DB Seminar Schedule Seminar Schedule ================================================================= Chui Chun Kit30/11/07 Gong Jian Jim7/12/07 Loo Kin.

DB Seminar Schedule Seminar Schedule ================================================================= Chui Chun Kit30/11/07 Gong Jian Jim7/12/07 Loo Kin Kong14/12/07 Ngai Wang Kay Jackie21/12/07 Siu Wing Yan Angela4/1/08 Tam Ming Wai11/1/08 Tsang Pui Kwan Smith18/1/08 U Leong Hou Kamiru25/1/08 Wong Wai Kit1/2/08 Cui Yingjie Jason15/2/08 LEE King For22/2/08 Lin Zhifeng Arthur7/3/08 Yuan Wenjun Clement14/3/08 Zhang Shiming Simon28/3/08 Zhang Yiwei Kelvin11/4/08 LEE Yau Tat18/4/08 Pan Guodong Delvin25/4/08 Please send the abstract to dbgroup@cs.hku.hk one week before your talk dbgroup@cs.hku.hk

Refreshing the Sky: The Compressed Skycube with Efficient Support for Frequent Updates Authors :Tian Xia, Donghui Zhang Northeastern University Published in : SIGMOD 2006 Presenter : Chun-Kit Chui (Kit)

Presentation Outline Introduction  What is skylines?  Motivation for subspace skyline queries.  Skycube. Compressed skycube (CSC) How to use the compressed skycube to answer skyline queries? How to handle object updates in compressed skycube? Experimental evaluation Conclusion

Introduction What is skyline?

Skyline query: Find the hotels with both low price and close to the beach. t 7 1 4 Price Dist. To Beach t 1 3 2 t 2 4 7 t 3 9 5 t 4 9 1 t 5 2 3 t 6 6 1 Hotels in Hawaii Ranked the price of the hotels (the smaller the cheaper). Ranked the distance of the hotel to the beach (the smaller the closer).

What is skyline? Skyline query: Find the hotels with both low price and close to the beach. t5t5 t6t6 t7t7 t1t1 t4t4 t2t2 t3t3 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 Dist. To Beach Price t 7 1 4 Price Dist. To Beach t 1 3 2 t 2 4 7 t 3 9 5 t 4 9 1 t 5 2 3 t 6 6 1 Hotels in Hawaii Ranked the price of the hotels (the smaller the cheaper). Simple plot of the hotels dataset with x-axis as the price rank, and y-axis as the rank of the dist. to beach. Ranked the distance of the hotel to the beach (the smaller the closer).

The hotels highlighted in red are the hotels with both low price and close to the beach among the others. They are the skylines in this query space. An object t which is not dominated by any objects in a set of dimensions U is called the skyline of U. i.e. t  sky(U). Here, we say hotel t 6 dominates hotel t 3 in terms of price and distance to the beach. An object a dominates object b in a set of dimensions U if the a is smaller than or equal to b in all dimensions. And a has a smaller value in at least one dimension. What is skyline? Skyline query: Find the hotels with both low price and close to the beach. t5t5 t6t6 t7t7 t1t1 t2t2 t3t3 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 Dist. To Beach Price t4t4 Simple plot of the hotels dataset with x-axis as the price rank, and y-axis as the rank of the dist. to beach.

The hotels highlighted in red are the hotels with both low price and close to the beach among the others. They are the skylines in this query space. An object t which is not dominated by any objects in a set of dimensions U is called the skyline of U. i.e. t  sky(U). Here, we say hotel t 6 dominates hotel t 3 in terms of price and distance to the beach. An object a dominates object b in a set of dimensions U if the a is smaller than or equal to b in all dimensions. And a has a smaller value in at least one dimension. What is skyline? Skyline query: Find the hotels with both low price and close to the beach. t5t5 t6t6 t7t7 t1t1 t2t2 t3t3 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 Dist. To Beach Price t4t4 Without loss of generality, we will use the MIN operation to evaluation skylines in this talk.

t 7 1 3 4 1 u 1 u 2 u 3 u 4 t 1 3 4 2 5 t 2 4 6 7 2 t 3 9 7 5 6 t 4 4 3 6 1 t 5 2 2 3 1 t 6 6 1 1 3 Subspace Skyline Query In many applications, users may issue skyline queries based on arbitrary subsets of dimensions.  Price, dist. to the beach, dist. to the shopping center … etc Results of subspace skylines can be very different! u1u1 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 u3u3 t2t2 t1t1 t3t3 t4t4 t5t5 t6t6 t7t7 t5t5 Skyline in  u 1, u 3  u4u4 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 u3u3 t5t5 t6t6 t7t7 t1t1 t2t2 t3t3 t4t4 Skyline in  u 3, u 4  Objects of 4 -dimensions

t 7 1 3 4 1 u 1 u 2 u 3 u 4 t 1 3 4 2 5 t 2 4 6 7 2 t 3 9 7 5 6 t 4 4 3 6 1 t 5 2 2 3 1 t 6 6 1 1 3 Subspace Skyline A d -dimensional space contains 2 d -1 subspaces, and the subspaces of various users ’ interests are unpredictable. On-the-fly computation (compute from scratch upon each query) does not achieve fast response time for an online system. u1u1 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 u3u3 t2t2 t1t1 t3t3 t4t4 t5t5 t6t6 t7t7 t5t5 Skyline in  u 1, u 3  u4u4 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 u3u3 t5t5 t6t6 t7t7 t1t1 t2t2 t3t3 t4t4 Skyline in  u 3, u 4  Objects of 4 -dimensions

t 7 1 3 4 1 u 1 u 2 u 3 u 4 t 1 3 4 2 5 t 2 4 6 7 2 t 3 9 7 5 6 t 4 4 3 6 1 t 5 2 2 3 1 t 6 6 1 1 3 Subspace Skyline Skycube (proposed by Yuan, et al., in VLDB 2005) is the collection of all subspace skyline results.  To answer subspace query, simply retrieve the skylines of the corresponding cuboids. Objects of 4 -dimensions t7t7 Cuboid Skyline u1u1 u2u2 u3u3 u4u4 u 1, u 2 u 1, u 3 u 1, u 4 u 2, u 3 u 2, u 4 u 3, u 4 u 1, u 2, u 3 u 1, u 2, u 4 u 1, u 3, u 4 u 2, u 3, u 4 u 1, u 2, u 3, u 4 t 5, t 6 t 1, t 5, t 6, t 7 t 5, t 6, t 7 t 5, t 6 t6t6 t7t7 t 1, t 5, t 6, t 7, t 9 t 5, t 6, t 7, t 9 t 5, t 7, t 4 t6t6 t6t6 t 1, t 5, t 6, t 7, t 9 The cuboid w.r.t. all dimensions is called the full space cuboid. Objects in the full space cuboid are called the full space skyline objects sky(D). Complete Skycube

