Floorplanning arranges the blocks of a chip and decides their shapes under objectives that press blocks together, short wirelength and a small outline, and constraints that hold them apart, non-overlap, clusters, MIB shapes and boundary blocks. Recent diffusion placers train on reference layouts alone and leave this coupled system to guidance, post-hoc loops and a legalizer, reporting only the endpoint, which hides what the generator contributes. We re-implement four of them under one recipe on FloorSet, score raw, refined and legalized layouts on one scale, and propose Trinity, a flow-matching floorplanner whose six differentiable functions for the constraints and objectives are its training loss term, the energy of a closed-form refiner after sampling and the base of a soft cost for every stage. The network thus learns the correction prior placers apply in their samplers, and sampling needs no guidance. Stage by stage, the training term lowers a plain transformer's raw soft cost by 26% and matters most at short budgets, the shared refiner decides more of the final cost than the generator and matches a ported placer's loop in 16 to 660 times fewer steps, Trinity's refined soft cost is 36% below the best ported pipeline, the soft cost ranks settings as the contest's hard cost does, and on the FloorSet val set the pipeline reaches a mean hard cost of 1.014 in 1.63 s per case.
Figures & tables
Figure 1: The network beside each pipeline is the generator that training updates and sampling evaluates. Left: In prior diffusion placers the constraints are absent from training (dashed), differ in form between stages ( = ), and the layouts in between are not measured. Middle: Trinity carries one set of constraint and objective functions, its differentiable physics, through training, refinement, and scoring. Right: Soft cost against refinement budget, each curve starting at the raw sample with zero refinement steps. Faster : the step at which each prior generator under our refiner reaches the soft cost of its own full loop (triangle). Lower : the refined soft cost of the best prior pipeline against ours.
Figure 2: Architecture (left): L blocks of the information mover on the wire graph, graph-biased attention and a feed-forward layer. Pipeline (right): each draw passes the generator, the refiner and the legalizer, top to bottom in each column. Key components: one physics term U enters the loss on x^0 and serves as the refiner’s energy, and the legalizer is one linear program with a contact snap. c constraints, t time step, pos spectral coordinates, xt noisy layout, v^ its implied velocity, Lfm the flow-matching loss, FFW the feed-forward layer, proj the projection, Pre Proc the relation extraction.
Figure 3
Raw sample
Refined
Legalized
Method
Overlap
WL
Area
Overlap
WL
Area
Soft
WL
Area
Vrel
Hard cost
s/draw
Trinity
0.023
−0.1
+0.2
0.002
+0.3
+1.1
1.06
+9.3
+10.5
0.045
1.234 / 1.089
0.28
FlowPlace
0.044
−0.2
+0.9
–
–
–
–
+18.5
+21.6
0.114
1.567 / 1.343
0.36
ChipDiffusion
0.051
−0.1
+1.2
0.007
+5.3
+11.7
2.05
+15.5
+18.2
0.114
1.521 / 1.300
0.41
MacroDiff+
0.064
+0.2
+2.4
0.003
+7.2
+12.7
1.66
+15.9
+18.3
0.122
1.546 / 1.308
0.39
DiffPlace
0.113
−0.3
+1.2
0.002
+6.7
+29.9
3.91
+13.9
+19.1
0.121
1.537 / 1.313
0.36
Table 3: Every method through its own pipeline at NFE=32 : the raw sample, the refined layout (own loop, released setting) and the legalized first draw; overlap ratio, WL and area gaps in percent, refined soft cost, Vrel , hard cost as first / best of 4, and seconds per draw. FlowPlace and PARSAC run no loop. ‡ PARSAC on the FloorSet val set, one worker / best of 4, seconds per worker.
Figure 4: One raw draw of a 67-block case under ChipDiffusion’s loop (top) and our refiner (bottom), early and late, the boxed region magnified beside each layout: left where the loop’s early state keeps the most overlap, right the cluster it opens most. Clusters coloured, pre-placed hatched, boundary-coded dashed, overlap red. Below each layout, the overlap ratio and the cluster gap in 10−3 .
Table 6Table 7
Figure 5: (a) The gap between Trinity and its λ=0 backbone along −∇U at three physics weights, with the seed floor (grey), the proximal prediction γt∥∇U∥ and a t2 guide. (b, c) The 12,000 reference layouts through each placer’s loop, with the original references dotted in (c).
Figure 6: The four constraint quantities, overlap, cluster gap, MIB variance and boundary distance, along our refiner at NFE=32 , hollow markers each ported loop at its released count.
Figure 7: (a) FloorSet val set, every Trinity setting of NFE , steps and N against PARSAC at 104 – 107 steps per worker. (b) One point per setting of NFE and steps. (c, d) Along our refiner, NFE=32 : the legalizer’s median expansion factor and the per-draw soft-hard correlation.
Table 11
Appendix figures & tables113 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 8: Top: the three stages, drawn schematically: floorplanning arranges blocks and shapes the soft ones, placement fills rows with fixed-height cells around the macros, routing draws every net as wires on metal layers. Bottom: a 60-block validation case of FloorSet: the reference layout with its annotations, nets and outline pins, the raw sample of Trinity at NFE=8 with overlap in red, and the legalized sample, each with its overlap ratio and wirelength and area gaps.
Suite
Year
Task
Size per instance
Soft shapes
Constraints
MCNC
1991
floorplanning
9 to 49 blocks
yes
no
GSRC
2000s
floorplanning
10 to 300 blocks
yes
no
IBM mixed-size
2004
mixed-size placement
10k to 200k cells
no
no
ISPD 2005 and 2006
2005
placement
0.2M to 2.2M cells
no
density
DAC 2012
2012
placement
0.1M to 1M cells
no
routability
CircuitNet
2022
placement and routing
10k to 100k cells
no
no
Appendix
Table 11: The benchmark suites of floorplanning and placement, each cited in the text. Sizes are per instance; “constraints” means design constraints beyond non-overlap, fixed shapes and the outline.
Data
Training set
FloorSet-Lite, MIB-consistent variant (Appendix R )
Training layouts
996,000 of the 1,008,000, with 21 to 120 blocks each
Our 12,000 val set
12,000: 100 per block count plus 2,000 random rows
Test set
the FloorSet val set of 100 cases, used once
Latent and conditioning
Latent per block
(u,v,ρ) : center over s and log(w/h) at the exact target area
Appendix
Table 12: The frozen training recipe of every trained model; the two rows marked Trinity are the components the composed arms add.
Stage
Jobs
GPU-hours
CPU-core-hours
Training
Trained runs of the study
68
710
–
Earlier attempts, not used
–
≈61
–
Evaluation
Generation and scoring
68
≈22
–
Guided sampling of three placers
5
≈11
≈5
Appendix
Table 13: Compute of the study. One RTX 5090 per job, the CPU workers of each stage as given in the text. "Held" counts the hours the GPU is reserved while the CPU workers legalize.
Schedule, peak rate
Rule, Trinity
Rule, λ=0
Trinity, 100 steps
Trinity, 200 steps
λ=0 , 100 steps
λ=0 , 200 steps
linear 10−3
1.2698
1.2688
1.1625 / 0.0105
1.1617 / 0.0118
1.1549 / 0.0096
1.1506 / 0.0107
cosine 10−3
1.2747
1.2713
1.1578 / 0.0105
1.1581 / 0.0117
1.1501 / 0.0095
1.1471 / 0.0107
linear 3×10−4
1.3182
1.3750
1.1786 / 0.0110
1.1594 / 0.0108
1.1755 / 0.0097
1.1486 / 0.0094
cosine 3×10−4
1.3188
1.3746
–
–
–
–
constant 3×10−4
1.3370
1.3597
1.2347 / 0.0120
1.2356 / 0.0131
1.2230 / 0.0104
1.2193 / 0.0115
linear 3×10−3
1.3529
1.3349
1.1910 / 0.0112
1.1876 / 0.0131
1.1859 / 0.0106
1.1801 / 0.0125
Appendix
Table 14: Refiner schedule and peak learning rate, all weights 1, betas (0,0.99) , on Trinity and on Trinity at λ=0 : the rule (the mean soft cost over the ten refined cells of NFE∈{1,32}× steps ∈{10,25,50,100,200} ), and for the six schedules run further the soft cost / overlap ratio at NFE=32 after 100 and 200 steps (raw: 1.5137 / 0.0225 and 2.0151 / 0.0222).
Arm
Trinity
λ=0
Arm
Trinity
λ=0
overlap ×10
1.2215
1.2332
all weights 1
1.2698
1.2688
overlap ×30
1.2270
1.2471
cluster ×3
1.2936
1.2938
overlap ×3
1.2434
1.2476
squared cluster gap
1.2938
1.3000
wirelength ×0.3
1.2604
1.2599
boundary ×0.3
1.3003
1.2972
cluster ×0.3
1.2616
1.2590
area ×3
1.3033
1.2927
area ×0.3
1.2624
1.2654
wirelength ×3
1.3368
1.3346
Appendix
Table 15: Weights and variants at the pick, by the rule of Table 14 .
Overlap weight
Steps
Soft cost
Overlap
HPWL gap
Area gap
×1
100
1.1625
0.0105
− 0.0058
0.0060
×3
100
1.1344
0.0079
− 0.0002
0.0087
×10
100
1.1114
0.0048
0.0040
0.0137
×30
100
1.1174
0.0032
0.0076
0.0207
×1
200
1.1617
0.0118
− 0.0094
− 0.0006
×3
200
1.1153
0.0077
− 0.0017
0.0048
Appendix
Table 16: The overlap-weight ladder at NFE=32 on Trinity after 100 and 200 steps.
Figure 9: Generator × refiner at NFE=32 with every ported loop on every generator’s own samples, the full version of Table 8 . Left: soft cost, where the DiffPlace loop makes every generator worse than its raw sample. Right: the hard cost of the first legalized draw (Table 50 ), where the same loop is the best of the three.
Refiner
Overlap
Boundary
Group
HPWL gap
Area gap
sevrel
Soft cost
raw
0.0225
0.8
2.4
− 0.0008
0.0022
0.0851
1.5131
closed-form, 100 steps
0.0105
0.2
0.9
− 0.0058
0.0060
0.0152
1.1625
ChipDiffusion, standard
0.0016
2.8
6.6
0.0470
0.1052
0.2289
1.7472
ChipDiffusion, scheduled
0.0016
3.0
6.5
0.0449
0.1046
0.2420
1.7908
ChipDiffusion, dual ascent
0.0096
12.0
4.2
0.0171
0.0940
0.4010
2.6232
DiffPlace
0.0009
18.3
4.6
0.0379
0.1588
0.4945
3.0349
Appendix
Table 17: The prior-art loops on the NFE=32 samples of Trinity: the ChipDiffusion loop in its standard, scheduled and Adam-with-dual-ascent variants (5,000 iterations), the DiffPlace and MacroDiff+ loops (500), and the closed-form refiner at all weights 1. Units as in Section 4.1 .
