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Instead of finding HSIZE by hand or from a table of primes, it can be dynamically computed and stored in a macro variable in order to size the table: %LET LOAD0.5; DATANULL; PCEILSIZEDO UNTIL JU1 ; P1; UCEILSQRTP; DOJ2 TOUUNTIL MODP, J0; END; END; CALL SYMPUTHSIZE, LEFTPUTP, BEST. ; STOP; RUN; As we already know, selecting a decent hash function is just part of the deal. No matter how good the function is, it is practically guaranteed that some addresses will be burdened with more than one key. Therefore, we have to devise a method to resolve the collisions. This is another point where hashing deviates from key - indexing and bitmapping, where keys are not stored in the table. With hashing, the keys themselves have to reside in the table because they will have to be compared to a search key unless it hashes to an empty node. Various collision resolution policies differ in the ways colliding keys are stored, linked as related to the same hash address, and traversed. 1. Separate Chaining One way of resolving collisions naturally suggests itself once we look at the distribution of our 10 keys among the addresses of the 17-node hash table shown in the previous section. The keys hashing to the same address form a visible chain. Using such chains to resolve collisions is thus logically called separate chaining. There are two ways such chains can be utilized. First, they can be stored outside the table by placing them in the occurrences of a two-dimensional array. A huge drawback of this method is poor memory utilization. If we have 100, 000 keys in SMALL and a single bad address colliding 10 keys, we will be forced to create a 2- dimensional array sized as 0:10, 0:100000 to resolve the collisions. Even with the load factor 1, it requires 10 times the memory the keys would occupy by themselves. On the positive side, the 2-dimensional chaining is quite fast, and it can work with load factors greater than 1. So, if good memory utilization is not too paramount, the method could be well recommended. Feel free to contact the author for the details of implementation. However, the idea of chaining can be exploited in a much neater fashion! Once the philosophy of allocating the storage for colliding keys is changed from sequential to linked, we arrive at an extremely elegant collision resolution policy, both very fast and reasonably memory-efficient. 2. Coalesced Chaining The core idea here is to place all chains of colliding keys into the hash table itself and combine the keys hashing to the same node into a linked list with the head residing at the colliding address. Setting the last link of each chain to null designates the end of the chain, thus helping us tell where to stop when the list is traversed serially. Since the linked lists are thus allowed to overlap in the hash table sharing the same storage locations, this approach is termed coalesced list chaining, or coalesced chaining for short. To make it possible, all we need is a numeric array of link items, say, LINK, parallel to the hash table. Note that in order for this method to work, at least one entry in the table must be empty. Otherwise if the table is full, there would be no room to place the final null link needed to terminate the traversing loop. Since a table allocated as 0:HSIZE has HSIZE1 entries, but the modulo-based hash function only addresses HSIZE nodes from 0 to HSIZE-1, this requirement will always be satisfied. Let us agree to always leave the address 0 empty by hashing a key as MOD KEY, HSIZE 1. That is, if KEY modulo HSIZE is 05, it will map to the address 06. As stated above, we now need two parallel arrays, one for the hash table itself and another one to hold the links: ARRAYHKEY 0: ARRAY LINK 0: Now we are ready to spell a detailed plan of loading the table: STEP 1. Set a counter variable R to the top address: RHSIZE. STEP 2. Hash: HMOD KEY,HSIZE 1. STEP 3. If LINKH., the node is empty, no list is attached to it. Go to step 7 and insert the key. A. Otherwise traverse the chain to find if the key is already in the table. If KEY HKEYH the key is duplicate. Get the next key and return to step 2. B. Else If HKEYH is not 0, it is not the end of the list yet. Set H HLINKH and repeat step from A. STEP 4. Find an empty address closest to the top: Decrement R until LINKR. STEP 5. Store the key at this address: HKEYR KEY. STEP 6. Memorize where KEY actually belongs: LINKHR; HR. STEP 7. Insert KEY into the address H and set its link to null: HKEYH KEY ; LINKH0. Now the node is occupied. Applied to our 10 sample keys and HSIZE17, this process results in the following loaded table colliding keys are shown in bold: H HLINK HKEY 00. 01. 0200970 0300971 04. 0500922 0615260 07. 08. 09. 1000400 11. 12. 1300067 1413050 1500532 1600185 1714543 Data Warehousing
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