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n13 - CS 70 Discrete Mathematics and Probability Theory...

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CS 70 Discrete Mathematics and Probability Theory Fall 2010 Tse/Wagner Note 13 A Killer Application In this lecture, we will see a “killer app” of elementary probability in Computer Science. Suppose a hash function distributes keys evenly over a table of size n . How many (randomly chosen) keys can we hash before the probability of a collision exceeds (say) 1 2 ? As we shall see, this question can be tackled by an analysis of the balls-and-bins probability space which we have already encountered. Application: Hash functions As you may already know, a hash table is a data structure that supports the storage of sets of keys from a (large) universe U (say, the names of all 250 million people in the US). The operations supported are ADD ing a key to the set, DELETE ing a key from the set, and testing MEMBER ship of a key in the set. The hash function h maps U to a table T of modest size. To ADD a key x to our set, we evaluate h ( x ) (i.e., apply the hash function to the key) and store x at the location h ( x ) in the table T . All keys in our set that are mapped to the same table location are stored in a simple linked list. The operations DELETE and MEMBER are implemented in similar fashion, by evaluating h ( x ) and searching the linked list at h ( x ) . Note that the efficiency of a hash function depends on having only few collisions —i.e., keys that map to the same location. This is because the search time for DELETE and MEMBER operations is proportional to the length of the corresponding linked list. The question we are interested in here is the following: suppose our hash table T has size n , and that our hash function h distributes U evenly over T . 1 Assume that the keys we want to store are chosen uniformly at random and independently from the universe U . What is the largest number, m , of keys we can store before the probability of a collision reaches 1 2 ? Let’s begin by seeing how this problem can be put into the balls and bins framework. The balls will be the m keys to be stored, and the bins will be the n locations in the hash table T . Since the keys are chosen uniformly and independently from U , and since the hash function distributes keys evenly over the table, we can see each key (ball) as choosing a hash table location (bin) uniformly and independently from T . Thus the probability space corresponding to this hashing experiment is exactly the same as the balls and bins space. We are interested in the event A that there is no collision, or equivalently, that all m balls land in different bins. Clearly Pr [ A ] will decrease as m increases (with n fixed). Our goal is to find the largest value of m such that Pr [ A ] remains above 1 2 . Let’s fix the value of m and try to compute Pr [ A ] . Since our probability space is uniform (each outcome has probability 1 n m ), it’s enough just to count the number of outcomes in A . In how many ways can we arrange m balls in n bins so that no bin contains more than one ball? Well, there are n places to put the first ball, then n -

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n13 - CS 70 Discrete Mathematics and Probability Theory...

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