at dix, i.e., it defines the graph containing all the nodes and edges reachable from di-x Elements(E). Each dimension di in a cube is represented by a set of elements Er. For instance dimension Movie in our example is represented by all the movies available for the users to rate. For simplicity and without loss of generality, we use the domain of the key dimension attribute to represent the set of elements of d, i.e., E;: dom(root(Hi). An example of the elements' set for the User dimension would be a set of all user IDs available in the data. Let E=( El,..., Eni Measures(M). M=(mI, m2, ., mk) represents a set of measures, where each m, is a different type of a rating from domain dom(mi). The measures can either be numeric or Boolean. A numeric measure usually represents a discrete finite ordered value, e.g, a movie rating on the scale of (1,., N.A Boolean measure can be used to represent a"status flag denoting the state of a rating or its specific characteristic, e.g., indicating whether a given movie has been seen by a given user Example 1. Consider the application for recommending movies to users that has the following dimensions, each dimension defined by the attributes specified in parentheses Movie: the set of all the movies that can be recommended it is defined as Movie(MovieID Title, Length, Release Y ear, Director, Genre) User: the people to whom movies are recommended; it is defined as User(UserlD, Name, Address, Age, Gender, Profession) Theater: the movie theaters showing the movies; it is defined as Theater(TheaterlD, Name Address, Capacity, City, Time: the time when the movie can be or has been seen; it is defined as Time(Date, DayOfWeek, TimeofWeek, Month, Quarter, Year) ompanion: represents a person or a group of persons with whom one can see the movie. It defined as Companion( companion Type), where attribute companionType has values and“ others We also use three rating measures in this example: PublicRating, a numeric measure specifying how much the general public liked the movie; Personal Rating, a numeric measure specifying how much a particular person liked or is predicted to like the movie in the settings specified by the Time, Theater, and Companion dimensions; and Consumed, a Boolean measure specifying whether or not a given user has actually seen a given movie in a given context. The Personal Rating assigned to a movie by a person depends on where and how the movie has been7 at di.x, i.e., it defines the graph containing all the nodes and edges reachable from di.x. Elements (E). Each dimension di in a cube is represented by a set of elements Ei. For instance, dimension Movie in our example is represented by all the movies available for the users to rate. For simplicity and without loss of generality, we use the domain of the key dimension attribute to represent the set of elements of di, i.e., Ei := dom(Root(Hi)). An example of the elements’ set for the User dimension would be a set of all user IDs available in the data. Let E = { E1, …, En }. Measures (M). M = {m1, m2, …, mk} represents a set of measures, where each mi is a different type of a rating from domain dom(mi). The measures can either be numeric or Boolean. A numeric measure usually represents a discrete finite ordered value, e.g., a movie rating on the scale of {1, …, N}. A Boolean measure can be used to represent a “status flag” denoting the state of a rating or its specific characteristic, e.g., indicating whether a given movie has been seen by a given user. Example 1. Consider the application for recommending movies to users that has the following dimensions, each dimension defined by the attributes specified in parentheses: • Movie: the set of all the movies that can be recommended; it is defined as Movie(MovieID, Title, Length, ReleaseYear, Director, Genre). • User: the people to whom movies are recommended; it is defined as User(UserID, Name, Address, Age, Gender, Profession). • Theater: the movie theaters showing the movies; it is defined as Theater(TheaterID, Name, Address, Capacity, City, State, Country). • Time: the time when the movie can be or has been seen; it is defined as Time(Date, DayOfWeek, TimeOfWeek, Month, Quarter, Year). • Companion: represents a person or a group of persons with whom one can see the movie. It is defined as Companion(companionType), where attribute companionType has values “alone”, “friends”, “girlfriend/boyfriend”, “family”, “co-workers”, and “others”. We also use three rating measures in this example: PublicRating, a numeric measure specifying how much the general public liked the movie; PersonalRating, a numeric measure specifying how much a particular person liked or is predicted to like the movie in the settings specified by the Time, Theater, and Companion dimensions; and Consumed, a Boolean measure specifying whether or not a given user has actually seen a given movie in a given context. The PersonalRating assigned to a movie by a person depends on where and how the movie has been