Data Warehousing modified by Donghui Zhang ©Jiawei Han and Micheline Kamber http://www-sal.cs.uiuc.edu/~hanj/bk2/ Chp 3 modified by Donghui Zhang 2018年9月18日星期二 Data Mining: Concepts and Techniques

Chapter 3: Data Warehousing and OLAP Technology for Data Mining What is a data warehouse? A multi-dimensional data model Data warehouse architecture Data warehouse implementation Further development of data cube technology 2018年9月18日星期二 Data Mining: Concepts and Techniques

Data Mining: Concepts and Techniques What is Data Warehouse? Defined in many different ways, but not rigorously. A decision support database that is maintained separately from the organization’s operational database Support information processing by providing a solid platform of consolidated, historical data for analysis. “A data warehouse is a subject-oriented, integrated, time-variant, and nonvolatile collection of data in support of management’s decision-making process.”—W. H. Inmon Data warehousing: The process of constructing and using data warehouses 2018年9月18日星期二 Data Mining: Concepts and Techniques

Data Warehouse—Subject-Oriented Organized around major subjects, such as customer, product, sales. Focusing on the modeling and analysis of data for decision makers, not on daily operations or transaction processing. Provide a simple and concise view around particular subject issues by excluding data that are not useful in the decision support process. 2018年9月18日星期二 Data Mining: Concepts and Techniques

Data Warehouse—Integrated Constructed by integrating multiple, heterogeneous data sources relational databases, flat files, on-line transaction records Data cleaning and data integration techniques are applied. Ensure consistency in naming conventions, encoding structures, attribute measures, etc. among different data sources E.g., Hotel price: currency, tax, breakfast covered, etc. When data is moved to the warehouse, it is converted. 2018年9月18日星期二 Data Mining: Concepts and Techniques

Data Warehouse—Time Variant The time horizon for the data warehouse is significantly longer than that of operational systems. Operational database: current value data. Data warehouse data: provide information from a historical perspective (e.g., past 5-10 years) Every key structure in the data warehouse Contains an element of time, explicitly or implicitly But the key of operational data may or may not contain “time element”. 2018年9月18日星期二 Data Mining: Concepts and Techniques

Data Warehouse—Non-Volatile A physically separate store of data transformed from the operational environment. Operational update of data does not occur in the data warehouse environment. Does not require transaction processing, recovery, and concurrency control mechanisms Requires only two operations in data accessing: initial loading of data and access of data. 2018年9月18日星期二 Data Mining: Concepts and Techniques

Data Warehouse vs. Heterogeneous DBMS Traditional heterogeneous DB integration: Build wrappers/mediators on top of heterogeneous databases Query driven approach When a query is posed to a client site, a meta-dictionary is used to translate the query into queries appropriate for individual heterogeneous sites involved, and the results are integrated into a global answer set Complex information filtering, compete for resources Data warehouse: update-driven, high performance Information from heterogeneous sources is integrated in advance and stored in warehouses for direct query and analysis 2018年9月18日星期二 Data Mining: Concepts and Techniques

Data Warehouse vs. Operational DBMS OLTP (on-line transaction processing) Major task of traditional relational DBMS Day-to-day operations: purchasing, inventory, banking, manufacturing, payroll, registration, accounting, etc. OLAP (on-line analytical processing) Major task of data warehouse system Data analysis and decision making 2018年9月18日星期二 Data Mining: Concepts and Techniques

Data Mining: Concepts and Techniques OLTP vs. OLAP 2018年9月18日星期二 Data Mining: Concepts and Techniques

Why Separate Data Warehouse? High performance for both systems DBMS— tuned for OLTP: access methods, indexing, concurrency control, recovery Warehouse—tuned for OLAP: complex OLAP queries, multidimensional view, consolidation. Different functions and different data: missing data: Decision support requires historical data which operational DBs do not typically maintain data consolidation: DS requires consolidation (aggregation, summarization) of data from heterogeneous sources data quality: different sources typically use inconsistent data representations, codes and formats which have to be reconciled 2018年9月18日星期二 Data Mining: Concepts and Techniques

