SlowGuess/NeurIPS_2023_OCR huggingface dataset:
process,  $t$  data points are sampled according to the  $D^2$ -Sampling distribution to serve as the candidate set of centers for swapping in, which takes time  $O(t)$  if the distances from data points to their centers are already known. It takes  $O(k^t)$  time to enumerate each subset with size at most  $t$  of the set of current centers opened. It takes  $O(ndk^t)$  time to recalculate the clustering cost after each swap if  $t$  nearest centers of each data point are maintained during the whole process. Thus, the total running time of Algorithm 2 is  $O(ndk^{2t + 1}\log (\epsilon^{-1}\log k))$ .
B Complementary Experiments
B.1 Experiments with Different Parameter Settings
In this section, we present the experiments on the performances of our proposed MLSP algorithm with different parameters  $T$ ,  $R'$ ,  $\epsilon$  and  $\eta$ . We also compare our MLS and MLSP algorithms with LS++ algorithm using different iteration rounds to show the parameter robustness of different algorithms.
Tables 3, 4 and 5 show the comparison results of different algorithms with varying iteration rounds on dataset rds, KEGG and Urban_10, respectively. The results show that, our proposed MLSP algorithm always achieves the best clustering cost compared with LS++ and MLS algorithms. It can be seen that the number of iteration rounds has little impact on our MLSP algorithm. For MLS algorithm, it performs better on clustering cost compared with LS++ algorithm for most cases, which indicates that larger iteration rounds will not influence the performance of our algorithms.
Tables 6, 7, and 8 show the results of MLSP algorithm with varying number of failure upper bound  $R'$  for fixed  $\epsilon = 0.5$ ,  $\eta = 0.5$  and  $T = 400$  on dataset rds, KEGG and Urban_10, respectively. It can be seen that a larger failure upper bound will lead to smaller deviation, and the running time becomes higher. In general, the failure upper bound does not influence the performance of our proposed MLSP algorithm. Tables 9, 10, and 11 show the results of MLSP algorithm with varying parameters  $\epsilon$  and  $\eta$  for fixed  $T = 400$  and  $R' = 5$  on dataset rds, KEGG, and Urban_10, respectively. It can be seen that smaller values of  $\epsilon$  and  $\eta$  result in better performances on clustering costs with smaller deviation, and the running time becomes higher. There is no significant difference between different choices of parameters  $\epsilon$  and  $\eta$ , which indicates that our proposed MLSP algorithm ensures the parameter robustness.
Method	Rounds	Best Cost	Average Cost	Time(s)	Method	Rounds	Best Cost	Average Cost	Time(s)
LS++	133.4074	139.6724±3.5172	2.56	LS++	133.1909	13582±2.0054	3.81
MLS	100	133.9806	139.6108±1.3052	1.12	MLS	200	132.5915	138.6551±2.3461	2.27
MLSP	131.7410	132.9932±0.9546	29.02	MLSP	131.7410	133.1258±2.2659	30.25
LS++	132.0117	137.2619±2.8543	5.49	LS++	132.1966	138.2526±2.0892	8.14
MLS	300	133.5915	138.6551±2.3461	3.55	MLS	400	131.8264	136.9465±3.1638	4.48
MLSP	131.7410	132.3278±0.8694	42.67	MLSP	131.7410	132.0323±0.4511	58.61
LS++	133.6918	140.7157±2.0026	12.26	LS++	131.7681	137.7633±3.0206	17.24
MLS	500	131.8298	138.3528±5.2943	5.57	MLS	600	131.8264	137.1135±3.6302	6.90