Figure 10: The four constraint quantities at NFE=32 along the closed-form refiner for the five generators, and each ported placer’s own post-hoc loop at its published setting as a hollow marker (ChipDiffusion 5,000 iterations, MacroDiff+ and DiffPlace 500). The loops match the refiner on overlap and leave the other three quantities one to four orders of magnitude above it.
Step
Rule
Kind
A
anchors
pre-placed blocks to their target box, fixed blocks to their shape, soft blocks to their exact area, an all-soft MIB group to one shape
rule
B
expansion
every free block moved outward from the layout centre by the smallest λ≥1 (cap 10) at which no pair overlaps; λ recorded per draw
rule
C
relations
per pair, the axis of larger separation (least penetration if overlapping) and the order along it, read from the unexpanded layout
sequence pair ( Murata et al., 1996 ) , TCG ( Lin and Chang, 2001 )
C ′
rewrites
a block ordered between an abutting pair, or outside a coded block on its coded side, re-ordered on the other axis; pinned pairs never rewritten
rule
D
abutment tree
per cluster a spanning tree over member pairs ranked by gap, missing perpendicular overlap, blocks in between and pre-placed collisions
rule
E
packing LP
one linear program over positions at fixed shapes
LP packing with pre-placed modules ( Murata and Kuh, 1998 )
Appendix
Table 18: The legalizer, one stage of eight steps. Every step is a one-line rule or a standard constraint-graph method; the program of step E and the knobs are below.
Figure 11: The raw x^0 of Trinity against its backbone at λ=0 on the same xt , 2,000 cases. (a) The RMS gap, its component along −∇U and across it, and the seed floor (two λ=0 seeds against the third). (b) The excess of the along-component over the floor for three physics weights, each with its proximal prediction γt∥∇U∥ , γt=λt2/2c , and a t2 guide.
t
0.001
0.005
0.01
0.03
0.1
0.3
1
RMS gap, seed floor
0.0069
0.0057
0.0052
0.0062
0.0150
0.0337
0.0510
RMS gap, Trinity
0.0077
0.0060
0.0050
0.0061
0.0150
0.0344
0.0533
along −∇U , seed floor
0.0058
0.0055
0.0053
0.0053
0.0062
0.0080
0.0100
along −∇U , Trinity
0.0076
0.0070
0.0066
0.0069
0.0090
0.0136
0.0216
excess of Trinity over the floor
0.0018
0.0015
0.0012
0.0016
0.0028
0.0056
0.0116
across ∇U , Trinity
0.0077
0.0060
0.0050
0.0061
0.0150
0.0343
0.0533
Appendix
Table 19: Trinity against its λ=0 backbone on the same xt , 2,000 cases, latent units, one column per t : the RMS gap and its part along −∇U for the seed floor (two λ=0 seeds against the third) and for Trinity, the excess over the floor and its part across ∇U , the prediction with γt=cλt2/2 , their ratio, and the excess at two other weights.
Figure 12: The 12,000 reference layouts through every correction. (a) RMS displacement of the movable blocks against steps. (b) Soft cost against steps, the reference layouts dotted. (c) The six terms of U at the end, every weight 1.
Correction
Steps
RMS
Max
Soft cost
Rose
Overlap
Boundary
Cluster
WL
Area
reference layouts
–
–
–
1.065
–
0.0000
0.75
0.00
−0.0
+0.0
ChipDiffusion loop
5,000
0.0272
0.0497
1.762
100%
0.0001
2.92
5.15
+4.0
+10.2
DiffPlace loop
500
0.0252
0.0576
2.584
100%
0.0008
14.41
2.25
+2.8
+12.5
MacroDiff+ loop
500
0.0290
0.0560
1.536
100%
0.0009
1.73
4.57
+4.4
+10.2
Trinity’s refiner
400
0.0055
0.0246
1.007
15%
0.0000
0.05
0.00
−0.2
+0.0
Trinity’s refiner, ω=1
400
0.0089
0.0315
1.055
36%
0.0048
0.04
0.00
−1.1
−1.1
Appendix
Table 20: The 12,000 reference layouts through each correction, means at the end: RMS and largest displacement of movable blocks ( s units), soft cost and share of cases whose soft cost rose, overlap, boundary distance and cluster gap ( 10−3 ), wirelength and area gaps in percent of the reference.
Figure 13: The 67-block case of Figure 4 , every correction from a raw draw (the row labels) at an early and a late budget, the box on each layout magnified in the window beside it: left, where the early state keeps the most overlap; right, the cluster the correction moves most, its gap under the window. Marks as in Figure 4 .
Figure 14: A 106-block case, the rows of Figure 13 .
Figure 15: A 31-block case, the rows of Figure 13 .
Checkpoint
Draws
n10
n30
n50
n100
n200
n300
Trinity
16
50.6 (0.92)
150.6 (0.91)
178.1 (0.98)
294.7 (0.36)
554.9 (1.00)
681.3 † (0.00)
64
50.3 (0.93)
150.0 (0.89)
177.9 (0.99)
294.0 (0.38)
553.1 (1.00)
681.8 † (0.00)
Trinity at λ=0
16
51.4 (0.70)
155.6 (0.81)
179.2 (0.81)
293.9 (0.80)
560.1 † (0.00)
679.8 † (0.00)
64
51.3 (0.61)
154.2 (0.80)
179.1 (0.73)
294.4 (0.65)
549.2 (0.01)
679.4 † (0.00)
Trinity, seed 1
16
49.4 (0.75)
150.1 (0.81)
178.2 (0.95)
297.0 (0.98)
548.1 (1.00)
694.3 † (0.00)
64
49.2 (0.85)
150.0 (0.84)
177.6 (0.97)
294.7 (0.98)
548.2 (1.00)
692.9 † (0.00)
Appendix
Table 21: GSRC soft blocks in a 10% square outline, three checkpoints of one recipe, one pipeline (4 seeds, 16 or 64 draws per seed, NFE 8, 400 steps, legalized): best fitting wirelength in mm over the seeds, in brackets the share of draws × seeds that fit. PARSAC’s best of 16 workers: 52.0 / 155.7 / 183.4 / 303.5 / 576.0 / 705.2. † Oversized: no draw fits, the best legal layout’s box against the outline at 16 draws Trinity on n300 1.29 W × 0.82 H; Trinity at λ=0 on n200 1.00 W × 1.13 H; Trinity at λ=0 on n300 1.18 W × 0.92 H; Trinity, seed 1 on n300 1.02 W × 1.03 H.
Loop
Iterations
μ s / step
MB
ms / layout
Less time
Trinity’s refiner, 400 steps
–
50.8 / 6.1 / 2.6
14 / 112 / 436
20.32 / 1.03
–
DiffPlace loop
500 / 9
60.7 / 7.5 / 3.1
19 / 153 / 597
30.34 / 1.54
67 × / 67 ×
MacroDiff+ loop
500 / 31
292.5 / 36.9 / 20.7
76 / 627 / 2,450
146.25 / 10.35
92 × / 129 ×
ChipDiffusion loop
5,000 / 8
69.6 / 8.8 / 5.3
33 / 272 / 1,062
348.01 / 26.43
904 × / 1,359 ×
Appendix
Table 22: Each loop against our refiner: iterations to one soft cost on the same samples (loop / ours, Figure 1 right), microseconds per step and layout and peak memory at 32 / 256 / 1,000 layouts per batch, milliseconds per layout and our time advantage at 32 / 1,000.
Head
NFE
Overlap
Group
HPWL gap
Area gap
Soft cost
FLD8b
FLD8f
FPD
MMD8b
p
x0 head (the base)
32
0.0262
2.7
− 0.0024
0.0105
2.106
0.108
0.0045
0.0013
17.7
–
v head
32
0.0373
2.9
− 0.0047
0.0123
2.597
0.275
0.0097
0.0028
33.7
1.00
ϵ head
32
0.0365
3.2
− 0.0008
0.0149
2.387
0.416
0.0184
0.0025
33.0
1.00
x0 head
1
0.0383
2.2
− 0.015
0.0093
2.491
0.417
0.0184
0.0041
34.1
–
v head
1
0.2505
13
0.18
0.61
77.2
32.0
2.37
0.184
595
1.00
ϵ head
1
0.042
200
5.9
20.3
152
102.7
17.8
5.50
1,238
1.00
Appendix
Table 23: The three heads on the plain base at NFE=32 and 1: overlap ratio, cluster gap, the two gaps, the soft cost and the distribution metrics ( p against the base at the same NFE ). Boundary, MIB and fixed at NFE=32 : 3.2 / 10 / 7.3 ( x0 ), 4.0 / 15 / 4.6 ( v ), 3.0 / 6.3 / 2.6 ( ϵ ).
Arm
Overlap
Group
HPWL gap
Area gap
Soft cost
FLD8b
FLD8f
FPD
MMD8b
p (32)
p (1)
the base
0.0262
2.67
− 0.0024
0.0105
2.106
0.108
0.0045
0.0013
17.7
–
–
attention heads
0.0279
2.71
− 0.0027
0.0107
2.149
0.137
0.0040
0.0015
19.8
0.84
0.95
information mover
0.0239
2.67
− 0.0004
0.0148
2.050
0.113
0.0038
0.0017
20.1
1.00
0.99
Appendix
Table 24: Graph mechanisms on the plain base at NFE=32 : overlap ratio, cluster gap, the two gaps, the soft cost and the distribution metrics, p against the base at NFE=32 and at 1. Boundary / MIB / fixed: 3.22 / 10.5 / 7.3 (the base), 3.18 / 8.4 / 5.5 (attention heads), 3.11 / 6.7 / 8.4 (information mover); normalized wirelength 0.3014 / 0.3013 / 0.3020 and compactness 1.038 / 1.038 / 1.042.
Arm
Overlap
Group
HPWL gap
Area gap
Soft cost
FLD8b
FLD8f
FPD
MMD8b
p
the base (absolute)
0.0262
2.67
− 0.0024
0.0105
2.106
0.108
0.0045
0.0013
17.7
–
mild
− 0.8%
+ 5.0%
+ 66%
+ 43%
2.132
0.126
0.0045
0.0016
23.4
1.00
mixture (adopted, the plain control)
− 0.5%
− 0.7%
+ 10%
− 1.5%
2.092
0.103
0.0033
0.0011
15.6
0.005
heavy
0.0%
+ 2.7%
+ 71%
+ 32%
2.107
0.115
0.0050
0.0014
19.9
1.00
Appendix
Table 25: Wire-dropout mixtures on the plain base: the delta against the base at NFE=32 on overlap, cluster gap, HPWL gap and area gap (at NFE=1 the overlap deltas are +0.4 , +0.9 and +2.1% ; boundary / MIB / fixed move +1.7 / −19 / −5% , +0.2 / −16 / −20% and −1.0 / −22 / −16% ), the absolute soft cost and the distribution metrics.