Subspace Skyline Skycube (proposed by Yuan, et al., in VLDB 2005) is the collection of all subspace skyline results. A skycube can be viewed as a lattice of cuboids. t7t7 Cuboid Skyline u1u1 u2u2 u3u3 u4u4 u 1, u 2 u 1, u 3 u 1, u 4 u 2, u 3 u 2, u 4 u 3, u 4 u 1, u 2, u 3 u 1, u 2, u 4 u 1, u 3, u 4 u 2, u 3, u 4 u 1, u 2, u 3, u 4 t 5, t 6 t 1, t 5, t 6, t 7 t 5, t 6, t 7 t 5, t 6 t6t6 t7t7 t 1, t 5, t 6, t 7, t 9 t 5, t 6, t 7, t 9 t 5, t 7, t 4 t6t6 t6t6 t 1, t 5, t 6, t 7, t 9 u 1 u 2 u 3 u 4 u 1 u 2 u 3 u 1 u 2 u 4 u 1 u 3 u 4 u 2 u 3 u 4 u 1 u 2 u 3 u 4 u 1 u 2 u 1 u 3 u 1 u 4 u 2 u 3 u 2 u 4 u 3 u 4 Skycube of 4 -dimensions The cuboid w.r.t. all dimensions is called the full space cuboid. Complete Skycube

Motivations In many scenarios of the subspace skyline applications, the data are changing constantly.  In an online hotel-booking system, room prices change due to the availability. On-the-fly computation  Low update cost.  Slow query response time. Complete Skycube  Fast query response time.  High update cost The skycube contains a huge number of duplicates. t7t7 Cuboid Skyline u1u1 u2u2 u3u3 u4u4 u 1, u 2 u 1, u 3 u 1, u 4 u 2, u 3 u 2, u 4 u 3, u 4 u 1, u 2, u 3 u 1, u 2, u 4 u 1, u 3, u 4 u 2, u 3, u 4 u 1, u 2, u 3, u 4 t 5, t 6 t 1, t 5, t 6, t 7 t 5, t 6, t 7 t 5, t 6 t6t6 t7t7 t 1, t 5, t 6, t 7, t 9 t 5, t 6, t 7, t 9 t 5, t 7, t 4 t6t6 t6t6 t 1, t 5, t 6, t 7, t 9 Complete Skycube

Motivations In many scenarios of the subspace skyline applications, the data are changing constantly.  In an online hotel-booking system, room prices change due to the availability. On-the-fly computation  Low update cost.  Slow query response time. Complete Skycube  Fast query response time.  High update cost The skycube contains a huge number of duplicates. For example, object t 6 appears in 12 cuboids. Whenever t 6 is updated, at least 12 cuboids have to be updated. In addition, all affected cuboids have to be recomputed to reflect the correct result. Both waste of storage and difficult to maintain. t7t7 Cuboid Skyline u1u1 u2u2 u3u3 u4u4 u 1, u 2 u 1, u 3 u 1, u 4 u 2, u 3 u 2, u 4 u 3, u 4 u 1, u 2, u 3 u 1, u 2, u 4 u 1, u 3, u 4 u 2, u 3, u 4 u 1, u 2, u 3, u 4 t 5, t 6 t 1, t 5, t 6, t 7 t 5, t 6, t 7 t 5, t 6 t6t6 t7t7 t 1, t 5, t 6, t 7, t 9 t 5, t 6, t 7, t 9 t 5, t 7, t 4 t6t6 t6t6 t 1, t 5, t 6, t 7, t 9 Complete Skycube

The Compressed Skycube

In the paper, the authors proposed:  A new compressed model for the Skycube, which greatly reduces the storage.  A new object-aware update scheme, which avoids unnecessary disk access and cuboids' computation. By taking advantages of the compact structure and the update scheme, the Compressed Skycube achieves both fast query response and efficient update.

Minimum Subspace D EFINITION : Given an object t, the minimum subspaces of t, denoted as mss(t), satisfies the following two conditions: 1. For any subspace U in mss(t), t is in the skyline of U ; 2. For any subspace V  U, t is not in the skyline of V.

Minimum Subspace Consider object t 6 again, it appears in the skylines of 12 cuboids. The minimum subspaces of t 6 are cuboids u 2 and u 3. u 1 u 2 u 3 u 4 u 1 u 2 u 3 u 1 u 2 u 4 u 1 u 3 u 4 u 2 u 3 u 4 u 1 u 2 u 3 u 4 u 1 u 2 u 1 u 3 u 1 u 4 u 2 u 3 u 2 u 4 u 3 u 4 Cuboids that contain object t 6 t7t7 Cuboid Skyline u1u1 u2u2 u3u3 u4u4 u 1, u 2 u 1, u 3 u 1, u 4 u 2, u 3 u 2, u 4 u 3, u 4 u 1, u 2, u 3 u 1, u 2, u 4 u 1, u 3, u 4 u 2, u 3, u 4 u 1, u 2, u 3, u 4 t 5, t 6 t 1, t 5, t 6, t 7 t 5, t 6, t 7 t 5, t 6 t6t6 t7t7 t 1, t 5, t 6, t 7, t 9 t 5, t 6, t 7, t 9 t 5, t 7, t 4 t6t6 t6t6 t 1, t 5, t 6, t 7, t 9 Cuboids that contain object t 6 are highlighted in blue. Complete Skycube Based on the definition of minimum subspaces, mms ( t 6 ) are highlighted in red.

