SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Method	Size	BB(Cost)	Best	Mean	Time(s)
LS++	476.88	488.32±22.61	2.93
FLS	476.88	476.88±0	10.44
LLOYD	rds(50,000*3)	476.79(811s)	476.88	507.69±31.3	0.18
MLSP	476.79	476.8±0.07	15.52
MLS	476.88	482.77±16.93	2.26
LS++	4.9412E+08	4.9412E+08±0	1.04
FLS	4.9412E+08	4.9412E+08±0	7.27
LLOYD	KEGG(53,413*23)	4.9412E+08(3901s)	4.9412E+08	4.9412E+08±0	0.26
MLSP	4.9412E+08	4.9412E+08±0	12.89
MLS	4.9412E+08	4.9412E+08±0	1.97
LS++	1.15E+05	1.16E+05±1.7E+03	3.26
FLS	1.15E+05	1.15E+05±229	14.73
LLOYD	Urban_10(100,000*2)	1.15E+05(6834s)	1.15E+05	1.23E+05±1.1E+04	0.18
MLSP	1.15E+05	1.15E+05±72	26.14
MLS	1.15E+05	1.16E+05±1.6E+03	3.08
LS++	1.6396E+15	1.6398E+15±3.3E+11	8.27
FLS	1.6396E+15	1.6397E+15±1.3E+11	58.78
LLOYD	RNG_AGR(199,843*7)	1.64E+15(4h)	1.6396E+15	1.6398E+15±3.2E+11	0.42
MLSP	1.6396E+15	1.6396E+15±0	53.25
MLS	1.6396E+15	1.6398E+15±1.3E+11	7.73
LS++	414253	417302±4629	9.86
FLS	414253	414733±956	50.11
LLOYD	Urban_GB(360,177*2)	OOM	414253	423509±12613	0.32
MLSP	414253	414642±314	71.46
MLS	414253	417063±6123	8.87
LS++	2.2779E+07	2.2781E+07±824	13.93
FLS	2.2779E+07	2.2780E+07±769	142.15
LLOYD	SPNET_3D(434,874*3)	OOM	2.2781E+07	2.2782E+07±609	0.49
MLSP	2.2779E+07	2.2779E+07±0	121.03
MLS	2.2779E+07	2.2781E+07±1022	10.33
LS++	3.8720E+06	3.8722E+06±108.8	37.1
FLS	3.8720E+06	3.8721E+06±44.1	338.19
LLOYD	syn(1,000,000*2)	OOM	3.8720E+06	3.8724E+06±236.4	0.69
MLSP	3.8720E+06	3.8721E+06±32.9	223.97
MLS	3.8720E+06	3.8722E+06±119.2	22.93
LS++	6.9094E+08	6.9094E+08±0	286.41
FLS	6.9094E+08	6.9094E+08±0	3244.76
LLOYD	USC_1990(2,458,685*68)	OOM	6.9094E+08	6.9094E+08±0	18.57
MLSP	6.9094E+08	6.9094E+08±0	2443.28
MLS	6.9094E+08	6.9094E+08±0	227.59
LS++	4.6299E+07	4.7891E+07±1.6E+06	305.27
FLS	4.6299E+07	4.8013E+07±1.5E+06	3894.45
LLOYD	SUSY(5,000,000*17)	OOM	4.6299E+07	4.8261E+07±1.3E+06	29.2
MLSP	4.6299E+07	4.7349E+07±7.7E+04	2610.89
MLS	4.6299E+07	4.7612E+07±1.6E+06	262.61
LS++	2.2614E+08	2.2726E+08±1.1E+06	614.07
FLS	2.2614E+08	2.2720E+08±9.2E+05	8728.13
LLOYD	HIGGS(11,000,000*27)	OOM	2.2614E+08	2.2858E+08±2.6E+06	109.06
MLSP	2.2614E+08	2.2718E+08±8.8E+05	6492.59
MLS	2.2614E+08	2.2728E+08±1.2E+06	522.15
LS++	1.5774E+13	1.6098E+13±1.6E+11	31994
FLS	1.5649E+13	1.5856E+13±1.1E+11	18306
LLOYD	SIFT(100,000,000*128)	OOM	1.5718E+13	1.5793E+13±1.5E+11	401
MLSP	-	-	>48h
MLS	1.5718E+13	1.5848E+13±1.1E+11	12738
Table 18: Comparison results on clustering costs and running time with  $k = 3$  on datasets with sizes larger than 50,000, where OOM is short for out of memory
Method	Size	BB(Cost)	Best	Mean	Time(s)
LS++	284.27	287.39±2.78	3.34
FLS	290.15	290.15±0	18.86
LLOYD	rds(50,000*3)	282.65(4h)	284.22	287.50±8.67	0.86