MLSP	131.7410	132.8539±1.7382	66.73	MLSP	131.7410	132.1285±0.5928	72.64
LS++	131.8519	137.8405±3.3180	22.83	LS++	131.8238	139.0674±2.5677	29.23
MLS	700	131.8493	136.1385±2.9510	7.66	MLS	800	131.7709	135.7753±4.0815	7.98
MLSP	131.7410	132.1379±0.8516	81.88	MLSP	131.7410	132.2103±0.7338	91.52
Table 3: Comparison results on dataset rds with varying iteration rounds
Method	Rounds	Best Cost	Average Cost	Time(s)	Method	Rounds	Best Cost	Average Cost	Time(s)
LS++	6.1871E+07	6.2712E+07±1.3E+06	2.13	LS++	6.1619E+07	6.3341E+07±1.8E+06	5.64
MLS	6.1587E+07	7.2593E+07±1.1E+07	2.11	MLS	200	6.1684E+07	6.4154E+07±2.2E+06	4.00
MLSP	6.1534E+07	6.1559E+07±4.9E+04	22.22	MLSP	6.1534E+07	6.1534E+07±0	36.08
LS++	6.1642E+07	6.2732E+07±1.7E+06	10.41	LS++	6.1547E+07	6.3276E+07±1.9E+06	16.58
MLS	6.1939E+07	6.5379E+07±7.2E+06	6.32	MLS	400	6.1547E+07	6.3255E+07±1.8E+06	7.98
MLSP	6.1534E+07	6.1559E+07±4.9E+04	46.18	MLSP	6.1534E+07	6.1546E+07±3.7E+04	57.13
LS++	6.1547E+07	6.2891E+07±1.6E+06	24.15	LS++	6.1547E+07	6.2878E+07±1.6E+06	32.97
MLS	6.1661E+07	6.3595E+07±1.7E+06	10.54	MLS	600	6.1534E+07	6.4042E+07±2.9E+06	12.95
MLSP	6.1534E+07	6.1559E+07±4.9E+04	63.61	MLSP	6.1534E+07	6.1572E+07±5.7E+04	77.49
LS++	6.1534E+07	6.2874E+07±1.3E+06	43.03	LS++	6.1547E+07	6.2261E+07±5.9E+05	54.28
MLS	6.1587E+07	6.5113E+07±3.4E+06	13.97	MLS	800	6.1953E+07	6.5383E+07±3.5E+06	16.22
MLSP	6.1534E+07	6.1547E+07±3.7E+04	86.74	MLSP	6.1534E+07	6.1559E+07±4.9E+04	96.73
Table 4: Comparison results on dataset KEGG with varying iteration rounds
Method	Rounds	Best Cost	Average Cost	Time(s)	Method	Rounds	Best Cost	Average Cost	Time(s)
LS++	24659.3733	25255.4895±613.5016	3.08	LS++	24660.9511	25024.9944±356.9319	8.36
MLS	100	24787.2318	25022.6100±348.1652	1.88	MLS	200	24659.3393	25022.5701±332.6797	3.34
MLSP	24659.0733	24726.3142±134.6180	40.81	MLSP	24659.0733	24806.2115±158.9739	48.55
LS++	24659.3733	25594.2717±914.0252	15.99	LS++	24659.0733	25383.4974±577.4748	25.79
MLS	300	24659.3733	25107.5051±420.6469	4.85	MLS	400	24659.3732	25166.5432±526.5936	6.31
MLSP	24659.0733	24761.5254±176.6838	54.50	MLSP	24659.0733	24793.3057±239.687	58.74
LS++	24659.3732	25217.6713±463.9049	37.94	LS++	24659.0733	25282.5551±727.8617	52.35
MLS	500	24659.3732	24835.8816±211.3455	7.79	MLS	600	24659.3732	24797.6361±28.3684	9.18
MLSP	24659.0733	24762.4264±178.6638	73.87	MLSP	24659.0733	24742.3513±134.4212	76.58
LS++	24659.3733	24852.2682±446.1449	69.19	LS++	24659.0844	25611.1218±891.9473	88.15
MLS	700	24659.4008	25073.7242±295.8309	10.61	MLS	800	24659.5079	25001.2324±351.8913	12.01
MLSP	24659.0733	24784.8027±134.1987	81.40	MLSP	24659.0733	24818.3757±219.2623	88.92
Table 5: Comparison results on dataset Urban_10 with varying iteration rounds
Failure	Best Cost	Average Cost	Time(s)
3	131.7408	133.8048±2.5306	39.49
4	131.7408	132.6853±1.5759	48.01
5	131.7408	132.8304±1.6403	59.41
6	131.7408	132.7752±1.1013	55.65
7	131.7408	132.2285±0.8758	60.03
8	131.7408	132.4954±0.6819	69.60
9	131.7408	132.3064±0.6889	72.11
10	131.7408	132.2246±0.6323	85.36
Table 6: Results of MLSP Algorithm on dataset rds with varying number of failure upper bound  ${R}^{\prime }$
Failure	Best Cost	Average Cost	Time(s)
3	6.1534E+07	6.1559E+07±4.9E+04	44.52
4	6.1534E+07	6.1559E+07±4.9E+04	59.61
5	6.1534E+07	6.1559E+07±4.9E+04	64.45
6	6.1534E+07	6.1559E+07±4.9E+04	70.69
7	6.1534E+07	6.1534E+07±0	76.37
