# ILP-Based and Heuristic Scheduling Techniques for Variable-Cycle Approximate Functional Units in High-Level Synthesis

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## Abstract

**:**

## 1. Introduction

- Scheduling for approximate computing circuits with accuracy-controllable approximate multipliers is mathematically derived using an ILP formulation. Our proposed scheduling takes account of exact and approximate computations, and determines that each arithmetic operation is scheduled as either exact or approximate under resource and time constraints such that the error at the output is minimized.
- A list-scheduling algorithm is proposed to solve the proposed scheduling problem in polynomial time, which can solve faster than the ILP method.

## 2. Related Work

## 3. ILP-Based Scheduling for Variable-Cycle Approximate Functional Units in High-Level Synthesis

#### 3.1. A Scheduling Example

#### 3.2. ILP Formulation

#### 3.3. Chaining

## 4. Heuristic Scheduling Algorithms Based on List Scheduling

#### 4.1. List-Scheduing Algorithm

Algorithm 1 List-Scheduling Algorithm | |

1 | ListScheduling(G(V,E)) begin |

2 | for $i\in \mathrm{V}$ do |

3 | ${t\_\mathit{alap}}_{i}$ ← ALAP_schedule |

4 | end for |

5 | for $i\in \mathrm{V}$ do |

6 | ${e}_{i}\leftarrow {\mathrm{Obtain}\mathit{res}}_{i}{,\mathit{err}}_{i}\mathrm{from}\mathrm{G}\left(\mathrm{V},\mathrm{E}\right)\mathrm{and}\mathrm{any}\mathrm{input}$ |

7 | end for |

8 | for n in 1..|M|+1 do |

9 | for $i\in \mathrm{V}$ do |

10 | if $i\in \mathrm{A}{\displaystyle \cup}i\in \mathrm{Apx}$ then |

11 | ${p}_{i}{=t\_\mathit{alap}}_{i}$ |

12 | else ${p}_{i}{=t\_\mathit{alap}}_{i}-{(T}_{\mathit{res}}-{T}_{\mathit{apx}})$ end if |

13 | end for |

14 | $t$ = 0,$\pi =\{\varnothing \},\tau =\{\varnothing \}$ |

15 | while $\pi \ne \mathrm{V}$ do |

16 | ${N}_{\mathit{mul}}{=\mathit{Const}}_{\mathit{mul}},t++$ |

17 | for $i\in \mathrm{V}$ do |

18 | for $i\in \tau $ do |

19 | if $i\in \mathrm{M}$ then |

20 | ${\mathit{tr}}_{i}{=\mathit{tr}}_{i}-{1,N}_{\mathit{mul}}{=N}_{\mathit{mul}}-1$ |

21 | if $t{r}_{i}=0$ then |

22 | $\pi =\pi \cup i$, $\tau =\tau \cap {\neg i,\mathit{tf}}_{i}=t$ |

23 | $\sigma \leftarrow \mathrm{successor}\mathrm{nodes}\mathrm{of}\mathrm{operation}i\mathrm{in}\mathrm{V}$ |

24 | end if |

25 | end if |

26 | if $i\in \mathrm{A}$ then |

27 | ${\mathit{tr}}_{i}{=\mathit{tr}}_{i}-1$ |

28 | if ${\mathit{tr}}_{i}=0$ then |

29 | $\pi =\pi \cup i$, $\tau =\tau \cap \neg i$, ${\mathit{tf}}_{i}=t$ |

30 | $\sigma \leftarrow \mathrm{successor}\mathrm{nodes}\mathrm{of}\mathrm{operation}i\mathrm{in}\mathrm{V}$ |

31 | end if |

32 | end if |

33 | end for |

34 | if ${N}_{\mathit{mul}}\ge 1\cap \left\{i\right|i\in \left(\sigma \cap M\right)\cap \mathrm{min}({p}_{i}\left)\right\}$ then |

35 | $t{s}_{i}=t$, ${N}_{\mathit{mul}}{=N}_{\mathit{mul}}-1,\sigma =\sigma \cap \neg i$ |

