Lecture 17: Shortest Paths

Transcription

1 Lecture 7: Shortet Path CSE 373: Data Structure and Algorithm CSE WI - KASEY CHAMPION

2 Adminitrivia How to Ace the Technical Interview Seion today! - 6-8pm - Sieg 34 No BS CS Career Talk Thurday - 5:30-6:30 - Bag 3 CSE WI - KASEY CHAMPION

3 Shortet Path How doe Google Map figure out thi i the fatet way to get to office hour? 3

4 Repreenting Map a Graph How do we repreent a map a a graph? What are the vertice and edge? 4

5 Repreenting Map a Graph K R S H D 3 P 5

6 Shortet Path The length of a path i the um of the edge weight on that path. Shortet Path Problem Given: a directed graph G and vertice and t Find: the hortet path from to t y 4 u 5 4 w v 3 6 x 5 t 6

7 Unweighted graph Let tart with a impler verion: the edge are all the ame weight (unweighted) If the graph i unweighted, how do we find a hortet path? 7

8 Unweighted Graph If the graph i unweighted, how do we find a hortet path? u w y t v x What the hortet path from to? - Well.we re already there. What the hortet path from to u or v? - Jut go on the edge from From to w,x, or y? - Can t get there directly from, if we want a length path, have to go through u or v. 8

9 Unweighted Graph: Key Idea To find the et of vertice at ditance k, jut find the et of vertice at ditance k-, and ee if any of them have an outgoing edge to an undicovered vertex. Do we already know an algorithm that doe omething like that? Ye! BFS! bfshortetpath(graph G, vertex ource) toviit.enqueue(ource) ource.dit = 0 while(toviit i not empty){ current = toviit.dequeue() for (v : current.outneighbor()) { if (v i unknown){ v.ditance = current.ditance + v.predeceor = current toviit.enqueue(v) mark v a known 9

10 Unweighted Graph Ue BFS to find hortet path in thi graph. bfshortetpath(graph G, vertex ource) toviit.enqueue(ource) ource.dit = 0 mark ource a viited while(toviit i not empty){ current = toviit.dequeue() for (v : current.outneighbor()){ if (v i not yet viited){ v.ditance = current.ditance + v.predeceor = current toviit.enqueue(v) mark v a viited u v w x y t

11 Unweighted Graph If the graph i unweighted, how do we find a hortet path? bfshortetpath(graph G, vertex ource) u toviit.enqueue(ource) ource.dit = 0 while(toviit i not empty){ current = toviit.dequeue() v for (v : current.outneighbor()) { if (v i unknown){ v.ditance = current.ditance + v.predeceor = current toviit.enqueue(v) mark v a known w x y t 3

12 What about the target vertex? Shortet Path Problem Given: a directed graph G and vertice,t Find: the hortet path from to t. BFS didn t mention a target vertex It actually find the hortet path from to every other vertex.

13 Weighted Graph Each edge hould repreent the time or ditance from one vertex to another. Sometime thoe aren t uniform, o we put a weight on each edge to record that number. The length of a path in a weighted graph i the um of the weight along that path. We ll aume all of the weight are poitive - For GoogleMap that definitely make ene. - Sometime negative weight make ene. Today algorithm doen t work for thoe graph - There are other algorithm that do work. 3

14 Weighted Graph: Take BFS work if the graph i unweighted. Maybe it jut work for weighted graph too? w 0 x u v t

15 Weighted Graph: Take BFS work if the graph i unweighted. Maybe it jut work for weighted graph too? w x 0 0 u v t 0 3 What went wrong? When we found a horter path from to u, we needed to update the ditance to v (and anything whoe hortet path went through u) but BFS doen t do that. 5

16 Weighted Graph: Take Reduction (informally) Uing an algorithm for Problem B to olve Problem A. You already do thi all the time. In Homework 3, you reduced implementing a hahet to implementing a hahmap. Any time you ue a library, you re reducing your problem to the one the library olve. Can we reduce finding hortet path on weighted graph to finding them on unweighted graph?

17 Weighted Graph Take Given a weighted graph, how do we turn it into an unweighted one without meing up the path length?

18 Weighted Graph: A Reduction u v t u Tranform Input v t u Unweighted Shortet Path v t u Tranform Output v t

19 Weighted Graph: A Reduction What i the running time of our reduction on thi graph? 5000 u 00 v O( V + E ) of the modified graph, which i low. t Doe our reduction even work on thi graph? 0.5 u 3! t v 5000 Ummm. tl;dr: If your graph weight are all mall poitive integer, thi reduction might work great. Otherwie we probably need a new idea.

20 Weighted Graph: Take 3 So we can t jut do a reduction. Intead figure out why BFS worked in the unweighted cae, try to make the ame thing happen in the weighted cae. How did we avoid thi problem: w x u v t

21 Weighted Graph: Take 3 In BFS When we ued a vertex u to update hortet path we already knew the exact hortet path to u. So we never ran into the update problem If we proce the vertice in order of ditance from, we have a chance.

22 Weighted Graph: Take 3 Goal: Proce the vertice in order of ditance from Idea: Have a et of vertice that are known -(we know at leat one path from to them). Record an etimated ditance -(the bet way we know to get to each vertex). If we proce only the vertex cloet in etimated ditance, we won t ever find a horter path to a proceed vertex. - Thi tatement i the key to proving correctne. - It nice if you want to practice induction/undertand the algorithm better.

23 Dijktra Algorithm Dijktra(Graph G, Vertex ource) initialize ditance to mark ource a ditance 0 mark all vertice unproceed while(there are unproceed vertice){ let u be the cloet unproceed vertex foreach(edge (u,v) leaving u){ if(u.dit+weight(u,v) < v.dit){ v.dit = u.dit+weight(u,v) v.predeceor = u mark u a proceed w Vertex Ditance Predeceor Proceed w x u v t x u v t 0

24 Dijktra Algorithm Dijktra(Graph G, Vertex ource) initialize ditance to mark ource a ditance 0 mark all vertice unproceed while(there are unproceed vertice){ let u be the cloet unproceed vertex foreach(edge (u,v) leaving u){ if(u.dit+weight(u,v) < v.dit){ v.dit = u.dit+weight(u,v) v.predeceor = u w mark u a proceed Vertex Ditance Predeceor Proceed 0 -- Ye w Ye x w Ye u 0 3 x Ye v 4 u Ye t 5 v Ye x u v t 0