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(IJCSIS) International Journal of Computer Science and Information Security, Vol. 8, No. 5, 2010 Density Based Clustering Algorithm using Sparse Memory Mapped File J. Hencil Peter A. Antonysamy Department of Computer Science Department of Mathematics St. Xavier’s College, Palayamkottai , India. St. Xavier’s College, Kathmandu, Nepal. hencilpeter@hotmail.com fr_antonysamy@hotmail.com Abstract: and section 4 explains the proposed solution. After the new algorithm’s explanation, section 5 shows the Experimental The DBSCAN [1] algorithm is a popular algorithm in Data Results and final section 6 presents the conclusion and future Mining field as it has the ability to mine the noiseless arbitrary work associated with this algorithm. shape Clusters in an elegant way. As the original DBSCAN algorithm uses the distance measures to compute the distance between objects, it consumes so much processing time and it’s computation complexity comes as O(N2). In this paper we have II RELATED WORK proposed a new algorithm for mining the density based clusters using Sparse Memory Mapped File (Spares MMF) [3]. All the The DBSCAN (Density Based Spatial Clustering of given objects are initially loaded into their corresponding Sparse Memory Mapped File’s locations and during the Application with Noise) [1] is the basic clustering algorithm SparseMemoryRegionQuery operation each objects’ to mine the clusters based on objects density. In this surrounding cells will be visited for the neighbour objects algorithm, first the number of objects present within the instead of computing the distance between each of the objects neighbour region (Eps) is computed. If the neighbour objects in the data set. Using the Sparse MMF approach, it is proved count is below the given threshold value, the object will be that the DBSCAN algorithm can process huge amount of marked as NOISE. Otherwise the new cluster will be formed objects without having any runtime issues and the new from the core object by finding the group of density algorithm’s performance analysis shows that proposed solution connected objects that are maximal w.r.t density-reachability. is super fast than the existing algorithm. The OPTICS [4] algorithm adopts the original Keywords: Sparse Memory Mapped File; Sparse MMF; DBSCAN algorithm to deal with variance density clusters. Sparse Memory; Neighbour Cells; Sparse Memory DBSCAN. This algorithm computes an ordering of the objects based on the reachability distance for representing the intrinsic hierarchical clustering structure. The Valleys in the plot I. INTRODUCTION indicate the clusters. But the input parameters ξ is critical for identifying the valleys as ξ clusters. Data mining is a fast growing field in which clustering plays a very important role. Clustering is the The DENCLUE [5] algorithm uses kernel density process of grouping a set of physical or abstract objects into estimation. The result of density function gives the local classes of similar objects [2]. Among the many algorithms density maxima value and this local density value is used to proposed in the clustering field, DBSCAN is one of the most form the clusters. If the local density value is very small, the popular algorithms due to its high quality of noiseless output objects of clusters will be discarded as NOISE. clusters. A Fast DBSCAN (FDBSCAN) Algorithm[6] has The most of the Density Based Clustering been invented to improve the speed of the original DBSCAN algorithms requires O (N2) computation time and requires algorithm and the performance improvement has been huge amount of main memory to process in the real time achieved through considering only few selected scenario. Since the seed object list grows during run time, it representative objects belongs inside a core object’s is very difficult to predict the required memory to process the neighbour region as seed objects for the further expansion. entire objects present in the data set. If the memory is Hence this algorithm is faster than the basic version of insufficient to process the growing seed objects, the DBSCAN algorithm and suffers with the loss of result DBSCAN algorithm will crash in the run time. So to get rid accuracy. of the instability problem and improve the performance, a new solution has been proposed in this paper. The MEDBSCAN [7] algorithm has been proposed recently to improve the performance of DBSCAN algorithm, Rest of the paper is organised as follows. Section 2 at the same time without loosing the result accuracy. In this gives the brief history about the related works in the same algorithm totally three queues have been used, the first queue