Shortest Path Problems

Transcription

1 Shortet Path Problem How can we find the hortet route between two point on a road map? Model the problem a a graph problem: Road map i a weighted graph: vertice = citie edge = road egment between citie edge weight = road ditance Goal: find a hortet path between two vertice (citie) 1

2 Shortet Path Problem Input: Directed graph G = (V, E) Weight function w : E R Weight of path p = v, v 1,..., v k w( p) k i1 w( v i 1, v i ) Shortet-path weight from u to v: t x δ(u, v) = min w(p) : u p v if there exit a path from u to v otherwie Note: there might be multiple hortet path from u to v

3 Variant of Shortet Path Single-ource hortet path G = (V, E) find a hortet path from a given ource vertex to each vertex v V Single-detination hortet path Find a hortet path to a given detination vertex t from each vertex v Revering the direction of each edge ingle-ource 3

4 Variant of Shortet Path (cont d) Single-pair hortet path Find a hortet path from u to v for given vertice u and v All-pair hortet-path Find a hortet path from u to v for ever pair of vertice u and v 4

5 Negative-Weight Edge Negative-weight edge ma form negative-weight ccle If uch ccle are reachable from 3 5 a b -4 4 c 6 d g the ource, then δ(, v) i not properl defined! Keep going around the ccle, and get w(, v) = - for all v on the ccle e -6 f 5

6 Negative-Weight Edge a: onl one path δ(, a) = w(, a) = 3 b: onl one path δ(, b) = w(, a) + w(a, b) = -1 c: infinitel man path, c,, c, d, c,, c, d, c, d, c 3 5 a b c 6 d g e -6 f ccle ha poitive weight (6-3 = 3), c i hortet path with weight δ(, c) = w(, c) = 5 6

7 Negative-Weight Edge e: infinitel man path:, e,, e, f, e,, e, f, e, f, e ccle e, f, e ha negative weight: 3 + (- 6) = -3 can find path from to e with arbitraril large negative weight δ(, e) = - no hortet path exit between and e Similarl: δ(, f) = -, δ(, g) = a b c 6 d g e -6 f h -8 j δ(, h) = δ(, i) = δ(, j) = i 3 h, i, j not reachable from

8 Ccle Can hortet path contain ccle? Negative-weight ccle Shortet path i not well defined Poitive-weight ccle: B removing the ccle, we can get a horter path Zero-weight ccle No! No! No reaon to ue them Can remove them to obtain a path with ame weight 8

9 Optimal Subtructure Theorem Given: v j A weighted, directed graph G = (V, E) v 1 p jk A weight function w: E R, p 1i p ij p ij v k A hortet path p = v 1, v,..., v k from v 1 to v k v i A ubpath of p: p ij = v i, v i+1,..., v j, with 1 i j k Then: p ij i a hortet path from v i to v j p 1i p ij p jk Proof: p = v 1 v i v j v k w(p) = w(p 1i ) + w(p ij ) + w(p jk ) Aume p ij from v i to v j with w(p ij ) < w(p ij ) w(p ) = w(p 1i ) + w(p ij ) + w(p jk ) < w(p) contradiction! 9

10 Triangle Inequalit For all (u, v) E, we have: δ (, v) δ (, u) + δ (u, v) u v - If u i on the hortet path to v we have the equalit ign u v 1

11 Algorithm Bellman-Ford algorithm Negative weight are allowed Negative ccle reachable from the ource are not allowed. Dijktra algorithm Negative weight are not allowed Operation common in both algorithm: Initialiation Relaxation 11

12 Shortet-Path Notation For each vertex v V: δ(, v): hortet-path weight d[v]: hortet-path weight etimate Initiall, d[v]= d[v]δ(,v) a algorithm progree [v] = predeceor of v on a hortet path from If no predeceor, [v] = NIL induce a tree hortet-path tree t x

13 Initialiation Alg.: INITIALIZE-SINGLE-SOURCE(V, ) 1. for each v V. do d[v] 3. [v] NIL 4. d[] All the hortet-path algorithm tart with INITIALIZE-SINGLE-SOURCE 13

14 Relaxation Step Relaxing an edge (u, v) = teting whether we can improve the hortet path to v found o far b going through u If d[v] > d[u] + w(u, v) we can improve the hortet path to v d[v]=d[u]+w(u,v) [v] u u v 5 9 u v 5 6 After relaxation: d[v] d[u] + w(u, v) RELAX(u, v, w) RELAX(u, v, w) u v 5 u v 5 6 no change 14

15 Bellman-Ford Algorithm Single-ource hortet path problem Compute δ(, v) and [v] for all v V Allow negative edge weight - can detect negative ccle. Return TRUE if no negative-weight ccle are reachable from the ource Return FALSE otherwie no olution exit 15

16 Bellman-Ford Algorithm (cont d) Idea: Each edge i relaxed V 1 time b making V-1 pae over the whole edge et. To make ure that each edge i relaxed exactl V 1 time, it put the edge in an unordered lit and goe over the lit V 1 time. (t, x), (t, ), (t, ), (x, t), (, x), (, ), (, x), (, ), (, t), (, ) 6 t x 9 16

