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JOURNAL OF COMPUTERS, VOL. 6, NO. 8, AUGUST 2011
Improvement of Escape Time Algorithm by NoEscape-Point
LIU Shuai, CHE Xiangjiu, WANG Zhengxuan
College of Computer Science and Technology
Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education
Jilin University, Changchun, China
Email: [email protected]
Abstract—Escape time algorithm is a universal algorithm
when to create fractal image. A class of algorithms based on
escape time algorithm is wasting-calculation. In this essay,
when combined with the feature of eventually periodic point
of functions, we define a kind of points as no-escape point.
To analyze the shortcomings of the classic algorithm, we
improve the escape time algorithm base on the no-escape
points. We analyze the algorithm and put forward the best
application scope for it. By create lots of fractal figures, we
find the figures created by the two algorithms are consistent
with each other except a few escape points. We compare the
complexity between the two algorithms and find the
iteration times by the improved algorithm are less than
escape time algorithm when creating the fractal images. We
do several experiments and find the improved algorithm is
universal and it reduces the time-consuming.
Index Terms—fractal; escape time; compute efficiency;
universal algorithm; no escape point
I. INTRODUCTION
Since Mandelbrot constructed the dynamic system
fc(z)=z2+c on the computer, which is called M-set, it has
considered to be a representative logo of chaotic
dynamics[1] and it has been the subject of intense research
in the world of fractal geometry and graphics. The
thinking and algorithm based on fractal has been used in
many sections of computer science soon.
There are two basic general algorithms when we create
fractal figures. One is escape time algorithm, the other is
iterated function system short for IFS. Since a lot of
fractal iteration is uncertain, the escape time algorithm is
becomes universal algorithm when we create fractal
figures of complex initial functions.
Nowadays, escape time algorithm and the derivative
algorithm of it are the main algorithms when we create
complex fractal figures. The escape time algorithm is
such an algorithm that it use escape thresholds and the
max iteration number to draw iteration trajectory.
Specifically, it colors every point in C-plane with
different colors by the difference of the iteration number.
We show the escape time algorithm as Algorithm a[2]:
Algorithm a (Escape Time Algorithm)
Step1. Set N as escape threshold number. Set M as the
max iteration number. Set mz=0 as the iteration number
of point z. Set Nz=z as fn(z) of point z.
Step2. For all points z in complex plane,
© 2011 ACADEMY PUBLISHER
doi:10.4304/jcp.6.8.1648-1653
While mz≤M and |Nz|<N
mz=mz+1. Nz=f(Nz).
Step3. Color all points z by mz.
As basic of algorithm a, if mz>M, the iterations of z is
convergence; else is divergence.
We know the algorithm needs lots of iterations to
create figures. Especially, when the structure of the initial
function is complex, the algorithm often requires lots of
computation.
In this essay, we find two flaws of escape time
algorithm. We call it flaw a and b.
Flaw a. The convergence points have to do max
iterations.
Flaw b. When the iteration is in a calculated region, the
algorithm does not use the known results. It is a calculate
waste.
To consider these two flaws, we put forward a new
algorithm to improve it by use no-escape points. At first,
we put forward the concept of no-escape point and the
thinking of the algorithm. Secondly, we put forward the
improved algorithm. We also prove the improved
algorithm creates better figures and fewer calculations to
the classic one. Finally, by create several fractal figures
by use both the improved and classic algorithm, we
validate the improvement of the new algorithm.
II. NO-ESCAPE POINT AND THE IMPROVED THINKING
To consider the definition of Julia set (short for J set
since), we know that the J set of function f is the repellers
of dynamic system of the function. In another words, the
J set divided dynamic system into two parts. They are
C1={z|fi(z) → ∞ } and C2={z|fi(z) →/ ∞ }. So C1 is
divergence and C2 is convergence.
When a point of the iterative array is in C1, the
iteration is limit to ∞. On the contrary, the iteration is
convergence. So it will avoid the repeat calculate if the
iteration can be predicted.
In addition, when we use the escape time algorithm to
create fractal figures by computer, we actually only create
the figures in displayed area. It is generally part of the
fractal figure. To considered both the display area and
iteration number and to extend the definition of escape
point, we create a new kind of point called no-escape
point and escape point with i order.
