2.3 Optimal paths. Optimal (shortest or longest) paths have a wide range of applications:

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1 . Optimal paths Optimal (shortest or longest) paths have a wide range of applications: Google maps, GPS navigators planning and management of transportation, electrical and telecommunication networks project planning VLSI design subproblems of more complex problems... E. Amaldi Foundations of Operations Research Politecnico di Milano

2 .. Shortest path problem Problem Given a directed graph G = (N, A) with a cost c ij for each arc (i, j) A, and two nodes s and t, determine a minimum cost (shortest) path from s to t. s is the origin s t 5 t is the destination c ij represents the cost (length, travel time, ) of arc (i, j) A E. Amaldi Foundations of Operations Research Politecnico di Milano

3 .. Dijkstra s algorithm Assumption c ij 0 (i, j) A Edsger Dijkstra (90-00) Definition: A path (i, i ), (i, i ),, (i k-, i k ) is simple if i u i v for all u v (a node is visited at most once) i i k i Property: If c ij 0 for all (i, j) A, every shortest path is simple. E. Amaldi Foundations of Operations Research Politecnico di Milano

4 input G = (N, A) with n = N and m = A, a node s N, c ij 0 (i, j) A with c ij = + if (i, j) A output Shortest paths from s to all other nodes of G s t 5 E. Amaldi Foundations of Operations Research Politecnico di Milano

5 Idea: Consider the nodes in order of increasing cost of the shortest path from s to any one of the other nodes. To each node j N, we assign a label L[j] which corresponds, at the end of the algorithm, to the cost of a minimum cost path from s to j. # 0 Greedy w.r.t. paths from s to j! 0 s # 6 8 # # # t # 5 E. Amaldi Foundations of Operations Research Politecnico di Milano 5

6 Example s t Two labels associated with each node j in S: [ L[j], pred[j] ] where L[j] = cost of a shortest path from s to j, pred[j] = predecessor of j in the shortest path from s to j [ 0, ] + ( S ) S * 0 + < 0 + [, ] s S E. Amaldi Foundations of Operations Research Politecnico di Milano 6

7 S * [ 0, ] [, ] 0 + < + [, ] 0 + < ( S ) 8 5 S [ 0, ] [, ] + ( S ) [, ] 0 + < + 0 * 5 [ 6, ] [ 0, ] S 0 + ( S ) [, ] [, ] 5 E. Amaldi Foundations of Operations Research Politecnico di Milano 7 [ 6, ] * [ 8, 5 ] 6 + < + 0

8 [ 0, ] [, ] 0 + ( S ) * [ 8, 5 ] S [, ] [ 6, ] 5 S [, ] [ 8, 5 ] + ( S ) [ 0, ] * 6 [, ] [ 6, ] [, ] E. Amaldi Foundations of Operations Research Politecnico di Milano 8 5

9 A set of shortest path from s to every other node j can be retrieved backwards: pred[j], pred[ pred [j] ],..., s [ 0, ] s [, ] 6 8 [, ] [ 8, 5 ] 6 [ 6, ] t [, ] j= t resulting path from s to t resulting path from s to node E. Amaldi Foundations of Operations Research Politecnico di Milano 9

10 Dijkstra s algorithm Data structure S N subset of nodes whose labels are final L[ j] =... cost of a shortest path from s to j, j S min{ L[i] + c ij : (i, j) + (S) }, j S E. Amaldi Foundations of Operations Research Politecnico di Milano 0

11 s v L[v]= h c vh = L[h] + ( S ) i S c ij = L[i]=5 L[j] j Given a directed graph G and the current subset of nodes S N, consider the outgoing cut + (S) and select (v,h) + (S) such that: L[v] + c vh = min { L[i] + c ij : (i, j) + (S) } thus L[v] + c vh L[i] + c ij (i,j) + (S) E. Amaldi Foundations of Operations Research Politecnico di Milano