Minimum Subspace u 1 u 2 u 3 u 4 u 1 u 2 u 3 u 1 u 2 u 4 u 1 u 3 u 4 u 2 u 3 u 4 u 1 u 2 u 3 u 4 u 1 u 2 u 1 u 3 u 1 u 4 u 2 u 3 u 2 u 4 u 3 u 4 Cuboids that contain object t 6 t7t7 Cuboid Skyline u1u1 u2u2 u3u3 u4u4 u 1, u 2 u 1, u 3 u 1, u 4 u 2, u 3 u 2, u 4 u 3, u 4 u 1, u 2, u 3 u 1, u 2, u 4 u 1, u 3, u 4 u 2, u 3, u 4 u 1, u 2, u 3, u 4 t 5, t 6 t 1, t 5, t 6, t 7 t 5, t 6, t 7 t 5, t 6 t6t6 t7t7 t 1, t 5, t 6, t 7, t 9 t 5, t 6, t 7, t 9 t 5, t 7, t 4 t6t6 t6t6 t 1, t 5, t 6, t 7, t 9 u 1 u 2 u 3 u 4 u 1 u 2 u 3 u 1 u 2 u 4 u 1 u 3 u 4 u 2 u 3 u 4 u 1 u 2 u 3 u 4 u 1 u 2 u 1 u 3 u 1 u 4 u 2 u 3 u 2 u 4 u 3 u 4 Cuboids that contain object t 5 Cuboids that contain t 5. Complete Skycube Cuboids that contain object t 6 are highlighted in blue. Based on the definition of minimum subspaces, mms ( t 6 ) are highlighted in red.

Minimum Subspace t7t7 Cuboid Skyline u1u1 u2u2 u3u3 u4u4 u 1, u 2 u 1, u 3 u 1, u 4 u 2, u 3 u 2, u 4 u 3, u 4 u 1, u 2, u 3 u 1, u 2, u 4 u 1, u 3, u 4 u 2, u 3, u 4 u 1, u 2, u 3, u 4 t 5, t 6 t 1, t 5, t 6, t 7 t 5, t 6, t 7 t 5, t 6 t6t6 t7t7 t 1, t 5, t 6, t 7, t 9 t 5, t 6, t 7, t 9 t 5, t 7, t 4 t6t6 t6t6 t 1, t 5, t 6, t 7, t 9 u 1 u 2 u 3 u 4 u 1 u 2 u 3 u 1 u 2 u 4 u 1 u 3 u 4 u 2 u 3 u 4 u 1 u 2 u 3 u 4 u 1 u 2 u 1 u 3 u 1 u 4 u 2 u 3 u 2 u 4 u 3 u 4 Cuboids that contain object t 5 Cuboids that contain t 5. t 4  u 4  t 9  u 1, u 2 ,  u 1, u 3  t 7  u 1 ,  u 4  t 1  u 1, u 3  t 5  u 4 ,  u 1, u 2 ,  u 1, u 3  t 6  u 2 ,  u 3  Minimum Subspaces Similar for all other skyline objects, we store the minimum subspaces of all skyline objects in a table. Complete Skycube For easy processing in the later sections, we organize the full-space skyline objects together in the front.

Compressed Skycube t7t7 Cuboid Skyline u1u1 u2u2 u3u3 u4u4 u 1, u 2 u 1, u 3 u 1, u 4 u 2, u 3 u 2, u 4 u 3, u 4 u 1, u 2, u 3 u 1, u 2, u 4 u 1, u 3, u 4 u 2, u 3, u 4 u 1, u 2, u 3, u 4 t 5, t 6 t 1, t 5, t 6, t 7 t 5, t 6, t 7 t 5, t 6 t6t6 t7t7 t 1, t 5, t 6, t 7, t 9 t 5, t 6, t 7, t 9 t 5, t 7, t 4 t6t6 t6t6 t 1, t 5, t 6, t 7, t 9 t 4  u 4  t 9  u 1, u 2 ,  u 1, u 3  t 7  u 1 ,  u 4  t 1  u 1, u 3  t 5  u 4 ,  u 1, u 2 ,  u 1, u 3  t 6  u 2 ,  u 3  Minimum Subspaces t7t7 Cuboid Skyline u1u1 u2u2 u3u3 u4u4 u 1, u 2 u 1, u 3 t 1, t 5, t 9 t 5, t 9 t 5, t 7, t 4 t6t6 t6t6 Compressed Skycube The compressed skycube (CSC) consists of non-empty cuboids such that an object t is stored in a cuboid iff the cuboid is the minimum subspace of t. Compare with the complete skycube, CSC has fewer number of duplicates. Complete Skycube

Querying the Compressed Skycube

Querying CSC Overview example: query space U q =  u 2, u 3, u 4  To find the skylines in U q i.e. sky( U q ), we only need to:  LEMMA 1. search within the cuboids of the compressed skycube which are the subsets of U q. u1u1 u2u2 u3u3 u4u4 t1t1 3425 t5t5 2231 t6t6 6113 t7t7 1341 t4t4 4361 t9t9 2237 Raw dataset Compressed Skycube CuboidSkyline u1u1 t7t7 u2u2 t6t6 u3u3 t6t6 u4u4 t 5, t 7, t 4 u 1, u 2 t 5, t 9 u 1, u 3 t 1, t 5, t 9