JOURNAL OF COMPUTERS, VOL. 6, NO. 8, AUGUST 2011
To set the domain of function f is Z, the max iteration
number is I, the set of sequence {z, f(z),…, fi(z)} is Hi(z),
we give the definition of no-escape point and escape
point with i order by definition a.
Definition a. To set Q ⊂ Z and ∃ z∈Q make Hi(z)
⊂ Q. When i≥I, we call z as a non-escape point based
on f, Q, I. It is short for non-escape point. We called
Hi-I (z) is a non-escape sequence of z based on f, Q, I. It
is short for non-escape sequence of z. When fi+1(z) ∉ Q,
we called z as an escape point with order i+1 based on
f, Q, I. It is short for escape point when i=0. We call
Hi(z) as an escape sequence with order i+1 of z when i
≤I. We call Hi(z)-Hi-I+1(z) as an escape sequence with
order i+1 of z when i>I.
The function f is unchanged in escape time algorithm.
So when we set Q as the domain of f in the display area,
we can call z as a no-escape point with order i or an
escape point with order i+1 for short in definition a. As
the same, we call Hi(z) as a no-escape sequence with
order i or an escape sequence with order i+1. So we can
find the following characters easily.
Character a. When all attract domain by an
attractor of f(z) is complete in displayed area, all
points in attract domain are no-escape points.
Character b. When all J-set of f(z) is complete in
displayed area, all points in J-set are no-escape points.
Character c. No-escape sequences in displayed area
are convergence by escape time algorithm.
Character d. The escape sequence with order i+1 in
displayed area is divergence by escape time algorithm.
The iteration numbers of divergence are from i-I to 1.
In fact, the non-escape point and non-escape sequence
are extension of the definition of eventually periodic
point and the eventually periodic sequence[3]. It is an
corresponding approximate calculation to the computer
algorithm. It enhances the speed of the algorithm. In fact,
every Hi(z) of z can be calculated in displayed area. Thus,
we can predict all points in Hi(z)are escape or not by the
calculation of the last point of Hi(z). Therefore, to
compare with the escape time algorithm, the improved
thinking is reduce lots of iteration number of the points in
Hi(z).
We illustrate the superiority of the improved thinking
by example a and b. In the examples, we set Q as
displayed area and f(z) as iteration function of z.
Example a. When the no-escape sequence Hi+I(z) ⊂ Q,
we calculate Hi(z) by step a and b.
Step a. iterate i+I times of z to calculate Hi+I(z).
Step b. We affirm the i+1 points {z, f(z), …, fi(z)} are
convergence because fi+I(z) is not divergence.
We use i+I times iteration of f. It is easy to know that
we need (i+I)·(i+1) times iteration of f by escape time
algorithm to calculate the i+1 points {z, f(z), …, fi(z)}.
So it is less calculate by the improved thinking.
Example b. When the escape sequence Hi(z) ⊂ Q, we
calculate Hi-1(z) by case a and b.
Case a. i≤I+1.
Step a. iterate i times of z to calculate Hi-1(z) and fi(z).
Step b. We affirm the i+1 points {z, f(z), …, fi(z)} are
divergence. The escape order number of fk(z) is i-k.
© 2011 ACADEMY PUBLISHER
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We use i times iteration of f. It is easy to know that we
need i times iteration of f.
When we use escape time algorithm to solve the
problem, we need to use 1+2+…+i=i•(i+1)/2 times
iteration of f because the escape order number of fk(z) is
i-k. So it is less calculate by the improved thinking.
Case b. i>I+1.
The iteration times of the i points by improved
thinking is also i. The iteration times by escape time
algorithm is (i-I+1)•I+I•(I+1)/2=I(i-I/2+3/2) because HiI(z) in Hi-1(z) is considered to be convergence in the
algorithm.
In this case, there will be several points with different
color by escape time algorithm and the improve thinking.
All these points with I times iteration are less than the
threshold number N, but with I+1 to i times iteration are
more than N. They are considered as convergence by the
escape time algorithm and divergence by the improved
thinking. To considered algorithm a, the improved
thinking is equal to escape time algorithm with larger I by
solved these points.
