This document presents a paper on theoretical and practical bounds for the initial value of a skew-compensated clock used in clock skew compensation algorithms. It begins with background on clock skew compensation based on an extended Bresenham's algorithm. It then derives optimal error bounds for floating-point operations and uses these to extend the main theorem to provide tighter theoretical bounds on the initial clock value. It also loosens these bounds for practical implementation with limited precision. Numerical examples are provided to compare the derived bounds with results from single and double precision arithmetic. The paper aims to make clock skew compensation algorithms more robust to floating-point precision loss.

1. Theoretical and Practical Bounds on the Initial Value of
Skew-Compensated Clock for Clock Skew Compensation
Algorithm Immune to Floating-Point Precision Loss
Kyeong Soo (Joseph) Kim
(With Seungyeop Kang)
School of Advanced Technology
Xi’an Jiaotong-Liverpool University
2023 International Conference for Mathematics and Its Applications
Chungnam National University
January 16–18, 2023
1 / 36

3. Outline
Background
Review of Clock Skew Compensation Based on Extended Bresenham’s
Algorithm
Clock Skew Compensation with Optimal Error Bounds
Numerical Examples
Conclusions
3 / 36

4. Next . . .
Background
Review of Clock Skew Compensation Based on Extended Bresenham’s
Algorithm
Clock Skew Compensation with Optimal Error Bounds
Numerical Examples
Conclusions
4 / 36

5. Target Environment: WSN/IoT Nodes
▶ Measuring data and/or detect events
with sensors and connected to a
WSN/Internet only through wireless
channels.
▶ Limited in processing and
battery-powered.
5 / 36

6. Target Environment: WSN/IoT Nodes
▶ Measuring data and/or detect events
with sensors and connected to a
WSN/Internet only through wireless
channels.
▶ Limited in processing and
battery-powered.
5 / 36

7. Effect of Clock Skew on Time Synchronization
Ideal
Clock
Faster
Clock
Slower
Clock
Local
Time
Reference
Time
Time
Synchronization
Error
6 / 36

8. Next . . .
Background
Review of Clock Skew Compensation Based on Extended Bresenham’s
Algorithm
Clock Skew Compensation with Optimal Error Bounds
Numerical Examples
Conclusions
7 / 36

9. Hardware Clock Model
▶ We confine our discussions to a network with one head node and one
sensor node WLOG, where we describe the hardware clock T of the
sensor node with respect to the reference clock t of the head node
using the first-order affine clock model1
:
T(t) = (1 + ϵ) t + θ, (1)
where ϵ∈R and θ∈R denote the clock skew and offset, respectively;
(1 + ϵ)∈R+ in (1) is called clock frequency ratio in the literature.
▶ As we focus on the clock skew compensation, we simplify (1) by
setting θ to 0 and obtain
T(t) = (1 + ϵ) t. (2)
1
R. T. Rajan and A.-J. van der Veen, Proc. CAMSAP 2011, Dec. 2011, pp. 297–300.
8 / 36

10. Logical Clock Model
Compensating for the clock skew from the hardware clock T in (2), we can
obtain the logical clock t̂ of the sensor node—i.e., the estimation of the
reference clock t given the hardware clock T—as follows: For
ti<t≤ti+1 (i=0, 1, . . .),
t̂
T(t)
= t̂
T(ti)
+
T(t) − T(ti)
1 + ϵ̂i
, (3)
where ti is the reference time for the ith synchronization and ϵ̂i is the
estimated clock skew from the ith synchronization.2
2
The sensor node does not know the reference clock t as such but only the hardware
clock corresponding to t—i.e., T(t)—during the operation.
9 / 36