Chapter 3: Data Warehousing and OLAP Technology for Data Mining What is a data warehouse? A multi-dimensional data model Data warehouse architecture Data warehouse implementation Further development of data cube technology 2018年9月18日星期二 Data Mining: Concepts and Techniques

From Tables and Spreadsheets to Data Cubes A data warehouse is based on a multidimensional data model which views data in the form of a data cube A data cube, such as sales, allows data to be modeled and viewed in multiple dimensions Dimension tables, such as item (item_name, brand, type), or time(day, week, month, quarter, year) Fact table contains measures (such as dollars_sold) and keys to each of the related dimension tables In data warehousing literature, an n-D base cube is called a base cuboid. The top most 0-D cuboid, which holds the highest-level of summarization, is called the apex cuboid. The lattice of cuboids forms a data cube. 2018年9月18日星期二 Data Mining: Concepts and Techniques

Cube: A Lattice of Cuboids all 0-D(apex) cuboid time item location supplier 1-D cuboids time,item time,location item,location location,supplier 2-D cuboids time,supplier item,supplier time,location,supplier time,item,location 3-D cuboids time,item,supplier item,location,supplier 4-D(base) cuboid 2018年9月18日星期二 time, item, location, supplier Data Mining: Concepts and Techniques

Conceptual Modeling of Data Warehouses Modeling data warehouses: dimensions & measures Star schema: A fact table in the middle connected to a set of dimension tables Snowflake schema: A refinement of star schema where some dimensional hierarchy is normalized into a set of smaller dimension tables, forming a shape similar to snowflake Fact constellations: Multiple fact tables share dimension tables, viewed as a collection of stars, therefore called galaxy schema or fact constellation 2018年9月18日星期二 Data Mining: Concepts and Techniques

Data Mining: Concepts and Techniques Example of Star Schema time_key day day_of_the_week month quarter year time item_key item_name brand type supplier_type item Sales Fact Table time_key item_key branch_key branch_key branch_name branch_type branch location_key street city state_or_province country location location_key units_sold dollars_sold avg_sales Measures 2018年9月18日星期二 Data Mining: Concepts and Techniques

Example of Snowflake Schema time_key day day_of_the_week month quarter year time item_key item_name brand type supplier_key item supplier_key supplier_type supplier Sales Fact Table time_key item_key branch_key location_key street city_key location branch_key branch_name branch_type branch location_key units_sold city_key city state_or_province country dollars_sold avg_sales Measures 2018年9月18日星期二 Data Mining: Concepts and Techniques

Example of Fact Constellation time_key day day_of_the_week month quarter year time item_key item_name brand type supplier_type item Shipping Fact Table Sales Fact Table time_key item_key time_key shipper_key item_key from_location branch_key branch_key branch_name branch_type branch location_key to_location location_key street city province_or_state country location dollars_cost units_sold units_shipped dollars_sold avg_sales shipper_key shipper_name location_key shipper_type shipper Measures 2018年9月18日星期二 Data Mining: Concepts and Techniques

A Data Mining Query Language: DMQL Cube Definition (Fact Table) define cube <cube_name> [<dimension_list>]: <measure_list> Dimension Definition ( Dimension Table ) define dimension <dimension_name> as (<attribute_or_subdimension_list>) Special Case (Shared Dimension Tables) First time as “cube definition” define dimension <dimension_name> as <dimension_name_first_time> in cube <cube_name_first_time> 2018年9月18日星期二 Data Mining: Concepts and Techniques