From example a and b, we know the improved
thinking reduce the iteration times when at the same case
of character a~d. Though there may several points with
different color in character 4, we can ignore them as they
are same to escape time algorithm with larger I.
In fact, when we create fractal figures, the part of
figures we attend is always J set hereabout. There exist
lots of conditions at the same of character a~d. So the
improved thinking can improve the creating speed.
III. IMPROVE OF ESCAPE TIME ALGORITHM BY NOESCAPE-POINT
In this essay, To set the displayed area m×n, we put
forward an improved algorithm by definition a and
character a~d. The new algorithm needs O(m×n) space.
In the algorithm, we set (x,y) as the point x+yi,
Count(x,y) as the iteration times at (x,y), Result(x,y) as
the result of the mapping of (x,y) by f to Q, Explore(x,y)
as f(x,y), logic variable flag as if there exist points are not
iterate to the max iteration time, |z| as the module of z, B
as the iteration is more than the threshold.
In order to describe simple, we call “neighborhood
rectangular of a point z” as the unit pixel rectangle (one
pixel × one pixel) in Q that contains z.
A The Describe of the Algorithm
Procedure No-Escape-Point
Input. f as the iteration function, Q=(x1,x2)×(y1,y2)
as the displayed area, max as the max iteration time,
bailout as the threshold.
Output. Count(x,y) as the escape iteration times of
every pixel in Q. We use them to color all pixels when we
create the figure.
Step 1.
Set flag=0.
For all points (x,y) in Q do.
{
Set Count(x,y)=1, Explore(x,y)=f(x,y).
If f(x,y)=(x,y).
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Set Count(x,y)=max+1.
Else if |f(x,y)≥bailout.
Set Result(x,y)=B.
Else.
Set Result(x,y)=the nearest pixel of f(x,y).
}
Goto Step 2.
Step 2.
If flag=1.
Goto Step 3.
Else
Goto Step 4.
Step 3.
Set flag=0.
For all points (x,y) in Q do.
{
If Result(x,y)∈Q and Count(x,y)<max+1.
Set Result(x,y)=Result(Result(x,y)),
Count(x,y)=Count(x,y)+Count(Result(x,y)), flag=1.
If Count(x,y)>max.
Count(x,y)=max+1.
If Result(x,y)=B.
Count(x,y)=max.
}
Goto Step 2.
Step 4.
For all points (x,y) in Q do.
{
While Count(x,y)<max and Result(x,y)≠B
{
Set k=Explore(x,y).
While |k|<bailout and Count(x,y)<max
{
If k∈Q
Goto Step 3.
Set Count(x,y)=Count(x,y)+1, k=f(k).
}.
}.
}.
Goto Step 5。
Step 5.
For all points (x,y) in Q do.
{
Set z1~z4 as the pixels of neighborhood
rectangular of Explore(x,y).
If Count(zi) is not all equal (i=1~4)
{
Set k=Explore(x,y).
While |k|<bailout and Count(x,y)<max
Set Count(x,y)=Count(x,y)+1, k=f(k).
}.
}.
Procedure Finished.
B The Correctness of the Algorithm
We iterate all points in Q one time by step 1 and find
the fixed point and several escape point.
We find the iteration result of all approximate noescape sequence and escape sequence with order>max by
step 2 and 3. To color these points in next step, we set the
© 2011 ACADEMY PUBLISHER
JOURNAL OF COMPUTERS, VOL. 6, NO. 8, AUGUST 2011
iteration number max of the points with iteration
number>max.
We iterate the points which the iteration result is not in
Q for times until they sum up to case i~iii.
Case i. The iteration number is max.
Case ii. The iteration result of the point>threshold.
Case iii. The iteration result is in Q.
We can easily find whether the point is convergence or
divergence by case i and ii. The points by case iii are
either summed up to case i and ii or to case iii in cycle by
the iteration goes on. Because the iteration number of a
point (x,y) is Count(x,y) in Q by step 3, all points are sum
up to case i and ii when we iterate max-Count(x,y) times
by case iii at most.
We use step 5 to dispose J set and asymptote in fractal
figures. The definition of asymptote is definition b. It is
easy to known that the escape time for a point z is k+1
when the point f(z) is between k-asymptote and k+1asymptote of f. It is no relationship with the precise value
of fi(z).
