In-database connectivity components analysis of data

1. A computer-implemented method for determining connectivity components defined by an input set of relations over an input set of data elements, wherein a connectivity component is a subset of the input set data elements that are pair-wise connected such that no other data element of the input set can be added to the subset that is connected to any of the data elements of the subset, and wherein two data elements are connected whenever the two data elements are related by a relation in the set of relations or a path of relations between data elements exists that connects the two data elements, the method comprising:

performing the following actions of selecting and forming iteratively beginning with a set of data elements that is the input set of data elements and a set of relations that is the input set of relations, and at each subsequent iteration using the contracted set of data elements as the set of data elements and the contracted set of relations as the set of relations, until the contracted set of relations is empty:

selecting a respective representative for each data element, each representative being a data element in the set of data elements, to form a contracted set of data elements, the selecting comprising:

assigning a random number to each data element in the set of data elements;

identifying for each data element a respective group of data elements that includes the data element and all other data elements that are related to the data element;

selecting as the respective representative a particular data element in the respective group that has an assigned random number with a predetermined random number position among the data elements in the respective group; and

replacing each data element with the representative of the data element to form a contracted set of data elements;

forming a contracted set of relations over the contracted set of data elements, the forming comprising:

defining two representatives in the contracted set of data elements as related in the contracted set of relations if the two representatives represent data elements that were connected in the set of relations; and

defining all other pairs of data elements in the contracted set of data elements as not related in the contracted set of relations;

whereby, when the contracted set of relations is empty, each data element in the contracted set of data elements is an isolated data element, an isolated data element being a data element that is not related to any other data element in the contracted set of data elements; and

outputting each isolated data element as a respective representative of a respective connectivity component of the input set of data elements, each connectivity component of the input set of data elements being representing by a distinct isolated data element.

2. The method of claim 1 , wherein the random number is a pseudorandom number between zero and one.

3. The method of claim 1 , wherein the predetermined random number position is the highest random number.

4. The method of claim 1 , wherein each data element has a respective weight and a respective leader, the method further comprising:

before performing the actions of selecting and forming:

assigning a common weight to each data element in the input set of data elements; and

nominating each data element in the input set of data elements as its own leader.

5. The method of claim 4 , wherein selecting a respective representative comprises:

assigning as the weight of each respective representative a sum of the weights of the data elements represented by the respective representative; and

for each first data element in the set of data elements, identifying the data elements nominating the first data element and replacing the nominations of the first data element with nominations nominating the representative of the first data element.

6. The method of claim 5 , comprising:

outputting the weight of each representative as a size of the connectivity component represented by the representative.

7. The method of claim 6 , wherein the common weight is one.

8. The method of claim 6 , comprising:

outputting the leader of each data element of the input set of data elements as an identifier of the connectivity component of which the data element is a member.

9. The method of claim 1 , comprising:

after each iteration of selecting and forming, outputting each isolated data element in the contracted set of data elements and removing the isolated data element from the set of contracted data elements.

10. Non-transitory computer-readable storage media encoded with computer program instructions for determining connectivity components defined by an input set of relations over an input set of data elements, wherein a connectivity component is a subset of the input set data elements that are pair-wise connected such that no other data element of the input set can be added to the subset that is connected to any of the data elements of the subset, and wherein two data elements are connected whenever the two data elements are related by a relation in the set of relations or a path of relations between data elements exists that connects the two data elements, the instructions when executed by one or more computers causing the one or more computers to perform operations comprising:

performing the following actions of selecting and forming iteratively beginning with a set of data elements that is the input set of data elements and a set of relations that is the input set of relations, and at each subsequent iteration using the contracted set of data elements as the set of data elements and the contracted set of relations as the set of relations, until the contracted set of relations is empty:

selecting a respective representative for each data element, each representative being a data element in the set of data elements, to form a contracted set of data elements, the selecting comprising:

assigning a random number to each data element in the set of data elements;

identifying for each data element a respective group of data elements that includes the data element and all other data elements that are related to the data element;

selecting as the respective representative a particular data element in the respective group that has an assigned random number with a predetermined random number position among the data elements in the respective group; and

replacing each data element with the representative of the data element to form a contracted set of data elements;

forming a contracted set of relations over the contracted set of data elements, the forming comprising:

defining two representatives in the contracted set of data elements as related in the contracted set of relations if the two representatives represent data elements that were connected in the set of relations; and

defining all other pairs of data elements in the contracted set of data elements as not related in the contracted set of relations;

whereby, when the contracted set of relations is empty, each data element in the contracted set of data elements is an isolated data element, an isolated data element being a data element that is not related to any other data element in the contracted set of data elements; and

outputting each isolated data element as a respective representative of a respective connectivity component of the input set of data elements, each connectivity component of the input set of data elements being representing by a distinct isolated data element.