Querying CSC Overview example: query space U q =  u 2, u 3, u 4  To find the skylines in U q i.e. sky( U q ), we only need to:  LEMMA 1. search within the cuboids of the compressed skycube which are the subsets of U q. u 1 u 2 u 3 u 4 u 1 u 2 u 3 u 1 u 2 u 4 u 1 u 3 u 4 u 2 u 3 u 4 u 1 u 2 u 3 u 4 u 1 u 2 u 1 u 3 u 1 u 4 u 2 u 3 u 2 u 4 u 3 u 4 Cuboids that contain object t 5 u 1 u 2 u 3 u 4 u 1 u 2 u 3 u 1 u 2 u 4 u 1 u 3 u 4 u 2 u 3 u 4 u 1 u 2 u 3 u 4 u 1 u 2 u 1 u 3 u 1 u 4 u 2 u 3 u 2 u 4 u 3 u 4 Cuboids that contain object t 6 If an object is the skyline of, the object must appear in some cuboids in the CSC which are subset of. u1u1 u2u2 u3u3 u4u4 t1t1 3425 t5t5 2231 t6t6 6113 t7t7 1341 t4t4 4361 t9t9 2237 Raw dataset

Querying CSC Overview example: query space U q =  u 2, u 3, u 4  To find the skylines in U q i.e. sky( U q ), we only need to:  LEMMA 1. search within the cuboids of the compressed skycube which are the subsets of U q.  LEMMA 2. If an object t in a cuboid V ( V is subset of U q ) is not dominated in U q by other objects in the same cuboid, then t is a full space skyline object. u1u1 u2u2 u3u3 u4u4 t1t1 3425 t5t5 2231 t6t6 6113 t7t7 1341 t4t4 4361 t9t9 2237 Raw dataset Compressed Skycube CuboidSkyline u1u1 t7t7 u2u2 t6t6 u3u3 t6t6 u4u4 t 5, t 7, t 4 u 1, u 2 t 5, t 9 u 1, u 3 t 1, t 5, t 9 No comparison is needed for t 6. t 5, t 7, t 4 are only locally compared to each other. Why ?

Querying CSC LEMMA 2. If an object t in a cuboid V ( V is subset of U q ) is not dominated in U q by other objects in the same cuboid, then t is a full space skyline object. u1u1 u2u2 u3u3 u4u4 t1t1 3425 t5t5 2231 t6t6 6113 t7t7 1341 t4t4 4361 t9t9 2237 Raw dataset Compressed Skycube CuboidSkyline u1u1 t7t7 u2u2 t6t6 u3u3 t6t6 u4u4 t 5, t 7, t 4 u 1, u 2 t 5, t 9 u 1, u 3 t 1, t 5, t 9 No comparison is needed for t 6. t 5, t 7, t 4 are only locally compared to each other. Why ? t5t5 t7t7 t4t4 u4u4 There are NO objects in this area. Otherwise, t 5, t 7, and t 4 will not be the skylines of u 4. Therefore, no other objects can dominate t 5, t 7 and t 4 in the superset of u 4. e.g.If t 4 is not dominated by t 7 and t 8 in full space, no other objects can dominate t 4. u3u3 Plot of objects t 5, t 7 and t 4 in u 4. Since they have the smallest values in u 4, they are skylines of u 4.

The query system based on CSC

Query system based on CSC Query buffer Skyline query Queryresults The system consists of a query buffer. The query buffer stores the most frequently requested query results.

Query system based on CSC Query buffer Compressed Skycube Compressed Skycube CSC-based query system Skyline query Queryresults If the requested query results are not in the buffer, the query buffer issue a query miss request to CSC. Query miss The system consists of a query buffer. The query buffer stores the most frequently requested query results.

Query system based on CSC Query buffer Compressed Skycube Compressed Skycube Disk CSC-based query system Skyline query Queryresults If the requested query results are not in the buffer, the query buffer issue a query miss request to CSC. Query miss Updates CSC monitors the updates of objects. The system consists of a query buffer. The query buffer stores the most frequently requested query results.

Query system based on CSC Query buffer Compressed Skycube Compressed Skycube Disk CSC-based query system Skyline query Queryresults If the requested query results are not in the buffer, the query buffer issue a query miss request to CSC. Query miss Disk access Updates CSC monitors the updates of objects. According to different object updates, CSC decides whether it needs to access the disk to retrieve new objects that are not in CSC. Disk access should be minimized. The system consists of a query buffer. The query buffer stores the most frequently requested query results.

Query system based on CSC Query buffer Compressed Skycube Compressed Skycube Disk CSC-based query system Skyline query Queryresults If the requested query results are not in the buffer, the query buffer issue a query miss request to CSC. Query miss Invalidating Disk access Updates CSC monitors the updates of objects. Finally, if some cuboids are updated, results in the buffer may not be accurate anymore. CSC then invalidates the affected query results in the buffer. According to different object updates, CSC decides whether it needs to access the disk to retrieve new objects that are not in CSC. Disk access should be minimized. The system consists of a query buffer. The query buffer stores the most frequently requested query results.

Updating the Compressed Skycube

Updating CSC Intuitions:  Not all updates of objects need to access the disk.  Not all updates of objects need to re-compute the skyline of a cuboid. These intuitions are supported by the theorems in the paper. D : full-space; sky(D) : full-space skyline. t : object before update; t new : object after update. t  sky(D) No dataset (disk) access t new  sky(D) t new  sky(D) May access dataset (disk) t  sky(D) Insert new skyline objects Considering the proportion of full-space skyline objects in the whole dataset, the above covers most cases of the updates

Updating CSC t  sky(D) and t new  sky(D) Key points:  The existing objects in CSC are NOT affected.  No need to retrieve objects that are NOT in CSC (no disk access) An example dataset with 2D space. In this case, t a is the full- space skyline. tata tbtb uaua ubub

Updating CSC t  sky(D) and t new  sky(D) Key points  The existing objects in CSC are NOT affected.  No need to retrieve objects that are NOT in CSC (no disk access) tata tbtb uaua ubub In the compressed skycube (CSC), t b is the skyline of u a because t b and t a overlap on dimension u a. An example dataset with 2D space. In this case, t a is the full- space skyline.

Updating CSC t  sky(D) and t new  sky(D) Key points  The existing objects in CSC are NOT affected.  No need to retrieve objects that are NOT in CSC (no disk access) tata tbtb uaua ubub t t  sky(D) and t new  sky(D) If t  sky(D) and t new  sky(D), t new will fall within this area, which will NOT affect the existing skyline objects. As a result, the objects in CSC are NOT affected. An example dataset with 2D space. In this case, t a is the full- space skyline. In the compressed skycube (CSC), t b is the skyline of u a because t b and t a overlap on dimension u a.

