Accelerating Density Peak Clustering Algorithm
Abstract
:1. Introduction
2. Related Works
2.1. Clustering Methods
2.2. Density Peak Clustering Algorithm
Algorithm 1. DPC algorithm. 
Input: the set of data points $\mathrm{X}\in {\mathbb{R}}_{N\times M}$ and the parameters ${d}_{c}$ for defining the neighborhood, and ${d}_{r}$ for selecting density peaks 
Output: the label vector of cluster index $\mathrm{y}\in {\mathbb{R}}_{N\times 1}$ 
Algorithm:

3. Accelerating APC by Scanning Neighbors Only
Algorithm 2. ADPC1 algorithm. 
Input: the set of data points $\mathrm{X}\in {\mathbb{R}}_{N\times M}$ and the parameters ${d}_{c}$ for defining the neighborhood, and ${d}_{r}$ for selecting density peaks. 
Output: the label vector of cluster index $\mathrm{y}\in {\mathbb{R}}_{N\times 1}$ 
Algorithm:

4. Accelerating APC by Skipping NonPeaks
Algorithm 3. ADPC2 algorithm. 
Input: the set of data points $\mathrm{X}\in {\mathbb{R}}_{N\times M}$ and the parameters ${d}_{c}$ for defining the neighborhood, and ${d}_{r}$ for selecting density peaks 
Output: the label vector of cluster index $\mathrm{y}\in {\mathbb{R}}_{N\times 1}$ 
Algorithm:

5. Performance Study
5.1. Test Datasets
5.2. Experiment Setup
5.3. Experiment Results
5.3.1. Test 1: Use a Fixed Threshold for Local Density
5.3.2. Test 2: Use an Exponential Kernel for Local Density
6. Conclusions
Funding
Acknowledgments
Conflicts of Interest
Appendix A. Implementation Details for Calculating Separation Distance in DPC
Appendix B. Implementation Details for Calculating Separation Distance in ADPC1
Appendix C. Implementation Details for Calculating Separation Distance in ADPC2
Appendix D. Datasets
Dataset  Number of Clusters  Number of Points 

Spiral  3  312 
Flame  2  240 
Aggregation  7  788 
R15  15  600 
D31  31  3100 
A1  20  3000 
A2  35  5250 
A3  50  7500 
S1  15  5000 
S2  15  5000 
S3  15  5000 
S4  15  5000 
Appendix E. More Experimental Results
Dataset (N)  p = 0.5  p = 1  p = 1.5  p = 2  p = 2.5  p = 3  p = 3.5  p = 4 

Spiral (N = 312)  62  90  110  139  161  187  208  232 
Flame (N = 240)  75  143  177  208  213  221  226  231 
Aggregation (N = 788)  609  704  744  760  763  761  761  768 
R15 (N = 600)  348  490  537  558  563  570  564  567 
D31 (N = 3100)  2952  3028  3034  3037  3013  2726  2034  2482 
A1 (N = 3000)  2854  2955  2967  2968  2971  2973  2964  2962 
A2 (N = 5250)  5140  5201  5192  5185  5202  5214  5236  5237 
A3 (N = 7500)  7398  7418  7413  7456  7477  7483  7486  7490 
S1 (N = 5000)  4305  4806  4912  4936  4945  4956  4974  4973 
S2 (N = 5000)  4351  4834  4934  4953  4950  4954  4968  4961 
S3 (N = 5000)  4474  4882  4942  4959  4977  4970  4977  4982 
S4 (N = 5000)  4355  4818  4930  4955  4964  4968  4972  4976 
Dataset (N)  p = 0.5  p = 1  p = 1.5  p = 2  p = 2.5  p = 3  p = 3.5  p = 4 

