Outline Introduction NP-Hardness Results and Heuristics

1. A Parallel Algorithm for the Degree-Constrained Minimum Spanning Tree Problem Using the Nearest-Neighbor Chains • Li-Jen Mao, Narsingh Deo, and Sheau-Dong Lang • University of Central Florida • Orlando • Email: {mao, deo, lang}@cs.ucf.edu

2. Outline • Introduction • NP-Hardness Results and Heuristics • Two Previous Parallel Approximate Algorithms • A New Algorithm using the Nearest-Neighbor Chains • Experimental Results • Conclusion and Future Work

3. The Degree-Constrained MST (d-MST) problem: • Given a connected, edge-weighted, undirected graph G and a positive integer d, find a spanning tree with the smallest weight among all possible spanning trees of G which contain no nodes of degree greater than d. • Applications include: •  backplane wiring among pins where no more than a fixed number of wire-ends can be wrapped around any pin on the wiring panel; •  telecommunication switches with a limited capacity; •  VLSI designs with limits on the number of transistors driven by the output current.

4. NP-Hardness of the d-MST Problem • The Hamiltonian Path problem which is NP-complete, is a special case of d-MST with d = 2 and all edge weights equal. • The d-MST is first introduced in [Deo and Hakimi, 1968], is NP-hard for d in the range 2 d (n  2). • Finding approximate solutions to d-MST within a constant factor (of the weight of an optimal tree) is NP-hard [Ravi, Marathe, Ravi, Rosenkrantz, and Hunt, 1993]. • The d-MST problem for the complete graphs of points in a plane is NP-hard for d = 3 [Papadimitriou and Vazirani, 1984], and conjectured NP-hard for d = 4.

5. Heuristic Algorithms • a branch-and-bound procedure based on Lagrangean Relaxation and edge exchanges [Volegnant, 1989], [Narula and Ho, 1980], • using subgradient optimization [Gavish, 1982], • using the minimum cycle basis of graph matroids [Yamamoto, 1978], • using neural networks, simulated annealing, greedy algorithms, and greedy random algorithms [Krishnamoorthy, Craig, and Palaniswami, 1996], • in general, heuristics have no guaranteed bounds on the quality of the solutions.

6. Two Approximate Parallel SIMD Algorithms • [Kumar, Mao, Deo, and Lang, 1997] • The iterative refinement approach (IR): • Alternately perform the following two steps until a spanning tree is produced in which every node satisfies the degree bound: •  MST phase -- compute MST using a parallel implementation of Prim’s algorithm; and •  Penalty Phase -- increase the weights of those tree edges that are incident to nodes with the degree exceeding the constraint d (this discourages the offending edges from appearing in the next MST).

7. Compute a 2-MST using the IR algorithm 1 1 1 6 6 5  7 5 6 6 6 7 6 7 4 6  8 4  6 4 The input graph Compute MST then penalize the offending edges: The final 2-MST: Max degree = 2 Total weight = 24 Max degree = 3 Total weight = 22 : penalized edges

8. The tree-construction, reciprocal nearest neighbor • (TC-RNN) approach to computing d-MST: • Adapt Sollin's MST algorithm to checking the degree constraint in each iteration: •  start with a forest F in which each node forms a single-node tree; •  each processor is assigned a node (tree) which simultaneously computes its nearest neighbor tree and merges with it if two trees are nearest neighbors of each other (RNNs); • This process continues until the forest contains (n 1) edges.

9. Compute 2-MST using the TC-RNN algorithm 1 1 6 5 6 5 7 6 4 4 Iteration 1 found 2 RNN pairs Iteration 2 found 1 RNN pair The input graph 6 7 Iteration 3 found 1 RNN pair Iteration 4 found 1 RNN pair

10. Comparison of the IR and TC-RNN Algorithms • Our empirical studies using randomly-generated, weighted graphs and the standard TSP benchmark problems demonstrate the following: • The IR algorithm is faster but for d = 2, it does not • terminate in most cases with a feasible solution; • The TC-RNN algorithm terminates with a feasible solution in most cases, even when d is 2, and it consistently finds a spanning tree with a weight lower than that of the IR algorithm.