8	6.1534E+07	6.1534E+07±0	91.05
9	6.1534E+07	6.1534E+07±0	100.82
10	6.1534E+07	6.1534E+07±0	105.89
Table 7: Results of MLSP Algorithm on dataset KEGG with varying number of failure upper bound  $R^{\prime}$
Failure	Best Cost	Average Cost	Time(s)
3	24659.0733	24824.3035±214.2367	41.44
4	24659.0733	24819.0528±238.5684	56.65
5	24659.0733	24755.9209±132.1597	59.97
6	24659.0733	24806.0814±158.7277	66.19
7	24659.0733	24829.4308±190.0205	80.24
8	24659.0733	24679.4947±58.5737	88.09
9	24659.0733	24703.7889±55.6132	92.57
10	24659.0733	24684.7586±51.1046	95.37
Table 8: Results of MLSP Algorithm on dataset Uran_10 with varying number of failure upper bound  $R'$
eta/epsilon	Best Cost	Average Cost	Time(s)
0.25/0.25	131.7408	132.6835±1.9397	61.57
0.5/0.25	131.7408	133.5867±2.0429	58.97
0.75/0.25	131.7408	133.0508±1.7254	63.59
0.25/0.5	131.7408	132.3304±0.6313	50.81
0.5/0.5	131.7408	132.3244±1.5983	51.40
0.75/0.5	131.7408	132.6404±2.3554	50.50
0.25/0.75	131.7693	134.4831±2.2116	52.86
0.5/0.75	131.7408	132.3092±0.9371	61.99
0.75/0.75	131.7408	133.2093±2.1465	60.73
Table 9: Results of MLSP Algorithm on dataset rds with varying parameters  $\epsilon$  and  $\eta$
eta/epsilon	Best Cost	Average Cost	Time(s)
0.25/0.25	6.1534E+07	6.1572E+07±5.7E+04	85.99
0.5/0.25	6.1534E+07	6.1546E+07±3.7E+04	78.57
0.75/0.25	6.1534E+07	6.1572E+07±5.7E+04	72.15
0.25/0.5	6.1534E+07	6.1572E+07±5.7E+04	68.67
0.5/0.5	6.1534E+07	6.1546E+07±3.7E+04	65.05
0.75/0.5	6.1534E+07	6.1559E+07±4.9E+04	62.09
0.25/0.75	6.1534E+07	6.1559E+07±4.9E+04	65.15
0.5/0.75	6.1534E+07	6.1546E+07±3.7E+04	63.55
0.75/0.75	6.1534E+07	6.1546E+07±3.7E+04	64.35
Table 10: Results of MLSP algorithm on dataset KEGG with varying parameters  $\epsilon$  and  $\eta$
eta/epsilon	Best Cost	Average Cost	Time(s)
0.25/0.25	24659.0733	24681.3945±240.9711	66.64
0.5/0.25	24659.0733	24837.5858±203.3399	72.86
0.75/0.25	24659.0733	24786.8942±168.9089	59.95
0.25/0.5	24659.0733	24671.9425±38.3418	60.31
0.5/0.5	24659.0733	24756.0949±134.5952	63.27
0.75/0.5	24659.0733	24746.6868±146.2990	56.24
0.25/0.75	24659.0733	24719.8526±61.3117	57.11
0.5/0.75	24659.0733	24804.6754±173.0019	54.95
0.75/0.75	24659.0733	24753.6144±204.2255	56.56
Table 11: Results of MLSP algorithm on dataset Urban_10 with varying parameters  $\epsilon$  and  $\eta$
B.2 Experiments on Small Datasets
In this section, we present the experiments of the performances of our proposed MLSP algorithm on other datasets with sizes smaller than 50,000 used in [15].
Table 12 and Table 13 show the results on 18 datasets with sizes smaller than 50,000 using  $k = 10$ . It can be seen that, on each dataset, the best clustering cost returned by our MLSP algorithm is smaller than BB method (note that BB method already guarantees a gap smaller than  $0.1\%$  to the optimal solutions on small datasets) and other local search methods. On each dataset, the average clustering cost returned by our MLSP algorithm nearly matches the result of the BB method. As for FLS, although the performance of FLS is better than that of LS++, our proposed MLSP algorithm improves the performance of FLS on clustering cost with smaller deviation for most datasets. As for running time, there is no significant difference on the running time among different local search algorithms for small datasets.