36 | if $i\in \mathrm{Apx}\cap {T}_{\mathit{apx}}=1$ then |

37 | $\pi =\pi \cup i$, ${\mathit{tf}}_{i}=t$ |

38 | $\sigma \leftarrow \mathrm{successor}\mathrm{nodes}\mathrm{of}\mathrm{operation}i\mathrm{in}\mathrm{V}$ |

39 | elif $i\in \mathrm{Apx}\cap {T}_{\mathit{apx}}\text{}1$ then |

40 | $\tau =\tau \cup i$, ${\mathit{tr}}_{i}{=T}_{\mathit{apx}}-1$ |

41 | else $\tau =\tau \cup i$, ${\mathit{tr}}_{i}{=T}_{\mathit{res}}-1$ end if |

42 | end if |

43 | if $\left\{i\right|i\in \left(\sigma \cap \mathrm{A}\right)\cap {\mathrm{min}(p}_{i}\left)\right\}$ then |

44 | ${\mathit{ts}}_{i}=t,\sigma =\sigma \cap \neg i$ |

45 | if $i\in \mathrm{A}\cap {T}_{\mathit{op}}=1$ then |

46 | $\pi =\pi \cup i$, ${\mathit{tf}}_{i}=t$ |

47 | |

48 | else $\tau =\tau \cup i$, ${\mathit{tr}}_{i}{=T}_{\mathit{op}}-1,$ end if |

49 | end if |

50 | end for |

51 | end while |

52 | if $\forall {i,\mathit{tf}}_{i}\le {\mathit{Const}}_{\mathit{time}}$ then |

53 | ${\mathit{best}\_\mathit{ts}}_{i}{=\mathit{ts}}_{i}$,${\mathit{best}\_\mathit{tf}}_{i}{=\mathit{tf}}_{i},\mathit{best}\_\mathrm{Apx}=\mathrm{Apx}$ |

54 | $\{{\mathrm{M}}^{\prime}={\mathrm{M}}^{\prime}\cap \neg i\}\cap \left\{i\right|i\in {\mathrm{M}}^{\prime}\cap {\mathrm{max}(e}_{i}\left)\right\}$ |

55 | else $\{\mathrm{Apx}=\mathrm{Apx}\cup i\}\cap \left\{i\right|i\in {\mathrm{M}}^{\prime}\cap {\mathrm{max}(e}_{i}\left)\right\}$ |

56 | $\{{\mathrm{M}}^{\prime}={\mathrm{M}}^{\prime}\cap \neg i\}\cap \left\{i\right|i\in {\mathrm{M}}^{\prime}\cap {\mathrm{max}(e}_{i}\left)\right\}$ |

57 | end if |

58 | $\{\mathrm{Apx}=\mathrm{Apx}\cap \neg i\}\cap \left\{i\right|i\in {\mathrm{M}}^{\prime}\cap {\mathrm{max}(e}_{i}\left)\right\}$ |

59 | end for |

60 | end |

#### 4.2. Proposed List-Scheduling Example

#### 4.3. Chaining

Algorithm 2 Add in List-Scheduling Algorithm ① | |

1 | if$i\in \mathrm{D}$then |

2 | ${\mathit{tr}}_{i}{=\mathit{tr}}_{i}-1,$ |

3 | if ${\mathit{tr}}_{i}=0$ then |

4 | $\pi =\pi \cup i$, $\tau =\tau \cap \neg i$, ${\mathit{tf}}_{i}=t$ |

5 | |

6 | End if |

7 | End if |

Algorithm 3 Add in List-Scheduling Algorithm ② | |

1 | if$\left\{i\right|i\in \left(\sigma \cap \mathrm{D}\right)\cap {\mathrm{min}(p}_{i}\left)\right\}$then |

2 | ${\mathit{ts}}_{i}=t,\sigma =\sigma \cap \neg i$ |

3 | if $i\in \mathrm{D}\cap {T}_{add}=1$ then |

4 | $\pi =\pi \cup i$, ${\mathit{tf}}_{i}=t$ |

5 | |

6 | else $\tau =\tau \cup i$, ${\mathit{tr}}_{i}{=T}_{add}-1,$ end if |

7 | End if |

Algorithm4 Chaining in List-Scheduling Algorithm | |

1 | if$\mathit{Condition}\_\mathit{chaining}\_\mathit{mul}=1$then |

2 | for $i\in \mathrm{V}$ do |

3 | if $\cap}\{i\in \mathrm{M},\pi \cap {\mathit{tf}}_{i}=t\cap i\mathrm{followed}\mathrm{by}j\in \mathrm{D}\$ then |

