Institute of Operating Systems and Computer Networks Algorithms Group. Network Algorithms. Tutorial 3: Shortest paths and other stuff

Transcription

1 Institute of Operating Systems and Computer Networks Algorithms Group Network Algorithms Tutorial 3: Shortest paths and other stuff Christian Rieck

2 Shortest paths: Dijkstra s algorithm 2

3 Dijkstra s algorithm v 6 15 v v 2 v 9 v 8 2 v v v v 5 11 v 3 Compute the shortest paths from v_0 to v_9! 3

4 Dijkstra s algorithm v_0 v_1 v_2 v_3 v_4 v_5 v_6 v_7 v_8 v_9 init v_0; infty -; infty -; infty -; infty -; infty -; infty -; infty -; infty -; infty -; infty 1 v_0; 5 v_0; 5 2 v_1; 18 3 v_2; 6 v_2; 8 4 v_4; 21 v_4; 14 5 v_7; 10 6 v_8; 17 v_8; 19 7 v_3; 28 8 v_6; 42 9 v_5; 41 4

5 Dijkstra s algorithm v_0 v_1 v_2 v_3 v_4 v_5 v_6 v_7 v_8 v_9 init v_0; infty -; infty -; infty -; infty -; infty -; infty -; infty -; infty -; infty -; infty 1 v_0; 5 v_0; 5 2 v_1; 18 3 v_2; 6 v_2; 8 the shortest path from v_0 to v_9 is: v_0 -> v_2 -> v_7 -> v_8 -> v_3 -> v_5 -> v_9 and has a total length of 41 4 v_4; 21 v_4; 14 5 v_7; 10 6 v_8; 17 v_8; 19 7 v_3; 28 8 v_6; 42 9 v_5; 41 4

6 Shortest paths: Moore-Bellman-Ford 5

7 Moore-Bellman-Ford S 8 10 E A Compute the shortest paths from S to all other vertices! D 2 B -1-2 C 6

8 Moore-Bellman-Ford S A B C D E init 0 infty infty infty infty infty S 10 8 A 12 C 10 E 9 D 5 8 A 7 C 5 there are no changes in left out rows and iterations the shortest path from S to C is for example S -> E -> D -> A -> C with total length of 7 7

9 Moore-Bellman-Ford After n-1 iterations, the algorithm gives the shortest paths from a source to all other vertices. After an additional iteration, the algorithm discovers a negative cycle, if one exist. 8

10 Shortest paths: Can we do something better? 9

11 Can we do something better? Bidirectional Dijkstra alternate between forward search from s and backward search from t algorithm terminates when some vertex w has been deleted from the queue of both searches this may reduce the search space Landmarks compute the shortest paths for some important vertices in a preprocessing step use these landmarks in shortest path algorithms 10

12 Can we do something better? A* uses a heuristic function to consider vertices that appear to lead most quickly to the target vertex first value at vertex x: f(x) = g(x) + h(x); where g(x) is the distance to that vertex from the source vertex and h(x) is the estimated distance from x to the target vertex Question: Which properties must h(x) have? 11

13 Can we do something better? Highway Hierarchies preprocessing hierarchical classification of the edges query local search from source vertex until the algorithm reaches an edge of higher rank go on at higher rank edges use this classification in shortest path algorithms 12

14 Hamiltonian Cycle Problem 13

15 Hamiltonian cycle Given: Graph G=(V,E), all edges have unit weight Wanted: A cycle of weight V This problem is NP-complete (nasty reduction from 3SAT). 14

16 Hamiltonian cycle still NP-complete for simple planar graphs with max degree 3 (Garey et al. 1974) planar non-alternating indegree-2 outdegree-2 (Demaine et al. 2018) there is always a Hamiltonian cycle in 4-connected planar graphs (Tutte 1956) the problem is polynomially solvable for solid square grid graphs (Umans et al. 1996) 15

17 Hamiltonian cycle Necessary conditions: G has a 2-factor, i.e., there is a set of disjoint cycles, covering all vertices G is 1-tough, i.e., remove a subset S of vertices; the resulting graph has at most S components Sufficient condition(s): for any two vertices u,v that are not adjacent, the following holds d(u)+d(v) V 3 there are many more sufficient conditions of this kind 16

18 Hamiltonian cycle Necessary conditions: G has a 2-factor, i.e., there is a set of disjoint cycles, covering all vertices G is 1-tough, i.e., remove a subset S of vertices; the resulting graph has at most S components This is a co-np-hard problem (Bauer et al. 1990) Sufficient condition(s): for any two vertices u,v that are not adjacent, the following holds d(u)+d(v) V 3 there are many more sufficient conditions of this kind 16

19 Traveling Salesman Problem 17

20 Traveling Salesman Problem Given: Weighted graph G=(V,E) Wanted: Hamiltonian cycle with minimum weight Trivial: this problem is NP-complete. It is one of the most studied optimization problems! Naive algorithm: check all O(n!) tours Better: use dynamic programming O(2 n n 2 ) There are a lot of different variants like MetricTSP, Bottleneck, 18

21 Traveling Salesman Problem in 1954, Danzig et al. are able to solve an instance consisting of 49 vertices nowadays we are able to solve instances with more than 85,000 vertices (Concorde, Applegate et al.) for metrictsp there is a 1.5-approximation (Christofides 1976) there is a PTAS for Euclidean TSP (Mitchell 1996; Arora 1996) 19

22 Hamiltonian Path Problem 20

23 Hamiltonian path Given: Graph G=(V,E) Wanted: A path that visits each vertex exactly once This problem is NP-complete. Reduction from Hamiltonian cycle problem: take an instance of HCP split an arbitrary vertex into two vertices there is a Hamiltonian cycle if and only if there is a Hamiltonian path between these two vertices! therefore, finding the longest path is NP-hard as well 21

24 Hamiltonian path What about shortest paths in general graphs? here we still want to visit each vertex exactly once! there might be cycles of negative weight! This problem is NP-hard as well! take an instance of the Hamiltonian path problem multiply each edge-weight by -1 solve shortest path get Hamiltonian path in original graph! 22

25 Questions? 23