11. Theory vs. Reality
5 10 15 20 25 30 35 40 45 50
Number of Messages
10−15
10−14
10−13
10−12
10−11
10−10
10−9
MSE
RLS
CR
Joint MLE
GMLLE (Two-Way)
LB for CR
CRLB for Joint MLE
LB for GMLLE
MSE of the estimated clock skews.a
a
K. S. Kim, S. Lee, and E. G. Lim, IEEE
Trans. Commun., vol. 65, no. 1, pp. 347–359,
Jan. 2017.
0 200 400 600 800 1000 1200 1400 1600 1800
Time [s]
-1E-04
-5E-05
0E+00
5E-05
1E-04
Measurement
Time
Estimation
Error
[s]
EE-ASCFR
Measurement time estimation errors of EE-ASCFR.a
a
X. Huan and K. S. Kim, Comput. Netw.,
vol. 166, pp. 1–8, Jan. 2020.
10 / 36

12. Direct Application of Bresenham’s Algorithm
Δ
Δ
1
̂
0,0 1 2 

1
2
− 1 + 1 
+ 1

∆
∆
−
− ( )
1 + ̂
11 / 36

13. Extension of Bresenham’s Algorithm I
Lemma (boundedness of ▽i)
▽i satisfies the following inequality:
|▽i| 2∆a. (4)
Definition (valid point set)
Given a destination point
(∆a, ∆b), we define a valid point
set V (∆a, ∆b) as a set of the points
on a valid path from (0, 0) to
(∆a, ∆b) according to Bresenham’s
algorithm.
(6,4)
0,0 1 2
1
2
3
4
3 4 5 6
All possible paths from the origin to a given destination point
based on M1 and M2 movements.
12 / 36

14. Extension of Bresenham’s Algorithm II
Definition (backward-reachable set)
Given a point (i, j)∈V (∆a, ∆b), we define a backward-reachable set B(i, j) as
a set of the points that can reach (i, j) by any combination of the movements
M1 and M2 as follows:
B(i, j) ≜
(k, l)|0≤ki, max(0, k−∆a+∆b)≤l≤ min(k, j) . (5)
Definition (extension of ▽i)
For a point (i, j)∈B(∆a, ∆b), we define ▽i(j) as follows:
▽i(j) ≜ 2(i∆b − j∆a). (6)
13 / 36

15. Extension of Bresenham’s Algorithm - III
Lemma (reachability of ▽i(j))
A point (i, j)∈V (∆a, ∆b) is reachable from any point (k, l)∈B(i, j) if we apply
Bresenham’s algorithm using ▽·(·) instead of ▽·.
14 / 36

16. Extension of Bresenham’s Algorithm - IV
Theorem (Clock Skew Compensation Based on Extended
Bresenham’s Algorithm)
Given the hardware clock i, we can obtain its skew-compensated clock j as follows:
Case 1. D
A
1: The skew-compensated clock j satisfies
i
D
A
− 1 j i
D
A
+ 1. (7)
Unless iD
A
is an integer, there are two values satisfying (7). Due to the effect of
limited floating-point precision, however, we cannot know the exact value of iD
A
.
15 / 36

17. Extension of Bresenham’s Algorithm - IV
Theorem (Clock Skew Compensation Based on Extended
Bresenham’s Algorithm)
In this regard, (7) can be extended to include the effect of the precision loss:
i
D
A
− 1 − ε j i
D
A
+ 1 + ε, (8)
where ε(0) is the error due to the precision loss.
Let k, . . ., k+l be the candidate values of j satisfying (8). We determine j by starting
from the point (i−l, k) and applying Bresenham’s algorithm with ▽i−l(k) and on; j
is determined by the y coordinate of the valid point whose x coordinate is i.
15 / 36

18. Extension of Bresenham’s Algorithm - IV
Theorem (Clock Skew Compensation Based on Extended
Bresenham’s Algorithm)
Case 2. D
A
1: In this case, we can decompose the skew-compensated clock j into
two components as follows:
j = i
D
A
= i + i
D − A
A
. (9)
Now that D−A
A
1, we can apply the same procedure of Case 1 to the second
component in (9) by setting ∆a and ∆b to A and D−A, respectively. Let ¯
j be the
result from the procedure. The skew-compensated clock j is given by i+ ¯
j as per (9).
15 / 36

19. Clock Skew Compensation Example
− 4 − 3
+ 1
− 2 − 1
+ 2
+ 3
+ 4
An example of common starting points reaching all possible candidate points for the case of ∆b
∆a ≤1.
16 / 36