Definition b. We call the J set with max iteration
times i-k as the k-asymptote of J set with iteration
times i. Generally, 0-asymptote is J set.
In fact, to consider the error of mapping z to f(z)=z1 by
function f, the error is near to an k-asymptote. Though
Result(z)≠fmax(z), it does not affect the escape number
when the error is between two asymptotes. So we can
ignore the errors when we create the fractal figures. In
another words, the algorithm is correct. We prove it as
theorem 1.
Theorem 1. The fractal figures created by escape time
algorithm and the improved one are almost the same
except several escape points.
Proof:
i> When z is a fixed point or escape by one iteration, it
is easy to know the iteration numbers are same by the two
algorithms. Theorem 1 is proved.
ii> When the result is in Q by the first m iteration
times, we prove theorem 1 by case a~d. To simplify the
proof, we set fm(z)=z1 and Count(z1)=n. So when we set
Result(z)= Result(z1), the Count(z) is equal to m+n.
Case a. When m+n≤max, the result of z is changed by
z1. It can sum up to case a~d in cycle.
Case b. When m+n>max and |f(z1)|<threshold, z is
convergence by iteration times max. So we set
Count(z)=max, theorem is proved by this case.
Case c. When m+n>max and |f(z1)|≥threshold, We set
Count(z)=max. The result is different of the two
algorithms. In escape time algorithm, this kind of points
is considered to be convergence, but the improved
algorithm is opposite.
However, to consider escape time with the max
iteration number>m+n, we find the result is same with the
improved algorithm. So it is still to be considered the
same by the two algorithms. Theorem is proved by this
case.
Case d. When z is between k-asymptote and k+1asymptote, we calculate it with step 5 by improved
algorithm. It is same to the escape time algorithm.
Theorem is proved by this case.
JOURNAL OF COMPUTERS, VOL. 6, NO. 8, AUGUST 2011
Based on i and ii, theorem is proved.
We know the figure created by the improved algorithm
is same to escape time algorithm except several points.
But when we increase the max iteration number of escape
time algorithm, these points are same by the two
algorithm. In fact, the iteration number should be infinity
in dynamic system and fractal. So it should be considered
escaped by the points of case c of ii>.
C. The Complexity of the Algorithm
We compare the improved algorithm with the classic
escape time algorithm. A basic operation time is to iterate
z one time.
We use theorem 2 to compare the basic operation times
of the two algorithms.
Theorem 2. The operation times of the improved
algorithm is not more than the classic escape time
algorithm.
Proof:
a − a1
b −b
To set m= 2
and n= 2 1 (Δx and Δy are
∆x
∆y
increment of row and line of displayed area), we assume
aij is the operation times of the point (a1+i·Δx, b1+j·Δy) by
the classic algorithm and bij by the improved one. So the
sum operation times is
and
m
n
∑ ∑b
ij
i =1
m
n
i =1
j =1
∑∑a
ij
of escape time algorithm
of improved one.
j =1
To consider every z(i,j), we need to iterate max times
when z is convergence by the classic algorithm. It is easy
to know the iteration number by improved algorithm is
not more than the classic one.
When z is divergence, we know there exist k<max
make |fk(z)|≥threshold. It means aij<max. To consider
theorem 1, we can divide the condition to case a~d.
Case a. The iteration of z is not in Q by one time.
When the iteration of z is always outside Q , z iterate
one time in Q and aij-1 times outside Q because the
iteration number aij make z escape. So bij=1+aij-1=aij.
When the iteration number k make fk(z)=z1 ∈ Q, the
iteration time reduce Count(Result(z1)). It is to say that bij
is nor more than aij.
Case b. The iteration of z is not in Q by times m.
The iteration time is one in the first m iteration times
by improved algorithm. It is the same with case a when
the iteration number>m. So bij is nor more than aij.
Case c. The iteration of z is in Q until escape.
The iteration time is one by improved algorithm and
max by the classic one. So bij is nor more than aij.
Case d. The neighborhood rectangular of the iteration
time k1 of z is cross with k2-asymptote. (k1, k2≤max)
It is easy to know that bij=aij=k1+k2 (k1+k2≤max) or
bij=aij=max (k1+k2>max).
