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The cardinality of a collection A (which might be an ordered or unordered list, a set, or what not) is basically the number of unique values in A. For example, the collections [1,2,3,4] and [1,2,1,3,1,4,3] have the same cardinality of 4 (and also correspond to the same set). Determing the Cardinality of a Collection: The Naive Approach Consider a collection A=[1,2,1,3,1,4,3]. How can we systematically determine the cardinality of A? Well, here are two of many ways to do this: • First sort A in ascending order. Then, we can perform a linear scan on A to remove duplicates. It’s pretty easy to see how this can be done.

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Finally, return the size of the possibly trickled-down collection (now a set) obtained. If the initial size of A is. Then, the cardinality of A, using this method, can be determined in (if we use merge-sort) time and extra space.

• Use a hash table: Perform a linear scan of A, hashing the values of A. It’s easy to see that cardinality of A is the number of keys in the hash table obtained. This uses time but extra space also. Notice that we can’t do any better (lower upper-bound) than because we have to look at the entire input (which is of size ). But can we determine the cardinality of A in time using sub-linear space (using strictly smaller space than )? That’s where probability comes in. Linear Probabilistic Counting This is a probabilistic algorithm for counting the number of unique values in a collection. It produces an estimation with an arbitrary accuracy that can be pre-specified by the user using only a small amount of space that can also be pre-specified. The accuracy of linear counting depends on the load factor (think hash tables) which is the number of unique values in the collection divided by the size of the collection.

The larger the load factor, the less accurate the result of the linear probabilistic counter. Correspondingly, the smaller the load factor, the more accurate the result. Nevertheless, load factors much higher than 1 (e.g. 16) can be used while achieving high accuracy in cardinality estimation (e.g.