Spiral (N = 312)  148  251  286  301  308  309  309  309 
Flame (N = 240)  118  188  219  231  232  234  234  235 
Aggregation (N = 788)  725  769  775  778  779  780  781  781 
R15 (N = 600)  439  527  556  572  576  581  584  584 
D31 (N = 3100)  2983  3044  3061  3067  3069  3070  3075  3085 
A1 (N = 3000)  2914  2967  2977  2978  2980  2980  2981  2983 
A2 (N = 5250)  5175  5212  5214  5216  5227  5236  5239  5243 
A3 (N = 7500)  7424  7447  7453  7475  7483  7491  7493  7493 
S1 (N = 5000)  4592  4880  4944  4963  4978  4983  4983  4984 
S2 (N = 5000)  4605  4910  4968  4980  4982  4984  4985  4985 
S3 (N = 5000)  4727  4932  4966  4979  4983  4985  4987  4988 
S4 (N = 5000)  4652  4910  4953  4971  4974  4980  4985  4987 
Dataset  Algorithm  p = 0.5  p = 1  p = 1.5  p = 2  p = 2.5  p = 3  p = 3.5  p = 4 

Spiral  DPC  0.038101  0.046879  0.079211  0.052139  0.04688  0.046879  0.046879  0.046879 
ADPC1  0.035094  0.041108  0.036096  0.032086  0.046879  0.031252  0.031252  0.046879  
ADPC2  0.04512  0.033088  0.04512  0.046878  0.031251  0.046845  0.04686  0.015646  
Flame  DPC  0.031253  0.031255  0.031273  0.031252  0.031253  0.031252  0.031252  0.031253 
ADPC1  0.015626  0.031253  0.015626  0  0.015599  0.015628  0  0.015628  
ADPC2  0.031255  0.015625  0.015605  0.015627  0.015627  0  0.0156  0.015606  
Aggregation  DPC  0.252673  0.265685  0.281293  0.296938  0.296938  0.296938  0.312564  0.281312 
ADPC1  0.081215  0.078132  0.062506  0.062507  0.062506  0.078133  0.062506  0.078133  
ADPC2  0.0781  0.062507  0.04688  0.04688  0.046901  0.062507  0.078133  0.062505  
R15  DPC  0.156265  0.171892  0.171892  0.171892  0.171892  0.187519  0.171891  0.171891 
ADPC1  0.125012  0.078131  0.078131  0.04688  0.046879  0.04688  0.046873  0.046878  
ADPC2  0.093759  0.07813  0.046879  0.046879  0.046878  0.031253  0.031253  0.031253  
D31  DPC  4.281704  4.344213  4.375495  4.375465  4.375464  4.234823  3.187837  3.625386 
ADPC1  0.883543  0.7657  0.742746  0.781333  0.812588  0.937631  1.687681  1.672092  
ADPC2  0.875124  0.71418  0.687573  0.7032  0.734489  0.828213  1.640798  1.578292  
A1  DPC  4.016051  4.078557  4.125437  4.141065  4.145534  4.160485  4.234813  4.187945 
ADPC1  0.81259  0.687572  0.687606  0.734455  0.781332  0.79696  0.828212  0.89072  
ADPC2  0.796961  0.656319  0.640726  0.67195  0.687573  0.687573  0.718826  0.762816  
A2  DPC  12.48567  12.4857  12.59509  12.59508  12.59509  12.68885  12.76698  12.7982 
ADPC1  2.078376  1.984586  2.093839  2.203357  2.328372  2.484632  2.564224  2.672159  
ADPC2  2.000502  1.922079  2.250205  2.000213  2.093969  2.187732  2.234609  2.297118  
A3  DPC  25.42454  25.53396  25.69023  25.72854  26.01839  26.11215  26.40587  26.33089 
ADPC1  3.926412  4.189912  4.47909  4.547357  4.769292  5.194029  5.486006  5.506579  
ADPC2  3.83774  3.802774  3.953544  4.1411  4.281703  4.420601  4.561412  4.742328  
S1  DPC  11.42309  11.81372  11.65747  11.71463  11.75125  11.79816  11.79813  11.82938 
ADPC1  3.922292  2.422134  2.140852  2.125225  2.140851  2.234612  2.344002  2.437762  
ADPC2  3.8584  2.328372  1.984588  1.93774  1.922111  1.953329  2.035082  2.125227  
S2  DPC  11.21994  11.45434  11.52083  11.5481  11.68874  11.64186  11.65748  11.67311 
ADPC1  3.687891  2.312742  2.031496  2.062718  2.15648  2.218986  2.344  2.469012  
ADPC2  3.641011  2.203357  1.906452  1.890826  1.922079  1.984584  2.047092  2.105681  
S3  DPC  11.29808  11.48559  11.56373  11.56373  11.62623  11.59498  11.65748  11.67311 
ADPC1  3.21909  2.14085  2.031468  2.031467  2.109567  2.234612  2.343999  2.469044  
ADPC2  3.187838  2.015836  1.890857  1.86247  1.906452  1.97208  2.031466  2.109599  
S4  DPC  11.25118  11.51685  11.57936  11.62623  11.56371  11.64183  11.71999  11.70437 
ADPC1  3.515999  2.344002  2.078346  2.093971  2.203359  2.265865  2.359625  2.437758  
ADPC2  3.484713  2.250242  1.937706  1.906451  1.968992  1.984586  2.062751  2.099317 
Dataset  Algorithm  p = 0.5  p = 1  p = 1.5  p = 2  p = 2.5  p = 3  p = 3.5  p = 4 

