Lecture 18 Graph-Based Algorithms. CSE373: Design and Analysis of Algorithms

Transcription

1 Lecture 18 Graph-Baed Algorithm CSE: Deign and Anali of Algorithm

2 Shortet Path Problem Modeling problem 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)

3 Shortet Path Problem What i hortet path? hortet length between two vertice for an unweighted graph: mallet cot between two vertice for a weighted graph: B 1 B A 6 A 19 4 E C D unweighted graph E C 1 D weighted graph

4 Shortet Path Problem Input: Directed graph G = (V, E) Weight funcon w : E R Weight of path p = v, v 1,..., v k w( p) k i1 w( v i 1, v i ) t x z Shortet path weight from u to v: p δ(u, v) = min w(p) : u v if there exit a path from u to v otherwie

5 Variant of Shortet Path Single ource hortet path Given 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 Revere the direction of each edge ingle ource Single pair hortet path Find a hortet path from u to v for given vertice u and v Solve the ingle ource problem All pair hortet path Find a hortet path from u to v for ever pair of vertice u and v

6 Shortet Path Repreentation For each vertex v V: d[v] = δ(, v): a hortet path etimate Iniall, d[v]= Reduce a algorithm progre [v] = predeceor of v on a hortet path from If no predeceor, [v] = NIL induce a tree hortet path tree Shortet path & hortet path tree are not unique

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

8 Relaxation 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 update d[v] and [v] u v 9 u v 6 RELAX(u, v, w) RELAX(u, v, w) u v u v 6 After relaxation: d[v] d[u] + w(u, v)

9 RELAX(u, v, w) 1. if d[v] > d[u] + w(u, v). then d[v] d[u] + w(u, v). [v] u All the ingle ource hortet path algorithm tart b calling INIT SINGLE SOURCE then relax edge The algorithm differ in the order and how man time the relax each edge

10 Dijktra Algorithm Single ource hortet path problem: No negative weight edge: w(u, v) > (u, v) E Maintain two et of vertice: S = vertice whoe final hortet path weight have alread been determined Q = vertice in V S: min priorit queue Ke in Q are etimate of hortet path weight (d[v]) Repeatedl elect a vertex u V S, with the minimum hortet path etimate d[v]

11 Dijktra (G, w, ) 1. INITIALIZE SINGLE SOURCE(V, ). S. Q V[G] 4. while Q. do u EXTRACT MIN(Q) 6. S S {u}. for each vertex v Adj[u] 8. do RELAX(u, v, w) 1 1 t t x 4 6 z x 4 6 z

12 Example 1 t 1 x t 1 x z z 1 t 1 x z 1 t 1 x z

13 Dijktra Peudo Code Graph G, weight function w, root relaxing edge

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

15 Dijktra Running Time Q T(Extract -Min) T(Decreae- Ke) Total arra (V) (1) (V ) binar heap (lg V) (lg V) (E lg V)

16 Negative Weight Edge What if we have negative weight edge? a b c 6 d g e -6 f h -8 j i

17 Negative Weight Edge a: onl one path (, a) = w(, a) = b: onl one path (, b) = w(, a) + w(a, b) = 1 a b c 6 d g e -6 f h -8 j i

18 Negative Weight Edge c: infinitel man path, c,, c, d, c,, c, d, c, d, c ccle c, d, c ha poitive weight (6 = ), c i hortet path with weight (, b) = w(, c) = a b c 6 d g e -6 f h -8 j i

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

20 Negative Weight Edge Negative weight edge ma form negative weight ccle If uch ccle are reachable from the ource: (, v) i not properl defined a b c 6 d g e -6 f h -8 j i

21 Ccle Can hortet path contain ccle? Negative weight ccle Poitive weight ccle: No! No! B removing the ccle we can get a horter path We will aume that when we are finding hortet path, the path will have no ccle

22 Bellman Ford Algorithm Single ource hortet path problem Compute d[v] and [v] for all v V Allow negative edge weight Return: TRUE if no negative weight ccle are reachable from the ource FALSE otherwie no olution exit Idea: Travere all the edge V 1 time, ever time performing a relaxation tep of each edge

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

24 Example 6 t x 9 z E: (t, x), (t, ), (t, z), (x, t), (, x), (, z), (z, x), (z, ), (, t), (, )

25 Example Pa 1 t x 6 Pa z Pa t x Pa z t x z t x z E: (t, x), (t, ), (t, z), (x, t), (, x), (, z), (z, x), (z, ), (, t), (, )

26 Detecting Negative Ccle for each edge (u, v) E do if d[v] > d[u] + w(u, v) then return FALSE return TRUE b b b c c c Oberve edge (, b): d[b] = -1, d[] + w(, b) = -4 d[b] > d[] + w(, b)

27 BELLMAN FORD(V, E, w, ) 1. INITIALIZE-SINGLE-SOURCE(V, ). for i 1 to V - 1. do for each edge (u, v) E 4. do RELAX(u, v, w). 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(E) O(E) O(VE) Running time: O(VE)

28 Single Source Shortet Path in DAG Given a weighted DAG: G = (V, E) olve the hortet path problem Idea: Topologicall ort the vertice of the graph Relax the edge according to the order given b the topological ort for each vertex, we relax each edge that tart from that vertex Are hortet path well defined in a DAG? Ye, (negative weight) ccle cannot exit

29 DAG SHORTEST PATHS(G, w, ) 1. topologicall ort the vertice of G. INITIALIZE-SINGLE-SOURCE(V, ) (V+E) (V). for each vertex u, taken in topologicall orted order 4. do for each vertex v Adj[u]. do RELAX(u, v, w) (V) (E) (V+E) Running time: (V+E)

30 Example 6 1 r t x z r t x z r t x z

31 Example 6 1 r t x z r t x z r t x z

32 Example 6 1 r t x z