12 pred[ j] = predecessor of j in the shortest path from s to j j S v such that L[v]+c vj = min {L[i] + c ij : i S} j S with c ij = if (i,j) A s L[i]=5 i c ij = j v c vj = L[v]= L[ j]=5 pred[ j]=v S + (S) E. Amaldi Foundations of Operations Research Politecnico di Milano

13 Pseudocode of Dijkstra s algorithm input output G = (N, A), n = N, m = A, s N, c ij 0 (i, j) A Shortest paths from s to all the other nodes BEGIN END S := {s}; L[s] := 0; pred[s] := s; WHILE S n DO Select (v,h) + (S) ={ (i,j) : (i,j)a, is, js} such that L[v] + c vh = min { L[i] + c ij : (i,j) + (S) }; L[h] := L[v] + c vh ; pred[h] := v; S:=S {h}; END-WHILE outgoing cut induced by S N.B.: If + (S)=, the algorithm ends: no h N \ S is reachable from s E. Amaldi Foundations of Operations Research Politecnico di Milano

14 Complexity Depends on how, at each iteration, the arc (v,h) is selected among those of the current outgoing cut + (S). If all m arcs are scanned and those that do not belong to + (S) are discarded, the overall complexity would be O(nm), hence O(n )for dense graphs. If all labels L[j] are determined by appropriate updates ( Prim s algorithm), we need to consider a single arc of + (S) for each node js overall complexity O(n ). E. Amaldi Foundations of Operations Research Politecnico di Milano

15 Proposition: Dijkstra s algorithm is exact. Proof At k-th step: S = {s, i,..., i k } and L[ j ] = cost of a minimum cost path from s to j, j S cost of a minimum cost path with all intermediate nodes in S j S By induction on the number k of steps : inductive basis : it is true for k = since S = {s}, L[s] = 0 and L[ j] = c sj j s. inductive step : if it is true at the k-th step, it is also true at the (k+)-th step. E. Amaldi Foundations of Operations Research Politecnico di Milano 5

16 (k+)-th step: Let h S be the node that is inserted in S and the path from s to h such that: L[v] + c vh L[i] + c ij (i, j) + (S). Let us verify that every path from s to h has c() c(). s S v L[v] i j h There exist is and js such that = (i, j) where (i, j) is the first arc + (S). by choice of (v,h) c() = c( ) + c ij + c( ) L[i] + c ij L[v] + c vh = c(). c( ) L[ i ]: by induction assumption c( ) 0 because c ij 0 E. Amaldi Foundations of Operations Research Politecnico di Milano 6

17 A set of shortest paths from s to all the nodes j can be retrieved via the vector of predecessors. [, ] [ 8, 5 ] [ 0, ] s t [, ] 5 [, ] [ 6, ] E. Amaldi Foundations of Operations Research Politecnico di Milano 7

18 Remarks ) Taking the union of a set of shortest paths from node s to all the other nodes of G, we obtain an arborescence rooted at s. s t 5 Such arborescences, which are referred to as shortest path trees, have nothing to do with minimum cost spanning trees! E. Amaldi Foundations of Operations Research Politecnico di Milano 8

19 ) Dijkstra s algorithm is not applicable when there are arcs with negative cost c ij. Example Dijkstra s algorithm yields the path (,),(,) of cost, - but path (,),(,),(,) has cost. Due to c < 0 the last step in the exactness proof fails! The minimum cost from to is not updated after the first step. According to a greedy choice on the arcs in + ({}), it is taken as c which is locally optimal (c < c ) even though the path (,),(,) has a strictly smaller cost because c < 0. E. Amaldi Foundations of Operations Research Politecnico di Milano 9

20 Exercise Determine the shortest paths form node 0 to all the other nodes of the following graph: E. Amaldi Foundations of Operations Research Politecnico di Milano 0