20. ̂
T
∆
∆
−
− ( )
1 + ̂
0,0
Eq. (8)
Clock skew compensation based on extended Bresenham’s algorithm.
17 / 36

21. Next . . .
Background
Review of Clock Skew Compensation Based on Extended Bresenham’s
Algorithm
Clock Skew Compensation with Optimal Error Bounds
Numerical Examples
Conclusions
18 / 36

22. Issues in The Clock Skew Compensation Based on
Extended Bresenham’s Algorithm
▶ The extended bounds on the value of skew-compensated clock, which
are closely related with the initial condition of the proposed algorithm,
include a term for floating-point operation error due to precision loss
(i.e., ε in (8)).
▶ The value of ε is not explicitly mentioned in the theorem but
approximately set for numerical examples based on the property of the
single-precision floating-point format as defined in the IEEE standard for
floating-point arithmetic (IEEE 754-2008) [1].
▶ We revisit the main theorem, therefore, and provide practical as well as
theoretical bounds based on a systematic analysis of the errors of
floating-point operations.
19 / 36

23. Relative Errors of Floating-Point Operations - I
▶ An associated set of floating point numbers F with a base β and a precision
p and no restriction on the exponent range is defined as follows:
F = {0} ∪
n
Mβe
M, e ∈ Z, βp−1
≤ |M| βp
o
. (10)
▶ Let fl : R → F denote a round-to-nearest function satisfying
|t − fl(t)| = min
f∈F
|t − f|, t ∈ R. (11)
Single-precision floating-point format based on IEEE 754 standard.3
3
“Single-precision floating-point format,” Wikipedia: The Free Encyclopedia,
Wikimedia Foundation, Accessed: Aug. 4, 2022.
20 / 36

24. Relative Errors of Floating-Point Operations - II
Based on F and fl, we can define two relative errors for t∈R and t,0:
E1(t) ≜
|t − fl(t)|
|t|
(12)
E2(t) ≜
|t − fl(t)|
|fl(t)|
, (13)
where E1 and E2 are the errors relative to t and fl(t), respectively.
21 / 36

25. Relative Errors of Floating-Point Operations - III
Table: Optimal relative error bounds 4.
t bound on E1(t) bound on E2(t)
real number u
1+u
u
xy u
1+u
u
x/y







u − 2u2
if β = 2,
u
1+u
if β 2







u−2u2
1+u−2u2 if β = 2,
u if β 2
▶ x, y∈F.
▶ u=1
2
β1−p
is the unit roundoff associated with fl and F.
4
C.-P. Jeannerod and S. M. Rump, Mathematics of computation, vol. 87, no. 310, pp.
803–819, 2018.
22 / 36

26. Bounds on The Initial Value of Skew-Compensated Clock
Lemma
For t=x
y
z
, where x, y, z∈R with x, y≥0 and z0, fl(t) satisfies
1 − u + 2u2
(1 + u)2(1 + 2u)
t ≤ fl(t) ≤
(1 + 2u)3
(1 + u − 2u2
)
(1 + u)2
t. (14)
▶ Note that ε (≥0) in (8) of the main theorem indicates the error due to
the precision loss in computing iD
A
.
▶ The above lemma provides the optimal bounds on the initial value of
skew-compensated clock based on the floating-point operations of
non-negative real numbers.
23 / 36

27. Extension of The Main Theorem
Now (8) of the main theorem can be extended as follows:
$
1−u+2u2
(1+u)2(1+2u)
t
%
≤ j ≤
(1 + 2u)3
(1+u−2u2
)
(1+u)2
t
'
, (15)
where t=iD
A
and u=2−p
.
24 / 36

28. Loosening Bounds for Practical Implementation I
▶ Though (15) provides theoretically-guaranteed lower and upper bounds
on the initial value of skew-compensated clock j, obtaining the exact
values of bounds—i.e., the lhs and the rhs—could be a challenge,
especially at resource-constrained sensor nodes with limited
floating-point precision.
▶ For its practical implementation based on limited floating-point
precision, therefore, we can loosen the lhs and the rhs of (15) as follows:
$
1−u+2u2
(1+u)2(1+2u)
t
%
≤ j ≤
(1 + 2u)3
(1+u−2u2
)
(1+u)2
t
'
,
$
1−u
(1+2u)3
t
%
≤ j ≤
l
(1 + 2u)3
t
m
. (16)
25 / 36