11. The media of claim 10 , wherein each data element has a respective weight and a respective leader, the operations further comprising:

before performing the actions of selecting and forming:

assigning a common weight to each data element in the input set of data elements; and

nominating each data element in the input set of data elements as its own leader.

12. The media of claim 11 , wherein selecting a respective representative comprises:

assigning as the weight of each respective representative a sum of the weights of the data elements represented by the respective representative; and

for each first data element in the set of data elements, identifying the data elements nominating the first data element and replacing the nominations of the first data element with nominations nominating the representative of the first data element.

13. The media of claim 12 , wherein the operations comprise:

outputting the weight of each representative as a size of the connectivity component represented by the representative.

14. The media of claim 13 , wherein the common weight is one.

15. The media of claim 13 , wherein the operations comprise:

outputting the leader of each data element of the input set of data elements as an identifier of the connectivity component of which the data element is a member.

16. The media of claim 10 , wherein the operations comprise:

after each iteration of selecting and forming, outputting each isolated data element in the contracted set of data elements and removing the isolated data element from the set of contracted data elements.

17. A database system comprising one or more computers programmed for determining connectivity components defined by an input set of relations over an input set of data elements, wherein a connectivity component is a subset of the input set data elements that are pair-wise connected such that no other data element of the input set can be added to the subset that is connected to any of the data elements of the subset, and wherein two data elements are connected whenever the two data elements are related by a relation in the set of relations or a path of relations between data elements exists that connects the two data elements, wherein the one or more computers of the database system are programmed perform operations comprising:

performing the following actions of selecting and forming iteratively beginning with a set of data elements that is the input set of data elements and a set of relations that is the input set of relations, and at each subsequent iteration using the contracted set of data elements as the set of data elements and the contracted set of relations as the set of relations, until the contracted set of relations is empty:

selecting a respective representative for each data element, each representative being a data element in the set of data elements, to form a contracted set of data elements, the selecting comprising:

assigning a random number to each data element in the set of data elements;

identifying for each data element a respective group of data elements that includes the data element and all other data elements that are related to the data element;

selecting as the respective representative a particular data element in the respective group that has an assigned random number with a predetermined random number position among the data elements in the respective group; and

replacing each data element with the representative of the data element to form a contracted set of data elements;

forming a contracted set of relations over the contracted set of data elements, the forming comprising:

defining two representatives in the contracted set of data elements as related in the contracted set of relations if the two representatives represent data elements that were connected in the set of relations; and

defining all other pairs of data elements in the contracted set of data elements as not related in the contracted set of relations;

whereby, when the contracted set of relations is empty, each data element in the contracted set of data elements is an isolated data element, an isolated data element being a data element that is not related to any other data element in the contracted set of data elements; and

outputting each isolated data element as a respective representative of a respective connectivity component of the input set of data elements, each connectivity component of the input set of data elements being representing by a distinct isolated data element.

18. The database system of claim 17 , wherein each data element has a respective weight and a respective leader, the operations further comprising:

before performing the actions of selecting and forming:

assigning a common weight to each data element in the input set of data elements; and

nominating each data element in the input set of data elements as its own leader;

wherein selecting a respective representative comprises:

assigning as the weight of each respective representative a sum of the weights of the data elements represented by the respective representative; and

for each first data element in the set of data elements, identifying the data elements nominating the first data element and replacing the nominations of the first data element with nominations nominating the representative of the first data element; and

wherein the operations comprise:

outputting the weight of each representative as a size of the connectivity component represented by the representative; and

outputting the leader of each data element of the input set of data elements as an identifier of the connectivity component of which the data element is a member.