Arm
Overlap
Boundary
Group
MIB
Fixed
HPWL gap
Area gap
Soft cost
the plain control
0.02602
3.227
2.651
8.75
5.82
− 0.00214
0.01037
2.0915
overlap 0.01
0.02423
3.179
2.631
10.6
5.56
− 0.00042
0.01180
2.0479
overlap 0.05
0.01948
2.929
2.529
12.6
5.15
− 0.00107
0.01235
1.9027
overlap 0.1
0.01796
3.034
2.570
11.5
5.37
− 0.00063
0.01348
1.8950
overlap 0.05t
0.02108
3.124
2.636
12.1
6.59
− 0.00088
0.01461
1.9864
cluster 0.01
0.02764
3.318
2.294
9.32
8.69
− 0.00280
0.00761
2.1048
Appendix
Table 26: One term at a time on the plain backbone, absolute at NFE=32 . Bold: the term’s own target. Normalized wirelength and compactness stay within ±0.6% of the plain control (0.3014 and 1.037) for every constraint arm; the wirelength arms move it to 0.3015 / 0.3008 / 0.3000 / 0.3013 and the area arms move compactness to 1.033 / 1.026 / 1.017 / 1.031 for 0.01/0.05/0.1/0.05t .
Arm
FLD8b
FLD8f
FPD
MMD8b
p
FLD8b
FLD8f
FPD
MMD8b
p
the plain control
0.1031
0.0033
0.0011
15.6
–
0.4325
0.0179
0.0036
32.6
–
overlap 0.01
0.0993
0.0078
0.0010
18.9
1.000
0.2238
0.0121
0.0020
28.4
0.015
overlap 0.05
0.0735
0.0026
0.0007
14.1
0.010
0.1229
0.0044
0.0008
19.3
0.005
overlap 0.1
0.0707
0.0032
0.0006
16.2
0.975
0.1221
0.0058
0.0007
21.9
0.005
overlap 0.05t
0.0957
0.0034
0.0010
19.3
1.000
0.1548
0.0059
0.0015
26.6
0.005
cluster 0.01
0.1022
0.0038
0.0012
14.8
0.751
0.4322
0.0196
0.0039
32.0
0.935
Appendix
Table 27: Distribution metrics of the single-term arms, the five columns at NFE=32 and then at NFE=1 ; p is the permutation test against the plain control at the same NFE .
Arm
Overlap
Boundary
Group
MIB
Fixed
HPWL gap
Area gap
Soft cost
the plain control
0.02602
3.227
2.651
8.75
5.82
− 0.00214
0.01037
2.0915
all six 0.01
0.02341
0.948
2.424
2.86
5.52
− 0.00151
0.00199
1.5565
all six 0.05
0.02627
0.759
2.348
0.667
6.72
− 0.00320
− 0.00014
1.5615
all six 0.1
0.02959
0.769
2.554
0.515
16.13
− 0.00063
0.00272
1.6312
all six 0.05t
0.02588
1.082
2.471
2.53
6.30
− 0.00253
0.00070
1.6224
all six 0.1t
0.02728
0.880
2.571
1.18
6.67
− 0.00180
0.00032
1.6175
Appendix
Table 28: All six terms at one common weight on the plain backbone: absolute at NFE=32 and the delta against the plain control. Normalized wirelength stays at 0.3011 to 0.3019 and compactness falls from 1.037 to 1.023 to 1.030.
Arm
FLD8b
FLD8f
FPD
MMD8b
p
FLD8b
FLD8f
FPD
MMD8b
p
the plain control
0.1031
0.0033
0.0011
15.6
–
0.4325
0.0179
0.0036
32.6
–
all six 0.01
0.0578
0.0037
0.0010
3.05
0.005
0.1947
0.0103
0.0022
9.98
0.005
all six 0.05
0.0564
0.0029
0.0012
3.08
0.005
0.1701
0.0097
0.0024
10.4
0.005
all six 0.1
0.0581
0.0052
0.0013
5.23
0.005
0.1530
0.0117
0.0023
11.4
0.005
all six 0.05t
0.0693
0.0042
0.0012
4.03
0.005
0.2144
0.0112
0.0025
12.6
0.005
all six 0.1t
0.0564
0.0044
0.0011
4.05
0.005
0.1764
0.0097
0.0022
10.7
0.005
Appendix
Table 29: Distribution metrics of the all-six arms, the five columns at NFE=32 and then at NFE=1 .
Arm
Overlap
Boundary
Group
MIB
Fixed
HPWL gap
Area gap
Soft cost
Best of 4
NFE=1
plain, λ=0 (the plain control)
0.02602
3.227
2.651
8.75
5.82
− 0.00214
0.01037
2.0915
1.9074
2.4900
plain + all six 0.01
0.02341
0.948
2.424
2.86
5.52
− 0.00151
0.00199
1.5565
1.4479
1.7198
plain + all six 0.05
0.02627
0.759
2.348
0.667
6.72
− 0.00320
− 0.00014
1.5615
1.4586
1.7001
spectral PE, λ=0
0.02433
3.310
2.942
10.5
7.21
+ 0.00116
0.01906
2.1305
1.9404
2.5024
spectral PE + all six 0.01
0.02343
1.029
2.554
3.52
7.18
− 0.00062
0.00307
1.5912
1.4763
1.7489
spectral PE + all six 0.05
0.02556
0.652
2.405
0.712
11.19
− 0.00177
0.00128
1.5373
1.4378
1.6649
Appendix
Table 30: The composed grid at NFE=32 : the constraint columns, the two gaps, the soft cost, the best-of-4 soft cost and the NFE=1 soft cost. Normalized wirelength is 0.3011 to 0.3028 on every row; compactness is 1.037 / 1.029 / 1.027 / 1.046 / 1.030 / 1.028 / 1.045 / 1.029 / 1.023 / 1.032 / 1.024 down the rows.
Arm
FLD8b
FLD8f
FPD
MMD8b
p
FLD8b
FLD8f
FPD
MMD8b
p
plain, λ=0 (the plain control)
0.1031
0.0033
0.0011
15.6
–
0.4325
0.0179
0.0036
32.6
–
plain + all six 0.01
0.0578
0.0037
0.0010
3.05
0.005
0.1947
0.0103
0.0022
9.98
0.005
plain + all six 0.05
0.0564
0.0029
0.0012
3.08
0.005
0.1701
0.0097
0.0024
10.4
0.005
spectral PE, λ=0
0.1358
0.0059
0.0015
28.4
1.000
0.4317
0.0186
0.0041
44.9
1.000
spectral PE + all six 0.01
0.0578
0.0036
0.0011
3.80
0.005
0.1854
0.0108
0.0025
12.7
0.005
spectral PE + all six 0.05
0.0581
0.0036
0.0013
4.56
0.005
0.1668
0.0099
0.0023
11.8
0.005
Appendix
Table 31: Distribution metrics of the composed grid, the five columns at NFE=32 and then at NFE=1 .
Arm
NFE
Overlap
Group
HPWL gap
Area gap
Soft cost
FLD8b
FLD8f
FPD
MMD8b
p
the plain control
32
0.02602
2.651
− 0.00214
0.01037
2.0915
0.1031
0.0033
0.0011
15.6
–
Trinity
32
0.02253
2.325
− 0.00069
0.00224
1.5137
0.0423
0.0075
0.0012
2.5
0.005
FlowPlace
32
0.04383
3.196
− 0.00231
0.00871
2.4564
0.2274
0.0088
0.0013
24.4
1.00
ChipDiffusion
32
0.05113
3.581
− 0.00076
0.01185
2.8020
0.2733
0.0098
0.0011
31.2
1.00
MacroDiff+
32
0.0639
4.33
+ 0.00236
0.02417
3.8458
0.624
0.0276
0.280
81.5
1.00
DiffPlace
32
0.1126
4.833
− 0.00291
0.01229
6.2293
2.855
0.191
0.0146
247
1.00
Appendix
Table 32: Every generator at NFE=32 and 1, the annealer and the regressors on their single layout: overlap ratio, cluster gap, the two gaps, the soft cost (the regressors’ as the per-case median) and the distribution metrics, p against the plain control at the same NFE . Normalized wirelength is 0.297 to 0.303 for every diffusion row at NFE=32 ; the boundary, MIB and fixed columns are in Table 33 ’s sources.
Setup
NFE=32
Best of 4
NFE=1
WL / Area (%)
Trinity : PE + mover + all six 0.01
1.5137 / 0.0225
1.4092
1.6785 / 0.0316
−0.07 / +0.22
PE + mover + all six 0.05
1.5192 / 0.0257
1.4320
1.6341 / 0.0329
−0.26 / −0.37
spectral PE + all six 0.05
1.5373 / 0.0256
1.4378
1.6649 / 0.0327
−0.18 / +0.13
plain + all six 0.01
1.5565 / 0.0234
1.4479
1.7198 / 0.0319
−0.15 / +0.20
plain + all six 0.05
1.5615 / 0.0263
1.4586
1.7001 / 0.0340
−0.32 / −0.01
PE + mover + all six 0.1t
1.5784 / 0.0246
1.4797
1.7064 / 0.0324
+0.00 / +0.49
Appendix
Table 33: Every model of the study on our 12,000 val set, sorted by the NFE=32 soft cost. Columns: soft cost / overlap at NFE=32 , the best-of-4 soft cost, soft cost / overlap at NFE=1 (or on the single layout, the regressors’ soft cost as the per-case median), and the wirelength / area gap at NFE=32 in percent. Unless stated, a model is on the adopted recipe (wire-dropout mixture); "plain base" means no wire dropout.