4 | ${\mathit{ts}}_{j}=t,\sigma =\sigma \cap \neg i$ |

5 | if $j\in \mathrm{D}\cap {T}_{add}=1$ then |

6 | $\pi =\pi \cup j$, ${\mathit{tf}}_{j}=t$ |

7 | $\sigma \leftarrow \mathrm{successor}\mathrm{nodes}\mathrm{of}\mathrm{op}j\mathrm{in}\mathrm{V}$ |

8 | else $\tau =\tau \cup j$, ${\mathit{tr}}_{j}{=T}_{add}-1,$ end if |

9 | end if |

10 | if ${\mathrm{N}}_{mul}\ge 1{\displaystyle \cap}\{i\in \mathrm{D},\pi \cap {\mathit{tf}}_{i}=t\cap i\mathrm{followed}\mathrm{by}j\in \mathrm{M}\}$ then |

11 | ${\mathit{ts}}_{i}=t$, ${\mathrm{N}}_{mul}{=\mathrm{N}}_{mul}-1,\sigma =\sigma \cap \neg i$ |

12 | if $i\in \mathrm{M}\cap {T}_{res}=1$ then |

13 | $\pi =\pi \cup i$, ${\mathit{tf}}_{i}=t$ |

14 | $\sigma \leftarrow \mathrm{successor}\mathrm{nodes}\mathrm{of}\mathrm{op}i\mathrm{in}\mathrm{V}$ |

15 | else $\tau =\tau \cup i$, ${\mathit{tr}}_{i}{=T}_{res}-1,$ end if |

16 | end if |

17 | end for |

18 | end if |

19 | if$\mathit{Condition}\_\mathit{chaining}\_\mathit{add}=1$then |

20 | for $i\in \mathrm{V}$ do |

21 | if $\{i\in \mathrm{D},\pi \cap {\mathit{tf}}_{i}=t\cap i\mathrm{followed}\mathrm{by}j\in \mathrm{D}\}$ then |

22 | ${\mathit{ts}}_{i}=t$,$\sigma =\sigma \cap \neg i$ |

23 | if $i\in \mathrm{D}\cap {T}_{add}=1$ then |

24 | $\pi =\pi \cup i$, ${\mathit{tf}}_{i}=t$ |

25 | |

26 | else $\tau =\tau \cup i$, ${\mathit{tr}}_{i}{=T}_{\mathit{add}}-1,$ end if |

27 | end if |

28 | end for |

29 | end if |

## 5. Experiment

#### 5.1. Exprimental Setup

- All-exact (AE): each of the multiplications is performed without approximation and takes two cycles.
- All-approximated (AA): each of the multiplications is approximated and performed in one cycle.
- Mixed: each multiplication is determined as being either exact or approximated in two cycles or one cycle, respectively.
- Mixed-chain: each multiplication is determined as being either exact or approximated in two cycles or one cycle, respectively, and considering chaining.

#### 5.2. Exprrimental Results

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**An example of scheduling for approximate computing circuits with variable-cycle approximate multipliers. (

**a**) Approximate multiplications. (

**b**) Exact multiplications. (

**c**) Variable-cycle multiplications.

**Figure 2.**Example of chaining in this work. (

**a**) Add to Add (no Chaining). (

**b**) Add to Add (Chaining). (

**c**) Exact Mult to Add (no Chaining). (

**d**) Exact Mult to Add (Chaining).

**Figure 3.**Proposed list-scheduling example. (

**a**) A given DFG. (

**b**) Approximate all multiplications (result of first list scheduling). (

**c**) Exact multiplication with the largest error(result of second list scheduling). (

**d**) Exact multiplication with the second largest error (result of third list scheduling). (

**e**) Exact multiplication with the smallest error (result of fourth list scheduling). (

**f**) Final output that satisfies resource and time constraints.

G(V,E) | Data Flow Graph (DFG) |
---|---|

V | Set of operations |

E | Set of data dependencies between operations |

M | Set of multiplications ($\mathrm{M}\subseteq \mathrm{V}$) |

A | Set of operations other than multiplication (A$\subseteq \mathrm{V}$) |

$\mathrm{Apx}$ | Set of approximate multiplications $(\mathrm{Apx}\subseteq \mathrm{M}$) |