Defining a Star Schema in DMQL define cube sales_star [time, item, branch, location]: dollars_sold = sum(sales_in_dollars), avg_sales = avg(sales_in_dollars), units_sold = count(*) define dimension time as (time_key, day, day_of_week, month, quarter, year) define dimension item as (item_key, item_name, brand, type, supplier_type) define dimension branch as (branch_key, branch_name, branch_type) define dimension location as (location_key, street, city, province_or_state, country) 2018年9月18日星期二 Data Mining: Concepts and Techniques

Defining a Snowflake Schema in DMQL define cube sales_snowflake [time, item, branch, location]: dollars_sold = sum(sales_in_dollars), avg_sales = avg(sales_in_dollars), units_sold = count(*) define dimension time as (time_key, day, day_of_week, month, quarter, year) define dimension item as (item_key, item_name, brand, type, supplier(supplier_key, supplier_type)) define dimension branch as (branch_key, branch_name, branch_type) define dimension location as (location_key, street, city(city_key, province_or_state, country)) 2018年9月18日星期二 Data Mining: Concepts and Techniques

Defining a Fact Constellation in DMQL define cube sales [time, item, branch, location]: dollars_sold = sum(sales_in_dollars), avg_sales = avg(sales_in_dollars), units_sold = count(*) define dimension time as (time_key, day, day_of_week, month, quarter, year) define dimension item as (item_key, item_name, brand, type, supplier_type) define dimension branch as (branch_key, branch_name, branch_type) define dimension location as (location_key, street, city, province_or_state, country) define cube shipping [time, item, shipper, from_location, to_location]: dollar_cost = sum(cost_in_dollars), unit_shipped = count(*) define dimension time as time in cube sales define dimension item as item in cube sales define dimension shipper as (shipper_key, shipper_name, location as location in cube sales, shipper_type) define dimension from_location as location in cube sales define dimension to_location as location in cube sales 2018年9月18日星期二 Data Mining: Concepts and Techniques

Measures: Three Categories distributive: if the result derived by applying the function to n aggregate values is the same as that derived by applying the function on all the data without partitioning. E.g., count(), sum(), min(), max(). algebraic: if it can be computed by an algebraic function with M arguments (where M is a bounded integer), each of which is obtained by applying a distributive aggregate function. E.g., avg(), min_N(), standard_deviation(). holistic: if there is no constant bound on the storage size needed to describe a subaggregate. E.g., median(), mode(), rank(). 2018年9月18日星期二 Data Mining: Concepts and Techniques

A Concept Hierarchy: Dimension (location) all all Europe ... North_America region Germany ... Spain Canada ... Mexico country Vancouver ... city Frankfurt ... Toronto L. Chan ... M. Wind office 2018年9月18日星期二 Data Mining: Concepts and Techniques

View of Warehouses and Hierarchies Specification of hierarchies Schema hierarchy day < {month < quarter; week} < year Set_grouping hierarchy {1..10} < inexpensive 2018年9月18日星期二 Data Mining: Concepts and Techniques

Multidimensional Data Sales volume as a function of product, month, and region Dimensions: Product, Location, Time Hierarchical summarization paths Region Industry Region Year Category Country Quarter Product City Month Week Office Day Product Pick one node from each dimension hierarchy, you get a data cube! Month 2018年9月18日星期二 Data Mining: Concepts and Techniques

Data Mining: Concepts and Techniques A data cube all 0-D(apex) cuboid country product quarter 1-D cuboids product, quarter product,country quarter, country 2-D cuboids product, quarter, country 3-D(base) cuboid 2018年9月18日星期二 Data Mining: Concepts and Techniques

Data Mining: Concepts and Techniques Question: Suppose a data cube has three dimensions: Product: TV, PC, VCR Quarter: 1 Qtr, 2 Qtr, 3 Qtr, 4 Qtr Country: USA, Canada, Mexico Let there be one measure: total sales. Questions: How many values does the base cuboid (Product, Quarter, Country) has? How many values does the cuboid (Quarter, Country) have? How many values does the complete cube have? 2018年9月18日星期二 Data Mining: Concepts and Techniques