MLSP	283.13	283.27±0.21	58.61
MLS	284.21	287.23±2.72	2.97
LS++	1.9201E+08	1.9205E+08±2.1E+04	2.17
FLS	1.9201E+08	1.9205E+08±1.6E+04	10.92
LLOYD	KEGG(53,413*23)	1.92E+08(3901s)	1.9201E+08	1.9754E+08±1.6E+07	0.26
MLSP	1.9201E+08	1.9201E+08±0	24.83
MLS	1.9201E+08	1.9204E+08±2.3E+04	1.88
LS++	5.62E+04	56281±76	5.33
FLS	5.62E+04	56248±49	26.22
LLOYD	Urban_10(100,000*2)	56231(4h)	5.62E+04	57768±2762	0.14
MLSP	5.62E+04	56246±36	40.97
MLS	5.62E+04	56315±83	4.64
LS++	5.0721E+14	5.0721E+14±3.3E+11	15.65
FLS	5.0721E+14	5.0739E+14±3.0E+11	122.09
LLOYD	RNG_AGR(199,843*7)	5.0721E+14(4h)	5.0721E+14	5.0781E+14±3.2E+11	0.35
MLSP	5.0721E+14	5.0736E+14±3.1E+11	73.2
MLS	5.0721E+14	5.0759E+14±3.0E+11	14.24
LS++	201878	202554±954	16.48
FLS	201878	201879±2.5	97.19
LLOYD	Urban_GB(360,177*2)	OOM	201878	204943±7816	0.53
MLSP	201873	201874±2.03	103.94
MLS	201878	202166±629	14.89
LS++	8.8280E+06	8.8309E+06±1.4E+03	26.85
FLS	8.8280E+06	8.8289E+06±1.1E+03	331.24
LLOYD	SPNET_3D(434,874*3)	OOM	8.8280E+06	8.8315E+06±1.6E+03	0.83
MLSP	8.8273E+06	8.8273E+06±0.17	180.12
MLS	8.8277E+07	8.8313E+06±1.0E+03	22.94
LS++	1.8662E+06	1.8696E+06±3.2E+03	66.78
FLS	1.8765E+06	1.8674E+06±1.8E+03	527.22
LLOYD	syn(1,000,000*2)	OOM	1.8661E+06	1.9579E+06±1.1E+03	0.85
MLSP	1.8652E+06	1.8659E+06±1.1E+03	391.86
MLS	1.8658E+06	1.8694E+06±3.1E+03	31.62
LS++	3.9017E+08	3.9375E+08±1.1E+07	439.74
FLS	3.9017E+08	3.9017E+08±0	4802.6
LLOYD	USC_1990(2,458,685*68)	OOM	3.9017E+08	3.9861E+08±3.6E+07	20.11
MLSP	3.9017E+08	3.9017E+08±0	3612.99
MLS	3.9017E+08	3.9732E+08±1.4E+07	324.97
LS++	4.0623E+07	4.1035E+07±1.5E+06	451.28
FLS	3.9880E+07	4.0997E+07±4.5E+04	5329.33
LLOYD	SUSY(5,000,000*17)	OOM	3.9848E+07	4.0648E+07±6.4E+05	31.67
MLSP	3.8720E+07	3.9316E+07±1.4E+05	4137.19
MLS	3.9904E+07	4.0882E+07±3.6E+06	297.43
LS++	2.0829E+08	2.1016E+08±1.3E+06	1316.59
FLS	2.0829E+08	2.1012E+08±1.2E+06	18429.3
LLOYD	HIGGS(11,000,000*27)	OOM	2.0829E+08	2.1046E+08±5.7E+06	114.18
MLSP	2.0829E+08	2.1010E+08±1.1E+06	12713.2
MLS	2.0829E+08	2.1018E+08±1.4E+06	1019.6
LS++	1.5035E+13	1.5194E+13±8.8E+10	41376
FLS	1.4778E+13	1.4994E+13±9.1E+10	27908
LLOYD	SIFT(100,000,000*128)	OOM	1.4805E+13	1.5021E+13±8.0E+10	531
MLSP	-	-	>48h
MLS	1.4778E+13	1.4992E+13±8.9E+10	12931
Table 19: Comparison results on clustering costs and running time with  $k = 5$  on datasets with sizes larger than 50,000, where OOM is short for out of memory
B.4 Experiments on Different Local Search Algorithms with Fixed Time Limits
In this section, we conduct some additional experiments on different datasets with fixed time limits. The experimental results in Table 20 demonstrate that our proposed MLSP algorithm achieves the best clustering quality with fixed time limits.
