A Linear Time Algorithm for Finding Three Edge-Disjoint Paths in Eulerian Networks

Transcription

1 A Linear Time Algorithm for Finding Three Edge-Disjoint Paths in Eulerian Networks Maxim A. Babenko Ignat I. Kolesnichenko Ilya P. Razenshteyn Moscow State University 36th International Conference on Current Trends in Theory and Practice of Computer Science Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

2 General Problem Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

3 General Problem Given an undirected graph G = (V, E) and k pairs of distinct terminals {s i, t i }, i = 1,...,k, find a collection of k paths P 1,..., P k such that P i connects s i with t i. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

4 General Problem Given an undirected graph G = (V, E) and k pairs of distinct terminals {s i, t i }, i = 1,...,k, find a collection of k paths P 1,..., P k such that P i connects s i with t i. Two versions: Paths P i are node-disjoint. Paths P i are edge-disjoint. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

5 General Problem Given an undirected graph G = (V, E) and k pairs of distinct terminals {s i, t i }, i = 1,...,k, find a collection of k paths P 1,..., P k such that P i connects s i with t i. Two versions: Paths P i are node-disjoint. Paths P i are edge-disjoint. P 1 s 1 t 1 P 2 s 2 t 2 P 3 s 3 t 3 Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

6 Known results Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

7 Known results The node-disjoint version is a more general one. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

8 Known results The node-disjoint version is a more general one. Both versions are solvable in polynomial time [Robertson, Seymour 1995]. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

9 Known results The node-disjoint version is a more general one. Both versions are solvable in polynomial time [Robertson, Seymour 1995]. The algorithms are complicated and have a high running time. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

10 Eulerian Case Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

11 Eulerian Case We require paths P i to be edge-disjoint. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

12 Eulerian Case We require paths P i to be edge-disjoint. For each node v V we require that deg v be odd iff v is a terminal (Eulerianess). Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

13 Eulerian Case Remark We require paths P i to be edge-disjoint. For each node v V we require that deg v be odd iff v is a terminal (Eulerianess). This definition only makes sense if all the terminals are distinct. If some of them may coincide then one must adjust it as follows: deg v is odd iff v coincides with an odd number of terminals. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

14 Eulerian Case: k = 1 Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

15 Eulerian Case: k = 1 Let k = 1, i.e. the network has a single pair of terminals {s 1, t 1 }. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

16 Eulerian Case: k = 1 Let k = 1, i.e. the network has a single pair of terminals {s 1, t 1 }. A problem instance is feasible iff the terminals are in the same connected component. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

17 Eulerian Case: k = 1 Let k = 1, i.e. the network has a single pair of terminals {s 1, t 1 }. A problem instance is feasible iff the terminals are in the same connected component. Checking for feasiblity and constructing P 1 and be carried out in linear time (trivial). Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

18 Eulerian Case: k = 2 Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

19 Eulerian Case: k = 2 Let k = 2, i.e. the network has two pairs of terminals: {s 1, t 1 } and {s 2, t 2 }. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

20 Eulerian Case: k = 2 Let k = 2, i.e. the network has two pairs of terminals: {s 1, t 1 } and {s 2, t 2 }. Suppose s 1 and t 1 are in distinct connected components. Then the instance is clearly infeasible. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

21 Eulerian Case: k = 2 Let k = 2, i.e. the network has two pairs of terminals: {s 1, t 1 } and {s 2, t 2 }. Suppose s 1 and t 1 are in distinct connected components. Then the instance is clearly infeasible. Otherwise, let P 1 be an arbitrary s 1 t 1 path. Then G P is Eulerian (w.r.t. the remaining terminals {s 2, t 2 }), hence always feasible. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

