Distributed Cooperative Algorithm for 𝑘 - 𝑀 Set with Negative Integer 𝑘 by Fractal Symmetrical Property

. In recent years, fractal is widely used everywhere and escape time algorithm (ETA) became the most useful fractal creating method. However, ETA performs not so well because it needs huge computations. So, in this paper, we first present an improved fractal creating algorithm by symmetrical radius of 𝑘 - 𝑀 set. Meanwhile, we use distributed cooperative method to improve classic ETA into parallel system, which is called distributed cooperative ETA (DCETA). Secondly, we present the proof of fractal property in 𝑘 - 𝑀 set 𝑓 c (𝑧) = 𝑧 𝑘 + 𝑐 with exponent 𝑘 (𝑘 < 0 ), which concludes its threshold and symmetrical property. Finally, computational result shows correctness of the novel DCETA, which shows better computational effectiveness and lower waste.


Introduction
Since Mandelbrot has presented  set (Mandelbrot's set), fractals have been soon used everywhere [1].Nowadays, generalized  sets become representatives of chaotic dynamics.We present  set in Definition 1 and - set in Definition 2 as follows [2].In Definition 1    () is defined in (1).
Then, when we reform    () to    () in (2), definition of - set is presented in Definition 2. Consider    () = ( −1  ())  +  where  1  () =   + . (2) Nowadays, many scholars research deeply in - set with many kinds of exponents.In fact, we know that the dynamics system - set is generalization of  set, and classic  set is actually 2- set.
Then, because the structure of - set is complex, we use escape time algorithm (ETA) to create its fractal.The algorithm ETA compares |   ()| to a given large number  of all points .Admittedly,  point  is in - set when the iteration time is more than another given number .To explain it, we present the escape time algorithm in Algorithm 1.
In Algorithm 1, if   > , the iterations of  are convergent; else are divergent.
In fact, Algorithm 1 is an approximate algorithm because  and  cannot be determined.For example, if there exists  point  makes |   ()(0)| <  and | +1  ()(0)| >  or |   ()(0)| >  and | +1  ()(0)| < , we will find different results when  = .So we cannot make sure the point  is convergent or not for ETA in these two conditions.Another problem is that when the iteration exponent is negative or complex number with negative real part, the iteration value is often changed from greatly large to greatly low or opposite.It is hardly to use ETA in this condition.So a strict proof is necessary to be given, which is used to prove convergence and divergence of - set in complex plane with  < 0.
Then, there is another problem in fractal creating algorithms, that is, the huge time cost.Assuming the displayed area contains  ×  points, which need to compute  times 2 International Journal of Distributed Sensor Networks Step 1.
Assuming   = 0 is iterated times of points .

Step 2.
For all points c in complex plane, While   ≤  and           < Color all points  by different   .
in the worst condition ( is maximum iteration times), we know that the computational times  =  ×  × .This is a huge number and it needs huge time to be processed.So we have to use a distributed cooperative algorithm to decrease the computational times.
For years, distributed cooperative algorithms are widely used in many areas, which indeed decrease time cost by using multiprocessors instead of single processor.Today, researchers usually use multicomputers to construct a cooperative system.Furthermore, parallel and distributed system is always used for transportation [3], estimation [4], optimization [5], and localization [6] in engineering area.Admittedly, we call them distributed cooperative transportation/estimation/optimization/localization algorithms.
In fractal area, there are many initial iterated functions with complex structure [7,8].So we have to use novel iterated method to process them [9].Admittedly, distributional method is an effective method to decrease the computational time, just like parallel and cloud environment [10,11].
In this paper, in order to decrease huge time cost which comes from fractal creating algorithms of - set, we present a distributed cooperative algorithm under a parallel system at first.We create - set with both classic ETA and novel DCETA to validate the correctness of DCETA.Besides, in order to validate the positive of DCETA, the parallel experimental results are also presented and analyzed.
Second, we prove that - set has strongly symmetrical properties with phase angles.In this case, we decrease the iterated time further.Meantime, we analyze these results and find the parallel properties of this distributional algorithm.It is also experimented in DCETA and the experimental results also validate our conclusions.
In conclusion, in this paper, we firstly present a separate method to reform ETA into distributional environment.Meanwhile, we analyze this novel algorithm with experimental results.Secondly, we research in properties of - set and find self-symmetrical regions with phase angles.Then, we present a separate method to reform ETA into distributional environment.Meanwhile, we analyze this novel algorithm with experimental results.The results also validate properties of - set.Finally, some Julia sets are created to validate containable properties between - set and the corresponding Julia sets.
The remainder of the paper is organized as follows.We proved properties of - set when  < 0 in Section 2.Moreover, we have their parallel experimental results and analysis in Section 3. Finally, Section 4 summarizes the main results of the paper.
In this paper,  is a negative integer,  and  are positive integers,  is a complex number,  and  are positive real numbers.

