CS 557 Lecture IX. Drexel University Dept. of Computer Science

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1 CS 7 Lectre IX Dreel Uniersity Dept. of Compter Science Fall 00 Shortest Paths Finding the Shortest Paths in a graph arises in many different application: Transportation Problems: Finding the cheapest way to trael between two locations. Motion Planning: What is the most natral way to trael a robot in an enironment. Commnication Problems: The shortest set of hbs to get a message between two nodes in a network. Which two locations are farthest apart, i.e., what is the diameter of a network.

2 The Single Sorce Shortest Path We are gien a graph G(V,E) and a real weight fnction w from E to R, define the weight of a path p< 0,,, k > as the sm of weight of its edges: k w( p) i w(, i i ). We define the shortest-path weigh between and by: d (, ) min{ w( p) : p if there is a otherwise path between and The Single Sorce Shortest Path Gien a graph G(V,E), we want to find a shortest path from a sorce s to eery erte in V. Variants: Single destination shortest path. Single pair shortest path. All-pairs shortest path. We will make different assmptions abot edge weights!

3 Uniqeness s 7 y s 9 s y y Sb-optimality Gien a weighted graph G(V,E) and weight fnction w on edges. Let P<,,, k > be a shortest path from to k and for any i and j sch that i j k let P ij < i, i+,, j > be the sb path of P from i to j. Then, P ij is the shortest path from i to j. P ij i j k

4 Sb-optimality Let G(V,E) be a weighted, directed graph with weight fnction w. Sppose that a shortest path P from sorce s to a erte can be decomposed into a path P and an edge (,) as follows: s P Then the weight of shortest path from s to is d( s, ) d( s, ) + w(, ) Triangle Ineqality Let G(V,E) be a weighted, directed graph with weight fnction w. Then for all edges (,), we hae d( s, ) d( s, ) + w(, ) s d( s, ) d( s, ) w(, )

5 Relaation Techniqes For a erte in V, we maintain an attribte ], which is an pper bond on the shortest path from s to. We call ] a shortest path estimate. Initially, all the shortest path estimates are infinity. As algorithms proceeds this ales gets closer and closer to actal ale d( s, ) of shortest path between s and. Relaation The process of relaing an edge (,) consists of testing whether we can improe the shortest path fond so far to, by etending it to. A relaation step may decrease the of shortest path estimate ] sing the triangle ineqality: Rela(,,w) End if ]> ]+w(,) the ] ]+ w(,)

6 The Effect of relaation 9 Rela(,) Rela(,) 7 Properties of Relaation Let G(V,E) be a weighted graph with weight fnction w. Then Immediately after relaing (,), we hae d [ ] ] + w(, ) For for eery erte in G: ] d( s,) If there in no path in G between s and : d [ ] d( s, ) Let p<s,,,> be the shortest path between s and. If after the relaation. at any time prior to relaation of (,) then d [ ] d( s, ) d [ ] d( s, )

7 Bellman-Ford Algorithm This is the most basic single-sorce shortest path algorithm: The algorithm finds the shortest path from sorce s to eery erte in the graph. The actal shortest path can be constrcted easily. Starts with an estimate of shortest distance and eentally conerges to shortest weight paths. Algorithm SSSP(G) for each V ] do Initialize ] to infinity, which will conerge to shortest-path ale. s] for i 0 to V do Relaation: will rela each edge V - times. for each edge(, ) E do if ] > ] + w(, ) then ] ] + w(, ) for each edge(, ) E do if ] > ] + w(, ) then At the end, test to see if a soltion is fond (gets soltion if no negatieweight cycles eits. Otpt " NoSoltion!" 7

8 Well-definedness If the graph contains a negatie cycle then some shortest path may not eist. Consider the following eample, as we go arond the cycle we always get shorter path. <0 k i Eample Initialization: A] 0 st Relaation: Process edges in order (A,B), (A,C), (B,C), (B,D), (D,C), (E,D), (B,E). B] C] D] E] B B A E 0 A E - - C D C D 8

9 Eample st Relaation: Same order of process for edges, i.e. (A,B), (A,C), (B,C), (B,D), (D,C), (E,D), (B,E). (there are more relations bt the ] ales will not change). B - 0 A E - C D Rnning time and Correctness Rnning time: O(nm), where n V and m E. After V - iteration the ] represent the shortest path between s and each erte : Initially: s]0 Let s k denote the shortest path between s and k then: After st path ] is correct, since ] s]+w(s, ). After nd path ] is correct, since ] ]+w(, ). After k th path k ] is correct, since k ] k- ]+w( k-, k ). This holds for all ertices, since the longest path in the graph has length V -. 9

10 Dijkstra s Algorithm This algorithm works only for the graphs with non-negatie edge weights The reslt of this algorithm is similar to BFS if the graph is nweighted. Like Prim s algorithm ses a priority qee. Has better rnning time than Bellman-Ford. Algorithm SSSP(G) Initialize of ] is similar to for each V do Bellman-Ford. ] s] 0 Relaation: each erte can be S sbject to relaation as many Q V times as its in-degree. while Q do The changes de to relaation Etract - Min( Q) will be handled by Decrease- S S { } Key algorithm. for each Adj[ ] do if ] > ] + w(, ) then ] ] + w(, ) 0

11 Eample A 0 B 0 D Q: (A,0) ( B, ) ( C, ) C ( D, ) A 0 0 B C 0 D Etract-Min: A Decrease-Key: B,C Q: (C,) ( D, ) (B,0) Eample 9 A 0 B C 0 D Etract-Min: C Decrease-Key: B,D Q: (B,9) (D,) 8 A 0 B 0 D Etract-Min: D Decrease-Key: B Q: (B,9) C

12 Eample B Etract-Min: B A D Q: EMPTY C Rnning Time Etract-Min: will be eected V times. Decrease-Key: will be eected E times. T ( n, m) O( nlog n + m log n)

13 Correctness At the termination of algorithm ] d( s,). Assme not, i.e. ] d( s,). Right before is added to S. Let p be the shortest path between bewteen s and. Let y be the first erte on p otside S and let be the erte on p right before y.clearly when is inserted to S, y] d( s,y). Bt this will reslt in a contradiction. s P y] d( s,y) S d( s,) P y ] Shortest Path in DAGs SSSP is well defined for DAGs, since DAGs can not hae negatie cycles. We are looking for a fast algorithm (as opposed to Dijkstra's and Bellman-Ford). Obsere that, if there is a path from to, then precedes in topological sort. Which means to find the SSSP we can jst pass once oer the ertices in the topologically sorted ordered.

14 Algorithm DAG-Shortest-Path(G,w,s) Topologically Sort the ertices of G Initialize the ] for all the ertices. For each erte taken in topologically sorted order do For each erte in Adj[] do Rela(,,w) Eample r s t r s t

15 Eample r s t r s t Eample r s t r s t

16 Eample r s t Rnning Time Since eery erte will DAG-Shortest-Path(G,w,s) Topologically Sort the ertices of G Initialize the ] for all the ertices. For each erte taken in topologically sorted order do For each erte in Adj[] do Rela(,,w) be looked at at most once, the oter loop will be eected O( V ) times. The inner loop will be eected only O( E ) times. The oer all rnning time is O( V + E ).

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

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