Data Mining: Concepts and Techniques A Sample Data Cube Total annual sales of TV in U.S.A. Quarter Product Country All, All, All sum TV VCR PC 1Qtr 2Qtr 3Qtr 4Qtr U.S.A Canada Mexico 2018年9月18日星期二 Data Mining: Concepts and Techniques

Data Mining: Concepts and Techniques Browsing a Data Cube Visualization OLAP capabilities Interactive manipulation 2018年9月18日星期二 Data Mining: Concepts and Techniques

Typical OLAP Operations Roll up (drill-up): summarize data by climbing up hierarchy or by dimension reduction Drill down (roll down): reverse of roll-up from higher level summary to lower level summary or detailed data, or introducing new dimensions Slice and dice: project and select Pivot (rotate): reorient the cube, visualization, 3D to series of 2D planes. 2018年9月18日星期二 Data Mining: Concepts and Techniques

Chapter 3: Data Warehousing and OLAP Technology for Data Mining What is a data warehouse? A multi-dimensional data model Data warehouse architecture Data warehouse implementation Further development of data cube technology 2018年9月18日星期二 Data Mining: Concepts and Techniques

Design of a Data Warehouse: A Business Analysis Framework Four views regarding the design of a data warehouse Top-down view allows selection of the relevant information necessary for the data warehouse Data source view exposes the information being captured, stored, and managed by operational systems Data warehouse view consists of fact tables and dimension tables Business query view sees the perspectives of data in the warehouse from the view of end-user 2018年9月18日星期二 Data Mining: Concepts and Techniques

Data Warehouse Design Process Top-down, bottom-up approaches or a combination of both Top-down: Starts with overall design and planning (mature) Bottom-up: Starts with experiments and prototypes (rapid) From software engineering point of view Waterfall: structured and systematic analysis at each step before proceeding to the next Spiral: rapid generation of increasingly functional systems, short turn around time, quick turn around Typical data warehouse design process Choose a business process to model, e.g., orders, invoices, etc. Choose the grain (atomic level of data) of the business process Choose the dimensions that will apply to each fact table record Choose the measure that will populate each fact table record 2018年9月18日星期二 Data Mining: Concepts and Techniques

Data Mining: Concepts and Techniques Multi-Tiered Architecture Operational DBs other sources Monitor & Integrator OLAP Server Metadata Extract Transform Load Refresh Analysis Query Reports Data mining Serve Data Warehouse Data Marts Data Sources Data Storage OLAP Engine Front-End Tools 2018年9月18日星期二 Data Mining: Concepts and Techniques

Three Data Warehouse Models Enterprise warehouse collects all of the information about subjects spanning the entire organization Data Mart a subset of corporate-wide data that is of value to a specific groups of users. Its scope is confined to specific, selected groups, such as marketing data mart Independent vs. dependent (directly from warehouse) data mart Virtual warehouse A set of views over operational databases Only some of the possible summary views may be materialized 2018年9月18日星期二 Data Mining: Concepts and Techniques

Data Warehouse Development: A Recommended Approach Multi-Tier Data Warehouse Distributed Data Marts Enterprise Data Warehouse Data Mart Data Mart Model refinement Model refinement Define a high-level corporate data model 2018年9月18日星期二 Data Mining: Concepts and Techniques

OLAP Server Architectures Relational OLAP (ROLAP) Use relational or extended-relational DBMS to store and manage warehouse data and OLAP middle ware to support missing pieces Include optimization of DBMS backend, implementation of aggregation navigation logic, and additional tools and services greater scalability Multidimensional OLAP (MOLAP) Array-based multidimensional storage engine (sparse matrix techniques) fast indexing to pre-computed summarized data Hybrid OLAP (HOLAP) User flexibility, e.g., low level: relational, high-level: array Specialized SQL servers specialized support for SQL queries over star/snowflake schemas 2018年9月18日星期二 Data Mining: Concepts and Techniques