Dataset	Method	Time Point	Cost	Time Point	Cost	Time Point	Cost	Time Point	Cost
rds	LS++	137.96	137.96	137.93	137.93
FLS	138.98	137.26	134.68	134.62
MLSP	133.62	132.29	132.28	132.14
MLS	140.19	138.23	137.63	137.63
KEGG	LS++	62855194	62855194	62793065	62793065
FLS	61633499	61633499	61633499	61608744
MLSP	61534479	61534479	61534479	61534479
MLS	63419827	63419827	63419827	63419827
Urban_10	LS++	25011.62	25011.62	25011.62	25011.62
FLS	25103.83	25066.53	25058.21	24973.42
MLSP	24767.85	24767.66	24751.62	24748.37
MLS	24920.38	24885.80	24885.80	24885.80
RNG_AGR	LS++	1.3833E+14	1.3833E+14	1.3833E+14	1.3833E+14
FLS	1.3822E+14	1.3813E+14	1.3800E+14	1.3797E+14
MLSP	1.3743E+14	1.3703E+14	1.3690E+14	1.3689E+14
MLS	1.3826E+14	1.3826E+14	1.3826E+14	1.3826E+14
Urban_GB	LS++	89234.23	89234.23	89234.23	89234.23
FLS	89691.46	89691.46	89688.77	89406.39
MLSP	89314.46	89311.78	88946.64	88407.99
MLS	88955.80	88955.80	88955.80	88955.80
SPNET_3D	LS++	2576396	2576396	2575784	2575784
FLS	2573606	2572637	2571369	2570776
MLSP	2570513	2570292	2570160	2569484
MLS	2574266	2574266	2574266	2574266
syn	LS++	561444.2	561444.2	561444.2	561444.2
FLS	561914.9	561914.9	561914.9	561914.9
MLSP	560502.5	560502.4	560502.4	560502.4
MLS	562385.8	562385.8	562385.8	562385.8
USC_1990	LS++	270910366	270910366	270910366	270910366
FLS	270910366	270910366	270825616	270825616
MLSP	270735773	270735773	270735773	270735773
MLS	270825616	270825616	270825616	270825616
SUSY	LS++	32437956	32437956	32437956	32437956
FLS	31693128	31693128	31693128	31693128
MLSP	31756413	31650548	31650548	31629757
MLS	32548680	32548680	32548680	32548680
HIGGS	LS++	189640888.6	188373729.5	188373729.5	187098096.4
FLS	189186932.5	189186932.5	189186932.5	189186932.5
MLSP	184084129.5	184084129.5	184084129.5	184084129.5
MLS	186689935.5	186689935.5	186689935.5	186689935.5
SIFT	LS++	1.82E+13	1.77E+13	1.67E+13	1.58E+13
FLS	1.79E+13	1.59E+13	1.55E+13	1.52E+13
MLSP	1.85E+13	1.85E+13	1.85E+13	1.85E+13
MLS	1.38E+13	1.38E+13	1.38E+13	1.38E+13
Table 20: Comparison results on clustering costs with  $k = {10}$  and fixed time limits