29. Loosening Bounds for Practical Implementation II
▶ Note that the lhs and the rhs of (16) consist only of the elements of F,
i.e.,
1 − u = (2p
− 1) 2−p
∈ F,
1 + 2u =
2p−1
+ 1
21−p
∈ F,
which eliminates the rounding errors for those terms not belonging to
F in (15).
26 / 36

30. Next . . .
Background
Review of Clock Skew Compensation Based on Extended Bresenham’s
Algorithm
Clock Skew Compensation with Optimal Error Bounds
Numerical Examples
Conclusions
27 / 36

31. Experimental Settings
▶ We fix D to 1, 000, 000 and generate one million samples of A whose
clock skews are uniformly distributed in the range of
[−100 ppm, 100 ppm].
▶ For the clock skew compensation by single and double-precision
floating-point arithmetic, we round down the results to obtain integer
values.
▶ The calculated bounds are compared to those based on binary512
floating-point format providing 489 precision in bits, which serve as a
reference for the comparison.
28 / 36

32. Comparison of Bounds
Bounds i
∆LB*
∆UB†
Min. Max. Avg. Min. Max. Avg.
1e6 0 0 0 0 0 0
Theoretical bounds of (15) 1e7 0 0 0 0 0 0
based on double precision 1e8 0 0 0 0 0 0
1e9 0 0 0 0 0 0
1e6 0 0 0 0 0 0
Theoretical bounds of (15) 1e7 −1 0 −5.0115e−1 0 1 2.2325e−1
based on single precision 1e8 -18 0 −7.9666 0 6 1.7206
1e9 -125 0 −4.7102e1 0 104 3.6533e1
1e6 0 0 0 0 0 0
Practical bounds of (16) 1e7 0 2 1.0023 0 1 2.2325e−1
based on single precision 1e8 0 12 5.9980 0 6 2.9912
1e9 0 120 5.9890e1 0 60 2.9880e1
1e6 0 1 4.9696e−1 0 0 0
Approximate bounds of (11) of [2] 1e7 −2 1 −5.0346e−1 −2 1 −2.8513e−1
based on single precision with ε=10−7
i 1e8 -20 10 −5.0063 -20 10 −4.8369
1e9 -199 100 −4.9669e1 -199 100 −4.9645e1
*
∆LB = LBbinary512 − LB, where LBbinary512 is the lower bound of (15) based on binary512.
†
∆UB = UB − UBbinary512, where UBbinary512 is the upper bound of (15) based on binary512.
29 / 36

33. Comparison of Algorithms
Algorithm i
Compensation error*
# of iterations
Min. Max. Avg. Min. Max. Avg.
Single precision†
1e6 0 0 0 – – –
1e7 0 0 0 – – –
1e8 −4 1 −1.9842 – – –
1e9 -19 44 1.2502e1 – – –
1e6 −1 0 −4.9663e−1 1 2 1.5034
Clock skew compensation with 1e7 −1 0 −4.9668e−1 1 8 4.5232
practical bounds of (16) 1e8 −1 0 −5.0220e−1 1 72 3.6710e1
1e9 −1 0 −4.9734e−1 1 832 4.1872e2
1e6 −1 0 −4.9663e−1 2 2 2
Clock skew compensation with 1e7 −1 0 −4.9668e−1 3 4 3.0050
approximate bounds of (11) of [2] 1e8 −1 0 −5.0220e−1 21 22 2.1005e1
1e9 −1 0 −4.9734e−1 201 202 2.0101e2
*
With respect to ⌊iD
A
⌋ based on double precision.
†
⌊iD
A
⌋ based on single precision.
30 / 36