Updating CSC t  sky(D) and t new  sky(D) Key points  The existing objects in CSC are NOT affected.  No need to retrieve objects that are NOT in CSC (no disk access) tata tbtb uaua ubub t t  sky(D) and t new  sky(D) If t  sky(D) and t new  sky(D), t new will fall within this area, which will NOT affect the existing skyline objects. As a result, the objects in CSC are NOT affected. Only when t new overlaps with some subspace (e.g. u a or u b ), t new will becomes the skyline of the corresponding cuboids. In this case, t new is added into CSC. t new An example dataset with 2D space. In this case, t a is the full- space skyline. In the compressed skycube (CSC), t b is the skyline of u a because t b and t a overlap on dimension u a.

Updating CSC t  sky(D) and t new  sky(D) Two steps approach to update CSC  Compare t new with existing full-space skyline objects (sky(D)).  Determine the minimum subspaces of t new Can be determined by ANY dominating object in sky(D) (Why?). tata tbtb uaua ubub t t  sky(D) and t new  sky(D) If t  sky(D) and t new  sky(D), t new will fall within this area, which will NOT affect the existing skyline objects. As a result, the objects in CSC are NOT affected. Only when t new overlaps with some subspace (e.g. u a or u b ), t new will becomes the skyline of the corresponding cuboids. In this case, t new is added into CSC. t new An example dataset with 2D space. In this case, t a is the full- space skyline. In the compressed skycube (CSC), t b is the skyline of u a because t b and t a overlap on dimension u a.

Updating CSC tata tbtb uaua ubub t t  sky(D) and t new  sky(D) Two steps approach to update CSC  Compare t new with existing full-space skyline objects (sky(D)).  Determine the minimum subspaces of t new Can be determined by ANY dominating object in sky(D) (Why?). Another example with t a and t b as the full-space skylines.

Updating CSC tata tbtb uaua ubub t t new t  sky(D) and t new  sky(D) Two steps approach to update CSC  Compare t new with existing full-space skyline objects (sky(D)).  Determine the minimum subspaces of t new Can be determined by ANY dominating object in sky(D) (Why?). t new will be the skyline of a subspace (e.g. u b ) only when t new overlaps with a full- space skyline in that subspace (e.g. overlaps with t a in u b ). That is, t new lies on the red lines. Another example with t a and t b as the full-space skylines.

Updating CSC tata tbtb uaua ubub t t new t  sky(D) and t new  sky(D) Two steps approach to update CSC  Compare t new with existing full-space skyline objects (sky(D)).  Determine the minimum subspaces of t new Can be determined by ANY dominating object in sky(D) (Why?). If t new is dominated by t b, the minimum subspaces of t new must be the minimum subspaces of t b. E.g. If t new is the skyline of u a, t b is also the skyline of u a as t b is a full-space skyline and is dominating t new. t new will be the skyline of a subspace (e.g. u b ) only when t new overlaps with a full- space skyline in that subspace (e.g. overlaps with t a in u b ). That is, t new lies on the red lines.

Updating CSC tata tbtb uaua ubub t t new t  sky(D) and t new  sky(D) Two steps approach to update CSC  Compare t new with existing full-space skyline objects (sky(D)).  Determine the minimum subspaces of t new Can be determined by ANY dominating object in sky(D) (Why?). Similarly, if t new is the skyline of u b, t a is also the skyline of u b because t a is a full-space skyline and is dominating t new. If t new is dominated by t b, the minimum subspaces of t new must be the minimum subspaces of t b. E.g. If t new is the skyline of u a, t b is also the skyline of u a as t b is a full-space skyline and is dominating t new.

Updating CSC tata tbtb uaua ubub t t  sky(D) and t new  sky(D) Two steps approach to update CSC  Compare t new with existing full-space skyline objects (sky(D)).  Determine the minimum subspaces of t new Can be determined by ANY dominating object in sky(D) (Why?). Similarly, if t new is the skyline of u b, t a is also the skyline of u b because t a is a full-space skyline and is dominating t new. If t new is dominated by t b, the minimum subspaces of t new must be the minimum subspaces of t b. E.g. If t new is the skyline of u a, t b is also the skyline of u a as t b is a full-space skyline and is dominating t new. Therefore, to determine the minimum subspaces of t new, we only need to consider the minimum subspaces of any full-space skylines that dominates t new. t new

Updating CSC t  sky(D) and t new  sky(D) Two steps approach to update CSC  Compare t new with existing full-space skyline objects (sky(D)).  Determine the minimum subspaces of t new Can be determined by ANY dominating object in sky(D). u1u1 u2u2 u3u3 u4u4 t1t1 3425 t5t5 2231 t6t6 6113 t7t7 1341 t4t4 4361 t9t9 2237 Objects in the Compressed Skycube Full-space skylines Update object Object t 9 is not a full- space skyline object.

Updating CSC t  sky(D) and t new  sky(D) Two steps approach to update CSC  Compare t new with existing full-space skyline objects (sky(D)).  Determine the minimum subspaces of t new Can be determined by ANY dominating object in sky(D). u1u1 u2u2 u3u3 u4u4 t1t1 3425 t5t5 2231 t6t6 6113 t7t7 1341 t4t4 4361 t9t9 2237 Objects in the Compressed Skycube Full-space skylines Update object We first compare object t 9 with existing full-space skyline objects. Object t 9 is not a full- space skyline object.

The full-space skyline object t 1 does NOT dominate object t 9, continue to compare the next full-space skyline object. Updating CSC t  sky(D) and t new  sky(D) Two steps approach to update CSC  Compare t new with existing full-space skyline objects (sky(D)).  Determine the minimum subspaces of t new Can be determined by ANY dominating object in sky(D). u1u1 u2u2 u3u3 u4u4 t1t1 3425 t5t5 2231 t6t6 6113 t7t7 1341 t4t4 4361 t9t9 2237 Objects in the Compressed Skycube Full-space skylines Update object We first compare object t 9 with existing full-space skyline objects. Object t 9 is not a full- space skyline object.