area. Section 3 gives the introduction of original DBSCAN will store the neighbours of the core object which belong 122 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 8, No. 5, 2010 inside Eps distance, the second queue is used to store the An Object p is density connected to another object q if there neighbours of the core object which belong inside 2*Eps is an object o such that both, p and q are density reachable distance and the third queue is the seeds queue which store from o w.r.t Eps and MinObjs. the unhandled objects for further expansion. This algorithm guarantees some notable performance improvement if Eps Definition 6: Cluster value is not very sensitive. A Cluster C is a non-empty subset of a Database D w.r.t Eps and MinObjs which satisfying the following conditions. Though the DBSCAN algorithm’s complexity can be reduced to O(N * logN) using some spatial trees, it is an For every p and q, if p ∈ cluster C and q is density reachable extra effort to construct, organize the tree and the tree from p w.r.t Eps and MinObjs then q ∈ C. requires an additional memory to hold the objects. In this For every p and q, q ∈ C; p is density connected to q w.r.t new algorithm different new complexity O (N * 2Eps) has Eps and MinObjs. been achieved and it is proved that the new complexity better than the previous version of DBSCAN algorithms when the Eps value is minimal. Definition 7: Noise II. INTRODUCTION TO DBSCAN ALGORITHM An object which doesn’t belong to any cluster is called noise. The working principles of the DBSCAN algorithm The DBSCAN algorithm finds the Eps are based on the following definitions: Neighbourhood of each object in a Database during the clustering process. Before the cluster expansion, if the Definition 1: Eps Neighbourhood of an object p algorithm finds any non core object, it will be marked as NOISE. With a core object, algorithm initiate a cluster and The Eps Neighbourhood of an object p is referred as surrounding objects will be added into the queue for the NEps(p), defined as further expansion. Each queue objects will be popped out NEps(p) = {q ∈ D | dist(p,q) <=Eps}. and find the Eps neighbour objects for the popped out object. When the new object is a core object, all its neighbour Definition 2: Core Object Condition objects will be assigned with the current cluster id and its unprocessed neighbour objects will be pushed into queue for An Object p is referred as core object, if the neighbour further processing. This process will be repeated until there objects count >= given threshold value (MinObjs). i.e. is no object in the queue for the further processing. |NEps(p)|>=MinObjs IV. PROPOSED SOLUTION Where MinObjs refers the minimum number of neighbour A new algorithm has been proposed in this paper to objects to satisfy the core object condition. In the above improve the performance as well as to process huge amount case, if p has neighbours which are exist within the Eps of data. This algorithm is totally relying on Sparse MMF and radius count is >= MinObjs, p can be referred as core object. the Sparse MMF concept has been explained below briefly: Definition 3: Directly Density Reachable Object A. Sparse Memory Mapped File (Sparse MMF) An Object p is referred as directly density reachable from The Sparse MMF [3] is the derived mechanism of another object q w.r.t Eps and MinObjs if Memory Mapped File. The Memory Mapped File [3] is like p ∈ NEps(q) and virtual memory and it allows reserving a region of address space and committing physical storage to the region. The difference is that the physical storage comes from a file that |NEps(q)|>= MinObjs (Core Object condition) is already on the disk instead of the system’s paging file. The memory mapped file can be used to access the data file on Definition 4: Density Reachable Object disk (even very huge files), load and execute executable files and libraries and allowing multiple processes running on the An object p is referred as density reachable from another same machine to share data with each other. The Sparse object q w.r.t Eps and MinObjs if there is a chain of objects MMF is similar to Memory Mapped File but it occupies only p1,…,pn, p1=q, pn=p such that pi+1 is directly density the required storage space in the physical file. If we use reachable from pi. Memory Mapped File to reserve the region of memory, while committing the changes to the file on disk, the file size will Definition 5: Density connected object be equivalent of the created Memory Mapped File size. Instead if we replace the same with Sparse MMF, final file’s size will be equivalent to the e non-zero element which is 123 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 