Some Tree Representations for Solving the Degree Constrained Minimum Spanning Tree Problems
Wamiliana
Jurusan Matematika, FMIPA, Universitas Lampung Jl. Soemantri Brojonegoro No.1 Bandar Lampung, Indonesia
Email : [email protected] Abstract
To solve the Degree Constrained Minimum Spanning Tree (DCMST) Problem using computer we need to represent the tree in the most efficient way in order to reduce the processing time, especially the initialization process. The DCMST Problem is concerned with finding, in a given edge weighted graph G (all weights are non- negative), the minimum weight spanning tree T satisfying specified degree restrictions on the vertices. This problem arises naturally in communication networks where the degree of a vertex represents the number of line interfaces available at a terminal (center). In this paper we will discuss some tree representations that can be used for solving the DCMST Problem.
Keywords: Minimum spanning tree, degree constrained, tree representation.
Introduction
The Degree Constrained Minimum Spanning Tree (DCMST) problem is concerned with finding a minimum- weight spanning tree whilst satisfying degree requirements on the vertices. The applications of the Degree Constrained Minimum Spanning Tree problems that may arise in real-life include: the design of telecommunication, transportation, and energy networks. It is also used as a subproblem in the design of networks for computer communication, transportation, sewage and plumbing. Gavish7, for example, used the DCMST as a sub- problem in the design of a centralized computer network; and Gavish8 also provides several examples of optimi- zation problems that are faced in the process of designing computer commu- nication networks.
Typically, the DCMST can be applied in cases where n vertices (or terminals/servers/road intersections) need
to be connected with a minimum length of an underlying transportation mode (wires, pipes, canals or roads). However, the handling capacity of each of the vertices imposes a restriction on the number of edges (or wires/roads) that can be connected to a vertex. The DCMST may be used in the design of the road system, which has to serve a collection of suburbs/towns, and has the additional restriction that no more than certain number of roads (example: four roads) are allowed to meet at an intersection. A degree constraint in a communication network also limits the liability in the case of vertex failure. In computer networks, the degree restrictions can be used to cater for the number of line interfaces available at a server/terminal.
In this paper we consider undirected, finite graph G = (V,E), where V represents the vertices/nodes/centers in the networks and E as the set of edges/connections. G is assumed to have no multiple edges. By letting
V = {1,2,…,n} we can identify the edges of G as unordered pairs (i,j). The weight (cost or distance) of edge (i,j) is denoted by cij. The cij’s are assumed to be non negative.
The minimum Spanning Tree (MST) is one of fundamental problem in network design that aim is to select a subset of edges in G so that the resulting network is connected and the total weight of the edges selected is as small as possible. To find a minimum-spanning tree, there are two well-known algorithms: Kruskal’s and Prim’s . However, the earliest algorithm for finding a minimum spanning tree according to Graham and Hell (1982) was suggested by Boruvka in 1926 who developed an algorithm for finding the most economical layout for a power-line network. Degree Constrained Minimum Spanning Tree (DCMST) problem is to find a minimum spanning tree T of G such that the degree of vertex i in T is at most bi , 1≤ i ≤n.
The DCMST problem has been considered by a number of authors and both heuristic and exact methods have been investigated. In order to do the computational work of the methods proposed, first we need to represent the tree so that it can be easily coded. This process is very important because it will determine the computational time in general.
In this paper we will discuss some tree representations for solving the DCMST problem.
Results and Discussion
The Methods and Tree Representations There are many methods that had been investigated for the DCMST problem, including exacts and heuristics. The
methods developed include: Langrangean relaxation by Gavish and Volgenant6,16 , branch and bound by Narula and Ho13, Savelsbergh and Volgenant15. and Volge- nant16; the branch and cut by Caccetta et al.2 variations of Prim’s and Kruskal’s algorithms13,18, genetic Algo-rithm by Zhou and Gen,22 Simulated Annealing by Krishnamoorthy10,11, Irerative Refine- ment1,5; Modified Penalty by Wamil- iana17, and Tabu Search3,4,19,20.
In order to test the methods and to do the computation, it is necessary for us to manipulate trees on a continual basis. A large majority of heuristics approaches assume that one can move from one solution (vector) to another on its neighborhood by merely alternating, swapping or moving elements in and out of the solution vector. For example, for the DCMST problem we can move from one solution (tree) to another by simply delete one leaf and connect the isolated vertex using the other edge that not in the tree but in G that connect to that isolated vertex.
However, such representation may not work well for optimization problems in trees. The main criteria that are critical in designing efficient encodings (or repre- sentations) of trees for use in neigh- borhood search heuristics are:
(1) The representation should lend itself to the neighborhood structure and neighborhood operators. (2) The enco- ding should ensure that the result of this neighborhood operator is also a tree.
(3) The encoding should be efficient with respect to the computations that are required to transverse back and forth between an encoding and the tree that it represents.
(4) The encoding should be unbiased, in the sense that it should not prefer any one region of the solution search space over another.
Below are some tree representations that can be used for encoding and decoding tree.
Edge list representation
Consider a set U = {1,2,…,n-1} as one possible representation of a tree in which the ith element of V is the edge index of the ith edge that in tree. Thus U represents the set of {E\U} edges of G that are not in the tree.