22 Eulerian Case: k = 2 Let k = 2, i.e. the network has two pairs of terminals: {s 1, t 1 } and {s 2, t 2 }. Suppose s 1 and t 1 are in distinct connected components. Then the instance is clearly infeasible. Otherwise, let P 1 be an arbitrary s 1 t 1 path. Then G P is Eulerian (w.r.t. the remaining terminals {s 2, t 2 }), hence always feasible. Checking for feasiblity and constructing P 1, P 2 and be carried out in linear time (trivial). Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

23 Our Contribution Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

24 Our Contribution Let k = 3. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

25 Our Contribution Let k = 3. We give a linear-time algorithm for constructing P 1, P 2, P 3 (if they exist). Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

26 Our Contribution Let k = 3. We give a linear-time algorithm for constructing P 1, P 2, P 3 (if they exist). Three seems to be the largest value of k where some elementary but non-trivial combinatorics works. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

27 Some Easy Observations Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

28 Some Easy Observations Graph G may be assumed to be connected. (Components containing no terminal may be dropped. If a component contains some but not all terminals then the instance either infeasible or reduces to k = 2 and k = 1.) Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

29 Some Easy Observations Graph G may be assumed to be connected. (Components containing no terminal may be dropped. If a component contains some but not all terminals then the instance either infeasible or reduces to k = 2 and k = 1.) It suffices to construct a pair of edge-disjoint paths P 1 and P 2 (out of three). The remaining problem reduces to k = 1 and, hence, is always feasible. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

30 Feasibility Criterion Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

31 Feasibility Criterion Consider the demand graph H with nodes T := {s 1, s 2, s 3, t 1, t 2, t 3 } and edges s 1 t 1, s 2 t 2, s 3 t 3. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

32 Feasibility Criterion Consider the demand graph H with nodes T := {s 1, s 2, s 3, t 1, t 2, t 3 } and edges s 1 t 1, s 2 t 2, s 3 t 3. Remark An instance is Eulerian iff G + H is an Eulerian graph. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

33 Feasibility Criterion Consider the demand graph H with nodes T := {s 1, s 2, s 3, t 1, t 2, t 3 } and edges s 1 t 1, s 2 t 2, s 3 t 3. Remark An instance is Eulerian iff G + H is an Eulerian graph. For a subset U V let d G (U) (resp. d H (U)) be the number of edges connecting U with V U in graph G (resp. H). Theorem For k = 3 a problem instance is feasible iff d G (U) d H (U) for each U V (cut conditions). Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

34 Feasibility Criterion Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

35 Feasibility Criterion Remark Proof of the above criterion was given by Schrijver. His method applies reductio ad absurdum to the smallest counterexample and is non-algorithmic. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

36 Feasibility Criterion Remark Proof of the above criterion was given by Schrijver. His method applies reductio ad absurdum to the smallest counterexample and is non-algorithmic. Remark In a general problem cut conditions are necessary but not sufficient. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

37 Checking for Feasibility Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

38 Checking for Feasibility There are exponentally many subsets U V. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

39 Checking for Feasibility There are exponentally many subsets U V. Put U := U T and call U the signature of U. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

40 Checking for Feasibility There are exponentally many subsets U V. Put U := U T and call U the signature of U. There are finitely many signatures. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

41 Checking for Feasibility There are exponentally many subsets U V. Put U := U T and call U the signature of U. There are finitely many signatures. We check feasibility for each signature separately. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

42 Checking for Feasibility Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

43 Checking for Feasibility Recap: check that d G (U) d H (U) for each U V. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

44 Checking for Feasibility Recap: check that d G (U) d H (U) for each U V. Suppose the singature U = U T is fixed. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

45 Checking for Feasibility Recap: check that d G (U) d H (U) for each U V. Suppose the singature U = U T is fixed. ν := d H (U) = d H (U ), hence the right-hand side is fixed. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

46 Checking for Feasibility Recap: check that d G (U) d H (U) for each U V. Suppose the singature U = U T is fixed. ν := d H (U) = d H (U ), hence the right-hand side is fixed. Checking that d G (U) ν for all U V obeying U T = U reduces to a max-flow computation (takes linear time). Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