Chapter 3: Data Warehousing and OLAP Technology for Data Mining What is a data warehouse? A multi-dimensional data model Data warehouse architecture Data warehouse implementation Further development of data cube technology 2018年9月18日星期二 Data Mining: Concepts and Techniques

Efficient Data Cube Computation Data cube can be viewed as a lattice of cuboids The bottom-most cuboid is the base cuboid The top-most cuboid (apex) contains only one cell How many cuboids in an n-dimensional cube with L levels? Materialization of data cube Materialize every (cuboid) (full materialization), none (no materialization), or some (partial materialization) Selection of which cuboids to materialize Based on size, sharing, access frequency, etc. 2018年9月18日星期二 Data Mining: Concepts and Techniques

Data Mining: Concepts and Techniques Cube Operation Cube definition and computation in DMQL define cube sales[item, city, year]: sum(sales_in_dollars) compute cube sales Transform it into a SQL-like language (with a new operator cube by, introduced by Gray et al.’96) SELECT item, city, year, SUM (amount) FROM SALES CUBE BY item, city, year Need compute the following Group-Bys (date, product, customer), (date,product),(date, customer), (product, customer), (date), (product), (customer) () (item) (city) () (year) (city, item) (city, year) (item, year) (city, item, year) 2018年9月18日星期二 Data Mining: Concepts and Techniques

Cube Computation: ROLAP-Based Method Efficient cube computation methods ROLAP-based cubing algorithms (Agarwal et al’96) Array-based cubing algorithm (Zhao et al’97) Bottom-up computation method (Beyer & Ramarkrishnan’99) H-cubing technique (Han, Pei, Dong & Wang:SIGMOD’01) ROLAP-based cubing algorithms Sorting, hashing, and grouping operations are applied to the dimension attributes in order to reorder and cluster related tuples Grouping is performed on some sub-aggregates as a “partial grouping step” Aggregates may be computed from previously computed aggregates, rather than from the base fact table 2018年9月18日星期二 Data Mining: Concepts and Techniques

Cube Computation: ROLAP-Based Method (2) This is not in the textbook but in a research paper Hash/sort based methods (Agarwal et. al. VLDB’96) Smallest-parent: computing a cuboid from the smallest, previously computed cuboid Cache-results: caching results of a cuboid from which other cuboids are computed to reduce disk I/Os Amortize-scans: computing as many as possible cuboids at the same time to amortize disk reads Share-sorts: sharing sorting costs cross multiple cuboids when sort-based method is used Share-partitions: sharing the partitioning cost across multiple cuboids when hash-based algorithms are used 2018年9月18日星期二 Data Mining: Concepts and Techniques

Multi-way Array Aggregation for Cube Computation Partition arrays into chunks (a small subcube which fits in memory). Compressed sparse array addressing: (chunk_id, offset) Compute aggregates in “multiway” by visiting cube cells in the order which minimizes the # of times to visit each cell, and reduces memory access and storage cost. A B 29 30 31 32 1 2 3 4 5 9 13 14 15 16 64 63 62 61 48 47 46 45 a1 a0 c3 c2 c1 c 0 b3 b2 b1 b0 a2 a3 C 44 28 56 40 24 52 36 20 60 What is the best traversing order to do multi-way aggregation? 2018年9月18日星期二 Data Mining: Concepts and Techniques

Multi-way Array Aggregation for Cube Computation 29 30 31 32 1 2 3 4 5 9 13 14 15 16 64 63 62 61 48 47 46 45 a1 a0 c3 c2 c1 c 0 b3 b2 b1 b0 a2 a3 C 44 28 56 40 24 52 36 20 60 B 2018年9月18日星期二 Data Mining: Concepts and Techniques

