Myriad of applications

Transcription

1 Shortet Path

2 Myriad of application Finding hortet ditance between location (Google map, etc.) Internet router protocol: OSPF (Open Shortet Path Firt) i ued to find the hortet path to interchange package between erver (IP) Traffic information ytem Routing in VSLI etc...

3 Shortet ditance between two point not alway follow human intuition It may depend on many more contrain than the pure geometric one.

4 Shortet path problem in direct weighted graph Given a digraph G = (V, E) with edge weight w : E R, a path p = {v 0,..., v k } i a equence of conecutive edge, where (v i, v i+ ) E define w(p) = k i=0 w(v i, v i+ ). The hortet path between u and v a δ(u, v) = min p {w(p) u p v} v v v v 4 v δ(v, v 5 ) = If G i undirected, we can conider every edge a doubly directed. Unweighted, every edge of weight =.

5 Optimal ubtructure of hortet path Given G = (V, E), w : E R, for any hortet path p : u v and any i, j vertice in p, the ub-path p = i j in p ha the hortet ditance δ(i, j). i j u v Negative cycle u 4 5 v δ(u, v) =

6 Taxonomy of hortet path problem Single ource hortet path (SSSP): Given G = (V, E), w : E R and V, compute δ(, v), v V {v}. In the graph below we want to compute (, a), (, b), (, c), (, d) All path hortet path (APSP): Given = (V, E), w : E R compute δ(u, v) for every pair (u, v) V V. In the graph below we want to compute (, a), (a, ),... (d, b), (b, d), (d, c), (c, d) a b d c

7 Single ource hortet path Let u conider the particular cae of having a ource and a ink t Aume that w : e R + Brute-force(G, W,, t) for all imple p : t do compute w(p) end for Compare all p return the p with mallet w(p) The number of path could be O( n ) a d 4 n b t t

8 Shortet Path Tree SSSP algorithm have the property that at termination the reulting path form a hortet path tree. Given G = (V, E) with edge weight w e and a ditinguihed V, a hortet path tree i a directed ub-tree T = (V, E ) of G,.t. T i rooted at, V i the et of vertice in G reachable from, v V the path v in T i the hortet path δ(, v). a d b c f a d b c f

9 Triangle Inequality Given G = (V, E), W, if u, v, z V, notice the hortet path u v i any other path between u and v. Therefore. Theorem For all u, v, z V δ(u, v) δ(u, z) + δ(z, v). u v u v δ(u, z) Want minimum z δ(z, y) δ(u, v) - - z Notice, in thi cae δ(u, v) =

10 Baic technique for SSSP: Relaxation Given G = (V, E), W. v V we maintain a SP-etimate d[v], which i an UB on δ(, v). Initially, tart with d[v] = +, v V {} and d[] = 0. Repeatedly improve etimate toward the goal d[v] = δ(, v). For (u, v) E, Relax(u, v, w(u, v)) if d[v] > d[u] + w(u, v) then d[v] = d[u] + w(u, v) end if d[u] d[v] u w(u,v) v

11 Generic Relaxation algorithm Relaxation(G, W, ) for all v V {} do d[v] = + end for d[] = 0 while (u, v) with d[v] > d[u] + w(u, v) do Relax(u, v, w(u, v)) end while Lemma For all v V, Relaxation(G, W, ) maintain the invariant that d[v] δ(, v). Proof (Induction) I.H. when applying Relax(u, v, w(u, v)) we get d[u] δ(, u) By the triangle ineq. δ(, v) δ(, u) + δ(u, v) d[u] + w(u, v). Therefore, letting δ(u, v) = d[u] + w(u, v) i not a problem.

12 Generic Relaxation algorithm a b a b d c d c a b d c a d a b b c d c

13 Relaxation algorithm Note: If a negative-weight cycle i reachable from, then Relaxation(G, W, ) doe not top. If we apply Relaxation(G, W, ) to the graph in the graph below, we get that the number of tep follow T (n) = T (n ) +, T (n) = Θ( n/ ) n Prove that indeed the complexity of Relaxation on the above graph i T (n) = Θ( n/ )

