Degree Constrained Minimum Spanning Tree Problem Concrete Mathematics

Degree Constrained Minimum Spanning Tree Problem Concrete Mathematics Term Project TEAM 3 Cho Ji. Hyoung 20022138 Undarmaa 20022167 {jhcho , undraa @icu. ac. kr} 11/29/2020 1 TEAM 3

Outline Our Goal DC(Degree Constrained) Spanning Tree Our Heuristic Algorithm Simulation Model Example Conclusion & Future work References 11/29/2020 2 TEAM 3

Our Goal Access Network : Increase network size and Need MST DC MST problem : In the NP-Complete problem set Problems Searching Solutions : Inherently exponential problem Heuristic Algorithm 11/29/2020 3 TEAM 3

DC Spanning Tree To find MST for any given network is easy and wellunderstood problem. However, the problem becomes NP-complete when there is a constraint on degree of MST The NP-Completeness covers even the basic version of this problem which is known as the ‘Hamiltonian Path Problem’. For degree values greater than the Hamiltonian Path problem, the problem is proven to be NP-Hard. How to solve ? ? ? Heuristic approach is unavoidable Considered to be applied Ethernet based Access Network. 11/29/2020 4 TEAM 3

Our Approach : Heuristic Algorithm Corresponding MST is formed (using one of the well known algorithms given by Prim, Kruskal or Sollin) and taken as the lower bound for the goal cost Swap the edges Add an edge which is not included current MST cycle Remove an edge from MST no cycle May increase the degree of the new end edges, but decrease the degree of the old ones By swapping, we can adjust the degrees to a desired value Main operation is heuristic Which edge to add(least cost edge) which one to remove (largest cost edge) Obtaining these to conditions simultaneously is impossible Tradeoff is needed 11/29/2020 5 TEAM 3

Degree Adjustment Phase With Heuristic 1 Degree Adjustment Phase With Heuristic 2 (1) TO_ADJUST set. ={V>K} (2) Remove one vertex V 1 from TO_ADJUST set. (3) put them into CHILDREN set. (4) Assign E the value of C(E) - C(V 1, V 2) (5) DEGREE(V 1) and DEGREE(V 2) decreases by 1. (6) If DEGREE(V 1) <= K (7) Return the found MST. (2) Remove one vertex V 1 from TO_ADJUST set. (3) Find the least cost edge E (4) Remove the edge which has the larger cost among (V 1, V 2) and (V 1, V 3) (6) DEGREE(V 1) and DEGREE(V 2) or DEGREE(V 3) decreases by 1. (7) If DEGREE(V 1) <= K then go to (8), else go to (3) to iterate once more. (8 Return the found MST. 11/29/2020 6 TEAM 3

Heuristic Algorithm 1 # of node DCMST cost(d. MST)/cost(MST) 2 2083 1995 1. 044110276 3 1907 1830 1. 042076503 4 1985 1939 1. 023723569 5 1905 1879 1. 013837147 6 1782 1 11/29/2020 7 TEAM 3

Heuristic Algorithm 2 # of nodes DCMST cost(d. MST)/cost(MST) 2 1855 1737 1. 067933218 3 2017 1930 1. 04507772 4 1876 1805 1. 03933518 5 1794 1748 1. 026315789 6 1860 1 11/29/2020 8 TEAM 3

Simulation Model Range = 2 # of node = 15 11/29/2020 9 TEAM 3

Step 1 : Found MST (Prim’s Algorithm 11/29/2020 10 TEAM 3 )

Final Step : Result less than degree 2 11/29/2020 11 TEAM 3

Progress 1 … 7 8 9 10 11 12 13 14 Study basic STA Research degree constrained STA Simulation & Modeling Conclusion 11/29/2020 12 TEAM 3

Conclusions & Future work Two heuristics are proposed to attack the degree constrained MST problem This component generated networks and heuristic algorithms are run on these test cases In the coding phase, MATLAB is used Attacking to it with some heuristic techniques gives a lot of ideas about the structure of such network problems and how reasonable solutions can be achieved Future Work Gives somewhat reasonable answers but not the optimal ones Concentrated to optimize the algorithms To obtain results faster and closer to the optimality To adapt our research part : Ethernet PON 11/29/2020 13 TEAM 3

References [1] Source Routing Appendix to IEEE Standard 802. 1 d Media Access Control (MAC) Bridges for details on the spanning tree algorithm [2] Network Flows : Theory, Algorithms, and Applications by Ravindra K. Ahuja, Thomas L. Magnanti, and James B. Orlin. [3] R. L. Graham and P. Hell. On the history of the minimum spanning tree problem. Annals of the History of Computing, 7(1): 43 -57, 1985. [4]J. B. Kruskal. On the shortest spanning tree of a graph and the traveling salesman problem. Proceedings of the American Mathematical Society, 7: 48 -50, 1956. [5]R. C. Prim. Shortest connection networks and some generalizations. Bell Systems Technology Journal, 36: 1389 -1401, 1957. [6]K. H. Rosen. Discrete Mathematics and Its Applications. Mc. Graw-Hill, Inc. , New York, NY, third edition, 1995 11/29/2020 14 TEAM 3

Thank You ! 11/29/2020 15 TEAM 3