Multi-way Array Aggregation for Cube Computation 61 62 63 64 c2 45 46 47 48 c1 29 30 31 32 c 0 B 13 14 15 16 60 b3 44 B 28 b2 9 56 40 24 b1 5 52 36 20 b0 1 2 3 4 Order: ABC AB: plane AC: line BC: point a0 a1 a2 a3 A 2018年9月18日星期二 Data Mining: Concepts and Techniques

Multi-Way Array Aggregation for Cube Computation (Cont.) Let A: 40 values, B: 400 values, C: 4000 values. One chunk contains 10*100*1000 = 1,000,000 values. ABC needs how much memory? AB plane: 40*400=16,000 AC line: 40*(4000/4) = 40,000 BC point: (400/4)*(4000/4) = 100,000 total: 156,000 CBA needs how much memory? CB plane: 4000*400=1,600,000 CA line: 4000*(40/4) = 40,000 BA point: (400/4)*(40/4) = 1000 total: 1,641,000 --- 10 times more! 2018年9月18日星期二 Data Mining: Concepts and Techniques

Chapter 3: Data Warehousing and OLAP Technology for Data Mining What is a data warehouse? A multi-dimensional data model Data warehouse architecture Data warehouse implementation Further development of data cube technology 2018年9月18日星期二 Data Mining: Concepts and Techniques

Iceberg Cube Computing only the cuboid cells whose count or other aggregates satisfying the condition: HAVING COUNT(*) >= minsup Motivation Only a small portion of cube cells may be “above the water’’ in a sparse cube Only calculate “interesting” data—data above certain threshold Suppose 100 dimensions, only 1 base cell. How many aggregate (non-base) cells if count >= 1? What about count >= 2? 2018年9月18日星期二 Data Mining: Concepts and Techniques

Data Mining: Concepts and Techniques Example data P L M sale p1 l1 m1 * p2 p3 m2 p4 l2 p5 p6 l3 dimensions: P – 6 product L – 3 location, M – 2 month. Base cuboid: 6*3*2 = 36 cells. Here 30 are empty. Especially when #dimensions is large, should store in a compressed way. Iceburg cube query asks for non-empty cuboids! E.g. is cuboid (P,L) empty? SELECT P, L, M, COUNT(*) FROM Sales CUBE BY P, L, M HAVING COUNT(*)>=2 2018年9月18日星期二 Data Mining: Concepts and Techniques

Data Mining: Concepts and Techniques Naïve approach all P L M count(*) p1 l1 m1 1 p2 p3 m2 p4 l2 p5 p6 l3 M P L P, L P, M L, M P, L, M Given (P, L, M), calculate all cuboids bottom-up. In every cuboid, delete tuples whose count(*)<2. ?? Can we apply count(*)>=2 to (P, L, M) before calculating cuboids? Drawback: calculate the complete iceberg including underwater! 2018年9月18日星期二 Data Mining: Concepts and Techniques

Computing iceberg cube using BUC BUC (Beyer & Ramakrishnan, SIGMOD’99) Bottom-up vs. top-down?—depending on how you view it! Apriori property: Aggregate the data, then move to the next level If minsup is not met, stop! 2018年9月18日星期二 Data Mining: Concepts and Techniques

Computing iceberg cube using BUC L M count(*) p1 l1 m1 1 p2 p3 m2 p4 l2 p5 p6 l3 4 P,L,M 3 P,L 5 P,M 7 L,M 2 P 6 L 8 M 1 all p1: 1 p2: 1 p3: 1 p4: 1 p5: 1 p6: 1 Icerberg: all: ()6 2018年9月18日星期二 Data Mining: Concepts and Techniques

Computing iceberg cube using BUC L M count(*) p1 l1 m1 1 p2 p3 m2 p4 l2 p5 p6 l3 4 P,L,M 3 P,L 5 P,M 7 L,M 2 P 6 L 8 M 1 all m1: 2 m2: 1 Icerberg: all: ()6 L: (l1)3, (l2)2 L,M: (l1,m1)2 l1: 3 m1: 1 m2: 1 l2: 2 l3: 1 2018年9月18日星期二 Data Mining: Concepts and Techniques