Model
NFE
0 steps
10
25
50
100
200
400
Trinity
1
1.679 / 0.0316
1.517 / 0.0201
1.309 / 0.0152
1.208 / 0.0115
1.149 / 0.0083
1.104 / 0.0056
1.066 / 0.0034
−1.2 / −0.1
−1.0 / +1.0
−0.8 / +1.1
−0.6 / +1.1
−0.4 / +1.2
−0.3 / +1.2
−0.3 / +1.2
2
1.660 / 0.0305
1.480 / 0.0188
1.289 / 0.0141
1.194 / 0.0106
1.137 / 0.0076
1.096 / 0.0052
1.062 / 0.0031
−1.2 / −0.0
−0.9 / +1.0
−0.7 / +1.1
−0.6 / +1.1
−0.4 / +1.2
−0.4 / +1.2
−0.3 / +1.1
4
1.631 / 0.0288
1.460 / 0.0173
1.274 / 0.0128
1.182 / 0.0096
1.129 / 0.0068
1.090 / 0.0047
1.059 / 0.0029
−1.0 / +0.0
−0.7 / +1.1
−0.5 / +1.2
−0.4 / +1.2
−0.3 / +1.2
−0.2 / +1.2
−0.2 / +1.0
Appendix
Table 34: The full NFE× steps grid, Trinity and Trinity at λ=0 (the same backbone without the auxiliary loss), two lines per cell: soft cost / overlap ratio, then wirelength gap / area gap to the reference in percent (positive above the reference).
NFE
Steps
Overlap
Boundary
Group
MIB
HPWL gap
Area gap
sevrel
Soft cost
1
0
0.0316
0.8
1.8
1.7
− 0.0124
− 0.0013
0.0819
1.6785
1
10
0.0201
1.4
1.3
0.0080
− 0.0096
0.0100
0.0911
1.5170
1
25
0.0152
0.7
1.1
5.7×10−4
− 0.0075
0.0106
0.0464
1.3090
1
50
0.0115
0.4
0.9
5.1×10−5
− 0.0058
0.0113
0.0279
1.2084
1
100
0.0083
0.3
0.8
1.2×10−5
− 0.0043
0.0122
0.0206
1.1490
1
400
0.0034
0.1
0.5
7.0×10−7
− 0.0029
0.0116
0.0100
1.0661
Appendix
Table 35: Every soft column along the descent on Trinity at the setting of Appendix E (Trinity at λ=0 is within 0.005 on every column from 25 steps on). Units as in Section 4.1 .
Model
32 / 0
32 / 25
32 / 100
32 / 400
1 / 0
1 / 400
Rule
Trinity
1.514 / 0.0225
1.216 / 0.0084
1.111 / 0.0048
1.058 / 0.0023
1.679 / 0.0316
1.066 / 0.0034
–
−0.1 / +0.2
+0.3 / +1.4
+0.4 / +1.4
+0.3 / +1.1
−1.2 / −0.1
−0.3 / +1.2
PE + mover, λ=0
2.015 / 0.0222
1.212 / 0.0079
1.109 / 0.0044
1.058 / 0.0022
2.427 / 0.0357
1.065 / 0.0034
–
+0.2 / +1.8
+0.5 / +1.8
+0.5 / +1.7
+0.4 / +1.2
−1.1 / +1.6
−0.3 / +1.4
PE + mover + all six 0.05
1.519 / 0.0257
1.232 / 0.0101
1.125 / 0.0060
1.066 / 0.0028
1.634 / 0.0329
1.077 / 0.0038
1.1587
−0.3 / −0.4
+0.1 / +1.0
+0.3 / +1.3
+0.4 / +1.3
−1.5 / −0.9
−0.4 / +1.2
Appendix
Table 36: The reduced grid for every model, one column per NFE / refiner steps, two lines per model: soft cost / overlap ratio with the rule (the mean soft cost over the nine refined cells; the regressors’ raw soft cost is the per-case median, their means overflow), then wirelength gap / area gap to the reference in percent (positive above the reference).
Model
NFE
0 steps
25
100
400
Trinity
1
1.492 / 2.329
1.224 / 1.647
1.102 / 1.331
1.038 / 1.162
Trinity
32
1.423 / 1.876
1.174 / 1.421
1.079 / 1.244
1.033 / 1.144
Trinity, λ=0
1
2.014 / 3.704
1.203 / 1.817
1.091 / 1.365
1.034 / 1.163
Trinity, λ=0
32
1.888 / 2.611
1.161 / 1.426
1.074 / 1.245
1.032 / 1.146
FlowPlace
1
7.340 / 16.53
2.122 / 4.766
1.235 / 1.907
1.083 / 1.297
FlowPlace
32
2.246 / 3.352
1.319 / 1.809
1.163 / 1.463
1.072 / 1.238
Appendix
Table 37: Per-case median / 90th percentile of the soft cost along the descent, for the lines whose means are dominated by a few cases.
Placer
Inside its released sampler
The port
ChipDiffusion
Every step, ten Adam iterations at learning rate 8×10−3 on the predicted clean sample of α⋅legality+10−4⋅HPWL , with α adapted per sample from 0 while t<0.5 , and the diffusion step continues from the guided prediction.
The same loop on the post-hoc port’s legality and wirelength functions, the next state the Euler step plus the scaled correction.
DiffPlace
From 1−t≥0.85 on, the predicted noise is moved by λt∇overlap(xt) with λt=200(1−t)2 , the gradient rescaled per block to 0.03 and clipped at 0.08. The released update moves the sample up the overlap gradient.
The same on the port’s overlap term, the step continuing from the guided noise. A reference arm flips the sign of the update.
MacroDiff+
Every DDIM step, an Adam loop on the predicted clean sample with HPWL and overlap phases, clipping and a die clamp, 300 to 700 iterations per design in its table and 500 by default.
500 iterations at learning rate 0.01 each step, the post-hoc port’s settings, the step continuing from the guided prediction.
FlowPlace
A projection onto the hard constraints inside the sampler, disabled in its evaluation configuration.
Not reproduced.
Appendix
Table 38: What each released sampler does to the sample at every step, and how it is ported onto our Euler sampler. Every port moves positions only, keeps the pre-placed blocks fixed, works on the post-hoc ports’ canvas (the predicted layout’s bounding box plus 5%), and guides the final prediction as well.
Figure 16: The raw samples against NFE , free sampling solid, the placer’s own guidance dashed, DiffPlace’s update with the sign flipped dotted: the soft cost, the overlap ratio, the boundary distance, the cluster gap, the area gap and sevrel , log axes.
Placer
Sampling
NFE
Overlap
Boundary
Cluster
MIB
Fixed
HPWL gap
Area gap
sevrel
Soft cost
ChipDiffusion
free
1
0.0421
2,218
198
347,608
1,304
+5.931
+19.8
0.976
151.6
2
0.1961
316
38.3
13,364
216
+0.841
+1.870
0.840
97.81
4
0.1138
29.4
6.77
275
26.3
+0.053
+0.186
0.585
11.91
8
0.0595
5.59
3.86
38.1
4.40
+0.003
+0.034
0.327
3.829
16
0.0543
3.70
3.61
19.6
2.50
−0.002
+0.016
0.254
3.057
32
0.0511
3.30
3.58
16.0
2.20
−0.001
+0.012
0.236
2.802
Appendix
Table 39: The raw samples of the three placers at every NFE , free (our Euler sampler) and guided (their own in-sampler correction), and DiffPlace’s update with its sign flipped: the soft columns and the soft cost, means over four draws and the 12,000 cases. Boundary, cluster and fixed in 10−3 , MIB in 10−6 , the gaps as fractions of the reference.
Figure 17: The soft cost along the closed-form refiner from the free (solid) and the guided (dashed) samples of each placer at NFE=1 , 8 and 32.
Placer
Sampling
NFE
0 steps
25
100
400
ChipDiffusion
free
1
151.6 / 0.0421
115.9 / 0.0287
92.74 / 0.0111
72.63 / 0.0039
8
3.829 / 0.0595
1.662 / 0.0307
1.299 / 0.0162
1.137 / 0.0061
32
2.802 / 0.0511
1.516 / 0.0272
1.271 / 0.0158
1.131 / 0.0062
guided
1
88.29 / 0.0035
79.26 / 0.0069
72.99 / 0.0041
59.94 / 0.0038
8
3.974 / 0.0231
1.940 / 0.0138
1.305 / 0.0071
1.151 / 0.0028
32
3.782 / 0.0182
1.828 / 0.0107
1.291 / 0.0057
1.152 / 0.0025
Appendix
Table 40: The closed-form refiner on the free and the guided samples of each placer, the reduced grid of Appendix I.2 : soft cost / overlap ratio per cell of NFE and refiner steps.
Placer
Sampling
Refiner
Overlap
Boundary
Cluster
MIB
Fixed
HPWL gap
Area gap
sevrel
Soft cost
ChipDiffusion
free
ours, 400 steps
0.0062
0.192
1.13
0.0000
0.0001
+0.012
+0.030
0.017
1.131
ChipDiffusion loop
0.0069
4.56
6.68
15.7
2.19
+0.053
+0.117
0.273
2.048
DiffPlace loop
0.0008
29.1
6.10
15.7
2.19
+0.058
+0.246
0.563
3.614
MacroDiff+ loop
0.0032
2.21
6.80
15.7
2.19
+0.061
+0.116
0.174
1.602
guided
ours, 400 steps
0.0025
0.699
1.11
0.0000
0.0001
+0.024
+0.089
0.029
1.152
ChipDiffusion loop †
0.0022
15.5
11.5
17.2
2.66
+0.099
+0.273
0.533
3.582
Appendix
Table 41: Each placer’s NFE=32 samples, free and guided, after the closed-form refiner (400 steps) and after the three ported loops at their released iteration counts (ChipDiffusion 5,000 in its scheduled variant, DiffPlace and MacroDiff+ 500); the raw rows are the NFE=32 rows of Table 39 , whose units apply. A dagger marks guided sampling through the method’s own loop, its complete published method.
Placer
Sampling
NFE
FLD8b
FLD8f
FLD12b
FLD12f
FPD
MMD8b
p
ChipDiffusion
free
1
102
17.67
234
48.12
5.501
1,223
1.00
32
0.2733
0.009784
1.361
0.06643
0.001091
31.2
1.00
guided
1
101
15.71
230
43.13
7.418
1,171
1.00
32
3.1
0.2972
8.633
1.071
0.00531
512
1.00
DiffPlace
free
1
102
17.78
234
48.36
5.361
1,230
1.00
32
2.855
0.1911
10.73
1.139
0.01458
247
1.00
Appendix
Table 42: The distribution metrics of Appendix C for the free and the guided samples at NFE=1 and 32: the Fréchet distances on the density descriptors and on positions, the MMD in 10−3 , and the permutation p that the run is closer to the reference than the plain control.