8, No. 5, 2010 stored in the Sparse MMF. So Sparse MMF gives better Address(CenterObject)) and next time when the new object storage result and hence it has been used in our research. found, current object’s offset will be stored in the previous object’s NextSeedObject field and so on. Eventually last B. Object’s Structure object’s NextSeedObject field will be assigned with NULL. Thus the extra memory as well as buffer/queue requirement As this algorithm’s core is Spare MMF, the objects to store the seed objects has been removed in this solution. that needs to be processed by this algorithm are organized bit This function has been customized to update the neighbour differently and each objects’ structure will have three objects offset in the either field NextSeedObectOffset or additional fields NextObjectOffset, NextSeedObjectOffset NextTempObjectOffset. If this function receives an update and NextTempObjectOffset. flag UpdateMasterSeedOffset, neighbour objects offset will be stored in NextSeedObectOffset field and input update flag is UpdateTempSeedOffset then the NextTempObjectOffset will be updated with the neighbour object(s) offset. The DBSCAN algorithm’s computation complexity varies based on the RegionQuery function and it uses distance function to compute the neighbours present with in the certain radius (Eps). In this new approach, distance computation during the SparseMemoryRegionQuery function call has been removed and it visit’s the required number of neighbour cells from the center cell. Figure 1. Sparse Memory Mapped File Object’s Structure While loading all the objects in Sparse MMF, all the objects are chained in a sequence like linked list (but not exactly linked list). The first additional field NextObjectOffset will hold the Offset value of the next object, second object will hold the offset of its immediate successor object, etc and the final object’s NextObjectOffset Figure 2. Neighbour Cells Diagram will set to NULL to indicate that there are no more objects further to visit during the clustering process. So the first object’s address should be retained always to visit the entire In this proposed solution, we have selected two objects loaded in the Sparse MMF. The other two fields dimensional dataset for the experiment and the above NextSeedObjectOffset and NextTempObjectOffset fields are diagram shows the neighbour cells with different distance. used by SparseMemoryRegionQuery function call and it is The center cell has been painted in red colour and it’s explained in the below section. distance of object stored in the cell will be zero, next immediate neighbours whose distance is 1 from the center C. SparseMemoryRegionQuery function cell have been painted in blue colour, the yellow colour cells distance are greater than 1 and <=2 and so on. These neighbour cells offsets are pre-computed and stored in M X 2 The proposed algorithm doesn’t uses any extra buffer or dimensional array and it will be passed to the queue to store the seed objects as well as neighbour objects SparseMemoryRegionQuery function to visit only the during the run time, instead each object has the required number of neighbour cells to process. Thus the corresponding Offset field and in which the exact offset of distance computation between objects is not required. the next seed object will be stored. In the original DBSCAN algorithm, RegionQuery function has been used to retrieve the neighbour objects and in this new algorithm SpareMemoryRegionQuery function has been introduced instead of RegionQuery. This function visits all the required surrounding cells in memory and the non empty cell objects will be chained and return back as seed objects. i. e The function start from the center cell and visit the neighbour cells one by one. When the non empty object found in the first time, center object’s NextSeedOffset field will be assigned the Offset of new object (Address(NewObject) – 124 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 8, No. 5, 2010 based on the maximum possible Eps value supported by the algorithm and based on this K value also determined. So these two array values are populated with the required values before the actual clustering process. E. Algorithm 1) Input D, Eps, MinObjs. 2) Create SparseMemoryMapped File. 3) Load the pre-computed Neighbour Cells Offset Array “NCOArray” and Offset Index Array “OIArray” Values. D. Neighbour Cells and Index Offset Array 4) Initialize the SparseMemoryMapped file with the dataset D, assign ClusterID field of all objects with UNCLASSIFIED and preserve the First Object’s Address. 5) ClusterID = NOISE, CurrentObject = FirstObject. 