Method	Size	BB(Cost)	Best	Mean	Time(s)
LS++	30.01	30.45±0.53	0.04
FLS	29.79	29.91±0.14	0.15
LLOYD	Iris(150*4)	29.79(735s)	30.32	30.90±0.4	0.01
MLSP	29.74	29.76±0.33	6.61
MLS	29.93	30.21±0.19	0.17
LS++	216.36	219.43±2.84	0.37
FLS	214.52	216.68±2.83	1.78
LLOYD	SEEDS(210*7)	218.49(448s)	215.65	219.99±2.93	0.12
MLSP	214.52	215.08±0.49	8.07
MLS	214.95	219.29±2.82	0.46
LS++	251.86	254.17±1.71	0.37
FLS	251.86	252.05±0.56	1.61
LLOYD	GLASS(214*9)	251.86(2566s)	253.25	259.94±3.94	0.11
MLSP	251.86	251.99±0.54	7.25
MLS	253.29	254.00±0.55	0.46
LS++	377753	392980±4437	0.37
FLS	375974	377496±1934	1.71
LLOYD	BM(249*6)	375974(1204s)	376590	384475±5560	0.07
MLSP	375974	376276±265	6.81
MLS	378649	384982±4131	0.37
LS++	29.57	30.16±0.29	0.38
FLS	29.27	29.38±0.15	1.73
LLOYD	UK(258*5)	29.28(4h)	29.94	30.33±0.26	0.08
MLSP	29.27	29.29±0.01	7.14
MLS	29.41	30.00±0.27	0.39
LS++	6.96E+10	7.01E+10±5.7E+08	0.39
FLS	6.96E+10	7.06E+10±3.1E+09	1.77
LLOYD	HF(299*12)	6.96E+10(4h)	6.96E+10	6.97E+10±2.3E+08	0.09
MLSP	6.96E+10	6.96E+10±0	8.56
MLS	6.96E+10	7.01E+10±4.3E+08	0.4
LS++	3.36E+10	3.45E+10±6.0E+08	0.4
FLS	3.36E+10	3.41E+10±4.5E+08	2.01
LLOYD	WHO(440*8)	3.40E+10(4h)	3.44E+10	3.49E+10±7.4E+08	0.11
MLSP	3.36E+10	3.37E+10±4.6E+07	9.17
MLS	3.42E+10	3.52E+10±5.6E+08	0.4
LS++	1.1315E+06	1.1454E+06±8.7E+03	0.45
FLS	1.1312E+06	1.1412E+06±2.4E+04	1.81
LLOYD	HCV(572*12)	1.1315E+06(4h)	1.1329E+06	1.1538E+06±2.9E+04	0.15
MLSP	1.1311E+06	1.1410E+06±2.2E+04	11.05
MLS	1.1505E+06	1.2135E+06±4.2E+04	0.9
LS++	1.0786E+06	1.0983E+06±1.1E+04	0.51
FLS	1.0786E+06	1.0816E+06±5.3E+03	2.11
LLOYD	Abs(740*21)	1.0786E+06(4h)	1.0824E+06	1.1209E+06±2.1E+04	0.12
MLSP	1.0786E+06	1.0786E+06±0	13.42
MLS	1.0789E+06	1.0979E+05±1.1E+04	0.47
Table 12: Comparison results on clustering costs and running time with  $k = 10$  on datasets with sizes ranging from 150 to 740
Method	Size	BB(Cost)	Best	Mean	Time(s)
LS++	776.04	793.74±7.3	0.51
FLS	762.16	766.38±4.9	2.14
LLOYD MLSP	TR(980*10)	772.47(4h)	770.72	776.48±5.0	0.12
762.16	764.71±1.9	11.89
MLS	785.01	805.89±11.5	0.56
LS++	1.1734E+08	1.2124E+08±3.4E+05	0.58
FLS	1.1734E+08	1.2092E+08±3.9E+05	2.35
LLOYD MLSP	SGC(1000*21)	1.1742E+08(4h)	1.1734E+08	1.1749E+08±1.7E+05	0.09
1.1734E+08	1.1752E+08±5.1E+04	13.45
MLS	1.1735E+08	1.1846E+08±9.6E+04	0.62
LS++	2.7116E+06	2.7657E+06±4.4E+04	0.58
FLS	2.7073E+06	2.8519E+06±1.2E+04	2.51
LLOYD MLSP	HEMI(1995*7)	2.7421E+06(4h)	2.7144E+06	2.7349E+06±6.7E+03	0.18
2.7070E+06	2.7123E+06±6.7E+03	13.39
MLS	2.7292E+06	2.7886E+06±6.9E+04	0.59
LS++	5.3689E+09	5.4772E+09±1.0E+08	0.58
FLS	5.3610E+09	5.4256E+09±5.6E+07	2.65
LLOYD MLSP	pr2392(2392*2)	5.3578E+09(4h)	5.3599E+09	5.3624E+09±2.3E+07	0.14