Distributed Cooperative ETA with Fractal Symmetrical Property
In order to run ETA in distributed system, we use SIMC in the distributional method in Algorithm 2.Not as usual as other cooperative algorithm, DCA creates the final image with the hidden cooperative multitask nodes, which provides a part of the final image.So we cannot find the cooperative relation in DCT.In our experiment, by these two distributed cooperative methods which are presented, the parallel system is constructed with 3 same PCs.In this system, one PC is the primary node and other two are task nodes.Then, all subresults are connected in the primary node as the final result.
In this paper, we use  = −3 and −4 to process the experiment and compute a sector with center (0, 0) and radius ( √ 2/2) (s is a side of displayed area).For example, when  = −3, the original and improved fractal is presented in Figure 1 (with no larger than threshold 10 and no lower than threshold 0.1).Similarly, the original and improved fractal is presented in Figure 2 (with no larger than threshold 10 and no lower than threshold 0.1) when  = −4.In these figures, Theorems 4 and 6, Lemma 5, and Inference 2 are validated.In detail, we find the symmetrical property from the general views of Figures 1-2, and they validate Lemma 5 and Inference 2.Then, the escape areas are all similar-round with the escape points in Figures 1-2.They validate Theorem 4. Additionally, we have the farthest fractal area to validate Theorem 6.
In this paper, the two worker nodes compute results of the area (, ) with modality of polar coordinates.In the area,  = ( √ 2/2), and  = [2/||, 2(1 + )/||), where  ∈ {0, 1} denotes number of the two worker nodes.We present Figures 3-4 to validate the conclusion we given.In Figure 3, we present the two fractal image created by two worker nodes.In Figure 4, we present the remainder of whole - set.Then, we find that the pseudoescape lines are the symmetry axes of - set.Moreover, Figure 5 is similar to Figures 3-4.It is presented to validate our conclusion with  = −4.
Then, the basic computational area is presented in Figures 6(a     In this paper, we study properties when  is a negative integer from (2).At first, we give threshold and convergent properties of - set by Theorem 3.
Theorem 3. - set contains whole complex plane except countable points.
Inference 1. Origin is the only escape point of - set when  = −1.
In this way, we can analyze these countable escape points of - set.So when we let  = {Set of escape points}, we know that  = {  },  = 1 ⋅ ⋅ ⋅ ∞ is a partition of , where It is because of the following: Then, when we reach Theorem 4 to find all escape points which are attracting, we will found the structure of - set.Proof.Firstly, (4) is applied to compute all escape points of - set: Generalizing (4) to infinite, we have (5).It means that  is an escape point.However, using 1/∞ = 0, we have (6).It means that  is similar to a periodic point.Meantime, in this paper,  point  is called pseudoperiodic point when it reaches (4)-( 6): So when applying (7), we have (8) to reach its derivative: Then we have (9) Admittedly, infinite is attracting fixed point, and zero is pseudo-2 periodic point.
So when applying ( 10) into ( 9), we have (11) when  − 1 is even or (12) when  − 1 is odd: So when   = 0 or   = ∞, (10) is infinite, and computational result of ( 11) is zero.Moreover, it is admittedly that computational result of ( 12) is zero.So we know these pseudoperiodic points are all attracting.
Theorem 4 is proved.
Moreover, we know that escape points of - set are all attracting from Theorem 4. So when we use a threshold  as the threshold, we know that 1/ can also be used as a threshold in the fractal generation.
When assuming that radius of the approximately circle is .So we have (13) from the newer threshold: Then, it reforms to (14) In this case, we prove the novel threshold of - set.Then, the symmetrical fractal property of -Mandelbrot set is presented in the following section.

