initialized to at the start Relaxing an edge (,) means testing whether we can improe the shortest path to fond so far by going throgh 9 Relax(,) 7 6 Relax(,) 6 Relax (,,G) if.d() >.d()+g.w(,) then.setd(.d()+g.w(,)).setparent() ECE0 8

9 Dijkstra's Algorithm Non-negatie edge weights Greedy, similar to Prim's algorithm for MST Like breadth-first search (if all weights = 1, one can simply se BFS) Use Q, a priority qee ADT keyed by.d() (BFS sed FIFO qee, here we se a PQ, which is reorganized wheneer some d decreases) Basic idea Ø maintain a set S of soled ertices Ø at each step select "closest" ertex, add it to S, and relax all edges from ECE0 9

10 Dijkstra s Psedo Code Dijkstra(G,s) 01 for each ertex G.V() 0.setd( ) 03.setparent(NIL) 04 s.setd(0) 0 S // Set S is sed to explain the algorithm 06 Q.init(G.V()) // Q is a priority qee ADT 07 while not Q.isEmpty() 08 Q.extractMin() 09 S S {} 10 for each.adjacent() do 11 Relax(,, G) 1 Q.modifyKey() relaxing edges ECE0 10

11 Dijkstra s Example Dijkstra(G,s) 01 for each ertex G.V() 0.setd( ) 03.setparent(NIL) s 04 s.setd(0) 0 S 06 Q.init(G.V()) 07 while not Q.isEmpty() 08 Q.extractMin() 09 S S {} 10 for each.adjacent() do 11 Relax(,, G) 1 Q.modifyKey() s ECE x y 10 3 x y

12 Dijkstra s Example (cont ) Dijkstra(G,s) 01 for each ertex G.V() 0.setd( ) 03.setparent(NIL) s 04 s.setd(0) 0 S 06 Q.init(G.V()) 07 while not Q.isEmpty() 08 Q.extractMin() 09 S S {} 10 for each.adjacent() do 11 Relax(,, G) 1 Q.modifyKey() s ECE x y x y

13 Dijkstra s Example (cont ) Dijkstra(G,s) 01 for each ertex G.V() 0.setd( ) s 03.setparent(NIL) 04 s.setd(0) 0 S 06 Q.init(G.V()) 07 while not Q.isEmpty() 08 Q.extractMin() 09 S S {} 10 for each.adjacent() do 11 Relax(,, G) 1 Q.modifyKey() s ECE x y x y

14 Dijkstra s Psedo Code Dijkstra(G,s) 01 for each ertex G.V() 0.setd( ) 03.setparent(NIL) 04 s.setd(0) 0 S // Set S is sed to explain the algorithm 06 Q.init(G.V()) // Q is a priority qee ADT 07 while not Q.isEmpty() 08 Q.extractMin() 09 S S {} 10 for each.adjacent() do 11 Relax(,, G) 1 Q.modifyKey() relaxing edges ECE0 14

15 Dijkstra s Rnning Time Extract-Min exected V time Decrease-Key exected E time Time = V T extractmin + E T modifykey T depends on different PQ implementations Q T(extractMin) T(modifyKey) Total array Ο(V) Ο(1) Ο(V ) binary heap Ο(lg V) Ο(lg V) Ο(E lg V) ECE0 1