5.3578E+09	5.3668E+09±2.3E+07	10.76
MLS	5.3629E+09	5.4203E+09±6.2E+07	0.58
LS++	1.3902E+05	1.4155E+05±1.5E+03	0.62
FLS	1.3829E+05	1.4018E+05±1.1E+03	3.43
LLOYD MLSP	TRR(5456*24)	1.3796E+05(4h)	1.3868E+05	1.4177E+05±2.1E+03	0.18
1.3796E+05	1.3829E+05±2.5E+02	25.48
MLS	1.4591E+05	1.4939E+05±2.0E+03	0.64
LS++	1167.6	1184.01±15.88	0.58
FLS	1163.7	1174.1±13.06	5.06
LLOYD MLSP	AC(7195*22)	1181.7(4h)	1169	1172.1±6.05	0.36
1163.7	1168.2±9.01	22.32
MLS	1165.5	1181.7±12.8	0.64
LS++	1.6104E+06	1.6508E+06±2.4E+05	0.24
FLS	1.6099E+06	1.6196E+06±7.3E+04	2.26
LLOYD MLSP	rds_cnt(10000*4)	1.6119E+06(4h)	1.6009E+06	1.6247E+06±1.4E+05	0.11
1.6009E+06	1.6105E+06±7.2E+02	18.67
MLS	1.6146E+06	1.6520E+06±2.5E+05	0.42
LS++	1.8286E+07	1.8421E+07±1.1E+04	0.28
FLS	1.8269E+07	1.8404E+07±1.3E+04	0.94
LLOYD MLSP	HTRU2(17898*8)	1.8723E+07(4h)	1.8266E+07	1.8380E+07±2.0E+05	0.15
1.8266E+07	1.8312E+07±4.2E+03	24.43
MLS	1.8275E+07	1.8569E+07±1.7E+04	0.3
LS++	9.0239E+06	9.1784E+06±9.3E+04	7.06
FLS	8.9904E+06	9.0384E+06±5.4E+04	19.87
LLOYD MLSP	GT(36733*11)	8.9909E+06(4h)	9.0001E+06	9.1229E+06±7.7E+04	0.33
8.9904E+06	9.0131E+06±3.6E+04	47.51
MLS	9.0239E+06	9.1784e+06±9.3E+04	6.29
Table 13: Comparison results on clustering costs and running time with  $k = 10$  on datasets with sizes ranging from 980 to 36733
B.3 Experiments with Different Values of  $k$
In this section, we present the experiments of the performances of our proposed MLSP and MLS algorithms on different datasets with different values of  $k$ . Tables 14 and 15 present the performances of different local search algorithms with  $k = 3$  on small datasets. Table 18 presents the performances of different local search algorithms with  $k = 3$  on large datasets. Tables 16 and 17 present the performances of different local search algorithms with  $k = 5$  on small datasets. Table 19 presents the performances of different local search algorithms with  $k = 5$  on large datasets.
It can be seen that, compared with other local search methods, our proposed MLSP algorithm achieves the best clustering performance on most datasets with different values of  $k$ . On each dataset, the best clustering cost returned by our MLSP algorithm matches the clustering cost of BB method (note that BB method already guarantees a gap of  $0.1\%$  to the optimal solution). On each dataset, the average clustering cost returned by our MLSP algorithm nearly matches the result of the BB method. As for running time, there is no significant difference on running time among all local search algorithms on small datasets. However, as the data sizes grow, our proposed MLS algorithm becomes much faster than other local search algorithms.