47 Checking for Feasibility Recap: check that d G (U) d H (U) for each U V. Suppose the singature U = U T is fixed. ν := d H (U) = d H (U ), hence the right-hand side is fixed. Checking that d G (U) ν for all U V obeying U T = U reduces to a max-flow computation (takes linear time). Overall, checking the instance for feasiblity takes linear time. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

48 Local Moves Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

49 Local Moves For a terminal t T and an edge e = tx E one may carry one a local move along e, i.e. replace t with x and remove e from E. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

50 Local Moves For a terminal t T and an edge e = tx E one may carry one a local move along e, i.e. replace t with x and remove e from E. Local moves preserve Eulerianess. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

51 Local Moves For a terminal t T and an edge e = tx E one may carry one a local move along e, i.e. replace t with x and remove e from E. Local moves preserve Eulerianess. If an instance is feasible then either s 1 = t 1 or s 1 admits a local move. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

52 Naive Approach Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

53 Naive Approach Trace a path P 1 from s 1 to t 1 by applying local moves to s 1 that preserve feasiblity. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

54 Naive Approach Trace a path P 1 from s 1 to t 1 by applying local moves to s 1 that preserve feasiblity. Stop when s 1 = t 1. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

55 Naive Approach Trace a path P 1 from s 1 to t 1 by applying local moves to s 1 that preserve feasiblity. Stop when s 1 = t 1. The remaining problem corresponds to k = 2 and hence is solvable in linear time. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

56 Naive Approach How fast is this method? Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

57 Naive Approach How fast is this method? Each feasibility check costs O(E) time. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

58 Naive Approach How fast is this method? Each feasibility check costs O(E) time. If a certain move was found to be infeasible at some stage then it remains infeasbile. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

59 Naive Approach How fast is this method? Each feasibility check costs O(E) time. If a certain move was found to be infeasible at some stage then it remains infeasbile. Each edge is checked at most once. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

60 Naive Approach How fast is this method? Each feasibility check costs O(E) time. If a certain move was found to be infeasible at some stage then it remains infeasbile. Each edge is checked at most once. O(E) feasibility checks are totally necessary. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

61 Naive Approach How fast is this method? Each feasibility check costs O(E) time. If a certain move was found to be infeasible at some stage then it remains infeasbile. Each edge is checked at most once. O(E) feasibility checks are totally necessary. The overall running time is O(E 2 ). Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

62 Naive Approach How fast is this method? Each feasibility check costs O(E) time. If a certain move was found to be infeasible at some stage then it remains infeasbile. Each edge is checked at most once. O(E) feasibility checks are totally necessary. The overall running time is O(E 2 ). Can one do better? Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

63 Algorithm Outline Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

64 Algorithm Outline Check for feasibility of the initial problem instance (takes linear time). Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

65 Algorithm Outline Check for feasibility of the initial problem instance (takes linear time). Construct an arbitrary node-simple path P 1 from s 1 to t 1 (takes linear time). Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

66 Algorithm Outline Check for feasibility of the initial problem instance (takes linear time). Construct an arbitrary node-simple path P 1 from s 1 to t 1 (takes linear time). Apply the sequence of moves along P 1 thus bringing s 1 to t 1. Consider the corresponding sequence of networks: (G, H) = (G 0, H 0 ),(G 1, H 1 ),...,(G m, H m ) (here H i are the demand graphs) Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

67 Algorithm Outline Check for feasibility of the initial problem instance (takes linear time). Construct an arbitrary node-simple path P 1 from s 1 to t 1 (takes linear time). Apply the sequence of moves along P 1 thus bringing s 1 to t 1. Consider the corresponding sequence of networks: (G, H) = (G 0, H 0 ),(G 1, H 1 ),...,(G m, H m ) (here H i are the demand graphs) If (G m, H m ) is feasible then we are done (problem reduces to k = 2) (takes linear time). Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

