Hash tables are very popular data structures. They store values that can be looked up by keys in constant time. When the number of keys is smaller than the total number of possible keys, this becomes very effective for lookup, insertion and deletion. If we are able to allocate an arraywhere is there one position for every possible key, we have what we call direct addressing. When the array index is computed from the key, we may have collisions which we solve by chaining values against the same key. Open addressing is another way to deal with collisions. When the search for a value takes constant time even in worst case, we have perfect hashing. The set of keys don't change once stored. Fixed hash functions are vulnerable  to malicious attacks where no keys hash to the same slot and degrading the performance to O (n) for lookups. This is resolved with universal hashing which picks out a hashing function at random from a set of hash functions at the beginning of the execution.  One such family is given by (((ak+b) mod p) mod m) where p is a prime number to denote the number of keys and is much larger than m the number of slots. Next, with open addressing the elements are not chained and they are stored in empty slots of the hash table itself. The sequence of positions probeddepend on the keys because they are associated with a permutation of the slots. We  make the assumption that each key is just as likely as the other based on the permutation. This is called uniform hashing. In practice the distribution of keys is not uniform and keys may not even be independent of each other.Three techniques are commonly used to compute the probe sequence required open addressing. These are linear, quadratic and double hashing. Double hashing gives the maximum number of probe sequences and hence better results. It takes theform h (k,i) = (h1 (k) +ih2 (k)) mod m. On order to improve the hashing to the constant time even for worst case such as for perfect hashing, we could use two levels of hash tables.

#codingexercise 
Double  GetNthRootProductEachTermRaisedPDividedQ (Double [] A,Double  p, Double q)
{

If ( A== null) return 0;

Return A.NthRootProductEachTermRaisedPDividedQ(p, q);

}