17 BELLMAN-FORD(V, E, w, ) 6 t x Pa 1 6 t x 9 9 E: (t, x), (t, ), (t, ), (x, t), (, x), (, ), (, x), (, ), (, t), (, ) 1

18 Example (t, x), (t, ), (t, ), (x, t), (, x), (, ), (, x), (, ), (, t), (, ) Pa 1 (from previou lide) 6 t 5 x Pa Pa 3 t 5 x Pa t 5 x t 5 x

19 Detecting Negative Ccle (perform extra tet after V-1 iteration) for each edge (u, v) E do if d[v] > d[u] + w(u, v) then return FALSE return TRUE b b t pa nd pa 5 c 3-8 (,b) (b,c) (c,) 5 c 3 b -8 Look at edge (, b): d[b] = -1 d[] + w(, b) = -4 d[b] > d[] + w(, b) c 3 19

20 BELLMAN-FORD(V, E, w, ) 1. INITIALIZE-SINGLE-SOURCE(V, ). for i 1 to V do for each edge (u, v) E 4. do RELAX(u, v, w) 5. for each edge (u, v) E 6. do if d[v] > d[u] + w(u, v). then return FALSE 8. return TRUE (V) O(V) O(VE) O(E) O(E) Running time: O(V+VE+E)=O(VE)

21 Shortet Path Propertie Upper-bound propert We alwa have d[v] δ (, v) for all v. The etimate never goe up relaxation onl lower the etimate 6 v 5 x Relax (x, v) 6 v 5 x

22 Dijktra Algorithm Single-ource hortet path problem: No negative-weight edge: w(u, v) >, (u, v) E Each edge i relaxed onl once! Maintain two et of vertice: d[v]=δ (, v) d[v]>δ (, v)

23 Dijktra Algorithm (cont.) Vertice in V S reide in a min-priorit queue Ke in Q are etimate of hortet-path weight d[u] Repeatedl elect a vertex u V S, with the minimum hortet-path etimate d[u] Relax all edge leaving u 3

24 Dijktra (G, w, ) S=<> Q=<,t,x,,> S=<> Q=<,t,x,> 1 5 t 3 1 x t x 4 6 4

25 Example (cont.) 1 t 1 x t 1 x S=<,> Q=<,t,x> S=<,,> Q=<t,x> 5

26 Example (cont.) S=<,,,t> Q=<x> S=<,,,t,x> Q=<> 1 5 t 1 x t 1 x

27 Dijktra (G, w, ) 1. INITIALIZE-SINGLE-SOURCE(V, ). S 3. Q V[G] 4. while Q 5. do u EXTRACT-MIN(Q) 6. S S {u} O(V) build min-heap. for each vertex v Adj[u] 8. do RELAX(u, v, w) Executed O(V) time O(lgV) 9. Update Q (DECREASE_KEY) (V) O(VlgV) O(E) time (total) O(lgV) O(ElgV) Running time: O(VlgV + ElgV) = O(ElgV)

28 Problem 1 Write down weight for the edge of the following graph, o that Dijktra algorithm would not find the correct hortet path from to t. 1 u 1 w 1 t iteration d[]= d[u]=1 d[v]=1 1-1 v nd iteration 3 rd iteration 4 th iteration d[w]= d[u]= S={} Q={u,v,w} S={,u} Q={v,w} S={,u,v} Q={w} d[w] i not correct! d[u] hould have converged when u wa included in S! S={,u,v,w} Q={} 8

29 Problem (Exercie 4.3-4, page 6) We are given a directed graph G=(V,E) on which each edge (u,v) ha an aociated value r(u,v), which i a real number in the range r(u,v) 1 that repreent the reliabilit of a communication channel from vertex u to vertex v. We interpret r(u,v) a the probabilit that the channel from u to v will not fail, and we aume that thee probabilitie are independent. Give an efficient algorithm to find the mot reliable path between two given vertice. 9

30 Problem (cont.) Solution 1: modif Dijktra algorithm Perform relaxation a follow: if d[v] < d[u] w(u,v) then d[v] = d[u] w(u,v) Ue EXTRACT_MAX intead of EXTRACT_MIN 3

31 Problem (cont.) Solution : ue Dijktra algorithm without an modification! r(u,v)=pr( channel from u to v will not fail) Auming that the probabilitie are independent, the reliabilit of a path p=<v 1,v,,v k > i: r(v 1,v )r(v,v 3 ) r(v k-1,v k ) We want to find the channel with the highet reliabilit, i.e., max r( u, v) p ( u, v) p 31

32 Problem (cont.) But Dijktra algorithm compute ( u, v) p min w( u, v) p Take the lg lg(max r( u, v)) max lg( r( u, v)) p ( u, v) p p ( u, v) p 3

33 Problem (cont.) Turn thi into a minimiation problem b taking the negative: p p ( u, v) p ( u, v) p min lg( r( u, v)) min lg( r( u, v)) Run Dijktra algorithm uing w( u, v) lg( r( u, v)) 33