The full-space skyline object t 5 dominates object t 9, we then retrieve the minimum subspaces of t 5. The retrieved subspaces are the candidates for the minimum subspaces of object t 9. The full-space skyline object t 1 does NOT dominate object t 9, continue to compare the next full-space skyline object. Updating CSC t  sky(D) and t new  sky(D) Two steps approach to update CSC  Compare t new with existing full-space skyline objects (sky(D)).  Determine the minimum subspaces of t new Can be determined by ANY dominating object in sky(D). u1u1 u2u2 u3u3 u4u4 t1t1 3425 t5t5 2231 t6t6 6113 t7t7 1341 t4t4 4361 t9t9 2237 Objects in the Compressed Skycube Full-space skylines Update object We first compare object t 9 with existing full-space skyline objects. Object t 9 is not a full- space skyline object.

The full-space skyline object t 5 dominates object t 9, we then retrieve the minimum subspaces of t 5. The retrieved subspaces are the candidates for the minimum subspaces of object t 9. Updating CSC t  sky(D) and t new  sky(D) Two steps approach to update CSC  Compare t new with existing full-space skyline objects (sky(D)).  Determine the minimum subspaces of t new Can be determined by ANY dominating object in sky(D). u1u1 u2u2 u3u3 u4u4 t1t1 3425 t5t5 2231 t6t6 6113 t7t7 1341 t4t4 4361 t9t9 2237 Objects in the Compressed Skycube Object t 9 is not a full- space skyline object. t1t1  u 1, u 3  t5t5  u 4 ,  u 1, u 2 ,  u 1, u 3  t6t6  u 2 ,  u 3  t7t7  u 1 ,  u 4  t4t4 u4u4 t9t9  u 1, u 2 ,  u 1, u 3  Minimum subspace  u 4 ,  u 1, u 2 ,  u 1, u 3  are the candidates of the minimum subspaces of Object t 9. Minimum subspaces of t 9 are  u 1, u 2 ,  u 1, u 3 ..

Updating CSC t  sky(D) and t new  sky(D) Two steps approach to update CSC  Compare t new with existing full-space skyline objects (sky(D)).  Determine the minimum subspaces of t new Can be determined by ANY dominating object in sky(D). u1u1 u2u2 u3u3 u4u4 t1t1 3425 t5t5 2231 t6t6 6113 t7t7 1341 t4t4 4361 t9t9 2237 Objects in the Compressed Skycube Object t 9 is not a full- space skyline object. t1t1  u 1, u 3  t5t5  u 4 ,  u 1, u 2 ,  u 1, u 3  t6t6  u 2 ,  u 3  t7t7  u 1 ,  u 4  t4t4 u4u4 t9t9  u 1, u 2 ,  u 1, u 3  Minimum subspace Compressed Skycube CuboidSkyline u1u1 t7t7 u2u2 t6t6 u3u3 t6t6 u4u4 t 5, t 7, t 4 u 1, u 2 t 5, t 9 u 1, u 3 t 1, t 5, t 9 Finally, we update the compressed skycube and insert object t 9 into the corresponding cuboids. Minimum subspaces of t 9 are  u 1, u 2 ,  u 1, u 3 ..

Updating CSC t  sky(D) and t new  sky(D) Two steps approach to update CSC  Compare t new with existing full-space skyline objects (sky(D)).  Determine the minimum subspaces of t new Can be determined by ANY dominating object in sky(D). u1u1 u2u2 u3u3 u4u4 t1t1 3425 t5t5 2231 t6t6 6113 t7t7 1341 t4t4 4361 t9t9 2237 Objects in the Compressed Skycube Object t 9 is not a full- space skyline object. t1t1  u 1, u 3  t5t5  u 4 ,  u 1, u 2 ,  u 1, u 3  t6t6  u 2 ,  u 3  t7t7  u 1 ,  u 4  t4t4 u4u4 t9t9  u 1, u 2 ,  u 1, u 3  Minimum subspace Compressed Skycube CuboidSkyline u1u1 t7t7 u2u2 t6t6 u3u3 t6t6 u4u4 t 5, t 7, t 4 u 1, u 2 t 5, t 9 u 1, u 3 t 1, t 5, t 9 Finally, we update the compressed skycube and insert object t 9 into the corresponding cuboids. Minimum subspaces of t 9 are  u 1, u 2 ,  u 1, u 3 .. We don’t need to retrieve the objects that are NOT in the compressed skycube throughout the whole update process. i.e. No disk access in this case.

Updating CSC t  sky(D) and t new  sky(D) Key points  The objects that are previously dominated by skylines are still dominated by the skylines after update. tata tbtb uaua ubub t An example with t a and t b as the full-space skylines.

Updating CSC t  sky(D) and t new  sky(D) Key points  The objects that are previously dominated by skylines are still dominated by the skylines after update. tata tbtb uaua ubub t An example with t a and t b as the full-space skylines. Since t new is full-space skyline, it must falls in the red area.

Updating CSC t  sky(D) and t new  sky(D) Key points  The objects that are previously dominated by skylines are still dominated by the skylines after update. tata tbtb uaua ubub t An example with t a and t b as the full-space skylines. Since t new is full-space skyline, it must falls in the red area. The purple objects, which are not in the compressed skycube, will not become skyline of any dimension after update of t. (they are still dominated by some skylines) In another words, no need to retrieve these objects from disks, no disk access in this case.

Updating CSC t  sky(D) and t new  sky(D) Key points  The objects that are previously dominated by skylines are still dominated by the skylines after update.  Existing skylines may be dominated by t new. tata tbtb uaua ubub t Since t new is full-space skyline, it must falls in the red area. If t new is updated to here, t b will be dominated by t new, t b is then removed from the cuboid of full-space skylines, and the minimum subspaces of t b need to be updated. t new The purple objects, which are not in the compressed skycube, will not become skyline of any dimension after update of t. (they are still dominated by some skylines) In another words, no need to retrieve these objects from disks, no disk access in this case.