NFE
Steps
Soft cost
First draw
Total
WL / area
Best of 4
Total
Soft pick
ms / draw
1
0
1.6785
1.4903 / 1.2875 / 2.362
1.7111
+15.5 / +17.8
1.3763 / 1.2394 / 1.787
1.5513
1.4841
358
1
10
1.5170
1.3375 / 1.1177 / 2.148
1.5836
+15.1 / +17.3
1.2296 / 1.0747 / 1.516
1.4183
1.3309
351
1
25
1.3090
1.3239 / 1.1123 / 2.071
1.5612
+14.2 / +16.6
1.2171 / 1.0698 / 1.485
1.3976
1.3166
348
1
50
1.2084
1.3088 / 1.1046 / 1.975
1.5470
+13.6 / +15.8
1.2071 / 1.0651 / 1.450
1.3968
1.3038
342
1
100
1.1490
1.2932 / 1.0965 / 1.865
1.5226
+12.6 / +14.7
1.1903 / 1.0595 / 1.413
1.3557
1.2847
330
1
200
1.1042
1.2624 / 1.0838 / 1.632
1.4648
+10.9 / +12.9
1.1623 / 1.0498 / 1.335
1.3024
1.2527
314
Appendix
Table 43: Trinity through the legalizer on the NFE× steps grid (selected cells; every cell is in the supplementary data): the soft cost of the refined layout, the hard cost of the first legalized draw of every case as mean / median / 90th percentile, its hard-cost total, its wirelength / area gap in percent, the best of four legalized draws as mean / median / 90th percentile and total, the mean of the soft pick of four, and the legalizer’s time per draw. Every cell is feasible on all draws.
NFE
Steps
Hard LP
Penalties
Flips
Last rungs
λ p50
λ p90
ρ(λ,area gap)
ρ(λ,cost)
Area gap
HPWL gap
Vrel
1
0
0.114
0.443
0.325
0.117
2.54
8.21
0.33
0.19
+ 0.178
+ 0.155
0.104
1
400
0.248
0.491
0.189
0.073
1.39
3.65
0.46
0.42
+ 0.105
+ 0.088
0.041
8
0
0.113
0.467
0.302
0.118
2.40
8.24
0.37
0.22
+ 0.180
+ 0.159
0.107
8
400
0.252
0.514
0.168
0.066
1.35
3.48
0.50
0.46
+ 0.101
+ 0.086
0.043
32
0
0.108
0.470
0.303
0.120
2.17
6.54
0.33
0.20
+ 0.180
+ 0.164
0.109
32
400
0.245
0.518
0.170
0.068
1.32
2.80
0.49
0.48
+ 0.105
+ 0.092
0.045
Appendix
Table 44: Per cell: the share of draws that ended on each rung of the legalizer’s ladder, the expansion factor λ , the rank correlation of λ with the area gap and the cost, and the gaps after legalization.
Line
NFE
Soft
First
Median
Best 4
Total
Vrel
Area
λ
LP
ρ
HPWL
FlowPlace
4
1.116
1.270
1.091
1.119
1.214
0.047
+ 0.13
1.53
0.202
0.41
+0.12
FlowPlace
8
1.114
1.275
1.096
1.118
1.215
0.048
+ 0.14
1.53
0.194
0.42
+0.12
FlowPlace
16
1.109
1.287
1.101
1.121
1.215
0.050
+ 0.14
1.51
0.190
0.45
+0.12
FlowPlace
32
1.107
1.296
1.106
1.124
1.228
0.051
+ 0.15
1.48
0.186
0.46
+0.13
ChipDiffusion
4
1.226
1.325
1.154
1.143
1.232
0.054
+ 0.17
1.66
0.162
0.62
+0.15
ChipDiffusion
8
1.137
1.309
1.118
1.130
1.234
0.053
+ 0.16
1.54
0.174
0.48
+0.14
Appendix
Table 45: The prior arts and Trinity on the same pipeline at 400 refiner steps, per NFE (the ϵ -head models produce no layout at NFE=1 and 2, where their legalized costs are 9 to 33). Columns: the soft cost, the first draw’s mean and median cost, the best-of-4 mean and hard-cost total, Vrel and the area gap after legalization, the median expansion factor λ , the share of draws the legalizer’s hard packing program solved directly (LP), and the per-draw soft–hard Spearman ρ .
Line
Steps
Soft
First
Median
p90
Best 4
LP
Last
λ
Area
Vrel
HPWL
SP-SA
0
12.341
19.019
18.266
28.18
14.382
0.000
0.170
1.51
+ 3.28
0.755
+2.88
SP-SA
100
13.094
17.195
16.510
25.43
12.984
0.000
0.164
1.94
+ 3.10
0.721
+2.76
SP-SA
400
10.891
13.809
13.294
20.13
10.427
0.000
0.150
2.76
+ 2.61
0.660
+2.50
Reg- z
0
21.53 / 24,599
2.306
1.910
3.932
–
0.034
0.137
10.0 (cap)
+ 0.61
0.162
+0.52
Reg- z
100
2.25 / 55.1
1.766
1.442
3.018
–
0.065
0.137
8.50
+ 0.40
0.095
+0.37
Reg- z
400
1.37 / 3.97
1.552
1.278
2.587
–
0.096
0.125
4.56
+ 0.27
0.078
+0.28
Appendix
Table 46: The single-shard lines (the sequence-pair annealer and the two regressors) across refiner steps: the soft cost (median / p90 where the mean overflows), the first draw’s mean, median and 90th-percentile hard cost (the single layout for the regressors), the best-of-4 mean, the share of draws solved by the hard LP and by the last rungs, the median expansion factor λ , and the area gap, Vrel and HPWL gap after legalization.
Quantity
1 / 0
1 / 10
1 / 400
8 / 0
8 / 400
32 / 0
32 / 10
32 / 100
32 / 400
ρ over draws
0.198
0.491
0.495
0.261
0.580
0.267
0.551
0.572
0.603
τ over draws
0.134
0.338
0.345
0.177
0.412
0.180
0.382
0.401
0.430
ρ over case means
0.203
0.505
0.502
0.261
0.583
0.271
0.583
0.581
0.610
overlap ratio
0.223
0.365
0.418
0.278
0.506
0.272
0.450
0.460
0.531
sevrel
0.058
0.482
0.507
0.102
0.556
0.112
0.496
0.558
0.572
boundary distance
0.034
0.381
0.410
0.043
0.396
0.044
0.339
0.421
0.388
Appendix
Table 47: Trinity, one column per NFE / refiner steps: the rank correlation of the soft cost with the legalized cost over the 48,000 draws ( ρ , τ ) and over the 12,000 per-case means, then the Spearman of every soft column with the legalized cost.
Line
ρ
τ
r
R2
rlog
Rlog2
ρ
τ
r
R2
rlog
Rlog2
Trinity
0.938
0.754
0.899
0.808
0.899
0.809
0.472
0.326
0.238
0.057
0.311
0.097
FlowPlace
0.923
0.738
0.747
0.558
0.846
0.716
0.369
0.252
0.065
0.004
0.237
0.056
ChipDiffusion
0.989
0.919
0.925
0.855
0.935
0.875
0.805
0.619
0.843
0.711
0.901
0.811
DiffPlace
0.975
0.898
0.892
0.796
0.906
0.822
0.858
0.665
0.796
0.633
0.873
0.763
MacroDiff+
0.985
0.919
0.923
0.853
0.925
0.856
0.861
0.671
0.830
0.688
0.890
0.793
SP-SA
0.643
0.619
0.795
0.633
0.822
0.676
0.484
0.335
0.459
0.210
0.538
0.290
Appendix
Table 48: Every line: Spearman ρ , Kendall τ , Pearson r and R2 , log-log r and R2 of soft against hard cost, across the cell means (first six columns: 42 cells per diffusion line, 7 per single-shard line, 231 for all eight lines, 175 for the five lines with finite means) and pooled over every draw (last six: 2,016,000 per diffusion line, 336,000 for the annealer, 84,000 per regressor, 10,584,000 and 8,400,000 for the two unions); n/a where the regressors’ overflowing means make a linear fit meaningless.
Refiner
Soft
First
Median
p90
Best 4
Med. 4
LP
Last
λ
Area
Vrel
ρ / ms
HPWL
none (raw)
1.514
1.5125
1.302
2.451
1.3057
1.214
0.108
0.120
2.17
+ 0.180
0.109
0.267 / 341
+0.164
MacroDiff+ loop
1.564
1.4424
1.284
2.007
1.2588
1.200
0.107
0.078
1.48
+ 0.139
0.106
0.164 / 271
+0.121
ChipDiffusion loop
1.793
1.4563
1.284
2.075
1.2699
1.205
0.118
0.085
1.46
+ 0.150
0.106
0.171 / 278
+0.130
DiffPlace loop
3.035
1.4415
1.286
1.957
1.2581
1.200
0.090
0.073
1.53
+ 0.137
0.108
0.351 / 258
+0.114
ours, 10 steps
1.370
1.3561
1.132
2.219
1.1697
1.054
0.136
0.118
1.93
+ 0.175
0.058
0.551 / 337
+0.158
ours, 100 steps
1.111
1.3093
1.105
1.951
1.1353
1.044
0.170
0.098
1.54
+ 0.148
0.054
0.572 / 311
+0.132
Appendix
Table 49: The prior-art loops at their published iteration counts (MacroDiff+ 500, ChipDiffusion 5,000 in its scheduled variant, DiffPlace 500) and the closed-form refiner (ours) on Trinity’s NFE=32 samples, then the legalizer (the raw row reproduces the grid’s (32,0) cell): the soft cost, the first draw’s mean, median and p90 hard cost, the best-of-4 mean and median, the share of draws solved by the hard LP and by the last rungs, the median expansion factor λ , the area gap and Vrel after legalization, the per-draw soft–hard Spearman ρ and the milliseconds per draw, and the HPWL gap.
Generator
Raw sample
ChipDiffusion loop
DiffPlace loop
MacroDiff+ loop
Closed-form refiner
Trinity
1.513 / 1.306
1.456 / 1.270
1.442 / 1.258
1.442 / 1.259
1.234 / 1.089
+16.2 / +18.0
+13.0 / +14.9
+11.4 / +13.7
+12.1 / +13.9
+9.3 / +10.5
FlowPlace
1.567 / 1.343
1.495 / 1.284
1.454 / 1.265
1.471 / 1.267
1.296 / 1.124
+18.5 / +21.6
+14.5 / +16.9
+11.2 / +14.7
+13.1 / +15.2
+12.8 / +14.6
ChipDiffusion
1.609 / 1.364
1.521 / 1.300 †
1.483 / 1.280
1.492 / 1.285
1.322 / 1.137
+20.1 / +23.7
+15.5 / +18.2
+12.2 / +16.1
+13.7 / +16.3
+14.3 / +16.2
Appendix
Table 50: Generator × refiner at NFE=32 , then the legalizer, two lines per generator: the hard cost of the first draw / best of four, then the first draw’s wirelength gap / area gap to the reference in percent. Every prior-art loop runs at its published iteration count on every generator’s own samples, the closed-form refiner for 400 steps; every draw of the 13 loop rows is feasible, the raw cell of every row reproduces its grid’s (32,0) cell, and a dagger marks a method’s published generator and loop pair. Every loop moves the outline and the wires it did not model; the closed-form refiner leaves the smallest gaps on every generator.