6) WHILE CurrentObject <> NULL 7) If (CurrentObject.ClusterID == UNCLASSIFIED) Then 8) Call SparseMemoryRegionQuery function with CurrentObject, Eps, UpdateMasterSeedOffset, NCOArray and OIArray parameter and the function returns FirstSeedObject, LastSeedObject and SeedObjectsCount. 9) If (SeedObjectsCount >= MinObjs) Then// Core Object condition 10) ClusterID = GetNextID(ClusterID). 11) Assign the ClusterID to all the seed objects. 12) Move CurrentSeedObject to point its next seed object using the OffsetValue and assign NULL value to previous CurrentSeedObject’s NextSeedObjectOffset field. 13) WHILE CurrentSeedObject <> NULL 14) Call SparseMemoryRegionQuery function with CurrentSeedObject, Eps, UpdateTempSeedOffset, NCOArray and OIArray parameter and the function returns TempFirstSeedObject, TempLastSeedObject and TempSeedObjectsCount. 15) If (TempSeedObjectsCount >= MinObjs) Then 16) TempCurrentSeedObject = TempFirstSeedObject. 17) For I = 1 to TempSeedObjectsCount 18) If TempCurrentSeedObject .ClusterID IN {UNCLASSIFIED, NOISE} Then 19) If TempCurrentSeedObject.ClusterID == UNCLASSIFIED Then 20) Append the TempCurrentSeedObject to the Figure 3. NCOArray and IOArray LastSeedObject. 21) End If Two additional arrays are been used in this algorithm 22) TempCurrentSeedObject .ClusterID = to avoid the distance computation and improve the ClulsterID. performance. The first array Neighbour Cells Offset Array 23) End If (NCOArray) is an M X 2 array and it stores the offset values 24) Move TempCurrentSeedObject to point its next of neighbour cells from the center object. The Second Index seed object using the OffsetValue and assign Offset Array (IOArray) is K X 1 dimensional array and it NULL stores the NCOArray’s last index value for the corresponding value to previous TempCurrentSeedObject’s Eps value sequence starting from 0. For example if the Eps NextTempSeedObjectOffset field. value is 1 then IOArray[1] tells that NCOArray array 25) End For elements starting from 0 to 4 have the cells offset that need to 26) End If be visited by SparseMemoryRegionQuery during the neighbour objects computation. The value M will be decided 27) If (CurrentSeedObject. NextObjectOffset == 0) 125 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 8, No. 5, 2010 Then neighbour objects will be visited using 28) CurrentSeedObject =NULL. TempObjectNextSeedOffset instead of 29) Else ObjectNextSeedOffset and the UNCLASSIFIED cluster id 30) Move CurrentSeedObject to point its next seed object type objects present in the temporary seed chain will be using the OffsetValue and assign NULL value to appended to the LastSeedObject (main seed chain) for the previous CurrentSeedObject’s NextSeedObjectOffset further processing and all the UNCLASSIFIED and NOISE field. type objects present in the temporary seed list will be 31) End If assigned with the current Cluster ID. The LastSeedObject 32) END WHILE member will always point the last object in the seed chain. 33) Else //Non Core Object The entire object present in the main seed chain will be 34) CurrentObject.ClusterID = NOISE. processed one by one and cluster expansion will stop when 35) Assign NULL value to all the SeedObjects’ the traverse reaches the LastSeedObject and no more seed NextSeedOffset member. objects to process further. The complete clustering process 36) End If will stop once the initial loop process the entire objects 37) End If present in the data set. 38) If (CurrentObject. NextObjectOffset == 0) Then 39) CurrentObject=NULL. 40) Else 41) Move CurrentObject to point its next object using the OffsetValue. 42) End If 43) END WHILE This algorithm starts with creating the Sparse MMF with the required size and loads the Neighbour Cell Offset and Index Offset array values. The dataset D will be read one by one and each object will be placed in the corresponding memory locations. As mentioned in the section 4(B), while initializing the Sparse MMF with objects, each successive object’s memory offset will be stored in the Figure 4. Result of Dataset 1 previous objects NextObjectOffset field and last object’s NextObjectOffset field will be assigned with NULL value. F. Advantages Thus it is very essential to preserve the FirstObject’s address to visit all the remaining objects. The proposed algorithm is very stable. The main The algorithm starts the traverse from the first drawback of original DBSCAN algorithm is instability. object and visits the next objects one by one using the next Though all the objects present in the data set can be loaded object’s offset stored in the current object itself. When it by the DBSCAN algorithm, if we don’t have sufficient main finds the object and its cluster ID is UNCLASSIFIED, memory to hold the growing