A Parallel Algorithm for the DegreeConstrained Minimum Spanning Tree Problem Using the NearestNeighb

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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_at_cs.ucf.edu
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• 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

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• 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.

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• 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.

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• 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.

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• 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 Prims 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).

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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
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The input graph
The final 2-MST Max degree 2 Total weight
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Compute MST then penalize the offending edges
Max degree 3 Total weight 22
penalized edges
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• 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.

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Compute 2-MST using the TC-RNN algorithm

1
1
6
5
5
6
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
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• 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.

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• The New Tree-Construction, Nearest-Neighbor Chain
  (TC-NNC) Algorithm
• A nearest-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.

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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
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Algorithm TC-NNC No_of_roots n all processors
do par make a MIN-heap
out of n ?1 edges while (No_of_roots gt 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
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• 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.

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• 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
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• 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
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• 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
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• 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
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• 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 to improve the penalty function.