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Duplicate lookup keys are eliminated automatically. The speed of search does not depend on the number of keys. Because of its relatively wide key range and pure direct addressing nature, bitmapping operates with incredible speed in a niche, no other searching method can touch. Imagine SMALL file with 50 million 8-digit keys hardly small but let us stick with the name, and LARGE with mere 100 million records - figures not unusual in Data Warehousing today. Sorting either one for MERGE is not exactly painless, especially if LSAT tail is long. Storing the lookup keys in a format is practically hopeless, as already at 10 million keys, it usually tops 600 MB. Key - indexing would consume about 800 MB. A hash table described later would need at least 400 MB. On the other hand, a bitmap can be safely compiled with 120 million bits of temporary array storage in the worst case scenario M53; that is, it can be easily swallowed by mere 15 MB of memory! If we decide to store the bitmap on disk, it will take about 1.9 million 1- column rows; loading such a file into an array is a matter of several seconds. Moreover, the lookup speed and memory usage will remain exactly the same, no matter whether we have 100, 100 thousand, or 100 million keys to search. Thus, the bitmapping niche can be defined as no-matter-how- many-short- keys. But how can we capitalize on direct addressing in the case of keys with a huge, say 16-digit, range, that neither key - indexing or bitmapping can accommodate? Welcome to hashing! HASHING In order to fix the main drawback of key - indexing, bitmapping changes nothing principally - it simply expands the workable universe of keys about 53 times by using memory more efficiently. Hashing approaches the problem quite differently: It drops the requirement of a separate slot for each possible key and allows some amount of comparisons between keys. A simple example might be the easiest way of making the idea transparent. Let us suppose that SMALL contains just ten 3-digit keys : 185971400260922970543532050067 To use key - indexing we would have to allocate a table sized as 0:999 and map each key to the node corresponding to its value. Out of 1000 table nodes, only 10 end up occupied while the rest are simply wasted! Can we get away with a smaller table of a reasonable size, only slightly larger than the number of keys, and still be able to take advantage of direct addressing? Let us choose some number HSIZE somewhat greater than the number of keys NSMALL, for instance 17, and allocate an array sized as HKEY0:17. Let us agree to call the array HKEY the hash table, HSIZE - the hash table size, and the ratio NSMALL/HSIZE - the load factor. Thus, the load factor shows the number of lookup keys relative to the total number of nodes in the hash table, in other words, how sparse the hash table is. In our example, the load factor equals 0.588, that is, the hash table is about 41 percent sparse. Imagine some fast function H KEY taking a key as an argument and returning an address into HKEY, unique to each key supplied, so that H KEY maps each key to its own location one-to-one. Were such perfect hash function available, we would only have to plug it in the code for key - indexed search. However, although perfect hash functions are possible, they are quite difficult to discover, and once found, can only be used for the same set of keys. A lot more flexible method can be obtained by giving up the requirement of one-to-one correspondence between the keys and table addresses, and letting H KEY map two distinct keys to the same location in HKEY. However, if two different keys are sent to the same node, a phenomenon termed a collision occurs, and we must invoke some collision resolution policy in order to tell the keys apart in the process of inserting a key or searching for one. Thus, we arrive at the core concept behind hashing. If the hash function H KEY is good enough to map only a few keys to a particular hash address H, we can adopt the following strategy: 1. Use H KEY to hash KEY to some address H in the table. 2. If the address H is empty, the search is unsuccessful. 3. If the address is occupied, search the keys residing at H sequentially. Clearly, hashing is a typical hybrid algorithm: It combines direct addressing with sequential search, a comparison-based method. The average number of keys mapping to a hash address equals NSMALL/HSIZE, the load factor. If the hash table is not full and the keys are spread uniformly, the average number of key comparisons required to find or reject a key will thus be less than 1! Also, searching for a key should be the faster, the sparser the table is. So, in order to use of a hash table, we have to: 1. Choose a proper hash function H KEY. 2. Find an efficient way to resolve collisions. Before trying to formulate the requirements for a good hash function let us see first what it should not be. On one extreme, if a function is lightning fast but maps the majority of keys to the same hash address, it defies the very purpose of distributing keys among different addresses since then we would have to search all these keys attached to the same node sequentially! Hence, it is paramount that a good hash function should map the keys evenly across all hash table nodes without burdening some addresses with huge clusters and leaving the rest of them empty. On the other extreme, if a function maps the keys extremely uniformly but is computed slowly, it is of no good use, either, since in this case, direct addressing itself will become a bottleneck. Now we can have some rational requirements a good hash function should satisfy: 1. Its computation should be as fast as possible. 2. It should distribute the keys uniformly to minimize collisions. 3. It must return addresses in the range 0:HSIZE. There is a number of mapping methods conforming to these requirements 1. We will discuss the simplest technique called the division method. It makes use of the remainder modulo: HMOD KEY, HSIZE. It certainly fits 3, since this function always returns an integer in the range from 0 to HSIZE-1. It also satisfies 1, for although it incorporates a division, its computation is quite fast. However, to satisfy 2, the value for HSIZE must be chosen rather carefully. Suppose that R is the radix of the character set, and X and Y - small integers. It can be proven 1 that if HSIZE is a prime number such that RX is not equal to Y, the keys will be spread satisfactorily evenly. Let us see how it would work for our sample set of the ten keys. If we choose the target load factor as 0.625 and divide it into the number of keys, we will obtain 16. The first prime greater or equal to 16 is 17, so let us select it as HSIZE. The actual load factor is now 10/17 0.588. We should therefore allocate the hash table as ARRAYHKEY 0:17 TEMPORARY; To obtain the hash addresses, we divide each key by HSIZE17, compute the remainder H and insert KEY into H-th slot: 01 970. 02 971. 04 922. 05 260532. Data Warehousing
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