Spiral  DPC  0.217607  0.227606  0.24064  0.230642  0.218788  0.234399  0.218773  0.218806 
ADPC1  0.243648  0.210588  0.199559  0.177449  0.17189  0.218742  0.171892  0.187521  
ADPC2  0.201514  0.179477  0.177471  0.215608  0.171891  0.18752  0.171869  0.171893  
Flame  DPC  0.12501  0.125012  0.125014  0.14064  0.125013  0.14064  0.125014  0.14064 
ADPC1  0.109386  0.109386  0.109386  0.125014  0.093759  0.109406  0.109387  0.109387  
ADPC2  0.12501  0.109385  0.125013  0.109386  0.10938  0.109386  0.109384  0.109385  
Aggregation  DPC  1.299488  1.250166  1.250131  1.281362  1.281407  1.250169  1.250164  1.250162 
ADPC1  1.078868  1.062644  1.062612  1.062614  1.062644  1.062645  1.078269  1.078274  
ADPC2  1.11083  1.062613  1.046983  1.047022  1.047018  1.047019  1.047018  1.047018  
R15  DPC  0.779632  0.750111  0.750057  0.750113  0.750075  0.765738  0.734483  0.73445 
ADPC1  0.703232  0.671957  0.656351  0.640727  0.656351  0.640728  0.625097  0.625066  
ADPC2  0.703232  0.671978  0.656351  0.625098  0.640724  0.640691  0.625097  0.60947  
D31  DPC  20.34588  19.8146  19.90442  19.53329  19.65834  19.5177  19.54898  19.64271 
ADPC1  17.12682  16.74525  16.61117  16.39237  16.56423  16.39233  16.4581  16.59551  
ADPC2  17.11119  16.64242  16.36111  16.25173  16.28676  16.35301  16.53297  16.37674  
A1  DPC  18.58013  18.08004  17.95503  17.98625  17.8769  17.8988  17.95503  17.8769 
ADPC1  15.57978  15.26725  15.18911  15.15783  15.12657  15.15679  15.16565  15.19659  
ADPC2  15.42351  15.15789  15.04847  15.01722  14.90703  15.00159  14.92343  14.87658  
A2  DPC  56.31846  55.53715  55.05268  55.67778  55.25958  55.44339  55.5684  55.64653 
ADPC1  47.64571  46.62474  46.12989  46.31741  47.2863  46.45805  46.27054  46.87997  
ADPC2  47.3644  46.12753  45.97363  46.37992  46.12989  45.84858  45.91012  46.16624  
A3  DPC  115.6921  112.1025  112.9651  115.0125  112.3618  112.9651  112.1213  112.8714 
ADPC1  95.73069  93.50614  95.91672  95.73115  95.51016  94.68273  95.79348  96.21016  
ADPC2  96.21336  94.86226  94.79854  93.96594  94.73864  93.69754  94.52285  94.63775  
S1  DPC  53.11501  51.63048  50.92332  50.58346  50.44285  50.78661  50.19283  51.5836 
ADPC1  45.36419  43.80152  42.70859  42.8327  42.33265  42.0513  42.2279  42.50451  
ADPC2  45.77048  43.614  42.9889  41.89504  42.33735  41.86379  41.8169  41.92629  
S2  DPC  51.88047  50.97416  50.89606  50.00531  50.36472  50.08341  50.73976  50.56783 
ADPC1  44.75324  42.71937  42.00446  42.00442  42.22323  44.39533  42.66078  42.61389  
ADPC2  44.58285  42.48891  41.85598  41.86382  41.86418  41.7544  41.9107  41.75443  
S3  DPC  51.89613  50.14595  50.17026  49.89592  50.03655  50.63037  50.13029  50.03656 
ADPC1  43.45777  41.86382  41.75443  41.78568  41.64504  41.86382  42.22323  41.88364  
ADPC2  44.23907  41.84819  42.20761  41.77006  42.27011  41.65029  41.87944  41.89504  
S4  DPC  51.45296  50.1772  50.42723  50.5679  50.36469  50.05215  50.00527  50.23967 
ADPC1  44.00467  42.13493  41.84819  42.52013  42.05134  42.00442  42.274  42.28573  
ADPC2  44.47347  41.88291  42.08256  42.03568  42.17635  41.66067  41.86382  41.84976 
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Dataset  p = 0.5  p = 1  p = 1.5  p = 2  p = 2.5  p = 3  p = 3.5  p = 4 