Updating CSC t  sky(D) and t new  sky(D) Key points  The objects that are previously dominated by skylines are still dominated by the skylines after update.  Existing skylines may be dominated by t new. u1u1 u2u2 u3u3 u4u4 t1t1 3425 t5t5 2231 t6t6 6113 t7t7 1341 t4t4 4361 t9t9 2237 Objects in the Compressed Skycube t 10 1313 Object t 10 is updated. It was not a full-space skyline.

Updating CSC t  sky(D) and t new  sky(D) Key points  The objects that are previously dominated by skylines are still dominated by the skylines after update.  Existing skylines may be dominated by t new. u1u1 u2u2 u3u3 u4u4 t1t1 3425 t5t5 2231 t6t6 6113 t7t7 1341 t4t4 4361 t9t9 2237 Objects in the Compressed Skycube t 10 1313 t1t1  u 1, u 3  t5t5  u 4 ,  u 1, u 2 ,  u 1, u 3  t6t6  u 2 ,  u 3  t7t7  u 1 ,  u 4  t4t4 u4u4 t9t9  u 1,u 2   u 1,u 3  Minimum subspace We first compare t 1 with t 10. Since object t 10 dominates t 1 in the full- space, t 1 is no longer the skyline of. The minimum subspace of t 1 is removed. Object t 10 is updated. It was not a full-space skyline.

Updating CSC t  sky(D) and t new  sky(D) Key points  The objects that are previously dominated by skylines are still dominated by the skylines after update.  Existing skylines may be dominated by t new. u1u1 u2u2 u3u3 u4u4 t1t1 3425 t5t5 2231 t6t6 6113 t7t7 1341 t4t4 4361 t9t9 2237 Objects in the Compressed Skycube t 10 1313 We then compare t 5 with t 10. Since object t 10 dominates t 5 in only, we need to update the minimum subspaces of t 5. Object t 10 is updated. It was not a full-space skyline. t1t1  u 1, u 3  t5t5  u 4 ,  u 1, u 2 ,  u 1, u 3  t6t6  u 2 ,  u 3  t7t7  u 1 ,  u 4  t4t4 u4u4 t9t9  u 1,u 2   u 1,u 3  Minimum subspace

Updating CSC t  sky(D) and t new  sky(D) Key points  The objects that are previously dominated by skylines are still dominated by the skylines after update.  Existing skylines may be dominated by t new. u1u1 u2u2 u3u3 u4u4 t1t1 3425 t5t5 2231 t6t6 6113 t7t7 1341 t4t4 4361 t9t9 2237 Objects in the Compressed Skycube t 10 1313 We then compare t 5 with t 10. Since object t 10 dominates t 5 in only, we need to update the minimum subspaces of t 5. Object t 10 is updated. It was not a full-space skyline. u 1 u 2 u 3 u 4 u 1 u 2 u 3 u 1 u 2 u 4 u 1 u 3 u 4 u 2 u 3 u 4 u 1 u 2 u 3 u 4 u 1 u 2 u 1 u 3 u 1 u 4 u 2 u 3 u 2 u 4 u 3 u 4 Cuboids that contain object t 5 Remove as t 5 is no longer skyline of this subspace. t1t1  u 1, u 3  t5t5  u 4 ,  u 1, u 2 ,  u 1, u 3  t6t6  u 2 ,  u 3  t7t7  u 1 ,  u 4  t4t4 u4u4 t9t9  u 1,u 2   u 1,u 3  Minimum subspace

Updating CSC t  sky(D) and t new  sky(D) Key points  The objects that are previously dominated by skylines are still dominated by the skylines after update.  Existing skylines may be dominated by t new. u1u1 u2u2 u3u3 u4u4 t1t1 3425 t5t5 2231 t6t6 6113 t7t7 1341 t4t4 4361 t9t9 2237 Objects in the Compressed Skycube t 10 1313 We then compare t 5 with t 10. Since object t 10 dominates t 5 in only, we need to update the minimum subspaces of t 5. Object t 10 is updated. It was not a full-space skyline. u 1 u 2 u 3 u 4 u 1 u 2 u 3 u 1 u 2 u 4 u 1 u 3 u 4 u 2 u 3 u 4 u 1 u 2 u 3 u 4 u 1 u 2 u 1 u 3 u 1 u 4 u 2 u 3 u 2 u 4 u 3 u 4 Cuboids that contain object t 5 Update minimum subspaces of t 5. t1t1  u 1, u 3  t5t5  u 4 ,  u 1, u 2 ,  u 1, u 3  t6t6  u 2 ,  u 3  t7t7  u 1 ,  u 4  t4t4 u4u4 t9t9  u 1,u 2   u 1,u 3  Minimum subspace

Updating CSC t  sky(D) and t new  sky(D) Key points  The objects that are previously dominated by skylines are still dominated by the skylines after update.  Existing skylines may be dominated by t new. u1u1 u2u2 u3u3 u4u4 t1t1 3425 t5t5 2231 t6t6 6113 t7t7 1341 t4t4 4361 t9t9 2237 Objects in the Compressed Skycube t 10 1313 Similar for other objects, we update the skylines that are dominated by t new. Then we find the minimum subspaces of t new. The paper describes an algorithm to deduce the minimum subspaces of t new from the previous skylines. Object t 10 is updated. It was not a full-space skyline. t1t1  u 1, u 3  t5t5  u 4 ,  u 1, u 2 ,  u 1, u 3  t6t6  u 2 ,  u 3  t7t7  u 1 ,  u 4  t4t4 u4u4 t9t9  u 1,u 2   u 1,u 3  Minimum subspace t 10  u 1 ,  u 3  Minimum subspaces of t 10 are  u 1 , and  u 3 .

