Constrained Minimum Spanning Tree Algorithms

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1 December 8, 008

2 Introduction Graphs and MSTs revisited Minimum Spanning Tree Algorithms Algorithm of Kruskal Algorithm of Prim Constrained Minimum Spanning Trees Bounded Diameter Minimum Spanning Trees Degree Constrained Minimum Spanning Trees Minimum Steiner Trees Summary

3 Graphs and MSTs revisited Graphs, Trees and Spanning Trees Graph G = (V, E) Cost of an edge: c(e) R +, e E A tree T (V, E) is a connected graph containing no circles A spanning tree T (V, E ) of G is a tree containing all vertices of G, E E.

4 Graphs and MSTs revisited Minimum Spanning Trees A minimum spanning tree (MST) T MST (V, E MST ) of G is a spanning tree of G with minimum edge costs, i.e. e E MST c(e) min.

5 Minimum Spanning Tree Algorithms Used to model many real world problems Minimum cost transportation networks Minimum wire length network topologies Algorithms for solving travelling salesman problems Multicast routing algorithms main MST algorithms Algorithm of Kruskal [Kruskal, 9] Algorithm of Prim (Jarnik 90, Prim 97, Dijkstra 99) [Prim, 97]

6 Algorithm of Kruskal MST Algorithm of Kruskal

7 Algorithm of Kruskal MST Algorithm of Kruskal. E, L E. Sort all edges in L by increasing edge cost.. If E = V then M = (V, E ) is a MST of G, stop.. Select the lowest-cost edge e L. L L\{e}. If the graph G = (V, E {e}) contains no cycles, then E E {e} 7. Go to ().

8 Algorithm of Kruskal MST Algorithm of Kruskal - Implementation How to implement the check for cycles? Save for every node n in which cluster Cl(n) it is Cl(n) = set of connected nodes which contain n Update the clusters if an edge (i, j) is added: Cl(i) = Cl(j) = Cl(i) Cl(j) Do not add an edge (i, j) if Cl(i) = Cl(j)

9 Algorithm of Kruskal MST Algorithm of Kruskal - Complexity Time complexity Sorting the edges: O( E log E ) Updating the clusters: max. E times O(log V ) Sum: O( E log E + E log V ) O( E log E ) O( E log V ) Space complexity (Sorted) list of edges: O( E ) Disjoint-set data structure: O( V ) = O( V ) Sum: O( E + V )

10 Algorithm of Prim MST Algorithm of Prim 0

11 Algorithm of Prim MST Algorithm of Prim. Select start vertex v s V. U {v s }, E. Select the lowest-cost edge e = {i, j} E where i U and j V \U. E E {e}, U U {j}. If U = V then M = (V, E ) is a MST of G, stop. Otherwise, go back and continue from point ().

12 Algorithm of Prim MST Algorithm of Prim - Implementation Datastructure to determine next minimum cost edge: Adjacency list Binary heap Finonacci heap Binary heap Fibonacci heap Node distances as values of the heap structure Operation remove-min returns the least-distance node Operation decrease-key updates the distance of a node

13 Algorithm of Prim MST Algorithm of Prim - Complexity Time and space complexity dependent from implementation and used data structures Use of a heap structure: V times remove-min in O(log V ) max. E times decrease-key in O(log V ) or O() Min. edge cost data structure Time complexity Required memory adjacency matrix O( V ) O( V ) binary heap and adjacency list O( V log V + E log V ) O( V + V ) Fibonacci heap and adjacency list O( V log V + E ) O( V + V )

14 Constrained MST Problems General MST not satisfactory for all problem domains, e.g. Too long a path from one node to another Too high the degree of a node Capacity on a path between nodes insufficient 7 cost/capacity / / / / / /7 / 0/0 bandwidth requirement Bounded Diameter MST (BDMST) [Gruber et al., 00] Degree Constrained MST (DCMST) [Bui and Zrncic, 00] Capacitated MST (CMST) [Reimann and Laumanns, 00]

15 Constrained MST Problems Other MST problem formulations Generalized Minimum Spanning Tree (GMST) [Oencan et al., 008] Delay-Constrained Minimum Spanning Tree Hop-Constrained Minimum Spanning Tree (HMST)...