Spiral  19.87%  28.85%  35.26%  44.55%  51.60%  59.94%  66.67%  74.36% 
Flame  31.25%  59.58%  73.75%  86.67%  88.75%  92.08%  94.17%  96.25% 
Aggregation  77.28%  89.34%  94.42%  96.45%  96.83%  96.57%  96.57%  97.46% 
R15  58.00%  81.67%  89.50%  93.00%  93.83%  95.00%  94.00%  94.50% 
D31  95.23%  97.68%  97.87%  97.97%  97.19%  87.94%  65.61%  80.06% 
A1  95.13%  98.50%  98.90%  98.93%  99.03%  99.10%  98.80%  98.73% 
A2  97.90%  99.07%  98.90%  98.76%  99.09%  99.31%  99.73%  99.75% 
A3  98.64%  98.91%  98.84%  99.41%  99.69%  99.77%  99.81%  99.87% 
S1  86.10%  96.12%  98.24%  98.72%  98.90%  99.12%  99.48%  99.46% 
S2  87.02%  96.68%  98.68%  99.06%  99.00%  99.08%  99.36%  99.22% 
S3  89.48%  97.64%  98.84%  99.18%  99.54%  99.40%  99.54%  99.64% 
S4  87.10%  96.36%  98.60%  99.10%  99.28%  99.36%  99.44%  99.52% 
Dataset  Algorithm  p = 0.5  p = 1  p = 1.5  p = 2  p = 2.5  p = 3  p = 3.5  p = 4 