14 Recall: Dktra SSSP E.W.Dktra, A note on two problem in connexion with graph. Num. Mathematik, (959) Greedy algorithm. Relax edge in an increaing ball around. Ue a priority queue Q Dtra doe not work with negative weight Complexity: Fatet SSSP algorithm. Dktra(G, W, ) Initialize SP-etimate on V S =, Q = {V } while Q do u =EXT-MIN(Q) S = S {u} for all v Adj[u] do Relax(u, v, w(u, v)) end for end while Q implementation Wort-time complexity Array O(n ) Heap O(m lg n) Fibonacci heap O(m + n lg n)

15 Bellman-Ford-Moore SSSP R. Bellman (958) L. Ford (956) E. Moore (957) The algorithm BFM i ued for G with negative weight, but without negative cycle. Given G, w, V (G),with n vertice and m edge, the BFM algorithm doe n iteration: Each iteration i doe a relaxation on all edge than can be reached from in at mot i-tep, the remaining one are et to (e, e,..., e n ), (e }{{}, e,..., e n ), (e }{{}, e,..., e n ) }{{} i= i= i=n

16 BFM Algorithm Recall that given a graph G, V = n, E = n, and a et of edge weight w with a ource vertex v V, BFM (G, w, ) Initialize v, d[v] =, π[v] = u Initialize d[] = 0 for i = to n do for every (u, v) E do Relax(u, v, w(u, v)) end for end for for every (u, v) E do if d[v] > d[u] + w(u, v) then return Negative-weight cycle end if end for Relax(u, v, w(u, v)) if d[v] > d[u] + w(u, v) then d[v] = d[u] + w(u, v) π[v] = u end if

17 BFM Algorithm: Example a 4 g b f e c d Node i a b c d e f g

18 BFM Algorithm: Example a 4 g 8 b f e c d Node i a b c d e f g

19 BFM Algorithm: Example a 4 g 8 b f 9 c e d Node i a b c d e f g

20 BFM Algorithm: Example a b 4 g f 8 9 c e 8 d Node i a b c d e f g

21 BFM Algorithm: Example a b 4 g f 8 9 c e 7 d Node i a b c d e f g

22 BFM Algorithm: Example a b c 4 g f e d 4 Node i a b c d e f g

23 BFM Algorithm: Example a b c 4 g f e d 0 Node i a b c d e f g

24 BFM Algorithm: Example a b c 4 g f e d 9 Node i a b c d e f g

25 Complexity BFM O(n) O() O(m) O(nm) O(m) Complexity T(n)=O(nm)

26 Correctne of BFM Lemma In the BFM-algorithm, after the ith. iteration we have that d[v] the weight of every path v uing at mot i edge, v V. Proof (Induction on i) Before the ith iteration, d[v] min{w(p)} over all path p with at mot i edge. The relaxation only decreae d[v] The ith iteration conider all path with i edge when relaxing the edge to v. at mot i edge v

27 Correctne of BFM Theorem If G, w ha no negative weight cycle, then at the end of the BFM-algorithm d[v] = δ(, v). Proof Without negative-weight cycle, hortet path are alway imple. Every imple path ha at mot n vertice and n edge. By the previou lemma, the n iteration yield d[v] δ(, v). By the invariance of the relaxation algorithm d[v] δ(, v).

28 Correctne of BFM Theorem BFM will report negative-weight cycle if there exit in G. Proof Without negative-weight cycle in G, the previou theorem implie d[v] = δ(, v), and by triangle inequality d[v] δ(, u) + w(u, v), o BFM won t report a negative cycle if it doen t exit. If there i a negative-weight cycle, then one of it edge can be relaxed, o BFM will report correctly.

29 Shortet path in a direct acyclic graph (dag). Min-cot path in DAG INPUT: Edge weighted dag G = (V, E, w), V = n, w : E R together with given, t V. QUESTION: Find a path P : t of minimum total weight. Notice given a dag G = (V, E), W we wih to find a path P from to t.t. min P () P w. a c 6 b e g

30 Arranging dag into a line Arrange the dag in topological order, o that all edge go from left to right. Thi can be done in O(n) uing DFS. a c 6 b e f c a 6 b f e DFS

31 We want to find horter ditance from to v. Let d(v) = ditance v d(f ) = min{d(b)+, d(c)+} The chema i baed on the topological linearity of G. c a b f e 6 Shortet ditance in dag G Initialize d() := 0 and v V {}, d(v) := for all v V {} in linearized order do d(v) := min (u,v) E {d(u) + w uv } end for Complexity? T (n) = O(n)