NFE
Steps
N
Mean
Median
p90
Total
Soft pick (mean / median)
HPWL / Area (best of N )
32
0
1
1.5079
1.3003
2.427
1.7300
–
–
32
0
4
1.3014
1.2098
1.595
1.4423
1.4758 / 1.2775
–
32
0
16
1.2140
1.1696
1.459
1.2857
1.4548 / 1.2629
–
32
400
1
1.2331
1.0736
1.552
1.3857
–
+9.3 / +10.5
32
400
2
1.1412
1.0435
1.313
1.2452
1.2079 / 1.0597
+4.9 / +6.1
32
400
4
1.0885
1.0203
1.208
1.1622
1.1870 / 1.0500
+2.6 / +3.6
Appendix
Table 51: Best of N on the 12,000 cases from Trinity’s 16-draw caches (fresh noise), every draw refined 400 steps and legalized: the cheapest of the first N legalized draws per case (mean, median, 90th percentile, hard-cost total), and the soft pick (the legalized cost of the lowest-soft-cost draw among N , mean / median). Every one of the 768,000 legalized draws is feasible.
NFE
Steps
N
Total
Mean
Median
Mean s / case
Median s / case
Max s / case
HPWL gap
Area gap
8
400
48
1.0213
1.0144
1.0000
1.63
1.34
5.10
−0.004
+0.000
32
400
48
1.0223
1.0147
1.0000
1.86
1.59
5.61
−0.003
−0.001
8
400
16
1.0290
1.0193
1.0009
1.14
0.98
2.27
−0.002
+0.004
8
400
8
1.0469
1.0284
1.0035
1.04
0.88
2.21
−0.002
+0.008
8
400
4
1.1316
1.0896
1.0081
1.01
0.87
2.22
+0.032
+0.032
4
400
16
1.0351
1.0243
1.0007
1.08
0.99
2.34
−0.002
+0.007
Appendix
Table 52: The FloorSet val set per sample on an idle machine: the en/12 -weighted hard-cost total, the mean and median cost, and the wall time per case (mean, median, max).
NFE
Steps
N
Sample s
Refine s
Legalize s
Legalizer ms / draw
Solver ms / draw
8
400
48
0.10
0.61
0.91
240
87
32
400
48
0.32
0.61
0.93
244
88
8
400
16
0.07
0.62
0.45
258
92
8
400
8
0.07
0.61
0.35
240
87
8
400
4
0.06
0.62
0.33
249
89
4
400
16
0.04
0.62
0.42
264
94
Appendix
Table 53: The same settings: mean seconds per case by stage (legalization is the wall time over the N parallel draws on 16 workers), the legalizer’s milliseconds per draw and the solver’s share of it.
Line
Draws or workers
s / case
Core-s / case
Mean
Median
WL
Area
Trinity, NFE=8 , 400 steps
48 draws
1.62
12 + 0.72 GPU
1.014
1.000
−0.4
+0.0
Trinity, NFE=8 , 400 steps
16 draws
1.14
4 + 0.69 GPU
1.019
1.001
−0.2
+0.4
Trinity, NFE=4 , 100 steps
16 draws
0.63
4 + 0.20 GPU
1.042
1.006
+0.2
+1.8
Trinity, NFE=1 , 50 steps
16 draws
0.48
5 + 0.10 GPU
1.115
1.038
+3.6
+5.0
PARSAC, 107 steps
best of 16 workers
1,121
17,937
1.222
1.179
+26.0
+4.7
PARSAC, 107 steps
best of 4 workers
1,121
4,484
1.348
1.241
+32.9
+8.8
Appendix
Table 54: The FloorSet val set (100 cases), every line to its final legalized layout: seconds per case of wall time (for an annealer, one worker’s run), the CPU core-seconds per case (for Trinity, the legalizer’s, plus the GPU seconds of sampling and refinement), the mean and median hard cost, and the legalized layout’s wirelength and area gaps to the reference in percent. PARSAC workers are independent runs from random trees, one CPU core each. † Astra’s reported runtime on its own CPU.
Table 66
Figure 18: The FloorSet val set: the mean (left) and median (right) hard cost against seconds per case. PARSAC as the best legalized snapshot so far of one worker and of the best of 4 and 16 workers, against seconds per worker on one CPU core; Trinity at every setting of NFE , refiner steps and draws, on one GPU with 16 legalizer workers, the star at NFE=8 , 400 steps and 16 draws; ours → PARSAC is PARSAC started from the tree of the pipeline’s legalized layout, drawn from the pipeline’s time on. The sequence-pair annealer stays above 10 and is off the axis.
Case
Blocks
Seeds fitting
HPWL, 16
HPWL, 64
Dead space
s / case
GSRC
n10
10
4 / 4
50.6
50.3
3.4
0.84
n30
30
4 / 4
150.6
150.0
1.1
4.53
n50
50
4 / 4
178.1
177.9
1.6
1.10
n100
100
4 / 4
294.7
294.0
2.1
2.20
n200
200
4 / 4
554.9
553.1
0.7
8.60
Appendix
Table 57: Trinity on the soft-block benchmarks of the prior work: every block soft at its area, a square outline of 1.1 times the block area, terminals at their benchmark positions. Per case, how many of four sampling seeds give a fitting legalized layout with 16 draws, the best fitting wirelength over the seeds with 16 and 64 draws (HPWL of block centres and terminals, benchmark units / 1,000), the dead space of that layout in percent, and the seconds per case of one seed with 16 draws. † Oversized: no seed fits, the best legal layout.
Line
n100
n200
n300
Time
Trinity, 4 seeds, 16 / 64 draws per seed
294.7 / 294.0
554.9 / 553.1
681.3 ‡ / 681.8 ‡
2.2 / 8.6 / 7.0 s
PARSAC here, best of 16 / 4 workers
303.5 / 310.1
576.0 / 586.6
709.3 / 720.2
22 / 68 / 138 min
PARSAC (soft), 112 workers, best †
288
526
627
–
PARSAC (soft), 112 workers, worst †
301
553
663
–
PARSAC (hard), 112 workers, best †
291
530
628
–
GoodFloorplan †
319
575
705
–
Appendix
Table 58: GSRC soft blocks in a 10% square outline, per column the best fitting wirelength in bold and the second underlined: the wirelength in mm (benchmark units / 1,000) of Trinity with 16 and 64 draws per seed over 4 seeds, of PARSAC run here from random trees with its published schedule for 107 steps (the best final layout of 16 and of 4 workers, one CPU core each) and of the published rows. Seconds per case for Trinity with one seed and 16 draws (n100 / n200 / n300); minutes per worker for PARSAC. † From Table 2 of PARSAC ( Mostafa et al., 2024 ) , 112 workers of 107 steps; best and worst are over its reported runs. ‡ Oversized: no draw fits, the best legal layout.
Figure 19: GSRC soft blocks in a 10% square outline: the wirelength of fitting layouts in mm against seconds per case. PARSAC as the median over 16 workers of one worker’s best fitting snapshot so far and as the best of 16 workers; ours → PARSAC from the pipeline’s layout, drawn from the pipeline’s time on; Trinity with 16 draws, the best of four seeds; the published rows of Table 58 as dotted levels.
Model
NFE
Overlap
Boundary
Group
MIB
Fixed
Area gap
sevrel
Soft cost
abs. std
Trinity
32
2.8%
9.3%
3.2%
31%
40%
68%
8.2%
1.1%
0.017
Trinity
8
2.4%
9.8%
3.7%
31%
40%
91%
8.6%
1.1%
0.018
Trinity
1
2.9%
10.6%
4.4%
32%
36%
–
9.6%
1.0%
0.017
Trinity, λ=0
32
1.0%
1.0%
3.1%
16%
17%
12.5%
1.9%
0.7%
0.015
Trinity, λ=0
8
0.4%
1.1%
3.3%
12%
16%
14%
1.9%
1.0%
0.021
Trinity, λ=0
1
0.9%
1.3%
3.3%
12%
11%
14.5%
2.0%
1.6%
0.037
Appendix
Table 59: Std / mean per model and column at NFE=32 , 8 and 1, and the absolute std of the soft cost. Trinity’s area gap at NFE=1 has a mean near zero ( −0.0013 to +0.0043 ), so no ratio is given. MIB variance and fixed deviation are the wide columns everywhere.
Column
Original
Seed 1
Seed 2
Mean
Std
Original
Seed 1
Seed 2
Mean
Std
overlap
0.02253
0.02217
0.02132
0.02201
0.00062
0.03161
0.03054
0.02983
0.03066
0.00089
boundary
0.817
0.898
0.985
0.900
0.084
0.799
0.889
0.988
0.892
0.095
group gap
2.325
2.370
2.474
2.390
0.077
1.773
1.813
1.927
1.838
0.080
MIB variance
1.72
2.95
3.28
2.65
0.82
1.72
2.93
3.42
2.69
0.87
fixed
8.337
5.055
11.802
8.398
3.37
8.837
5.712
12.203
8.917
3.25
HPWL gap
− 0.00069
− 0.00054
+ 0.00014
− 0.00036
0.00044
− 0.01237
− 0.01216
− 0.01183
− 0.01212
0.00027
Appendix
Table 60: Trinity per column, three training seeds, the five columns at NFE=32 and then at NFE=1 (std / mean in Table 59 ). Units as in Section 4.1 .
Column
Original
Seed 1
Seed 2
Mean
Std
Original
Seed 1
Seed 2
Mean
Std
overlap
0.02220
0.02261
0.02255
0.02245
0.00022
0.03573
0.03545
0.03508
0.03542
0.00032
boundary
2.997
2.946
2.947
2.963
0.029
3.011
2.943
2.941
2.965
0.040
group gap
2.827
2.675
2.682
2.728
0.086
2.320
2.177
2.219
2.239
0.073
MIB variance
6.66
6.30
8.38
7.11
1.1
7.19
7.32
8.85
7.79
0.92
fixed
7.474
6.905
9.472
7.951
1.35
7.956
7.221
9.027
8.068
0.91
HPWL gap
+ 0.00219
− 0.00001
+ 0.00024
+ 0.00081
0.0012
− 0.01129
− 0.01308
− 0.01269
− 0.01235
0.00094
Appendix
Table 61: Trinity at λ=0 per column, the five columns at NFE=32 and then at NFE=1 . Units as in Section 4.1 .