34. Discussions: Comparison of Bounds
▶ Double precision is enough for the calculation of the theoretical
bounds of (15), while the results for both theoretical bounds of (15) and
the approximate bounds of [2] based on single precision violate the
reference bounds of (15) based on binary512 due to the floating-point
precision loss.
▶ This implies that the correctness of the results of the clock skew
compensation based on the main theorem with the approximate bounds
cannot be theoretically guaranteed.
▶ The loose practical bounds of (16) based on single precision do not
violate the reference bounds in spite of the limited floating-point
precision.
▶ It turns out that the loosening of the bounds counteracts the effect of
limited floating-point precision on the calculation of (16).
31 / 36

35. Discussions: Comparison of Bounds
▶ Double precision is enough for the calculation of the theoretical
bounds of (15), while the results for both theoretical bounds of (15) and
the approximate bounds of [2] based on single precision violate the
reference bounds of (15) based on binary512 due to the floating-point
precision loss.
▶ This implies that the correctness of the results of the clock skew
compensation based on the main theorem with the approximate bounds
cannot be theoretically guaranteed.
▶ The loose practical bounds of (16) based on single precision do not
violate the reference bounds in spite of the limited floating-point
precision.
▶ It turns out that the loosening of the bounds counteracts the effect of
limited floating-point precision on the calculation of (16).
31 / 36

36. Discussions: Comparison of Bounds
▶ Double precision is enough for the calculation of the theoretical
bounds of (15), while the results for both theoretical bounds of (15) and
the approximate bounds of [2] based on single precision violate the
reference bounds of (15) based on binary512 due to the floating-point
precision loss.
▶ This implies that the correctness of the results of the clock skew
compensation based on the main theorem with the approximate bounds
cannot be theoretically guaranteed.
▶ The loose practical bounds of (16) based on single precision do not
violate the reference bounds in spite of the limited floating-point
precision.
▶ It turns out that the loosening of the bounds counteracts the effect of
limited floating-point precision on the calculation of (16).
31 / 36

37. Discussions: Comparison of Bounds
▶ Double precision is enough for the calculation of the theoretical
bounds of (15), while the results for both theoretical bounds of (15) and
the approximate bounds of [2] based on single precision violate the
reference bounds of (15) based on binary512 due to the floating-point
precision loss.
▶ This implies that the correctness of the results of the clock skew
compensation based on the main theorem with the approximate bounds
cannot be theoretically guaranteed.
▶ The loose practical bounds of (16) based on single precision do not
violate the reference bounds in spite of the limited floating-point
precision.
▶ It turns out that the loosening of the bounds counteracts the effect of
limited floating-point precision on the calculation of (16).
31 / 36

38. Discussions: Comparison of Algorithms
▶ For the randomly-generated one million samples of A, the clock skew
compensation algorithm of [2] with both approximate and practical
bounds provides exactly the same bounded compensation errors
compared to the unbounded errors for the single-precision algorithm.
▶ The numbers of iterations with the practical bounds are larger than
those with the approximate bounds for i≥1e7.
▶ This indicates that the approximate bounds of [2] could provide valid
initial conditions for the cases considered in these examples with less
numbers of iterations than those for the practical bounds of (16) but at
the expense of the lack of theoretical guarantee.
32 / 36

39. Discussions: Comparison of Algorithms
▶ For the randomly-generated one million samples of A, the clock skew
compensation algorithm of [2] with both approximate and practical
bounds provides exactly the same bounded compensation errors
compared to the unbounded errors for the single-precision algorithm.
▶ The numbers of iterations with the practical bounds are larger than
those with the approximate bounds for i≥1e7.
▶ This indicates that the approximate bounds of [2] could provide valid
initial conditions for the cases considered in these examples with less
numbers of iterations than those for the practical bounds of (16) but at
the expense of the lack of theoretical guarantee.
32 / 36

40. Discussions: Comparison of Algorithms
▶ For the randomly-generated one million samples of A, the clock skew
compensation algorithm of [2] with both approximate and practical
bounds provides exactly the same bounded compensation errors
compared to the unbounded errors for the single-precision algorithm.
▶ The numbers of iterations with the practical bounds are larger than
those with the approximate bounds for i≥1e7.
▶ This indicates that the approximate bounds of [2] could provide valid
initial conditions for the cases considered in these examples with less
numbers of iterations than those for the practical bounds of (16) but at
the expense of the lack of theoretical guarantee.
32 / 36