16 Constrained MST Problems Most Constrained MST problems are NP-complete Different solution approaches, often mixed Ant-based algorithms / Ant Colony Optimization (ACO) Greedy Randomized Adaptive Search Procedure (GRASP) Evolutionary/Genetic Algorithms (EA) Variable Neighbourhood Search (VNS)

17 Bounded Diameter Minimum Spanning Trees Bounded Diameter Minimum Spanning Trees - BDMST Path p(v s, v t ) is a cycle-free way (e,..., e n ) E n from v s to v t. Diameter dia(g) is the length of the longest path between any nodes in G, dia(g) ( V ). 7 7 dia(g) = dia(g) = dia(g) = BDSMT problem is NP-hard for n [Gruber et al., 00]

18 Bounded Diameter Minimum Spanning Trees BDMST algorithms Analysis performed by [Gruber et al., 00] Definition of neighbourhoods Comparison of algorithmic approaches Variable Neighbourhood Search Evolutionary Algorithm Ant Colony Optimization

19 Bounded Diameter Minimum Spanning Trees BDMST algorithms - General remarks Algorithms start with a random tree T RTC - Randomized Tree Construction Center center(t ) of the tree center(t ) = {c}, c V if dia(t ) mod = 0 center(t ) = {c, c }, c, c V if dia(t ) mod = c c c 0 Tree height h(t ) is kept below D, D is diameter constraint

20 Bounded Diameter Minimum Spanning Trees BDMST neighbourhoods Neighbourhood = possible neighbouring solutions

21 Bounded Diameter Minimum Spanning Trees BDMST VNS algorithm

22 Bounded Diameter Minimum Spanning Trees BDMST algorithm comparison [Gruber et al., 00]

23 Bounded Diameter Minimum Spanning Trees BDMST algorithm comparison [Gruber et al., 00]

24 Degree Constrained Minimum Spanning Trees Degree Constrained Minimum Spanning Trees - DCMST Degree deg(v) is the number of edges connected to vertex v. DCMST with degree constraint n is a MST with deg(v) n v V MST 7 DCMST, max. deg(v) = 7

25 Degree Constrained Minimum Spanning Trees DCMST algorithms DCMST problem is NP-hard [Bui and Zrncic, 00] Heuristics calculate approximations to the optimal solution in polynomial time Many different algorithmic approaches Particle Swarm Optimization Genetic algorithms Ant-based algorithms (discussed later)...

26 Degree Constrained Minimum Spanning Trees An ant-based DCMST algorithm [Bui and Zrncic, 00] General idea Ants move along paths to find local optimums Ants leave pheromone traces on their paths Pheromone level P(e) of an edge e determines the probability that more ants will travel along it Algorithm outline. Determine degree constrained spanning tree T temp using ants. If e E temp c(e) < e E best c(e) then set globally best result T best = T temp. Enhance pheromone levels P(e) = δ P(e) e E best, δ >. Evaporate pheromone, P(e) = γ P(e) e E, γ <. Go to if T best has been set recently.

27 Degree Constrained Minimum Spanning Trees An ant-based DCMST algorithm - Iteration step Initialization:. Assign one ant to each vertex v V. IP(e) = (M c(e)) + (M m)/ where M, m are maximal and minimal edge costs in the graph. P(e) = IP(e) Exploration:. Ants randomly select and visit neighbouring nodes. For each traversed edge e: P(e) = P(e) + IP(e) Tree Construction:. Create edge list L sorted by decreasing P(e). Construct degree constrained spanning tree with highest-pheromone-level edges (head of L)

28 Degree Constrained Minimum Spanning Trees An ant-based DCMST algorithm [Bui and Zrncic, 00] - Iteration step 9 7 8

29 Minimum Steiner Trees MST extension - Minimum Steiner Trees MST which has to span only a vertex subset D V May contain additional nodes Steiner nodes Important application: Delay-Constrained Multicast Routing Problem (DCMR) [Skorin-Kapov and Kos, 00] Steiner node

30 Summary Minimum Spanning Trees Kruskal s algorithm O( E log V ) Prim s algorithm O( V log V + E ) Constrained MST problems Bounded Diameter MST VNS, ACO, EA Degree Constrained MST ant-based algorithm Capacitated MST Generalized MST... Minimum Steiner Trees

31 Bui, T. N. and Zrncic, C. M. (00). An ant-based algorithm for finding degree-constrained minimum spanning tree. In GECCO 0: Proceedings of the 8th annual conference on Genetic and evolutionary computation, pages 8, New York, NY, USA. ACM. Gruber, M., van Hemert, J., and Raidl, G. R. (00). Neighbourhood searches for the bounded diameter minimum spanning tree problem embedded in a vns, ea, and aco. In GECCO 0: Proceedings of the 8th annual conference on Genetic and evolutionary computation, pages 87 9, New York, NY, USA. ACM. Kruskal, J. (9).

32 On the shortest spanning subtree of a graph and the traveling salesman problem. In Proceedings of the American Mathematical Society. Oencan, T., Cordeau, J.-F., and Laporte, G. (008). A tabu search heuristic for the generalized minimum spanning tree problem. European Journal of Operational Research, 9():0 9. Prim, R. C. (97). Shortest connection networks and some generalizations. Bell System Tech. J.,. Reimann, M. and Laumanns, M. (00). A hybrid aco algorithm for the capacitated minimum spanning tree problem. Skorin-Kapov, N. and Kos, M. (00).

33 A grasp heuristic for the delay-constrained multicast routing problem. Telecommunication Systems, ().

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