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occupied addresses tend to aggregate forming clusters. Hence, if the table is not quite sparse, we will eventually have to travel through the entire table before finding an empty location to either insert a key or stop in the case of an unsuccessful search. To alleviate the problem, linear probing can be replaced by stepping through the table more than one node at a time in a pseudo-random manner, deterministic in the sense that for a given key hashing to a given address, the sequence of probes is the same each time it is invoked. That is, the size of the step called probe decrement should randomly depend on the key rather than being always 1. Also, the process must be able to examine all nodes in the table in an unbroken wrap-around cycle. Random sequencing virtually eliminates primary clustering by making it unlikely for a new empty location to be chosen next to an occupied one. One way to implement this plan is to hash a key, after it has been hashed once, again, this time using a slightly different hash function to obtain a probe decrement for the key. Such a policy of resolving collisions called double hashing has the overhead of the second hashing, but leads to separate probing sequences for different keys spreading empty and occupied nodes evenly across the table. The question is, how to select the second hash function to satisfy the requirements formulated above? It follows from the number theory that if the probe decrement C and the hash table size HSIZE are relatively prime, then if, starting at some address, we step through the table C nodes at a time in a wrap-around manner, each node will be covered exactly once. For instance, if HSIZE is prime, C can be any integer between 1 and HSIZE-1 inclusive, or if HSIZE is a power of 2, C can be any odd integer between 1 and logHSIZE-1. In the program below solving our sample file match problem by means of double hashing, HSIZE is left prime and C is calculated as C1MOD KEY, HSIZE-2. This has been proven to work satisfactorily in most cases. DOUBLE HASHING TABLE LOAD AND SEARCH; DATA MATCH KEEP KEYS SATLSAT; ARRAYHKEY 0: ARRAYHSAT 0: DO UNTIL EOF1; SET SMALL ENDEOF1; HMOD KEY, IF HKEYH EQ. THEN LINK INSERT; ELSE IF HKEYH NE KEY THEN DO; C1MOD KEY, IPROBE: H-C; IFHLT 0THENH IFHKEYH EQ. THEN LINK INSERT; ELSE IF HKEYH NE KEY THENGOTOIPROBE; END; END; DO UNTIL EOF2; SET LARGE ENDEOF2; HMOD KEY, IF HKEYH EQ. THEN FOUND0; ELSE IF HKEYH EQ KEY THEN FOUND1; ELSE DO; C1MOD KEY, SPROBE: H-C; IFHLT 0THENH IF HKEY H EQ. THEN FOUND0; ELSE IF HKEY H EQ KEY THEN FOUND1; ELSEGOTOSPROBE; END; IF NOT FOUND THEN CONTINUE; SSATHSATH; OUTPUT; END; STOP; INSERT: HKEYH KEY ; HSATH SSAT; RUN; Once again, those revering gotoless programming are encouraged to make the code prettier without hindering performance. If the satellites are not needed the array HSAT and instructions referring to it should be omitted. Let us take a look at the performance Table 1. With a 50 per cent sparse table, double hashing runs just a bit slower than coalesced chaining with 20 percent sparsity but uses less memory. So, double hashing is quite fast; it even loads an equally sparse table a tad faster than chaining. That a method based on stepping through the table before an empty node is found works so well, may seem surprising; however, this is a direct result of the double hashing probing methodology. In fact, independent experiments show that if the table is less than half full, double hashing makes on the average no more than 2 comparisons per miss, and no more than 1.3 comparisons per hit. HASHING WITH NON-INTEGER KEYS As the tests show, hashing performs admirably by any account regardless of the collision resolution policy used. To this end, however, we have only eliminated the range restriction of key - indexing and bitmapping, since the assumption of the keys being non-negative integers has been used in both implementations. Fortunately, with hashing, the nature of the keys is not critical because in its final stage, it is strictly comparison-based. Both hashes and traversals are used merely to minimize the number of comparisons necessary to carry out a search, yet the final decision if a hash address is not empty is made by comparing some keys in the table to the search key. Therefore, in order to be able to operate on keys of any type, we only have to figure out how to hash a key if it is not a positive integer. For the hash function to remain uniform and fast, it is critical to adhere to the following simple rules of efficient hashing: In order to minimize collisions, hashing a key should involve as many key characters as possible. String operations and conversions must be minimized. Let us consider a number of distinct situations. 1. Fractional Signed Keys In this case, we can simply rescale each key before hashing it by multiplying it by a suitable integer constant and adding another constant to the result if necessary. For instance, if our keys are in the decimal form X.Y, multiplying each key by Y would suffice. If, in addition, they can be negative, we would simply add an integer Z known to exceed the largest absolute value a negative key can assume. So, the entire change to the programs above needed to accommodate fractional signed keys would be using MOD KEY 2. Digital Strings It is a simple matter of using the INPUT function and an appropriate numeric informat. For example, if the keys were 16- digit account numbers stored in a character variable, we could simply choose Data WarehousingSharePoint for Project Management: How to Create a Project Management Information System (PMIS) with SharePoint