Based on above, we know bij≤aij for all i and j. So
m
n
m
n
i =1
j =1
∑ ∑b ≤ ∑ ∑ a
ij
i =1
j =1
ij
, theorem 2 is proved.
To study with case a~d, we find that the improved
algorithm is better than the classic one, especially in case
© 2011 ACADEMY PUBLISHER
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b and c. In fact, when we create the fractal figures, we are
always interested in the area of case a~d. So the improved
algorithm can reduce creating time.
Now we compare the complexity of the two.
Because the iterate function is uncertain and the
structure of fractal figures are complex, we can not solve
the complexity by normal method. So we give a new
method to compare the two by the creating figures.
We assume the displayed area has S pixels. When the
pixels are outside an unit rectangular of divergence
domain and crossed by an asymptote, we set the number
is S1. When the pixels are outside an unit rectangular of
divergence domain and not crossed by an asymptote, we
set the number is S2. Others’ number is S3. We set the
iteration times is E in S1, F in S2 and G in S3 by the
classic algorithm, and E* in S1, F* in S2 and G* in S3 by
the improved one. So the ratio of the two algorithm is
η=
E * + F * +G *
E + F +G
(1)
We set the max iteration number is max and the pixels
number is equal with every i-asymptote (i=1~max). So
we know that E=S1·max, E*=S1, F=S2·(max+1)/2, F*=S2
and G=G*.
To use them in formula (1), we gain formula (2).
η=
S1 + S 2 + G
S1 ⋅ max+ S 2 ⋅ (max+ 1) / 2 + G
(2)
When we create fractal figures, we are not interested in
other i-asymptotes except J set. Moreover, the area of J
set is small in displayed area. So η <<2/max generally. It
is to say that the reduced multiple is max/2.
Because we are concerned about J set, not i-asymptote,
we can change “If Count(zi) is not all equal (i=1~4)” to
“If Count(zi)=max and Count(zj)=max+1 (i, j=1~4)” in
step 5. It will be created the similar figures with great
speed.
In fact, we find most of the points are not changed with
step 5 by lots of experiment. We also find the figures are
similar to each other by the two algorithms. We can find
it in section 4.
IV. EXPERIMENT OF THE TWO ALGORITHMS
We compare the two algorithm by table 1 and figure
1~4. We create fractal figures of J set (The function is
f(z)=z2-1.25),
Newton
set
(The
function
is
f(z)=(2z3+1)/3z2, sine exponent function (The function is
f(z)= 3
sin 2
πz
2
and generalized 3x+1 function (The function
πz
sin
1
πz
is f(z)= ( x3 2 + sin 2 ) ) by both the two algorithms.
2
2
The displayed area, creating time and the number of
points with different Count() values are all in table 1. The
pixels in displayed area are 640×480.
We find the number of different points is small from
table 1. It means that it is hardly to find the difference
between the two figures created by the two algorithms.
Moreover, there are many different points are the points
in case iii of theorem 1. It is considered divergence by
2
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JOURNAL OF COMPUTERS, VOL. 6, NO. 8, AUGUST 2011
fractal and dynamic system. To find the difference
between the two kind figures intuitively, we set variable
array Iter(x,y) to memorize the iteration times for every
pixel (x,y) and instead of Count(x,y) when creating the
figures. Then we know the iteration times by the colors in
displayed area.
We create the fractal figures as figure 1. The threshold
is 10-5 in figure 1-a1and 1-a2. The threshold is 1000 in
other figures.
The figures are created by escape time algorithm in
figure 1-a1, 1-b1, 1-c1 and 1-d1, improved algorithm in
figure 1-a2, 1-b2, 1-c2 and 1-d2. The iterate function is
f(z)=(2z3+1)/3z2 in figure 1-a1 and 1-a2, f(z)=z2-1.25 in
sin 2
(Count() value=max and max+1 ), the improved
algorithm has less creating time.
The improved algorithm has large advantage when the
iteration function is complex and difficult to calculate.
For example, the time reduce greatly when the iteration
πz
πz
sin
sin
1
πz
functions are f(z)= 3 2 and f(z)= ( x3 2 + sin 2 ) .
