Parameters measuring the size of set families

Effective Range

Name: Effective Range

Symbol: $\mathrm{E}$

Definition:

The effective range $E(\mathcal{H})$ of a family $\mathcal{H}$ is the number of elements that can be distinguished by members of $\mathcal{H}$: $$ E(\mathcal{H}):= |{x:x\in\mathcal{X|,\, \exists h_1,h_2\in\mathcal{H},\, x\in h_1 \Delta h_2}} $$

Category: Basic

Symmetric: Yes

Monotonic: Yes

P-Monotonic: Yes

Doubly Monotonic: Yes

Strictly Monotonic: Yes

Values:

Effective Range of Trivial concepts
Effective Range of Tagged Singletons
Effective Range of Addressing
Effective Range of Majority
Effective Range of Half-intervals
Effective Range of Singletons plus empty set
Effective Range of Full Cube
Effective Range of Singletons

Relationships (as Parameter 1):

Effective Range / Yang dimension
Effective Range / Distinguishing Range
Effective Range / VC Dimension
Effective Range / Diameter
Effective Range / Hamming Radius
Effective Range / Log Shattering
Effective Range / Projected Membership Queries Complexity

Relationships (as Parameter 2):

Log Shattering / Effective Range