Title: A Parallel Algorithm for the DegreeConstrained Minimum Spanning Tree Problem Using the NearestNeighb
1A 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
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 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).
7Compute 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
The final 2-MST Max degree 2 Total weight
24
Compute MST then penalize the offending edges
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.
9Compute 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
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 - 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.
12An 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
13Algorithm 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
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 to improve the penalty function.