Method	Size	BB(Cost)	Best	Mean	Time(s)
LS++	83.96	84.82±1.05	0.26
FLS	83.96	83.96±0	0.5
LLOYD	Iris(150*4)	83.96(93s)	84.68	84.68±0	0.09
MLSP	83.96	83.96±0	3.46
MLS	83.96	84.52±0.73	0.43
LS++	598.29	600.83±5.59	0.27
FLS	598.29	598.29±0	0.51
LLOYD	SEEDS(210*7)	598.29(84s)	598.29	598.30±0.69	0..07
MLSP	598.29	598.29±0	3.34
MLS	598.29	600.70±5.63	0.67
LS++	629.02	629.51±0.73	0.27
FLS	629.02	629.02±0	0.62
LLOYD	GLASS(214*9)	692.02(107s)	629.02	629.02±0	0.05
MLSP	629.02	629.02±0	3.37
MLS	629.02	630.72±5.08	0.68
LS++	8.63E+05	8.69E+05±6.4E+03	0.27
FLS	8.63E+05	8.63E+05±0	0.57
LLOYD	BM(249*6)	8.63E+05(86s)	8.65E+05	8.65E+05±0	0.03
MLSP	8.63E+05	8.63E+05±0	3.69
MLS	8.63E+05	8.74E+05±7.3E+03	0.67
LS++	50.92	51.95±0.71	0.29
FLS	50.77	50.83±0.12	0.54
LLOYD	UK(258*5)	50.77(89s)	51.29	51.31±0.03	0.04
MLSP	50.77	50.77±0	3.67
MLS	50.77	51.78±0.81	0.67
LS++	7.83E+11	7.90E+11±6.5E+10	0.27
FLS	7.83E+11	7.84E+11±2.0E+10	0.56
LLOYD	HF(299*12)	7.83E+11(107s)	7.83E+11	7.83E+11±6.3E+07	0.05
MLSP	7.83E+11	7.83E+11±3.3E+07	4.02
MLS	7.83E+11	7.86E+11±3.4E+09	0.74
LS++	8.33E+10	8.43E+10±6.7E+08	0.29
FLS	8.33E+10	8.33E+10±0	0.58
LLOYD	WHO(440*8)	8.33E+10(117s)	8.40E+10	8.40E+10±0	0.06
MLSP	8.33E+10	8.33E+10±0	3.74
MLS	8.33E+10	8.46E+10±8.1E+08	0.71
LS++	2.75E+06	2.75E+06±0	0.32
FLS	2.75E+06	2.85E+06±1.8E+05	0.61
LLOYD	HCV(572*12)	2.75E+06(215s)	2.75E+06	2.79E+06±4.7E+04	0.08
MLSP	2.75E+06	2.75E+06±0	4.04
MLS	2.75E+06	2.79E+06±6.8E+04	0.78
LS++	2.63E+06	2.70E+06±5.8E+04	0.33
FLS	2.62E+06	2.62E+06±0	0.69
LLOYD	Abs(740*21)	2.62E+06(119s)	2.62E+06	2.62E+06±0	0.07
MLSP	2.62E+06	2.62E+06±0	5.4
MLS	2.62E+06	2.66E+06±2.1E+04	0.81
Table 14: Comparison results on clustering costs and running time with  $k = 3$  on datasets with sizes ranging from 150 to 740
Method	Size	BB(Cost)	Best	Mean	Time(s)
LS++	1145.77	1159.73±10.48	0.33
FLS	1134.45	1134.45±0	0.56
LLOYD	TR(980*10)	1134.45(126s)	1136.93	1138.09±0.94	0.09
MLSP	1134.45	1134.45±0	4.75
MLS	1134.45	1154.49±14.61	0.78
LS++	1.28E+09	1.28E+09±1.79E+06	0.35
FLS	1.28E+09	1.28E+09±0	0.82
LLOYD	SGC(1000*21)	1.28E+09(140s)	1.28E+09	1.28E+09±0	0.13
MLSP	1.28E+09	1.28E+09±0	5.18
MLS	1.28E+09	1.28E+09±1.1E+06	0.81
LS++	9.91E+06	9.91E+06±2.5E+03	0.53
FLS	9.91E+06	9.91E+06±0	0.92
LLOYD	HEMI(1995*7)	9.91E+06(97s)	9.91E+06	9.91E+06±0	0.09
MLSP	9.91E+06	9.91E+06±0	5.44
MLS	9.91E+06	9.91E+06±2.3E+03	0.79
LS++	2.13E+09	2.15E+09±1.7E+08	0.48
FLS	2.13E+09	2.14E+09±1.4E+08	0.89
LLOYD	pr2392(2392*2)	2.13E+09(123s)	2.13E+09	2.13E+09±0	0.09
MLSP	2.13E+09	2.13E+09±0	5.92
MLS	2.13E+09	2.15E+09±1.5E+08	0.76