Spiral  ADPC1  7.89%  12.31%  54.43%  38.46%  0.00%  33.33%  33.33%  0.00% 
ADPC2  –18.42%  29.42%  43.04%  10.09%  33.34%  0.07%  0.04%  66.62%  
Flame  ADPC1  50.00%  0.01%  50.03%  100.00%  50.09%  49.99%  100.00%  50.00% 
ADPC2  −0.01%  50.01%  50.10%  50.00%  50.00%  100.00%  50.08%  50.07%  
Aggregation  ADPC1  67.86%  70.59%  77.78%  78.95%  78.95%  73.69%  80.00%  72.23% 
ADPC2  69.09%  76.47%  83.33%  84.21%  84.21%  78.95%  75.00%  77.78%  
R15  ADPC1  20.00%  54.55%  54.55%  72.73%  72.73%  75.00%  72.73%  72.73% 
ADPC2  40.00%  54.55%  72.73%  72.73%  72.73%  83.33%  81.82%  81.82%  
D31  ADPC1  79.36%  82.37%  83.02%  82.14%  81.43%  77.86%  47.06%  53.88% 
ADPC2  79.56%  83.56%  84.29%  83.93%  83.21%  80.44%  48.53%  56.47%  
A1  ADPC1  79.77%  83.14%  83.33%  82.26%  81.15%  80.84%  80.44%  78.73% 
ADPC2  80.16%  83.91%  84.47%  83.77%  83.41%  83.47%  83.03%  81.79%  
A2  ADPC1  83.35%  84.11%  83.38%  82.51%  81.51%  80.42%  79.92%  79.12% 
ADPC2  83.98%  84.61%  82.13%  84.12%  83.37%  82.76%  82.50%  82.05%  
A3  ADPC1  84.56%  83.59%  82.57%  82.33%  81.67%  80.11%  79.22%  79.09% 
ADPC2  84.91%  85.11%  84.61%  83.90%  83.54%  83.07%  82.73%  81.99%  
S1  ADPC1  65.66%  79.50%  81.64%  81.86%  81.78%  81.06%  80.13%  79.39% 
ADPC2  66.22%  80.29%  82.98%  83.46%  83.64%  83.44%  82.75%  82.03%  
S2  ADPC1  67.13%  79.81%  82.37%  82.14%  81.55%  80.94%  79.89%  78.85% 
ADPC2  67.55%  80.76%  83.45%  83.63%  83.56%  82.95%  82.44%  81.96%  
S3  ADPC1  71.51%  81.36%  82.43%  82.43%  81.86%  80.73%  79.89%  78.85% 
ADPC2  71.78%  82.45%  83.65%  83.89%  83.60%  82.99%  82.57%  81.93%  
S4  ADPC1  68.75%  79.65%  82.05%  81.99%  80.95%  80.54%  79.87%  79.17% 
ADPC2  69.03%  80.46%  83.27%  83.60%  82.97%  82.95%  82.40%  82.06% 
Dataset  p = 0.5  p = 1  p = 1.5  p = 2  p = 2.5  p = 3  p = 3.5  p = 4 

Spiral  47.44%  80.45%  91.67%  96.47%  98.72%  99.04%  99.04%  99.04% 
Flame  49.17%  78.33%  91.25%  96.25%  96.67%  97.50%  97.50%  97.92% 
Aggregation  92.01%  97.59%  98.35%  98.73%  98.86%  98.98%  99.11%  99.11% 
R15  73.17%  87.83%  92.67%  95.33%  96.00%  96.83%  97.33%  97.33% 
D31  96.23%  98.19%  98.74%  98.94%  99.00%  99.03%  99.19%  99.52% 
A1  97.13%  98.90%  99.23%  99.27%  99.33%  99.33%  99.37%  99.43% 
A2  98.57%  99.28%  99.31%  99.35%  99.56%  99.73%  99.79%  99.87% 
A3  98.99%  99.29%  99.37%  99.67%  99.77%  99.88%  99.91%  99.91% 
S1  91.84%  97.60%  98.88%  99.26%  99.56%  99.66%  99.66%  99.68% 
S2  92.10%  98.20%  99.36%  99.60%  99.64%  99.68%  99.70%  99.70% 
S3  94.54%  98.64%  99.32%  99.58%  99.66%  99.70%  99.74%  99.76% 
S4  93.04%  98.20%  99.06%  99.42%  99.48%  99.60%  99.70%  99.74% 
Dataset  Algorithm  p = 0.5  p = 1  p = 1.5  p = 2  p = 2.5  p = 3  p = 3.5  p = 4 