seeds objects, DBSCAN SparseMemoryRegionQuery function will be called with algorithm will crash during run time. But the new algorithm required parameter. As the new cluster is not yet formed, doesn’t rely on the growing seeds and it will give guarantee SparseMemoryRegionQuery function needs to be called with to process all the objects as long as it is able to load. The UpdateMasterSeedField flag to update the seed objects’ second advantage of the new algorithm is capable of NextObjectSeedOffset field. The output of processing huge amount of objects. Since this algorithm is SparseMemoryRegionQuery will give FirstSeedObject, based on the Sparse MMF, it can support few GBs of data in LastSeedObject and SeedObjectsCount. If the current object a 32 bit Operating System where traditional approach is a non core object, the current object will be market as supports only few MBs of data in the real time scenario. NOISE and all its seed objects NextObjectSeedOffset field Also this algorithm can be customised to process very huge will be market with NULL value. Otherwise the cluster data set (e.g > 10 GB) using the Sparse MMF. Then the expansion will start with creating a new cluster ID as the beauty of Sparse MMF is, though we pre-allocate more current object is a core object. The new Cluster ID will be memory in the beginning, the real memory occupying is assigned to all the seed objects that are chained starting from based on the consumption. Eventually the performance is FirstSeedObject. Now the remaining objects (except really fast as the algorithm directly works on the memory. FirstSeedObject) present in the seed chain will be processed one by one and for all the remaining seed objects G. Limitations SparseMemoryRegionQuery will be called with UpdateTempSeedOffset flag to update the As this algorithm uses Sparse MMF and only very TempObjectNextSeedOffset field. This will avoid the few languages support this feature, scope for implementing overwriting of seed objects which are already exist in the this algorithm is limited. Second limitation is memory main seed list chain. So if the object is a core object, the 126 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 8, No. 5, 2010 customization. If we are planning to apply this algorithm to different sizes of 2 dimensional synthetic datasets were used support multidimensional dataset, memory needs to be and running time results are given below: customized accordingly and the computation complexity may vary. Also if the minimum distance between one object and TABLE 2. RUNNING TIME OF DBSCAN AND DBSCANSMMF IN SECONDS the immediate nearest object is greater than one unit or less than one unit, offset array values will change and it should be recomputed. Moreover creating and populating values in 1.DBSCANSMMF 2.DBSCANSMMF Offset arrays are an extra task. Last drawback of this algorithm is this doesn’t support duplicate objects. As the 1.DBSCAN 2.DBSCAN Number of object loaded in the corresponding memory location, it is not Objects possible to overwrite another object in the same location. These are the notable limitations of this algorithm. 1500 0.0007 0.3892 0.0005 0.2176 3000 0.0043 0.5395 0.0051 0.5684 H. Computation Complexity 6000 0.0081 1.8030 0.0094 1.8920 10000 0.0137 4.9124 0.0166 5.1122 The DBSCAN algorithm’s complexity has been 20000 0.0261 20.4426 0.0255 18.2351 calculated based on the number of RegionQuery function 30000 0.0377 43.3875 0.0269 41.1765 call. In which each RegionQuery function call need N 40000 0.0545 77.6204 0.0587 79.6543 distance computation and hence the computation complexity becomes O (N2) for processing all the N objects present in 60000 0.0799 195.8284 0.0676 181.8745 the dataset. As the new algorithm’s SparseRegionQuery process the neighbour cells, the complexity varies based on the Eps value and each SparseRegionQuery requires not more than 2(Eps+1) cells traversal. Eventually for processing all the N objects, our algorithm requires O (N * 2(Eps+1) ) time. The constant 1 can be removed as it is very small and the final complexity comes as O (N * 2Eps). This complexity is really a reduction when the Eps value is reasonable (e.g 1~10) and N value is very large. At the same time, if we have very less number of objects and the Eps value is too big, this new complexity won’t be an attractive one. However the real processing time will be very faster than the traditional RegionQuery function call as the SparseRegionQuery traverse the memory directly. TABLE 1. COMPARISON OF ALGORITHMS Fig 5. Scalability of Algorithm with different size of dataset Better Performance The above table and graph figures show that new Supports Duplicate Doesn’t depend on