Computing iceberg cube using BUC L M count(*) p1 l1 m1 1 p2 p3 m2 p4 l2 p5 p6 l3 4 P,L,M 3 P,L 5 P,M 7 L,M 2 P 6 L 8 M 1 all Icerberg: all: ()6 L: (l1)3, (l2)2 L,M: (l1,m1)2 M: (m1)4, (m2)2 m1: 4 m2: 2 2018年9月18日星期二 Data Mining: Concepts and Techniques

Range-sum query in a data cube Problem: compute range-sum, e.g. SELECT SUM(C.sales) FROM Customer C WHERE 37<=C.age<=52 and 3<=C.month<=7 Let each dim have n values. query cost = ? O(n2) update (one cell) cost = ? O(1) month 1 2 3 4 5 6 7 8 20 25 33 37 40 52 58 59 age 2018年9月18日星期二 Data Mining: Concepts and Techniques

Data Mining: Concepts and Techniques Prefix-sum solution original cube prefix-sum cube 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 10 12 14 16 9 15 18 21 24 20 28 32 25 30 35 40 36 42 48 49 56 64 Every cell stores the prefix-sum. Can answer a prefix-sum query with cost 1. Update cost? O(n2)! range-sum query cost? 2018年9月18日星期二 Data Mining: Concepts and Techniques

Data Mining: Concepts and Techniques Prefix-sum solution 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 – = 42 12 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 + – query cost = O(1) 21 6 2018年9月18日星期二 Data Mining: Concepts and Techniques

Data Mining: Concepts and Techniques What we have query cost update cost store original cube O(n2) O(1) store prefix-sum Can we do better? dynamic data cube (naïve version) O(log(n)) O(n) 2018年9月18日星期二 Data Mining: Concepts and Techniques

The Dynamic Data Cube [EDBT’00] 1..4 5..8 1..2 3..4 1..2 3..4 5..6 7..8 5..6 7..8 1..2 3..4 5..6 7..8 1 2 1 1 3 Organize the original cube into a tree structure with fanout = 4. Data is only stored in the leaf node. Each index entry is augmented with… 1 1 4 2018年9月18日星期二 Data Mining: Concepts and Techniques

The Dynamic Data Cube [EDBT’00] 1..4 5..8 4 8 12 16 4 8 12 16 1 4 8 12 16 4 8 12 16 1 1..4 4 8 12 16 4 8 12 16 1 4 8 12 16 4 8 12 16 1 5..8 Each index entry is augmented with an X-border (Y-border), which stores the prefix-sums that: only consider cells in the sub-tree; and cover all rows (columns). 2018年9月18日星期二 Data Mining: Concepts and Techniques

The Dynamic Data Cube [EDBT’00] 1..4 5..8 4 8 12 16 4 8 12 16 1 4 8 12 16 4 8 12 16 1 1..4 4 8 12 16 4 8 12 16 1 4 8 12 16 4 8 12 16 1 5..8 A prefix-sum range is broken down into (at most) four pieces. Three calculated by checking the borders at the root. Examine a single sub-tree! 2018年9月18日星期二 Data Mining: Concepts and Techniques

The Dynamic Data Cube [EDBT’00] 1..4 5..8 4 8 12 16 4 8 12 16 1 4 8 12 16 4 8 12 16 1 1..4 4 8 12 16 4 8 12 16 1 4 8 12 16 4 8 12 16 1 5..8 A prefix-sum range is broken down into (at most) four pieces. Three calculated by checking the borders at the root. Examine a single sub-tree! 2018年9月18日星期二 Data Mining: Concepts and Techniques