21 .. Shortest path problem with negative costs Observation: If the graph G contains a circuit of negative cost, the shortest path problem may not be well-defined. Example s t -6 circuit s cost: - Each time we go through the circuit, the cost decreases. There is no finite shortest path from s to t. E. Amaldi Foundations of Operations Research Politecnico di Milano

22 Floyd-Warshall s algorithm allows to detect the presence of circuits with negative cost, and hence identify the cases in which the problem is ill-defined. It provides a set of shortest paths between all pairs of nodes, even when there are arcs with negative cost. It is based on iteratively applying a triangular operation described below. E. Amaldi Foundations of Operations Research Politecnico di Milano

23 Floyd-Warshall s algorithm input output Directed graph G = (N, A) described via the n x n cost matrix C = [c ij ]. For each pair of nodes i, j N, the cost d ij of shortest path from i to j. Data structure: two n x n matrices D and P whose elements correspond, at the end of the algorithm, to d ij = cost of a shortest path from i to j p ij = predecessor of j in shortest path from i to j E. Amaldi Foundations of Operations Research Politecnico di Milano

24 Example D P For (i, j) A set d ij = c ij, for loops d ii = 0, and for (i, j) A set d ij =. The matrix of predecessors is initialized with p ij = i, for all i. E. Amaldi Foundations of Operations Research Politecnico di Milano Notation: from to, from to

25 Triangular operation with respect to node h: d ih h d hj j i d ij For each pair of nodes i, j, with i h and j h (including case i=j), check whether when going from i to j it is more convenient to go via h : d ih + d hj < d ij Cycle h= : Since there are no such arcs, the matrices D and P remain unchanged E. Amaldi Foundations of Operations Research Politecnico di Milano 5

26 Cycle for h= skip i = and j = D P = d < d +d = + = no update 6 = d < d +d = + = no update -0 = d < d +d = + = no update 8 = d < d +d = + = no update 0 = d < d +d = + = no update = d < d +d = + = no update = d < d +d = + = no update 9 = d < d +d = + = no update 0 = d < d +d = + = no update E. Amaldi Foundations of Operations Research Politecnico di Milano 6

27 Cycle for h= skip i = and j = D P HALT! 0 = d < d +d = + = no update = d < d +d = + 6 = 9 no update = d > d +d = - 0 = -7 p := p = = d = d +d = 8 + = no update 0 = d < d +d = = no update = d > d +d = 8-0 = - p := p = = d < d +d = + = 9 = d > d +d = + 6 = 7 0 = d > d +d = - 0 = -9 no update p := p = negative cost cycle found! E. Amaldi Foundations of Operations Research Politecnico di Milano 7

28 Pseudocode of Floyd-Warshall s algorithm BEGIN FOR i:= TO n DO FOR j:= TO n DO p ij := i; END-FOR END-FOR FOR h:= TO n DO /* triangular operation w.r.t. h */ FOR i:= TO n WITH i h DO FOR j:= TO n WITH j h DO IF (d ih + d hj < d ij ) THEN Triangular operation: d ij = d ih + d hj ; Update d ij if it is convenient in p ij := p hj ; terms of cost to reach j from i END-IF via h END-FOR END-FOR FOR i:= TO n DO IF d ii < 0 THEN STOP; END-IF /* a circuit with negative cost*/ END-FOR END-FOR END E. Amaldi Foundations of Operations Research Politecnico di Milano 8

29 Proposition: Floyd-Warshall s algorithm is exact. Proof s idea: Assume the nodes of G are numbered from to n. Just verify that, if the node index order is followed, after the h-th cycle the value d ij for any i and j corresponds to the cost of a shortest path from i to j with only intermediate nodes in {,, h}. Complexity Since in the worst case the triangular operation is executed for all nodes h and for each pair of nodes i and j, the overall complexity is O(n ). E. Amaldi Foundations of Operations Research Politecnico di Milano 9

30 Exercise Find the shortest paths between every pair of nodes of the following graph: E. Amaldi Foundations of Operations Research Politecnico di Milano 0