41. Next . . .
Background
Review of Clock Skew Compensation Based on Extended Bresenham’s
Algorithm
Clock Skew Compensation with Optimal Error Bounds
Numerical Examples
Conclusions
33 / 36

42. Conclusions
▶ Revisit the main theorem of [2] and derive theoretical and practical
bounds on the initial value of skew-compensated clock based on a
systematic analysis of the errors of floating-point operations.
▶ The theoretical bounds provide a reference framework for the analysis
and comparison of bounds on the initial value of skew-compensated
clock.
▶ The practical bounds could replace the approximate bounds suggested
in [2] as an implementation option for limited floating-point precision.
▶ Demonstrate through numerical examples that the proposed practical
bounds based on single-precision floating-point format do not violate
the theoretical bounds and thereby can guarantee the correctness of
the clock skew compensation even on resource-constrained computing
platforms like WSN/IoT sensor nodes.
34 / 36

43. Conclusions
▶ Revisit the main theorem of [2] and derive theoretical and practical
bounds on the initial value of skew-compensated clock based on a
systematic analysis of the errors of floating-point operations.
▶ The theoretical bounds provide a reference framework for the analysis
and comparison of bounds on the initial value of skew-compensated
clock.
▶ The practical bounds could replace the approximate bounds suggested
in [2] as an implementation option for limited floating-point precision.
▶ Demonstrate through numerical examples that the proposed practical
bounds based on single-precision floating-point format do not violate
the theoretical bounds and thereby can guarantee the correctness of
the clock skew compensation even on resource-constrained computing
platforms like WSN/IoT sensor nodes.
34 / 36

44. Conclusions
▶ Revisit the main theorem of [2] and derive theoretical and practical
bounds on the initial value of skew-compensated clock based on a
systematic analysis of the errors of floating-point operations.
▶ The theoretical bounds provide a reference framework for the analysis
and comparison of bounds on the initial value of skew-compensated
clock.
▶ The practical bounds could replace the approximate bounds suggested
in [2] as an implementation option for limited floating-point precision.
▶ Demonstrate through numerical examples that the proposed practical
bounds based on single-precision floating-point format do not violate
the theoretical bounds and thereby can guarantee the correctness of
the clock skew compensation even on resource-constrained computing
platforms like WSN/IoT sensor nodes.
34 / 36

45. Conclusions
▶ Revisit the main theorem of [2] and derive theoretical and practical
bounds on the initial value of skew-compensated clock based on a
systematic analysis of the errors of floating-point operations.
▶ The theoretical bounds provide a reference framework for the analysis
and comparison of bounds on the initial value of skew-compensated
clock.
▶ The practical bounds could replace the approximate bounds suggested
in [2] as an implementation option for limited floating-point precision.
▶ Demonstrate through numerical examples that the proposed practical
bounds based on single-precision floating-point format do not violate
the theoretical bounds and thereby can guarantee the correctness of
the clock skew compensation even on resource-constrained computing
platforms like WSN/IoT sensor nodes.
34 / 36

46. References
[1] IEEE Computer Society, IEEE Std 754™-2008, IEEE Standard for
floating-point arithmetic, Std., Aug. 2008.
[2] K. S. Kim and S. Kang, “Clock skew compensation algorithm immune
to floating-point precision loss,” IEEE Commun. Lett., vol. 26, no. 4, pp.
902–906, Apr. 2022.
[3] S. Kang and K. S. Kim, “Theoretical and practical bounds on the initial
value of skew-compensated clock for clock skew compensation
algorithm immune to floating-point precision loss,” ArXiv e-prints,
2022. [Online]. Available: https://arxiv.org/abs/2208.11520
35 / 36

47. Thanks for your attention!
If you have any question, please contact me at
Kyeongsoo.Kim@xjtlu.edu.cn.