Column
Original
Seed 1
Seed 2
Mean
Std
Original
Seed 1
Seed 2
Mean
Std
overlap
0.02602
0.02638
0.02656
0.02632
0.00028
0.04383
0.04602
0.04370
0.04452
0.0013
boundary
3.227
3.156
3.041
3.141
0.094
2.811
2.854
2.817
2.827
0.024
group gap
2.651
2.702
2.712
2.688
0.033
3.196
3.211
3.126
3.178
0.046
MIB variance
8.75
9.28
9.53
9.19
0.40
3.51
8.54
2.88
4.98
3.1
fixed
5.817
5.909
5.016
5.581
0.49
2.266
2.246
2.206
2.239
0.031
HPWL gap
− 0.00214
− 0.00089
− 0.00123
− 0.00142
0.00065
− 0.00231
− 0.00229
− 0.00279
− 0.00246
0.00028
Appendix
Table 62: The plain control (left five columns) and FlowPlace (right five) per column at NFE=32 . Units as in Section 4.1 .
Figure 20: Three training seeds per model at NFE=32 : the values, the mean and the standard deviation per soft column, for Trinity, Trinity at λ=0 , the plain control and FlowPlace. The term effect on boundary, sevrel and the soft cost is many times the spread.
Figure 76
Run
NFE
FLD8b
FLD8f
FPD
MMD8b
p
the plain control, original
1
0.4325
0.0179
0.0036
32.6
–
32
0.1031
0.0033
0.0011
15.6
–
the plain control, seed 1
1
0.4474
0.0181
0.0038
34.4
0.199
32
0.1112
0.0041
0.0013
17.2
1.000
the plain control, seed 2
1
0.4274
0.0161
0.0040
32.1
0.333
32
0.1066
0.0031
0.0013
15.9
0.955
Appendix
Table 63: Distribution metrics of the twelve caches at NFE=1 and 32; the reference is the plain control’s original run, p the permutation test against it. Units as in Section 4.1 .
Figure 23: Soft cost against refiner steps, one panel per NFE , on our 12,000 val set with four draws. Trinity starts lowest in every panel, every line of this work converges to the same endpoint, and the prior arts stay above at 400 steps; DiffPlace and MacroDiff+ at NFE≤2 start two decades higher.
Figure 24: The same data as Figure 23 on a linear axis from 1 to 3, where the lines of this work separate: the terms lift the raw sample and the first 25 steps, and the lead is largest at NFE=1 .
Figure 25: Refiner steps needed to reach a soft cost of τ against NFE , Trinity and the same backbone without the auxiliary loss. Trinity reaches each threshold one rung earlier at the low NFE values and the two agree from NFE=4 .
Figure 26: Every soft column of the raw sample against NFE for the nine generators of the composed grid (three backbones × three term settings). Boundary and MIB variance fall with the terms at every NFE , overlap falls with NFE on every model, and the fixed deviation is where the plain backbone leads.
Figure 27: Soft cost against NFE , one panel per step count. The NFE gain shrinks with refinement: 0.16 on the raw Trinity sample, 0.008 at 400 steps.
Figure 28: The soft grid as heatmaps, rows NFE , columns refiner steps, colour = log10 soft cost clipped at 10, one panel per line. Every grid is monotone in both axes.
Figure 29: Soft cost, first-draw hard cost and best-of-four hard cost against NFE at 400 refiner steps. The soft cost is monotone in NFE while the first draw has its minimum at NFE=2 and the best of four at NFE=8 .
Figure 30: Hard cost after legalization against refiner steps, one panel per NFE and selection (first draw, best of four), Trinity and the four prior arts. The refiner pays after legalization on every line, the largest move is from raw to 10 steps, and the order of the lines is the order of the raw soft cost.
Figure 31: Soft cost against first-draw hard cost, cell means, one connected path per line. Top: along the step axis (raw to 400 steps), one panel per NFE ; the paths are monotone along steps, and the prior arts’ paths end above Trinity’s. Bottom: along the NFE axis, one panel per step count; at 400 steps Trinity’s path runs upward, the soft cost keeps falling with NFE while the first-draw cost rises slightly.
Figure 32: The legalizer’s ladder against refiner steps at NFE∈{1,8,32} : the median expansion factor λ , the area gap and Vrel after legalization, the share of draws solved by the hard LP, and the milliseconds per draw. λ , the price of residual overlap, falls with refinement and Trinity needs the least; the two ϵ -head prior arts at NFE=1 have no layout to legalize.
Figure 33: The first-draw hard cost on the NFE× steps grid, colour = log10 cost clipped at 3.2, one panel per line; the annealer has no sampler and fills one row.
Figure 34: The best-of-four hard cost on the NFE× steps grid, same layout as Figure 33 .
Figure 35: Which rung of the legalizer solves each draw, as stacked shares against refiner steps at NFE=32 , one panel per line. The hard-LP share doubles from the raw sample to 400 steps and no draw is infeasible.
Figure 36: Per-draw correlations with the legalizer’s behaviour against refiner steps at NFE=32 : the soft cost predicts the rung a draw reaches and the time it takes, and the time predicts the hard cost on every line.
Figure 37: The same six cells as Figure 38 on the per-case means over four draws.
Figure 38: Soft cost against legalized hard cost per draw, a 2,000-draw subsample of six cells. The relation is a ranking, not a line: the Spearman correlation reaches 0.62 at Trinity’s (NFE=32,400) cell, and the hard cost keeps a heavy tail.
Figure 39: The per-draw Spearman correlation between soft cost and hard cost per cell, one heatmap per line, and its course against refiner steps at NFE=32 . The correlation rises with both axes on Trinity, from 0.20 to 0.60, and less on the prior arts.
Figure 40: Left: correlation measures per line, across cell means and pooled over draws. Right: the Spearman correlation of every soft column with the hard cost on Trinity at three cells; sevrel carries it, then overlap and boundary, and wirelength is negative because a shorter wirelength goes with a larger die.
Figure 41: (a) Figure 7 b with all eight lines: the regressors’ cells sit at a soft cost of 20 to 100, so their per-case median is drawn where the mean overflows. (b, c) The expansion factor λ against the hard cost per draw on Trinity at NFE=32 , raw and after 400 refiner steps. Refinement moves the mass to λ=1 , the draws that need no expansion.
Figure 42: Single-term ladders on the plain backbone: the term’s own target metric and the soft cost relative to the plain control against the weight, at NFE=32 (solid) and NFE=1 (dashed), with the 0.05t arm as a star. Each term moves its own target; boundary and area move the soft cost, the others hardly.
Figure 43: The term ladder on the three backbones, one panel per metric. The terms do the same work on every backbone, and the backbones differ in where they start: the information mover lowers overlap and the distribution distances, the spectral encoding alone raises the fixed deviation.
Figure 44: The side effects: every soft column relative to the plain control against the weight, one panel per term. The boundary term raises the fixed deviation, the area term lowers the boundary distance, and the MIB term raises nothing.
Figure 45: FLD (top) and MMD (bottom) against the weight per term at NFE=32 and 1, with a star where the arm is closer to the reference than the plain control at p<0.05 under the permutation test of Appendix C.1 . The distribution-closest weight is one step below the best raw target (overlap 0.05, boundary 0.01).
Figure 46: All six terms at one common weight, constant (left) and t -weighted (right): every constraint metric and the soft cost relative to the plain control. The weight 0.01 improves every constraint at once, 0.1 breaks the fixed deviation, and the t -weighted arms trade the same way at twice the weight.
Figure 47: Raw soft cost against NFE for the objective sweep, the architecture and recipe sweep, and the prior arts. The clean-sample head is flat in NFE ; the ϵ and v heads, ours and the prior arts’, are not.
Figure 48: Every model at NFE=32 : soft cost against FLD and against MMD, and overlap against soft cost. The two distribution metrics agree on the order, and the single-term overlap arms have the lowest overlap at the plain control’s soft cost.
Figure 49: Soft cost against FLD and MMD, one segment per model from NFE=1 to 32, coloured by sweep set. The term-trained arms occupy the lower left, and the segments of the ϵ -head and v -head models are long because they have no layout at NFE=1 .
Figure 50: Raw soft cost against NFE for every single-term arm and for the all-six and composed arms.
Figure 51: Left: Trinity’s best-of- N mean hard cost on the 12,000 cases with the median-to-p90 band, at NFE∈{32,8,1} with 400 steps, at NFE=32 raw, and the soft pick (the legalized cost of the draw with the lowest soft cost among N ). NFE=8 equals 32, NFE=1 gains less from N , and the soft pick does not replace legalizing every draw. Right: the N ladder on the FloorSet val set at NFE=8 and 400 steps.
Figure 52: The FloorSet val set solved one case at a time. Left: mean hard cost against seconds per case for every NFE× steps × draws setting with the envelope; N is the strongest lever and 16 draws at NFE=4 with 100 steps reach 1.043 in 0.63 seconds. Right: the seconds per case by stage; the refiner is a fixed 0.6 seconds at 400 steps and the legalizer grows with N .
Figure 53: Per FloorSet val set case at NFE=8 , 400 steps and 16 draws: seconds and hard cost against block count. The legalizer’s time grows with the block count and the cost does not.
Figure 54: First-draw and best-of-four hard cost at 400 steps against NFE for every line; the annealer and the regressors have no sampler and are drawn as levels.
Figure 55: Every soft column against refiner steps at NFE=32 for the models of this work and the prior arts. Overlap, the cluster gap, the MIB variance and the fixed deviation fall monotonically, the boundary distance after a rise at 10 steps; the HPWL and area gaps are the residual the refiner trades for them.
Figure 56: Every draw of three cases of our 12,000 val set (30, 70 and 120 blocks, four draws each) along the refiner at NFE=1 and 32. Overlap and sevrel fall on every draw, the MIB variance and the cluster gap reach zero within 10 to 200 steps, the boundary distance rises at 10 steps on most draws before it falls, and the HPWL and area gaps are the residual each draw trades for them. One draw of the 30-block case keeps a cluster gap of 0.02 at NFE=32 and ends at a soft cost of 1.28.
Figure 57: Left: the seventeen weighting arms at 200 steps; the overlap weight of 10 is the one change that moves the endpoint. Right: the overlap-weight ladder against steps.
Figure 58: The budget panel of Figure 1 in soft cost (left) and in the hard cost of the first legalized draw (right): Trinity at NFE=1 , 8 and 32, every prior-art generator at NFE=32 raw (cross) and under the closed-form refiner (dashed), and its own loop at the published iteration count (triangle). A horizontal arrow ends at the refiner budget that matches the loop, by linear interpolation between the measured budgets. The vertical arrow runs from the cheapest loop to our cost at 400 steps.