${\mathrm{M}}^{\prime}$ | Set of multiplications that have never been performed as exact multiplications $({\mathrm{M}}^{\prime}\subseteq \mathrm{M})$ |

$\sigma $ | Set of operations that can be executed |

$\tau $ | Set of operations that are being executed (set of operations currently running in multi-cycle) |

$\pi $ | Set of operations that have completed execution |

${T}_{\mathit{apx}}$ | Number of cycles required for approximate multiplication |

${T}_{\mathit{res}}$ | Number of cycles required for exact multiplication |

${T}_{\mathit{op}}$ | Number of cycles required for operations other than multiplication |

$i$ | i-th of all operations |

$t$ | Current clock cycle |

${\mathit{ts}}_{i}$ | Execution start time of the i-th operation |

${\mathit{tf}}_{i}$ | Execution finish time of the i-th operation |

${\mathit{tr}}_{i}$ | Remaining time until the end of the execution of the i-th operation |

${\mathit{Const}}_{\mathit{mul}}$ | The number of available multipliers (resource constraints) |

${\mathit{Const}}_{\mathit{time}}$ | The number of execution cycles for the entire circuit (time constraint) |

${N}_{\mathit{mul}}$ | Remaining number of multipliers available for the current clock cycle |

${t\_\mathit{alap}}_{i}$ | Execution time of the i-th operation in ALAP (result of ALAP) |

${\mathit{res}}_{i}$ | Exact value of the i-th operation |

${\mathit{err}}_{i}$ | Error of the i-th operation |

${e}_{i}$ | The magnitude of error given to the final output when approximating the i-th operation |

${p}_{i}$ | Priority based on ALAP results |

${T}_{add}$ | Number of cycles required for addition |

$j$ | i-th of addition |

$\mathit{Condition}\_\mathit{chaining}\_\mathit{mul}$ | Value indicating whether exact multiplication and addition can be performed chaining |

$\mathit{Condition}\_\mathit{chaining}\_\mathit{add}$ | Value indicating whether additions can be performed chaining |

$\mathrm{D}$ | Set of additions |

Benchmarks | Nodes (Mult) | Designs | Wins | Losses | Draws | ILP Exceeding 1 h |
---|---|---|---|---|---|---|

HAL | 11 (6) | 14 | 0 | 0 | 14 | 0 |

FIR filter | 21 (11) | 19 | 0 | 0 | 19 | 0 |

Auto Regression Filter | 28 (16) | 36 | 0 | 0 | 36 | 0 |

Motion Vectors Decoder | 32 (14) | 37 | 0 | 15 | 22 | 0 |

Elliptic Wave Filter | 34 (8) | 16 | 0 | 3 | 13 | 0 |

Cosine | 42 (14) | 52 | 0 | 1 | 51 | 0 |

Feedback Points | 53 (17) | 43 | 0 | 10 | 33 | 1 |

Matrix Multiplication | 109 (40) | 129 | 1 | 20 | 108 | 4 |

Smooth Triangle | 197 (69) | 257 | 35 | 63 | 159 | 55 |

Matrix Inversion | 333 (140) | 516 | 235 | 104 | 177 | 257 |

Benchmarks | ILP | List Scheduling | ||||
---|---|---|---|---|---|---|

Max | Min | Mean | Max | Min | Mean | |

HAL | 0.130 | 0.010 | 0.083 | 0.006 | 0.005 | 0.006 |

FIR filter | 0.860 | 0.080 | 0.214 | 0.020 | 0.013 | 0.017 |

Auto Regression Filter | 8.730 | 0.080 | 0.899 | 0.049 | 0.023 | 0.037 |

Motion Vectors Decoder | 35.380 | 0.080 | 1.269 | 0.050 | 0.024 | 0.036 |

Elliptic Wave Filter | 0.220 | 0.050 | 0.127 | 0.033 | 0.029 | 0.031 |

Cosine | 2387 | 0.050 | 46.490 | 0.087 | 0.039 | 0.058 |

Feedback Points | >3600 | 0.080 | 85.617 | 0.138 | 0.066 | 0.100 |

Matrix Multiplication | >3600 | 0.200 | 149.748 | 2.044 | 0.527 | 1.092 |

Smooth Triangle | >3600 | 0.300 | 1155 | 15.804 | 2.865 | 7.204 |

Matrix Inversion | >3600 | 1.020 | 2116 | 170.288 | 16.217 | 70.913 |

AE (12 Cycle) | Mixed-ILP (12 Cycle) | Mixed-List (12 Cycle) | |
---|---|---|---|