distance function. Ability to process growing Seed )? algorithm gives better performance when the algorithm’s Doesn’t Require huge dataset? input data set size grows. This is the expected obvious result extra Buffer (because of Algorithm as the new algorithm visits only the required neighbour cells Objects? Stability during the SparseMemoryRegionQuery function call instead of the computing distance between center and the entire objects in the data set. Another reason is directly accessing DBSCAN No No No Yes No No the memory is much faster than using the buffers to process the data that are usually used to implement the algorithms. DBSCANSMMF Yes Yes Yes No Yes Yes VI. CONCLUSION AND FUTURE ENHANCEMENT Above table show the comparison of some key features and In this paper we have proposed DBSCANSMMF DBSCANSMMF is superior in most of the features. algorithm to improve the performance as well as to process the huge amount of data using Sparse MMF. This new V. EXPERIMENTAL RESULTS algorithm doesn’t uses any growing seed list which causes the crash during the run time when there is no sufficient memory to store the seed objects. Instead the new algorithm The newly proposed algorithm and the original just maintains the seed list using the offset values and these DBSCAN algorithm have been implemented in Visual C++ values are stored in each objects corresponding offset field (2008) on Windows Vista OS and ran on PC with a 2.0 GHZ internally. So there is no need of creating duplicate objects processor and 4 GB RAM to observe the performance. The 127 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 8, No. 5, 2010 for processing the objects. Also this new algorithm takes O Tirunelveli. His interested research area is algorithms inventions in (N * 2Eps) computation complexity and this is better data mining. complexity as long as Eps value is reasonable. Email: hencilpeter@hotmail.com Future work will be to customize this algorithm to support duplicate objects. This can be achieved using the internal counter which will give the number of similar objects and the SparseMemoryRegionQuery also needs to be customized accordingly to support correct output. The next expansion will be customizing this algorithm to process super big data set (e.g. 50 GB). One of the real uses of Memory Mapped File is mapping the required portion of the file into memory to process and, un map the current mapped region and remap the next consecutive file region to process later. Like this we can process any big file and this algorithm needs to be customized to support this feature. REFERENCES [1] Ester M., Kriegel H.-P., Sander J., and Xu X. (1996) “A Density-Based Dr.A. Antonysamy is Principal of St. Xavier’s College, Algorithm for Discovering Clusters in Large Spatial Databases with Noise” Kathmandu, Nepal. He completed his Ph.D in Mathematics for the In Proceedings of the 2nd International Conference on Knowledge Discovery and Data Mining (KDD’96), Portland: Oregon, pp. 226-231 research on “An algorithmic study of some classes of intersection graphs”. He has guided and guiding many research students in [2] J. Han and M. Kamber, Data Mining Concepts and Techniques. Morgan Computer Science and Mathematics. He has published many Kaufman, 2006. research papers in national and international journals. He has organized Seminars and Conferences in state and national level. [3] Jeffrey Richter and Christophe Nasarre, WINDOWS VIA C/C++, Microsoft Press, 2008. Email: fr_antonysamy@hotmail.com. [4]M. Ankerst, M. Breunig, H. P. Kriegel, and J. Sander, “OPTICS: Ordering Objects to Identify the Clustering Structure, Proc. ACM SIGMOD,” in International Conference on Management of Data, 1999, pp. 49–60. [5] A. Hinneburg and D. Keim, “An efficient approach to clustering in large multimedia data sets with noise,” in 4th International Conference on Knowledge Discovery and Data Mining, 1998, pp. 58–65. [6]SHOU Shui-geng, ZHOU Ao-ying JIN Wen, FAN Ye and QIAN Wei- ning.(2000) "A Fast DBSCAN Algorithm" Journal of Software: 735-744 [7] Li Jian; Yu Wei; Yan Bao-Ping; , "Memory effect in DBSCAN algorithm," Computer Science & Education, 2009. ICCSE '09. 4th International Conference on , vol., no., pp.31-36, 25-28 July 2009. AUTHOR PROFILES J. Hencil Peter is Research Scholar, St. Xavier’s College (Autonomous), Palayamkottai, Tirunelveli, India. He earned his MCA (Master of Computer Applications) degree from Manonmaniam Sundaranar University, Tirunelveli. Now he is doing Ph.D in Computer Applications and Mathematics (Interdisciplinary) at Manonmaniam Sundranar University, 128 http://sites.google.com/site/ijcsis/ ISSN 1947-5500

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