2
2
We also find the number of different points is reduced. In
fact, the derivative is often large when the structure of
iteration function is complex. So the points in
convergence and divergence domain are less affected by
the initial value. Therefore, it does not affect the final
escape time though there is error by iteration.
We can see the iteration number in a2~d2 is much less
than a1~d1 from figure 1. In addition, the convergence
with large piece in c1 and d1 are corresponding to the
piece with escape time 1 in c2 and d2. We can find
similar conclusion in a1 and b1 corresponding to a2 and
b2.
So the new improved algorithm is advantage when the
condition is like case a~c.
Case a. There is self-mapping area in displayed area.
Case b. There is convergence attracted by attract point.
Case c. The structure of the iteration function and the
πz
1-b1 and 1-b2, f(z)= 3 2 in 1-c1 and 1-c2, f(z)=
πz
sin
1
πz
( x3 2 + sin 2 ) in 1-d1 and 1-d2[4-7]. The displayed
2
2
area is (-2,2)×(-2,2). We can see the iteration number of
the figures created by improved algorithm is smaller than
the classic one. It validate theorem 2.
We can see the creating time of improved algorithm is
more than the classic of J set and Newton set. The reason
is mostly for the simple iteration function. The time of
reduced iteration number is smaller than the query time in
step 3 and 5. But when we only consider J set in step 5
2
TABLE I.
Iterate Function f
2
2
RUNTIME OF THE TWO ALGORITHMS
f(z)=z2-1.25 f(z)=(2z3+1)/3z2
Displayed area
f(z)= 3
sin 2
πz
2
πz
sin
1
πz
( x3 2 + sin 2 )
2
2
2
f(z)=
Re∈(-2,2), Im∈(-2,2)
Escape Time Algorithm (s)
0.191
0.416
17. 582
145.677
Improved Algorithm(s)
0.292
0.520
3.243
6.449
Number with Different Count() value
0
0
4
8
Number of Changed Points with Step 5
3
15
2
3
Time by only J Set(s)
(Count=max、max+1)
0.183
0.408
1.943
3.747
Number of Different Points
3
11
2
0
Iteration times scale (└lnClassic/New┘)
-0.4
0
1.7
3.2
a1
a2
b1
b2
c1
c2
d1
d2
Fig.1 fractal image for Re∈(-2,2),Im∈(-2,2)
© 2011 ACADEMY PUBLISHER
JOURNAL OF COMPUTERS, VOL. 6, NO. 8, AUGUST 2011
calculation is complex.
Generally, The created fractal figures are contains
these cases. So it is great application value of the
improved algorithm. Meanwhile, this algorithm can used
into a series algorithms based on the classic escape time
algorithm.
V. CONCLUSION AND FUTURE WORK
In this essay, we have improved the classic escape time
algorithm by no-escape point. We have proved the
correctness of the algorithm. We have compared the two
algorithms by created some classic fractals and have
validated the advantage. We have found that the iteration
number is less than the classic algorithm and the creating
time is also less. We have solved the iteration number in
several conditions.
The next step we will do the two researches. One is
that we will improve the space complexity of this
algorithm. To study with figure 1, we find that the
judgment domain is convergence to the attract point when
the domain is small enough. So the other one is to find an
numerical algorithm to solve the attract periodic points.
ACKNOWLEDGMENT
The authors wish to thank the anonymous reviewers
for their helpful comments in reviewing this paper. This
work was supported by National Natural Science
Foundation of China (No. 60773098, 60905022).
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Liu Shuai, male, born in 1982, PhD in
College of Computer Science and
Technology, Jilin University. His main
research interests include computer
graphics and fractal.
Che Xiangjiu, male, born in 1969, PhD
and Professor in College of Computer
Science
and Technology,
Jilin
University. Senior membership of
China Computer Federation. His main
research interests include computer
graphics and CAGD.
Wang Zhengxuan, male, born in 1945,
professor and PhD supervisor in
College of Computer Science and
Technology, Jilin University.. He is a
member of the editorial board of
Journal of Computer-Aided Design &
Computer Graphics. His research
interests include computer graphics,
image processing and rendering of
fractal, computational geometry etc...