Updating CSC t  sky(D) and t new  sky(D) Key points  The objects that are previously dominated by skylines are still dominated by the skylines after update.  Existing skylines may be dominated by t new. u1u1 u2u2 u3u3 u4u4 t1t1 3425 t5t5 2231 t6t6 6113 t7t7 1341 t4t4 4361 t9t9 2237 Objects in the Compressed Skycube t 10 1313 Object t 10 is updated. It was not a full-space skyline. t1t1  u 1, u 3  t5t5  u 4 ,  u 1, u 2 ,  u 1, u 3  t6t6  u 2 ,  u 3  t7t7  u 1 ,  u 4  t4t4 u4u4 t9t9  u 1,u 2   u 1,u 3  Minimum subspace t 10  u 1 ,  u 3  Minimum subspaces of t 10 are  u 1 , and  u 3 . Compressed Skycube CuboidSkyline u1u1 t 7,t 10 u2u2 t6t6 u3u3 t 6,t 10 u4u4 t 5, t 7, t 4 u 1, u 2 t 5, t 9 u 1, u 3 t 1, t 5, t 9 Finally, we update the compressed skycube. Insert object t 10 into the corresponding cuboids and remove the dominated objects.

Experimental Evaluation

Storage Comparison Settings:  Dimensionality (Full-space) – [ 4, 8 ]; default = 6.  Cardinality – [ 100K, 500K ]; default = 300K.  Distribution: Independent, Corr, Anti-Corr.

Storage Comparison Due to less number of duplicates in the CSC structure, CSC is less affected by cardinality than Skycube. Logarithmic scale to reflect the exponential effect of the dimensionality. CSC is better than the Skycube in up to an other of magnitude.

Query Performance Queries on the complete skycube do not involve computations, their time is not reported. This set of experiments verifies that the query response of the CSC is indeed very fast.

Update Performance General update  Updates are from random objects in the whole dataset. Skycube is re-computed from scratch. CSC outperforms Skcube by several orders of magnitude. This is because the update scheme updates CSC incrementally and avoids many unnecessary computations when an objects’ update does not affect the CSC structure.

Update Performance General update  Updates are from random objects in the whole dataset. Skycube is re-computed from scratch. Full-space skyline update.  Updates are from random full-space skyline objects. For fair comparison, Skycube is re-computed from existing skylines plus new candidates.

Conclusion

Conclusions In the paper, the authours  addressed the update support of the skycube in dynamic environment, and provided an efficient and scalable solution for online skyline query system.  proposed a compact structure, the Compressed Skycube (CSC), with about 10% disk space of the Complete Skycube and fast query response.  proposed an object-aware update scheme, such that different updates trigger different amount of computation. The Compressed Skycube outperforms the Skycube in update by several orders of magnitude.

Thank you! Tian Xia and Donghui Zhang. Refreshing the Sky: the Compressed Skycube with Efficient Support for Frequent Updates. SIGMOD 2006.

DB Seminar Schedule Seminar Schedule ================================================================= Chui Chun Kit30/11/07 Gong Jian Jim7/12/07 Loo Kin Kong14/12/07 Ngai Wang Kay Jackie21/12/07 Siu Wing Yan Angela4/1/08 Tam Ming Wai11/1/08 Tsang Pui Kwan Smith18/1/08 U Leong Hou Kamiru25/1/08 Wong Wai Kit1/2/08 Cui Yingjie Jason15/2/08 LEE King For22/2/08 Lin Zhifeng Arthur7/3/08 Yuan Wenjun Clement14/3/08 Zhang Shiming Simon28/3/08 Zhang Yiwei Kelvin11/4/08 LEE Yau Tat18/4/08 Pan Guodong Delvin25/4/08 Please send the abstract to dbgroup@cs.hku.hk one week before your talk dbgroup@cs.hku.hk

Our Motivations (2) t 4 4 3 6 1 t 9 2 2 3 7 t 7 1 3 4 1 t 1 3 4 2 5 t 5 2 2 3 1 t 6 6 1 1 3 u 1 u 2 u 3 u 4 t 2 4 6 7 2 t 3 9 7 5 6 t 8 6 5 3 8 t7t7 Cuboid Skyline u1u1 u2u2 u3u3 u4u4 u 1, u 2 u 1, u 3 u 1, u 4 u 2, u 3 u 2, u 4 u 3, u 4 u 1, u 2, u 3 u 1, u 2, u 4 u 1, u 3, u 4 u 2, u 3, u 4 u 1, u 2, u 3, u 4 t 5, t 6 t 1, t 5, t 6, t 7 t 5, t 6, t 7 t 5, t 6 t6t6 t7t7 t 1, t 5, t 6, t 7, t 9 t 5, t 6, t 7, t 9 t 5, t 7, t 4 t6t6 t6t6 Corresponding Skycube Full-space skyline objects Other skyline objects (not in full-space) t 1, t 5, t 6, t 7, t 9

u 1 u 2 u 3 u 4 Querying CSC L EMMA 1: Given a query space U q and an object t, if for any subspace U i in mss(t), U i  U q, then t is not in the skyline of U q. Lemma 1 implies two important facts: 1) Only the existing cuboids that  U q need to be searched. 2) No other cuboids need to be accessed or computed in the query process. Example: U q =  u 2, u 3, u 4 , and t 9 can be safely pruned. t 4 4 3 6 1 t 9 2 2 3 7 t 7 1 3 4 1 t 1 3 4 2 5 t 5 2 2 3 1 t 6 6 1 1 3 t7t7 Cuboid Skyline u1u1 u2u2 u3u3 u4u4 u 1, u 2 u 1, u 3 t 1, t 5, t 9 t 5, t 9 t 5, t 7, t 4 t6t6 t6t6

Some properties of CSC The number of non-empty cuboids is solely decided by sky(D).  In other words, there does not exist a cuboid which only contains objects not in sky(D).  Each non-empty cuboid in CSC contains at least one object in sky(D).  Therefore, as long as the full-space skyline is unchanged, no new cuboid will be added to CSC.

DB Seminar Schedule Seminar Schedule ================================================================= Chui Chun Kit30/11/07 Gong Jian Jim7/12/07 Loo Kin.

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DB Seminar Schedule Seminar Schedule ================================================================= Chui Chun Kit30/11/07 Gong Jian Jim7/12/07 Loo Kin.

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