68 Algorithm Outline Check for feasibility of the initial problem instance (takes linear time). Construct an arbitrary node-simple path P 1 from s 1 to t 1 (takes linear time). Apply the sequence of moves along P 1 thus bringing s 1 to t 1. Consider the corresponding sequence of networks: (G, H) = (G 0, H 0 ),(G 1, H 1 ),...,(G m, H m ) (here H i are the demand graphs) If (G m, H m ) is feasible then we are done (problem reduces to k = 2) (takes linear time). Otherwise, find an index j such that (G j, H j ) is still feasible whereas (G j+1, H j+1 ) is infeasible (takes linear time, employs incremental reachability data structure). Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

69 Algorithm Outline Check for feasibility of the initial problem instance (takes linear time). Construct an arbitrary node-simple path P 1 from s 1 to t 1 (takes linear time). Apply the sequence of moves along P 1 thus bringing s 1 to t 1. Consider the corresponding sequence of networks: (G, H) = (G 0, H 0 ),(G 1, H 1 ),...,(G m, H m ) (here H i are the demand graphs) If (G m, H m ) is feasible then we are done (problem reduces to k = 2) (takes linear time). Otherwise, find an index j such that (G j, H j ) is still feasible whereas (G j+1, H j+1 ) is infeasible (takes linear time, employs incremental reachability data structure). Deal with (G j, H j ) (takes linear time). Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

70 Critical Instance Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

71 Critical Instance Recap: (G j, H j ) is feasible, (G j+1, H j+1 ) is infeasible. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

72 Critical Instance Recap: (G j, H j ) is feasible, (G j+1, H j+1 ) is infeasible. Lemma There exists and can be found in linear time a subset U V such that d Gj (U) = d Hj (U) = 2, G j [U] is connected and U T j = 2. U V U p 1 u 1 v 2 q 1 p 2 u 2 v 2 q 2 p 3 q 3 Here {p 1, q 1 },{p 2, q 2 },{p 3, q 3 } are the pairs of corresponding terminals. Terminal s 1 is one of q 1, q 2, q 3, p 3. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

73 Dealing with Critical Instance U V U p 1 u 1 v 2 q 1 p 2 u 2 v 2 q 2 p 3 q 3 Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

74 Dealing with Critical Instance U V U p 1 u 1 v 2 q 1 p 2 u 2 v 2 q 2 p 3 q 3 Construct a pair of paths from p 1, p 2 to q 1, q 2 in G j (takes linear time, existence follows from feasibility of (G j, H j )). Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

75 Dealing with Critical Instance U V U p 1 u 1 v 2 q 1 p 2 u 2 v 2 q 2 p 3 q 3 Construct a pair of paths from p 1, p 2 to q 1, q 2 in G j (takes linear time, existence follows from feasibility of (G j, H j )). If p 1 turns out connected to q 2 and p 2 to q 1 then apply a switch using the subgraph G j [U] (reduces to k = 2, takes linear time). Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

76 Dealing with Critical Instance Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

77 Dealing with Critical Instance Two out of three requested paths (say, P 1 and P 2 ) are ready. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

78 Dealing with Critical Instance Two out of three requested paths (say, P 1 and P 2 ) are ready. Remove the edge sets of P 1 and P 2. Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

79 Dealing with Critical Instance Two out of three requested paths (say, P 1 and P 2 ) are ready. Remove the edge sets of P 1 and P 2. Construct P 3 (reduces to k = 1, takes linear time). Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

80 Dealing with Critical Instance Two out of three requested paths (say, P 1 and P 2 ) are ready. Remove the edge sets of P 1 and P 2. Construct P 3 (reduces to k = 1, takes linear time). We are done! Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20

81 Thank you! Babenko, Kolesnichenko, Razenshtein (MSU) Finding Three Edge-Disjoint Paths SOFSEM / 20