Figure 59: Soft cost and overlap against refiner steps for the twelve schedule × peak-rate arms on Trinity at NFE=32 with every weight at 1. Linear and cosine decay at 10−3 coincide, and a constant rate over-refines: overlap turns up after 50 steps.
Figure 60: Three FloorSet val set cases with 21, 60 and 120 blocks through the pipeline, 48 draws at NFE=8 with 400 refiner steps and the shared legalizer, the draw with the cheapest legalized layout shown at every stage: the reference layout, the raw sample, the refined layout and the legalized layout. Blocks are coloured by constraint kind, every pairwise overlap region is red, block-to-block nets are faint lines between block centres, pins are dots on the outline with faint lines to their blocks, and the dashed box is the bounding outline. Under each panel: the signed area gap and HPWL gap to the reference, the severity sevrel and the overlap ratio of the soft vector of Appendix C , which unlike the hard cost keep a layout smaller or shorter than the reference visible.
Figure 61: Every generator on one 60-block FloorSet val set case from the same noise: the reference layout and the first draw of each line at NFE=8 , before any refinement. Colours as in Figure 60 . Under each panel, the overlap ratio and the soft cost of that layout.
Figure 62: A 120-block case under the protocol of Figure 61 , with one region magnified. The window is placed on the largest overlap of the worst line and is the same in all three columns.
Rung
d
L
Parameters
Width rungs
192 / 12
192
12
5.9M
256 / 12
256
12
10.6M
384 / 12
384
12
23.3M
512 / 12
512
12
42.1M
Depth rungs
Appendix
Table 64: The size rungs of Trinity recipe, defined by the hidden width d and the layer count L (head dimension 64 throughout). Everything not listed is Trinity’s.
d / L
Params
Overlap
Boundary
MIB
Fixed
Area gap
Soft cost 32 / 1
FLD
MMD
HPWL gap
Width rungs
192 / 12
5.9M
0.0347
1.00
13.8
7.2
+ 0.003
1.827 / 1.903
–
–
+0.001
256 / 12
10.6M
0.0297
0.97
3.7
6.6
+ 0.003
1.703 / 1.812
0.055
5.0
+0.001
384 / 12
23.3M
0.0277
0.86
2.0
6.2
− 0.004
1.612 / 1.759
0.044
3.7
−0.002
512 / 12
42.1M
0.0229
0.89
2.4
5.8
+ 0.000
1.534 / 1.682
0.042
2.0
−0.001
Depth rungs
Appendix
Table 65: The size rungs ( d / L ) on raw samples, NFE=32 unless stated: the constraint columns, the soft cost at NFE=32 / 1, and the two distribution distances against the reference, which the two rungs trained last do not have. Units as in Section 4.1 ; orange marks Trinity.
Figure 63: The width rungs ( d / L with L=12 ) against the width d and along the refiner. Top row: the soft cost raw and after 400 refiner steps at NFE=32 and 1, the hard cost of the first draw and of the best of four at 400 steps, and the two distribution distances. Middle row: the raw constraint columns at NFE=32 relative to Trinity, the legalizer’s statistics at NFE=32 with 400 steps relative to Trinity (area gap, Vrel , share of draws closed by the hard program, median expansion λ ), and the Spearman correlation between soft and hard cost per draw and across the cell means. Bottom row: the soft cost along the refiner at NFE=32 and 1, and the 400-step endpoint against NFE , one line per width.
Figure 64: The depth rungs ( d / L with d=768 ) against the layer count L and along the refiner, the panels of Figure 63 : the soft cost raw and after 400 refiner steps, the hard cost of the first draw and of the best of four at 400 steps, the two distribution distances, the raw constraint columns and the legalizer’s statistics relative to Trinity, the soft–hard Spearman correlation, and the soft cost along the refiner and against NFE , one line per depth.
d / L
32 / 0
32 / 25
32 / 100
32 / 400
1 / 0
1 / 100
1 / 400
256 / 12
1.703 / 0.0297
1.270 / 0.0123
1.150 / 0.0072
1.080 / 0.0033
1.812 / 0.0365
1.177 / 0.0101
1.083 / 0.0040
+0.1 / +0.3
+0.4 / +1.5
+0.6 / +1.8
+0.7 / +1.8
−1.2 / +0.0
−0.4 / +1.5
−0.1 / +1.7
384 / 12
1.612 / 0.0277
1.226 / 0.0103
1.123 / 0.0060
1.065 / 0.0028
1.759 / 0.0356
1.155 / 0.0094
1.071 / 0.0038
−0.2 / −0.4
+0.2 / +0.9
+0.4 / +1.2
+0.4 / +1.2
−1.5 / −0.8
−0.6 / +0.9
−0.3 / +1.2
512 / 12
1.534 / 0.0228
1.215 / 0.0088
1.115 / 0.0051
1.061 / 0.0024
1.682 / 0.0312
1.149 / 0.0084
1.068 / 0.0034
−0.1 / +0.0
+0.3 / +1.2
+0.4 / +1.3
+0.4 / +1.1
−1.3 / −0.3
−0.5 / +1.1
−0.3 / +1.2
Appendix
Table 66: The size rungs ( d / L ) along the closed-form refiner, one column per NFE / refiner steps, two lines per rung: soft cost / overlap ratio, then wirelength gap / area gap to the reference in percent (positive above the reference); orange marks Trinity.
Figure 65: The width rungs (top) and the depth rungs (bottom) through the legalizer against NFE : the hard cost of the first draw and of the best of four, raw (dashed) and after 400 refiner steps (solid), and the per-draw Spearman correlation between soft and hard cost at 400 steps.
d / L
NFE
Soft cost
First draw
p90
Best of 4
Total
Area gap
Vrel
ρ
HPWL gap
768 / 12
1
1.0661
1.2222 / 1.0682
1.486
1.1298
1.2337
+ 0.105
0.041
0.495
+0.087
512 / 12
1
1.0683
1.2221 / 1.0701
1.485
1.1263
1.2223
+ 0.104
0.042
0.480
+0.085
384 / 12
1
1.0708
1.2336 / 1.0738
1.514
1.1378
1.2494
+ 0.110
0.043
0.452
+0.092
256 / 12
1
1.0829
1.2483 / 1.0807
1.567
1.1459
1.2632
+ 0.119
0.045
0.471
+0.099
768 / 8
1
1.0716
1.2271 / 1.0709
1.505
1.1341
1.2352
+ 0.109
0.042
0.476
+0.090
768 / 6
1
1.0778
1.2463 / 1.0780
1.572
1.1452
1.2639
+ 0.119
0.044
0.441
+0.099
Appendix
Table 67: The size rungs ( d / L ) through the legalizer at 400 refiner steps and NFE=1 , 8 and 32: the soft cost of the refined layout, the hard cost of the first legalized draw as mean / median and its 90th percentile, the mean and the hard-cost total of the best of four, the legalized layout’s area gap and Vrel , and the per-draw Spearman correlation ρ between soft and hard cost. Every draw of every rung is feasible; orange marks Trinity.
Chip placement plays an important role in physical design. While generative models like diffusion models offer promising learning-based solutions, current methods have the following limitations: they use random synthetic data for pre-training, require long sampling times, and often result in overlaps due to their dependence on gradient-based solvers during the sampling process. To overcome these issues, we propose FlowPlace, which features mask-guided synthetic data generation, flow-based efficient training with flexible prior injection, and hard constraint sampling for overlap-free layouts. Experiments on OpenROAD and ICCAD 2015 benchmarks show FlowPlace achieves better PPA metrics, 10-50× faster sampling efficiency, and zero overlaps.
Peng Xie, Ke Xue, Yunqi Shi +6
1State Key Laboratory of Novel Software Technology, Nanjing University, China · School of Artificial Intelligence, Nanjing University, China · 3Huawei Noah’s Ark Lab, China
We propose using Vision-Language Models (VLMs) for macro placement in chip floorplanning, a complex optimization task that has recently shown promising advancements through machine learning methods. Because human designers rely heavily on spatial reasoning to arrange components on the chip canvas, we hypothesize that VLMs with strong spatial reasoning capabilities can effectively complement existing placement algorithms. We introduce VeoPlace (Visual Evolutionary Optimization Placement), a novel framework that uses a VLM, without any fine-tuning, to guide the actions of a base placer by constraining it to subregions of the chip canvas. The VLM proposals are iteratively optimized through an evolutionary search strategy with respect to resulting placement quality. Using global half-perimeter wirelength (gHPWL) after standard-cell placement and legalization as the evaluation metric, VeoPlace boosts ChiPFormer performance on 9 of 10 benchmarks with peak reductions exceeding 32%. We further demonstrate that VeoPlace generalizes to analytical placers, improving DREAMPlace on all 8 evaluated Superblue benchmarks with gains up to 4.4%. Our approach opens new possibilities for electronic design automation tools that leverage foundation models to solve complex physical design problems.
Ikechukwu Uchendu, Swati Goel, Karly Hou +5
Harvard University, Cambridge, MA, USA · Kempner Institute for the Study of Natural and Artificial Intelligence, Harvard University, Cambridge, MA, USA · Google DeepMind
Macro placement significantly affects a chip's post-route performance, power, and area (PPA). Most placement methods optimize half-perimeter wirelength (HPWL) as the primary objective. However, recent benchmarking shows a near-zero correlation between HPWL and post-route timing metrics such as the worst negative slack (WNS) and total negative slack (TNS). As a result, all six evaluated artificial intelligence (AI) placers degraded PPA relative to the hierarchical baseline. Recent efforts have tried to train cross-stage predictors to close this gap. However, existing methods focus on macro-only representations and use pre-route metrics as training labels. A label fidelity study of ten circuits at four design flow stages reveals that HPWL and pre-route timing poorly reflect final post-route timing rankings. In contrast, post-global-routing achieves the best balance between final timing fidelity and label generation cost-effectiveness. Based on this finding, PPAPlace is a timing-driven differentiable surrogate predicting post-route PPA from macro and standard-cell placements. The surrogate is a dual-stream predictor that combines graph attention over the chip netlist with spatial convolution over the placement grid. It is trained on post-global-routing labels. The predicted WNS and TNS gradients flow end-to-end back to cell coordinates. PPAPlace exploits these gradients in two ways: as a co-objective injected into an analytical placer's optimization loop (PPAPlace-CoOpt), and as a post-placement refinement step that adjusts macro positions via projected gradient descent (PPAPlace-Refine). On five ChiPBench test circuits excluded from training, PPAPlace improves average WNS and TNS by 22% and 51% over the hierarchical baseline while preserving power and routability, using the same predictor without test-circuit retraining. Code is available at https://github.com/ValleyC/PPAPlace.
Ruogu Chen, Jie Han
University of Alberta Department of Electrical and Computer Engineering Edmonton, AB, Canada