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An ideal key-to-address translation should satisfy two requirements, we mark the second one for further discussion: 1 The source key must be reduced in size to match the slot range 2 The slot addresses generated should be unique
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These two objectives con ict Although the source keys are presumably unique, it is impossible to compute smaller numbers which are unique for arbitrary source keys Hence, we settle for a lesser criterion 2! The slot addresses generated should be as unique as possible
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The degree of uniqueness of slot addresses is high if the addresses are uniformly distributed A uniform distribution of addresses will locate an equal fraction of the keys to each and every slot Techniques to generate uniformly distributed random numbers form the basis for such transformations There is a wide variety of choices for key-to-address transformations In this section we will use the remainder-of-division algorithm throughout We obtain relative addresses using a randomizing KAT by computing the remainder-of-division of the given key In Subsec 3-5-13 this method is de ned in more detail, and other methods, often more suitable, are surveyed and evaluated as well All the required principles for hashing can be developed using a single type of KAT Example 3-9 shows a small le where records are placed into slots using as KAT the remainder-of-division by 500
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A randomizing transformation for a personnel le uses the social security number as the key We assume that the value of the low-order digits of these numbers is evenly distributed and hence there is a high probability of deriving out of these digits a unique number for every employee If one wants to allow space for up to n = 500 employees, the value of the key may be divided by 500, leaving a remainder with values between 0 and 499
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Record for Al Joe Mary Pete
SSN 322-45-6178 123-45-6284 036-23-0373 901-23-4784
Slot address 178 284 373 284
Even though it was unlikely, given 4 out of 500 employees, by chance two identical slot addresses were generated: Joe and Pete both received slot number 284; the records for Joe and Pete will collide if both are placed directly into a le
The problem with randomizing transformations is that they will generate some identical addresses from di erent source keys, so that more than one record may be directed to the same place in storage Such an occurrence is called a collision Consider using the KAT from Example 3-9 The algorithm computed for Pete, with social security number = 901-23-4784, generated a slot number of 284, but this did not provide a free space for Pete s record, since Joe was assigned to that slot earlier As demonstrated in Example 3-9, before storing a new record into a slot of a direct le the content of the slot itself has to be tested The slot can be 1 empty 2 it can contain an older record for the same key 3 it can contain a colliding record for another key 4 if deletions don t rearrange records it can contain a tombstone
Insertion Procedure
Basic File-System Organization
The insertion procedure has to consider these three or four cases To determine on insertion whether a collision is occurring we check if the slot at the computed address is empty Insertion into an empty slot is straightforward If the slot is full, there is a collision Now we compare the key value of the record found in the slot with the key value of the new record If key elds match, then earlier a record with the same key was inserted into the le Perhaps the record found should be replaced by the new one; the implied operation is an update of all attributes but the key If the new and old key elds do not match, it is a true collision Then the rules for collision resolution, as set for the particular le organization, must be followed The most common choice is open addressing, where we search through successor slots until an empty slot is found The colliding record is inserted into that slot
Fetch Procedure When a record is to be fetched a similar process is followed An empty slot indicates that there is no record matching the search key If there is a record, then the search key is compared with the key of the record in the slot If they match, the record is retrieved from the slot If the new and old key elds do not match, then there had been a true collision earlier and we search further to try to locate a matching record The retrieval process depends on the collision resolution method in use For open addressing the scheme we will check the successor slots If the slot is empty we give up, if it is not empty, the key is checked and a matching record will be fetched If the record still does not match the search is continued through all slots Deletion of records from a direct le using open addressing must be handled with care, since just setting slots empty can confuse the retrieval process This problem is further addressed in Subsec 3-5-15 Termination Both insert and retrieval procedures are sure to terminate as long as the number of slots m is greater than the number of records n If m is su ciently greater than n the probability of nding an empty slot is good, and it will only be rarely neccessary to go to slots in a successor block These concepts will be expanded on in the performance evaluation of Sec 6-2 We summarize both the insert and retrieval algorithm as a decision rule in Table 3-5 below The Decision Rule The corresponding outcomes are now summarized in Table 35; we assume that all source keys are unique The precise handling of collisions is elaborated in Subsec 3-5-15
Table 3-5
Decision rules for collisions Insert Ok to insert Replace record if permitted Locate free successor slot Fetch No record found Record found Check successor slot
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