11. The New Tree-Construction, Nearest-Neighbor Chain (TC-NNC) Algorithm: • Anearest-neighbor chain consists of a sequence of nodes in which each node is followed by its nearest neighbor node; the chain must terminate with a pair of reciprocal nearest neighbors. • Each processor is in charge of one node throughout the algorithm execution. Initially, each node is in a tree by itself. • In each iteration, each tree is merged with its nearest neighbor tree while avoiding cycles and violation to degree constraints, resulting in a set of NN-chains. • This process continues until there is only one tree remaining.

12. An Example demonstrating the TC-NNC Algorithm: The first iteration: 4 NN chains : a nearest neighbor : reciprocal NNs The second iteration: 1 NN chain : a nearest neighbor : reciprocal NNs

13. Algorithm TC-NNC No_of_roots = n all processors do par make a MIN-heap out of n 1 edges while (No_of_roots > 1) do construct the NN-chains as follows: (a) each tree votes for an outgoing edge that links to another tree (b) each tree votes for the incoming edges selected from Step (a) (c) connect all winning edges of Step (b) to form NN-chains merge all trees along the NN-chains and update their roots to new roots update No_of_roots end while end do par

14. Experimental Results Comparing the IR, TC-RNN, and TC-NNC Algorithms: • All three algorithms were implemented on a SIMD parallel computer MasPar MP-1 with 8192 processors. • A biased-random weight-matrix generator was used to construct the input graphs for which the initial MST has a high value for the maximum node-degree. The random-graph generator takes the following input parameters: •  n – the size of the matrix; •  f – the number of nodes with large degree; and •  ld (ud) – lower (upper) bounds for the degree of the large-degree nodes.

15. Compute a 5-MST using randomly-weighted complete graphs with an MST forced to have max-degree 20: • The runtime for the TC-NNC algorithm ranges from 1.02 seconds (n = 500) to 3.11 seconds (n = 3500), which is much faster than the TC-RNN algorithm (4 seconds to 14 seconds), but is slightly slower than the IR algorithm (0.6 seconds to 4 seconds). MP-I Execution Time: n: between 500 and 3500; degree bounds ld = 15, ud = 20

16. Compute a 5-MST using randomly-weighted complete graphs with an MST forced to have max-degree 20: • compare the quality of the solutions, i.e., the (d-MST weight/MSTweight) ratios for the same input graphs. The ratios for algorithm IR range from 1.45 to 1.10, the ratios for algorithm TC-RNN range from 1.35 to 1.05, and the ratios for algorithm TC-NNC range from 1.37 to 1.07. Quality of Solutions: the ratio of (d-MST weight/MSTweight), for the same graph

17. Compute a d-MST using randomly-weighted complete graphs of 2000 Nodes with varying d Values: • the execution times of these algorithms decrease and approach the same limit the degree constraint increases from 2 to 10, with TC-RNN and TC-NNC having similar performance better than IR. MP-1 Execution time: d varies from 2 to 10

18. Compute a d-MST using randomly-weighted complete graphs of 2000 Nodes with varying d Values: • The quality-of-solutions is as follows: •  TC-RNN ranges from 1.04 to 1.33, •  IR from 1.08 to 1.21 (with minimum d = 4), •  TC-NNC from 1.04 to 1.33. Quality of Solutions

19. Conclusion and Future Work • We proposed a new parallel algorithm TC-NNC which improved upon the two earlier algorithms IR and TC-RNN for solving the d-MST problem. • The experimental results on randomly weighted graphs demonstrated the following: •  The speed of TC-NNC is better than that of TC- RNN, and is comparable to that of IR; and •  The quality-of solutions of TC-NNC is better than that of IR, and is very close to that of TC-RNN. • For further research, we plan to apply the ideas of iterative refinement and nearest neighbor chains to other constrained spanning tree problems, and toimprove the penalty function.