LS++	1.96E+05	1.98E+05±1.7E+03	0.49
FLS	1.96E+05	1.97E+05±6.8E+02	0.97
LLOYD	TRR(5456*24)	1.96E+05(325s)	1.96E+05	2.00E+05±3.3E+03	0.09
MLSP	1.96E+05	1.96E+05±3.8E+02	6.06
MLS	1.96E+05	1.98E+05±1.7E+03	0.78
LS++	2199.1	2234.68±41.91	0.92
FLS	2199.1	2201.90±8.42	2.02
LLOYD	AC(7195*22)	2199.10(222s)	2227.18	2227.18±0	0.16
MLSP	2199.1	2205.75±19.95	6.43
MLS	2199.1	2212.69±26.49	1.06
LS++	1.49E+07	1.49E+07±269.4	1.1
FLS	1.49E+07	1.49E+07±0	1.91
LLOYD	rds_cnt(10000*4)	1.49E+07(203s)	1.49E+07	1.49E+07±331.2	0.03
MLSP	1.49E+07	1.49E+07±0	4.96
MLS	1.49E+07	1.49E+07±226.1	0.91
LS++	8.21E+07	8.22E+07±1.6E+05	1.65
FLS	8.21E+07	8.21E+07±0	3.71
LLOYD	HTRU2(17898*8)	8.21E+07(1555s)	8.21E+07	8.21E+07±0	0.97
MLSP	8.21E+07	8.21E+07±0	8.44
MLS	8.21E+07	8.22E+07±2.0E+05	1.21
LS++	1.95E+07	1.97E+07±4.9E+05	4.38
FLS	1.95E+07	1.95E+07±6.0E+04	7.28
LLOYD	GT(36733*11)	1.95E+07(1936s)	1.95E+07	1.99E+07±5.4E+05	1.18
MLSP	1.95E+07	1.95E+07±7.3E+03	12.83
MLS	1.95E+07	1.97E+07±4.7E+05	3.82
Table 15: Comparison results on clustering costs and running time with  $k = 3$  on datasets with sizes ranging from 980 to 36733
Method	Size	BB(Cost)	Best	Mean	Time(s)
LS++	50.97	52.75±1.74	0.37
FLS	50.97	51.70±1.10	0.78
LLOYD	Iris(150*4)	50.92(355s)	51.19	51.19±0	0.1
MLSP	50.97	50.98±0.02	3.53
MLS	50.97	51.78±1.11	0.81
LS++	401.21	404.58±3.30	0.3
FLS	401.21	402.46±1.25	0.81
LLOYD	SEEDS(210*7)	401.21(376s)	401.21	406.69±2.23	0.11
MLSP	401.21	401.46±0.75	3.62
MLS	401.21	409.65±9.40	0.83
LS++	437.88	443.63±7.54	0.29
FLS	437.73	439.82±6.29	0.89
LLOYD	GLASS(214*9)	437.73(592s)	437.88	457.22±15.78	0.07
MLSP	437.73	437.73±0	3.43
MLS	437.73	439.37±2.49	0.86
LS++	6.0249E+05	6.0213E+05±1.8E+03	0.29
FLS	6.0249E+05	6.1293E+05±5.2E+03	0.83
LLOYD	BM(249*6)	6.0249E+05(389s)	6.0249E+05	6.1293E+05±4.9E+03	0.04
MLSP	6.0249E+05	6.0249E+05±0	3.71
MLS	6.0249E+05	6.2224E+05±2.0E+04	0.83
LS++	40.17	40.97±0.63	0.29
FLS	40.17	40.28±0.25	0.82
LLOYD	UK(258*5)	40.17(457s)	40.24	41.16±0.57	0.06
MLSP	40.17	40.25±0.25	3.72
MLS	40.34	41.55±0.85	0.83
LS++	3.0998E+11	3.1257E+11±2.5E+10	0.31
FLS	3.0998E+11	3.1083E+11±1.1E+10	0.89
LLOYD	HF(299*12)	3.0998E+11(1723s)	3.0998E+11	3.1001E+11±1.2E+08	0.06
MLSP	3.0998E+11	3.1021E+11±7.1E+08	4.65
MLS	3.0998E+11	3.1499E+11±5.6E+09	0.92
LS++	5.5982E+10	5.6642E+10±3.6E+08	0.33
FLS	5.5914E+10	5.6333E+10±2.1E+08	0.89
LLOYD	WHO(440*8)	5.5914E+10(1840s)	5.5914E+10	5.6219E+10±1.25E+08	0.09
MLSP	5.5914E+10	5.6034E+10±1.25E+08	4.84
MLS	5.5973E+10	5.6460E+10±3.5E+08	0.88
LS++	1.9716E+06	1.9722E+06±622.78	0.34
FLS	1.9716E+06	2.0044E+06±2.6E+04	0.91