Consider a swap-based neighborhood search heuristic in which an element of U is swapped with an element of U . While efficient in terms of memory requi- rements and while it lends itself nicely to a swap-based neighborhood search heu- ristic, such a procedure can not guarantee that the resulting set will continue remain a tree, as this swap may easily introduce a cycle.
Edge vector representation
Suppose that a vector U of length | E | = m, represents a tree.
Define: 1 , if edge e in treeij 0, otherwise
=
ij e
It can be seen that n-1 elements of U need to have value 1. Now consider search algorithms which produces a new solution in which k ( k=1,2,…) elements of U which have the value 0 are changed bto have a value of 1, while k another elements which have value 1 now acquire a value of 0.
This representation, as above, is not assured that the new solution will be a feasible tree. Palmer and Kershenbaum
(1994) stated that there is only a very small probability of
- n n log 2(n)
2 2
-
that a random representation of this type will be a tree.
Predecessor representation
To encode a tree using predecessor representation, first we designate a vertex r as the root vertex and then encode the tree by maintaining, for each i, p(i), which is the immediate predecessor in the path r to i. Consider a neighborhood operator that applies a random alteration to the value of p(i) for a certain (randomly chosen) i. Again, it is not necessary that such a a change will result in a feasible tree.
Oriented Tree representation
To encode a tree using tree orientation technique, first we choose a vertex r as a root. Associated with each node i, there are two predecessor pointers, LSon(i) and RBrth(i). These pointers store the left most son and the right most brother of vertex i. A leaf vertex has an LSon (i) value of the negative of itself. Similarly, if a particular vertex is the last vertex in one level of this structured tree, its RBrth(i) points to the negative ot its parent (predecessor) in the rooted tree.
With this tree encoding make easy for traversing and updating. However, if we use this for DCMST, the traversal operations are computationally expensive (time consuming). Furthermore, it is necessary to carry out such traversal operations to check the degree constraint and the connectivity constraints.
Adjacency Matrix’s representation
Suppose that wij is the weight of edge eij in tree T. Adjacecency matrix M = [mij] of T is defined as follow :
=
otherwise ,
0
j vertex to
adjacents i
vertex if
ij,
ij
m w
Because of its simplicity, many of the exact methods to represent a tree.
Cayley’s representation
One Efficient method for representing, encoding and decoding trees that lends itself extremely well to a neighborhood search type algorithm is based on Cayley’
Theorem. The resulting encoding is called Prufer number, due to Prufer (1918).
Moon (1967) gave a full description of the properties and Cayley’s method of representing a tree.
For problems that use tree as the key structure, like the DCMST problem, the tree representation by Prufer number is an advantage because in the Prufer number representation we can get information about the degree of the vertex in the tree.
The vertex in the tree that has degree r, in the Prufer number representation will appear r-1 times. Given a tree T, its Prufer number P is an encoding with n-2 indices.
Since the DCMST mainly concerns on the degree restriction and Prufer number representation has an advantage to be used, next we will discuss how to encode and decode a tree using Prufer number.
Encoding and Decoding Tree Using Prufer Number
Using Prufer number to represent a tree, especially for solving DCMST is one advantage because that number will also represent the degree of every vertex in the tree. The give the algorithm as follow:
Encoding tree
1. Let i be the lowest numbered leaf inT.
Let j be the predecessor vertex of i. Then j is the rightmost in the Prufer number representation P(T).
2. Remove i and (i,j) so that thus i is no longer be considered. If i is the only successor of j, j become a leaf.
3. If only two vertices remain to be considered, stop. P(T) is found. If not, go back to Step 1.
Decoding tree
1. Let P(T) be the Prufer number and let all vertices not part in P(T) be design- nated as eligible for consideration.
2. If no digits remains in P(T), there are exactly two vertices , i and j still eligi- ble for consideration. Add edge (i,j) to T and stop.
3. Let i be the lowest numbered eligible vertex. Let j be the leftmost digit of P(T). Add the edge (i,j) to T. Remove the leftmost digit from P(T).
Designate i as no longer eligible. If j does not occur anywhere in what remains of P(T), designate j as eligible.
4. Return to Step 2
Example: Suppose we have a tree as follow:
Figure. 1 The tree for example of encoding Prufer number
Using the encoding step, the Prufer number for that tree is P(T) = 37261. To
decode the tree with the similar example, we give details in the Table 1.
Table.1. Table of the Prufer number construction Vertices
to be
considered
Added edge
The tree construction
37261 4, 5 (3,4) T = {(3,4)}
7261 3,5 (3,7) T={(3,4),(3,7))
261 5,7 (2,5) T={(3,4),(3,7),(2,5)}
61 2,7 (2,6) T= {(3,4),(3,7),(2,5),(2,6)}
1 6,7 (1,6) T={(3,4),(3,7),(2,5),(2,6),(1,6)}
∅ 1,7 (1,7) T={(3,4),(3,7),(2,5),(2,6),(1,6),(1,7)}
Conclusion
To tackle the DCMST Problem, the decision to choose what tree representation is important. Most of exact methods use edge list and combine it with matrix adjacency. However, they have to make sure that in the process there is no cycle induced. For heuristics, especially for genetic algorithm, Cayley’s representation (prufer number) so far is the most efficient to represent tree. This is because it also represents the degree of each vertex in the tree, which implies the chromosome. However, this decision is also very time consuming because genetic algorithm in general requires more memory than the other heuristics do.
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