The Dynamic Data Cube [EDBT’00] 1..4 5..8 4 8 12 16 4 8 12 16 1 4 8 12 16 4 8 12 16 1 1..4 4 8 12 16 4 8 12 16 1 4 8 12 16 4 8 12 16 1 5..8 A prefix-sum range is broken down into (at most) four pieces. Three calculated by checking the borders at the root. Examine a single sub-tree! 2018年9月18日星期二 Data Mining: Concepts and Techniques

The Dynamic Data Cube [EDBT’00] 1..4 5..8 4 8 12 16 4 8 12 16 1 4 8 12 16 4 8 12 16 1 1..4 4 8 12 16 4 8 12 16 1 4 8 12 16 4 8 12 16 1 5..8 A prefix-sum range is broken down into (at most) four pieces. Three calculated by checking the borders at the root. Examine a single sub-tree! 2018年9月18日星期二 Data Mining: Concepts and Techniques

The Dynamic Data Cube [EDBT’00] 1..4 5..8 4 8 12 16 4 8 12 16 1 4 8 12 16 4 8 12 16 1 1..4 Query the sub-tree, get 6. 4 8 12 16 4 8 12 16 1 4 8 12 16 4 8 12 16 1 5..8 A prefix-sum range is broken down into (at most) four pieces. Three calculated by checking the borders at the root. Examine a single sub-tree! 2018年9月18日星期二 Data Mining: Concepts and Techniques

The Dynamic Data Cube [EDBT’00] 1..4 5..8 4 8 12 16 4 8 12 16 1 4 8 12 16 4 8 12 16 1 1..4 4 8 12 16 4 8 12 16 1 4 8 12 16 4 8 12 16 1 5..8 E.g. 16+12+8+6 = 42. Query cost: O(log(n)) Update cost? O(n) 2018年9月18日星期二 Data Mining: Concepts and Techniques

The Dynamic Data Cube [EDBT’00] 1..4 5..8 4 8 12 16 4 8 12 16 1 4 8 12 16 4 8 12 16 1 1..4 4 8 12 16 4 8 12 16 1 4 8 12 16 4 8 12 16 1 5..8 Update cost? At root, (up to) n border cells to modify. One level down, n/2 cells to modify…. total: O(n) 2018年9月18日星期二 Data Mining: Concepts and Techniques

Data Mining: Concepts and Techniques What we have query cost update cost store original cube O(n2) O(1) store prefix-sum Can we do better? dynamic data cube (naïve version) O(log(n)) O(n) Can we do better? dynamic data cube O(log2n) 2018年9月18日星期二 Data Mining: Concepts and Techniques

Key to reducing from O(n) to O(log2n) For each update, should update an X-border and a Y-border at each level. How to update an X-border in logarithmic time? More formally, given an array A[1..n] = (4, 8, 12, 16). How to add v to all elements from A[i] to A[n]? Idea: maintain a binary tree. 3 4 11 12 16 A[1] A[2] A[3] A[4] add 3 to A[2..4] 4 8 12 16 A[1] A[2] A[3] A[4] 2018年9月18日星期二 Data Mining: Concepts and Techniques

Dynamic Data Cube summary A balanced tree with fanout=4. The leaf nodes contains the original data cube. Each index entry stores an X-border and an Y-border. Each border is stored as a binary tree, which supports a 1-dim prefix-sum query and an update in O(log(n)) time. Overall, the DDC supports a range-sum query and an update both in O(log2n) time. 2018年9月18日星期二 Data Mining: Concepts and Techniques

Data Mining: Concepts and Techniques Summary Data warehouse A multi-dimensional model of a data warehouse Star schema, snowflake schema, fact constellations A data cube consists of dimensions & measures OLAP operations: drilling, rolling, slicing, dicing and pivoting OLAP servers: ROLAP, MOLAP, HOLAP Efficient computation of data cubes Partial vs. full vs. no materialization Multiway array aggregation Further development of data cube technology Iceberg Cube Dynamic Data Cube 2018年9月18日星期二 Data Mining: Concepts and Techniques