LUT | 2686 | 2126 | 2228 |

FF | 612 | 582 | 518 |

DSP | 16 | 12 | 12 |

PSNR | $\infty $) | 94.57 | 94.44 |

Power (uW) | 29,189 | 25,924 | 27,777 |

Benchmarks | Nodes (Mult) | Designs | Wins | Losses | Draws | ILP Exceeding 1 h |
---|---|---|---|---|---|---|

HAL | 11 (6) | 13 | 0 | 7 | 6 | 0 |

FIR filter | 21 (11) | 22 | 0 | 3 | 19 | 0 |

Auto Regression Filter | 28 (16) | 36 | 0 | 3 | 33 | 0 |

Motion Vectors Decoder | 32 (14) | 34 | 0 | 31 | 3 | 0 |

Elliptic Wave Filter | 34 (8) | 10 | 0 | 10 | 0 | 0 |

Cosine | 42 (14) | 47 | 0 | 35 | 12 | 0 |

Feedback Points | 53 (17) | 44 | 0 | 2 | 42 | 2 |

Matrix Multiplication | 109 (40) | 124 | 3 | 111 | 10 | 4 |

Smooth Triangle | 197 (69) | 248 | 35 | 91 | 122 | 61 |

Matrix Inversion | 333 (140) | 518 | 235 | 105 | 178 | 321 |

Benchmarks | ILP | List Scheduling | ||||
---|---|---|---|---|---|---|

Max | Min | Mean | Max | Min | Mean | |

HAL | 0.140 | 0.050 | 0.083 | 0.014 | 0.008 | 0.010 |

FIR filter | 17.530 | 0.090 | 1.141 | 0.044 | 0.020 | 0.032 |

Auto Regression Filter | 60.840 | 0.090 | 3.220 | 0.115 | 0.039 | 0.079 |

Motion Vectors Decoder | 43.250 | 0.130 | 2.071 | 0.117 | 0.039 | 0.073 |

Elliptic Wave Filter | 0.530 | 0.160 | 0.049 | 0.058 | 0.042 | 0.380 |

Cosine | 1675 | 0.090 | 38.162 | 0.192 | 0.078 | 0.115 |

Feedback Points | >3600 | 0.110 | 161.273 | 0.291 | 0.116 | 0.188 |

Matrix Multiplication | >3600 | 0.630 | 448.173 | 4.266 | 0.791 | 2.162 |

Smooth Triangle | >3600 | 0.360 | 1273 | 30.939 | 4.555 | 13.714 |

Matrix Inversion | >3600 | 8.750 | 2418 | 327.881 | 24.718 | 132.795 |

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## Share and Cite

**MDPI and ACS Style**

Ohata, K.; Nishikawa, H.; Kong, X.; Tomiyama, H. ILP-Based and Heuristic Scheduling Techniques for Variable-Cycle Approximate Functional Units in High-Level Synthesis. *Computers* **2022**, *11*, 146.
https://doi.org/10.3390/computers11100146

**AMA Style**

Ohata K, Nishikawa H, Kong X, Tomiyama H. ILP-Based and Heuristic Scheduling Techniques for Variable-Cycle Approximate Functional Units in High-Level Synthesis. *Computers*. 2022; 11(10):146.
https://doi.org/10.3390/computers11100146

**Chicago/Turabian Style**

Ohata, Koyu, Hiroki Nishikawa, Xiangbo Kong, and Hiroyuki Tomiyama. 2022. "ILP-Based and Heuristic Scheduling Techniques for Variable-Cycle Approximate Functional Units in High-Level Synthesis" *Computers* 11, no. 10: 146.
https://doi.org/10.3390/computers11100146