An Optimal Parallel Algorithm for Constructing a Spanning Tree on Proper Circle Trapezoid Graphs

Given a simple graph G with n vertices and m edges, the spanning tree problem is to find a spanning tree for a given graph G. This problem has many applications, such as electric power systems, computer network design and circuit analysis. For a simple graph, the spanning tree problem can be solved in ( ) log O n time with ( ) O n m + processors on the CRCW PRAM. In general, it is known that more efficient parallel algorithms can be developed by restricting classes of graphs. In this paper, we shall propose a parallel algorithm which runs ( ) log O n time with ( ) log O n n processors on the EREW PRAM for constructing on proper circle trapezoid graphs.


Introduction
Given a simple connected graph G with n vertices, the spanning tree problem is to find a tree that connects all the vertices of G.The spanning tree problem has applications, such as electric power systems, computer network design and circuit analysis [1].A spanning tree can be found in

( )
O n m + time using, for example, the depth-first search or breadth-first search.In recent years, a large number of studies have been made to parallelize known sequential algorithms.
For simple graphs, Chin
In general, it is known that more efficient algorithms can be developed by restricting classes of graphs.For instance, Wang et al. proposed an optimal parallel algorithm for constructing a spanning tree on permutation graphs that run in ( ) n processors on the EREW (Exclusive Read Exclusive Write) PRAM [4].Wang et al. proposed optimal parallel algorithms for some problems including the spanning tree problem on interval graphs that can be executed in ( ) n processors on the EREW PRAM [5].Bera et al. presented an optimal parallel algorithm for finding a spanning tree on trapezoid graphs that take in ( ) processors on the EREW PRAM [6].In addition, Honma et al. developed parallel algorithms for finding a spanning tree on circular permutation graphs [7] and circular trapezoid graphs [8].Both of them take in ( ) processors on the EREW PRAM.

2
O n time algorithm for solving maximum independent set problem and ( ) n time algorithm for solving maximum clique problem.Recently, Lin showed that circle trapezoid graphs are superclasses of trapezoid graphs [10].
In this study, we propose a parallel algorithm for spanning tree problem on a proper circle trapezoid graph.It can run in processors on the EREW PRAM.The rest of this paper is organized as follows.
Section 2 describes some definitions of circle trapezoid graphs and models and introduces the extended circle trapezoid model, as well as some notation.Section 3 presents some properties on circle trapezoid graphs, which are useful for finding a spanning tree in an efficient manner.Section 4 describes our parallel algorithm for the spanning tree problem and its complexity.Finally, Section 5 concludes the paper.

Circle Trapezoid Model and Graph
We first illustrate the circle trapezoid model before defining the circle trapezoid graph.There is a unit circle C such that the consecutive integer i, 1 4 i n ≤ ≤ are assigned clockwise on the circumference (n is the number of circle trapezoids).

Consider nonintersecting two arcs
along the circumference of C.  1(a).Table 1 shows the details of circle trapezoid model M of Figure 1.

Extended Circle Trapezoid Model
In the following, we introduce the extended circle trapezoid model EM constructed Moreover, copies i n I − of i I are created by shifting 4n to the left respectively, for each i I , 1 i n ≤ ≤ .Note that both interval pairs i I and i n I − in extended circle trapezoid model EM are corresponding to i CT in M. The following Algorithm CEM constructs an EM from a M. Figure 2 shows the EM constructed from the M illustrated in Figure 1.Table 2 Create copies i n I − by shifting 4n to the left for i I ; End

Definitions for Proper Circle Trapezoid Graph
In this study, we focus and treat a proper circle trapezoid graph.Graph G is a circle trapezoid graph corresponding to a circle trapezoid model M and an extended circle trapezoid model EM is constructed from M by executing Algorithm CEM.We consider circle trapezoid model M such that the extended Journal of Applied Mathematics and Physics     Here, some notations that form the basis of our algorithm are defined as follows.

The function ( )
nor i normalizes the interval pair number i in EM within the range 1 to n, which is expressed as For the example shown in Figure 2 3.

Property of Proper Circle Trapezoid Graph
We describe some properties on circle trapezoid graphs which are useful for constructing the algorithm for spanning tree problem on proper circle trapezoid graphs.
For two interval pairs i I and j I ( ) Figure 3 shows examples of the cases of disjoint and contain.The following Lemma 1 has been described in [9].

Parallel Algorithm
In this section, we propose an algorithm for constructing a spanning tree of a connected proper circle trapezoid graph p G .We assume that all trapezoids in the proper circle trapezoid model have been sorted by corner point a in ascending order, that is, Table 1 is given as an input of our algorithm.Algorithm CST returns a spanning tree if a given graph p G is connected.Instead of using a sophisticated technique, we propose simple parallel algorithms using only the parallel prefix computation [11] and Brent's scheduling principle [12].
Algorithm CST Input: i a and i b , 1 i n ≤ ≤ .
Output: A spanning forest F of G. Initially F be an empty set. Begin ing parallel prefix computation [11].
In Step 2, we set For the example shown in Figure 4, we set For the only case of 1 i = , we have ( ) We consider that T is added an edge form i to ( ) In the case of , T constructed in above way is connected graph that has n vertices.Thus, T is not a tree that has exactly one cycle C.
There exist some edge In Step 3, we consider the case of , this means that a given p G is disconnected, which is a contradiction to our hypothesis.Therefore, in the case of

Concluding Remarks
In this paper, we presented a parallel algorithm to solve the spanning tree problem on a proper circle trapezoid graph.This algorithm can be implemented in ( ) n processors on an EREW PRAM computation model using only parallel prefix computation [11] and Brent's scheduling principle [12] without using a sophisticated technique.Solutions to the spanning problem have applications in electrical power provision, computer network design, and circuit analysis, among others.For this reason, we think this paper is also worthy from both a theoretical and algorithmic point of view.In the future, we will continue this research by extending the results to other classes of graphs.
CRCW PRAM.In general, it is known that more efficient parallel algorithms can be developed by restricting classes of graphs.In this paper, we shall propose a parallel algorithm which runs ( ) log O n time with ( ) from a circle trapezoid model for making the problem easier.We first cut a circle trapezoid model M at point 1 on the circumference and next unroll onto the real horizontal line.Each circle trapezoid [ ]

Figure 3 .
Figure 3. Examples of disjoint and contain.(a) i I and j I are disjoint ( i j d a < ); (b) i I contains j I ( i j b a < can remove a cycle C.

Figure 4 .
Figure 4. Example of constructed spanning tree.
shows the details of EM of Figure2.
i CT in M. Begin For each non feedback circle trapezoid i CT pardo For each , ,

Table 1 .
Details of circle trapezoid model M.

Table 2 .
Details of extended circle trapezoid model EM.

Table 3 .
Details of extended circle trapezoid model EM.
Lemma 2 G is a circle trapezoid graph corresponding to a circle trapezoid model M, and extended circle trapezoid model EM is constructed from M. For two interval pairs , i j < be non-feedback circle trapezoids in circle trapezoid model M.Moreover, extended circle trapezoid model EM is constructed from M.i CT andThe following Lemma 2 generalizes Lemma 1.This is very useful to find the edges on circle trapezoid graph.<ij in EM.□We obtain the following Lemma 3 for a proper circle trapezoid model By definitions of d v and ld, if In addition, Algorithm CST can be executed on an EREW PRAM because neither concurrent read nor concurrent write are necessary.Thus, we have the subsequent theorem.
i ck and s are computed in ( )