LLOYD	HCV(572*12)	1.9716E+06(12768s)	1.9716E+06	1.9995E+06±3.3E+04	0.13
MLSP	1.9716E+06	1.9716E+06±0	5.02
MLS	1.9716E+06	2.0113E+06±3.1E+04	0.93
LS++	1.7472E+06	1.7630E+06±3.8E+04	0.36
FLS	1.7472E+06	1.7472E+06±0	0.98
LLOYD	Abs(740*21)	1.7472E+06(410s)	1.7472E+06	1.7502E+06±1.5E+03	0.09
MLSP	1.7472E+06	1.7472E+06±0	6.03
MLS	1.7472E+06	1.7676E+06±3.5E+04	1.16
Table 16: Comparison results on clustering costs and running time with  $k = 5$  on datasets with sizes
ranging from 150 to 740
Method	Size	BB(Cost)	Best	Mean	Time(s)
LS++	966.64	990.43±14.69	0.35
FLS	953.39	956.16±3.43	1.13
LLOYD MLSP	TR(980*10)	953.39(824s)	965.74	969.10+5.65	0.14
953.39	954.62±2.48	6.28
MLS	983.22	1000.86±10.34	0.93
LS++	4.6926E+08	4.7268E+08±2.8E+06	0.39
FLS	4.6919E+08	4.7197E+08±2.1E+06	1.06
LLOYD MLSP	SGC(1000*21)	4.6919E+08(3290s)	4.6919E+08	4.6949E+08±2.3E+05	0.09
4.6919E+08	4.6973E+08±1.0E+06	7.7
MLS	4.7204E+08	4.7529E+08±2.7E+06	1.22
LS++	5.3811E+06	5.4497E+06±5.6E+04	0.45
FLS	5.3811E+06	5.4090E+06±3.4E+04	1.24
LLOYD MLSP	HEMI(1995*7)	5.3811E+06(930s)	5.3811E+06	5.4262E+06±1.5E+04	0.13
5.3811E+06	5.3880E+06±2.1E+04	7.04
MLS	5.3811E+06	5.4781E+06±1.4E+04	0.86
LS++	1.1621E+09	1.1700E+09±6.3E+07	0.44
FLS	1.1619E+09	1.1636E+09±2.3E+07	1.31
LLOYD MLSP	pr2392(2392*2)	1.1619E+09(625s)	1.1620E+09	1.1620E+09±2.9E+05	0.14
1.1619E+09	1.1619E+09±0	7.46
MLS	1.1627E+09	1.1759E+09±1.4E+07	0.81
LS++	1.6920E+05	1.7055E+05±626.4	0.54
FLS	1.6894E+05	1.7012E+05±529.7	1.92
LLOYD MLSP	TRR(5456*24)	1.6870E+05(3094s)	1.7086E+05	1.7250E+05±3.0E+03	0.15
1.6870E+05	1.6903E+05±473.9	9.57
MLS	1.6914E+05	1.7027E+05±548.1	0.89
LS++	1636.1	1636.93±0.99	0.67
FLS	1636.1	1636.14±0	2.6
LLOYD MLSP	AC(7195*22)	1636.1(3552s)	1542.6	1642.56±0	0.24
1636.1	1636.14±0	7.68
MLS	1636.1	1637.25±1.26	1.57
LS++	5.3728E+06	5.3747E+06±2.2E+03	1.29
FLS	5.3725E+06	5.3725E+06±103	3.11
LLOYD MLSP	rds_cnt(10000*4)	5.3725E+06(7171s)	5.3744E+06	5.3751E+06±987	0.05
5.3725E+06	5.3728E+06±554	7.53
MLS	5.3728E+06	5.3742E+06±2.1E+03	1.58
LS++	4.2176E+07	4.2260E+07±4.6E+04	2.31
FLS	4.2152E+07	4.2160E+07±9.3E+03	5.03
LLOYD MLSP	HTRU2(17898*8)	4.2154E+07(4h)	4.2171E+07	4.2173E+07±2.9E+03	0.16
4.2152E+07	4.2153E+07±2.9E+03	11.01
MLS	4.2268E+07	4.2413E+07±1.4E+04	1.56
LS++	1.3351E+07	1.3643E+07±3.8E+05	5.34
FLS	1.3351E+07	1.3438E+07±2.6E+05	10.67
LLOYD MLSP	GT(36733*11)	1.3351E+07(4h)	1.3351E+07	1.3530E+07±3.4E+05	0.19
1.3351E+07	1.3351E+07±0	16.61
MLS	1.3358E+07	1.3673E+07±3.9E+05	4.93
Table 17: Comparison results on clustering costs and running time with  $k = 5$  on datasets with sizes ranging from 980 to 36733