Spiral  ADPC1  –11.97%  7.48%  17.07%  23.06%  21.44%  6.68%  21.43%  14.30% 
ADPC2  7.40%  21.15%  26.25%  6.52%  21.43%  20.00%  21.44%  21.44%  
Flame  ADPC1  12.50%  12.50%  12.50%  11.11%  25.00%  22.21%  12.50%  22.22% 
ADPC2  0.00%  12.50%  0.001%  22.22%  12.51%  22.22%  12.50%  22.22%  
Aggregation  ADPC1  16.98%  15.00%  15.00%  17.07%  17.07%  15.00%  13.75%  13.75% 
ADPC2  14.52%  15.00%  16.25%  18.29%  18.29%  16.25%  16.25%  16.25%  
R15  ADPC1  9.80%  10.42%  12.49%  14.58%  12.50%  16.33%  14.89%  14.89% 
ADPC2  9.80%  10.42%  12.49%  16.67%  14.58%  16.33%  14.89%  17.02%  
D31  ADPC1  15.82%  15.49%  16.55%  16.08%  15.74%  16.01%  15.81%  15.51% 
ADPC2  15.90%  16.01%  17.80%  16.80%  17.15%  16.21%  15.43%  16.63%  
A1  ADPC1  16.15%  15.56%  15.40%  15.73%  15.38%  15.32%  15.54%  14.99% 
ADPC2  16.99%  16.16%  16.19%  16.51%  16.61%  16.19%  16.88%  16.78%  
A2  ADPC1  15.40%  16.05%  16.21%  16.81%  14.43%  16.21%  16.73%  15.75% 
ADPC2  15.90%  16.94%  16.49%  16.70%  16.52%  17.31%  17.38%  17.04%  
A3  ADPC1  17.25%  16.59%  15.09%  16.76%  15.00%  16.18%  14.56%  14.76% 
ADPC2  16.84%  15.38%  16.08%  18.30%  15.68%  17.06%  15.70%  16.15%  
S1  ADPC1  14.59%  15.16%  16.13%  15.32%  16.08%  17.20%  15.87%  17.60% 
ADPC2  13.83%  15.53%  15.58%  17.18%  16.07%  17.57%  16.69%  18.72%  
S2  ADPC1  13.74%  16.19%  17.47%  16.00%  16.17%  11.36%  15.92%  15.73% 
ADPC2  14.07%  16.65%  17.76%  16.28%  16.88%  16.63%  17.40%  17.43%  
S3  ADPC1  16.26%  16.52%  16.77%  16.25%  16.77%  17.31%  15.77%  16.29% 
ADPC2  14.75%  16.55%  15.87%  16.29%  15.52%  17.74%  16.46%  16.27%  
S4  ADPC1  14.48%  16.03%  17.01%  15.91%  16.51%  16.08%  15.46%  15.83% 
ADPC2  13.56%  16.53%  16.55%  16.87%  16.26%  16.77%  16.28%  16.70% 
© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
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Lin, J.L. Accelerating Density Peak Clustering Algorithm. Symmetry 2019, 11, 859. https://doi.org/10.3390/sym11070859
Lin JL. Accelerating Density Peak Clustering Algorithm. Symmetry. 2019; 11(7):859. https://doi.org/10.3390/sym11070859
Chicago/Turabian StyleLin, JunLin. 2019. "Accelerating Density Peak Clustering Algorithm" Symmetry 11, no. 7: 859. https://doi.org/10.3390/sym11070859