Inference 2. 𝑐
In this case, we know that - sets also can be separated to 1 −  parts, which differ from each other with phase angle 2/(1 − ).
From Lemma 5 and Inference 2, we find the separation of   clearly by fractal symmetry of - set.In our distributional strategy,   is divided into 1 −  parts, which differ from each other with phase angle 2/(1 − ).In this case, we separate - set to  nodes.The th node computes area with phase angle (2( − 1)/(1 − ), 2/(1 − )) and we only compute area with phase angle (0, 2/(1 − )).The other area is same as the computational area and can be collaged with it.
Proof.Without loss of generality, we assume  point  =  ⋅   ( > 0) as a farthest escape point.Then, we find that |()| has the lowest modulus than any other points with modulus  in the following by using mathematical induction.(c) Let  =  + 1; |  ()| is lower than other points with modulus , where  = (2 + 1)/(1 − ) and phase angle of  +1 () is same to .Its proof is in the following.
Summarizing (a)-(c), we prove that   () is lowest when its modulus is lower than  in Algorithm 1. Then when we apply Theorem 3, we have threshold of these pseudoescape points which are determined by (  ()) .So  is farther from origin when   () is lower.
It means Theorem 6 is proved.
In this way, we know that - set has 1 −  farthest pseudoescape points, which are at lines  = (2 + 1)/(1 − ) for  = 1 ∼ 1−.In this paper, we use radii to name these 1− lines.Then, we use Theorem 7 to present their symmetrical properties.Proof.Using mathematical induction, first, let  = 1; we have Then, let  = , and assuming we have In this case, we know that these pseudoescape lines are all symmetrical lines of - set.So the ratio of computational area is reduced to 1/2(1 − ) with original displayed area.In conclusion of Section 3.2, we know the correctness of symmetrical property in - set.

Conclusion
In this paper, we analyze - set with formula   () =   +  ( = −1, −2, . ..) by using distributed cooperative algorithm.We present an algorithm DCETA to improve classic ETA to a distributed parallel system, and this novel fractal creating method has lower computations.With experimental results, we validate our conclusions.
Then, we proved the fractal property of - set, which contains the correctness of the novel threshold and the fractal symmetrical properties.These properties are improvements of [12,13].
Since we have already studied some special - set with many methods [14], and used - set in many other areas, in the next step, first, we will find some properties of - set, which can enhance understanding of - set.Then, we will use properties of - set into distributed parallel system to enhance its effectiveness and robustness.
Primary node (i) For all task nodesSend group message (area  , ) to th task nodes; in the message, area  = ( start ,  end ) × ( start  ,  end  ) denotes iterated area, and function f dotes the iterated mapping (ii) While all task nodes send its result Connect all sub-images and display it; (iii) Finished; Task nodes (i) If primary node send message (area  , )Use ETA to create sub-image by compute all points in this area with iterated function ; (ii) Send sub-image to primary node; Finished; Algorithm 2: DCA of ETA.

Figure 5 :
Figure 5: Original and corresponding fractal of - set by the two worker nodes ( = −4).

Figure 6 :
Figure 6: Fractal of the computational area by original and improved threshold ( = −3).

Theorem 4 .
All escape points of - set are attracting.