32 All pair hortet path: APSP Given G = (V, E), V = n, E = m and a weight w : E R we want to determine u, v V, δ(u, v). We aume we can have w < 0 but G doe not contain negative cycle. Naive idea: We apply O(n) time BFM or Dktra (if there are not negative weight) Repetition of BFM: O(n m) Repetition of Dktra: O(nm lg n) (if Q i implemented by a heap)

33 All pair hortet path: APSP Unlike in the SSSP algorithm that aumed adjacency-lit repreentation of G, for the APSP algorithm we conider the adjacency matrix repreentation of G. For convenience V = {,,... n}. The n n adjacency matrix W = (w(i, j)) of G, w: 0 if i = j w = w if (i, j) E if i j and (i, j) E

34 All pair hortet path: APSP The input i a n n adjacency matrix W = (w ) W = The output i a n n matrix D = (d ), where d = δ(i, j) For the implementation we alo need to compute the et of predeceor matrix Π k

35 Bernard-Floyd-Warhall Algorithm R. Bernard: Tranitivité et connexité C.R.Aca. Sci. 959 R. Floyd: Algorithm 97: Shortet Path. CACM 96 S. Warhall: A theorem on Boolean matrice. JACM, 96 The BFW Algorithm ued dynamic programming to compare all poible path between each pair of vertice in G. The algorithm work in O(n ) and the number of edge could be O(n ).

36 Optimal ubtructure of APSP Recall: Triangle inequality δ(u, v) δ(u, z) + δ(z, v). u δ(u, v) δ(u, z) δ(z, y) z Let p = p, p,..., p }{{ r, p } r and intermediate v. Let d (k) be the hortet i j.t. the intermediate vertice are in {,..., k}. So if k = 0, then d (0) = w. v

37 The recurrence Let p a hortet path i j with value d (k) If k i not an intermediate vertex of p, then d (k) If k i an intermediate vertex of p, then p = (i,..., k) (k,..., j) }{{}}{{} p p = d (k ) By triangle inequality p i a hortet path i k and p i a hortet path k j. { Therefore d (k) w if k = 0 = min{d (k ), d (k ) ik + d (k ) kj } if k

38 Bottom-up BFW-algorithm Given G = (V, E), w : E Z without negative cycle, the following DP algo. compute d (n), i, j V : BFW W = (w ) for k = to n do for i = to n do for j = to n do d (k) end for end for end for return d (n) = min{d (k ), d (k ) ik + d (k ) kj } Time complexity: T (n) = O(n ), S(n) = O(n ) but S(n) can be lowered to O(n ) How? Correctne follow from the recurrence argument.

39 Example D (0) = D() = D () = D() = D(4) = For intance, d, = (uing vertex ) = 4 (uing all vertice) d 4,

40 Example 0 NIL NIL NIL D (0) = Π (0) = NIL NIL NIL NIL 0 4 NIL NIL NIL 0 NIL NIL NIL D () = Π () = NIL NIL NIL NIL NIL NIL NIL NIL NIL D () = Π () = NIL NIL NIL NIL NIL NIL 0 D () = Π () = 0 0 NIL NIL NIL 4 NIL

41 Contructing the hortet path We want to contruct the matrix Π = (π ), where π = predeceor of j in hortet i j, we define a equence of matrice Π (0),..., Π (n).t. Π (k) = (π (k) ), i.e. the matrix of lat predeceor in the hortet path i j, which ue only vertice in {,..., k}. { If k = 0: π (k) NIL if i = j or w =, = i if i j and w. For k we get the recurrence: π (k) = { π (k ) π (k ) kj if d (k ) d (k ) ik otherwie. + d (k ) kj,

42 BFW with path BFW W d (0) = W for k = to n do for i = to n do for j = to n do if d (k) d (k) Π (k) d (k ) = d (k ) = Π (k ), d (k ) ik ele d (k) = d (k ) ik + d (k ) kj Π (k) end if end for end for end for return d (n) = Π (k ) kj Complexity: T (n) = O(n ) + d (k ) kj then

43 Concluion Dktra BFM BFW w 0 O(m lg n) O(nm) O(n ) w R NO O(nm) O(n ) There exit an algorithm by D. Johnon (978) that work in O(n lg n) for pare graph with negative edge. It ue Dktra a a function. For further reading on hortet path, ee chapter 4 and 5 of Cormen, Leieron, Rivet, Stein: Introduction to Algorithm.