Hashing

1. Hashing

2. Motivation • Sequential Searching can be done in O(N) access time, meaning that the number of seeks grows in proportion to the size of the file. • B-Trees improve on this greatly, providing O(Logk N) access where k is a measure of the leaf size (i.e., the number of records that can be stored in a leaf). • What we would like to achieve, however, is an O(1) access, which means that no matter how big a file grows, access to a record always takes the same small number of seeks. • Static Hashing techniques can achieve such performance provided that the file does not increase in time.

3. What is Hashing? • A Hash function is a function h(K) which transforms a key K into an address. • Hashing is like indexing in that it involves associating a key with a relative record address. • Hashing, however, is different from indexing in two important ways: • With hashing, there is no obvious connection between the key and the location. • With hashing two different keys may be transformed to the same address.

4. Collisions • When two different keys produce the same address, there is a collision. The keys involved are called synonyms. • Coming up with a hashing function that avoids collision is extremely difficult. It is best to simply find ways to deal with them. • Possible Solutions: • Spread out the records • Use extra memory • Put more than one record at a single address.

5. A Simple Hashing Algorithm • Step 1: Represent the key in numerical form • Step 2: Fold and Add • Step 3: Divide by a prime number and use the remainder as the address.

6. Hashing Functions and Record Distributions • Records can be distributed among addresses in different ways: there may be (a) no synonyms (uniform distribution); (b) only synonyms (worst case); (c) a few synonyms (happens with random distributions). • Purely uniform distributions are difficult to obtain and may not be worth searching for. • Random distributions can be easily derived, but they are not perfect since they may generate a fair number of synonyms. • We want better hashing methods.

7. Some Other Hashing Methods • Though there is no hash function that guarantees better-than-random distributions in all cases, by taking into considerations the keys that are being hashed, certain improvements are possible. • Here are some methods that are potentially better than random: • Examine keys for a pattern • Fold parts of the key • Divide the key by a number • Square the key and take the middle • Radix transformation

8. Predicting the Distribution of Records • When using a random distribution, we can use a number of mathematical tools to obtain conservative estimates of how our hashing function is likely to behave: • Using the Poisson Function p(x)=(r/N)xe-(r/N)/x! applied to Hashing, we can conclude that: • In general, if there are N addresses, then the expected number of addresses with x records assigned to them is Np(x)

9. Predicting Collisions for a Full File • Suppose you have a hashing function that you believe will distribute records randomly and you want to store 10,000 records in 10,000 addresses. • How many addresses do you expect to have no records assigned to them? • How many addresses should have one, two, and three records assigned respectively? • How can we reduce the number of overflow records?

10. Increasing Memory Space I • Reducing collisions can be done by choosing a good hashing function or using extra memory. • The question asked here is how much extra memory should be use to obtain a given rate of collision reduction? • Definition: Packing density refers to the ratio of the number of records to be stored (r) to the number of available spaces (N). • The packing density gives a measure of the amount of space in a file that is used.

11. Increasing Memory Space II • The Poisson Distribution allows us to predict the number of collisions that are likely to occur given a certain packing density. We use the Poisson Distribution to answer the following questions: • How many addresses should have no records assigned to them? • How many addresses should have exactly one record assigned (no synonym)? • How many addresses should have one record plus one or more synonyms? • Assuming that only one record can be assigned to each home address, how many overflow records can be expected? • What percentage of records should be overflow records?

12. Collision Resolution by Progressive Overflow • How do we deal with records that cannot fit into their home address? A simple approach: Progressive Overflow or Linear Probing. • If a key, k1, hashes into the same address, a1, as another key, k2, then look for the first available address, a2, following a1 and place k1 in a2. If the end of the address space is reached, then wrap around it. • When searching for a key that is not in, if the address space is not full, then an empty address will be reached or the search will come back to where it began.

13. Search Length when using Progressive Overflow • Progressive Overflow causes extra searches and thus extra disk accesses. • If there are many collisions, then many records will be far from “home”. • Definitions: Search length refers to the number of accesses required to retrieve a record from secondary memory. The average search length is the average number of times you can expect to have to access the disk to retrieve a record. • Average search length = (Total search length)/(Total number of records)

14. Storing More than One Record per Address: Buckets • Definition: A bucket describes a block of records sharing the same address that is retrieved in one disk access. • When a record is to be stored or retrieved, its home bucket address is determined by hashing. When a bucket is filled, we still have to worry about the record overflow problem, but this occurs much less often than when each address can hold only one record.

15. Effect of Buckets on Performance • To compute how densely packed a file is, we need to consider 1) the number of addresses, N, (buckets) 2) the number of records we can put at each address, b, (bucket size) and 3) the number of records, r. Then, Packing Density = r/bN. • Though the packing density does not change when halving the number of addresses and doubling the size of the buckets, the expected number of overflows decreases dramatically.

16. Making Deletions • Deleting a record from a hashed file is more complicated than adding a record for two reasons: • The slot freed by the deletion must not be allowed to hinder later searches • It should be possible to reuse the freed slot for later additions. • In order to deal with deletions we use tombstones, i.e., a marker indicating that a record once lived there but no longer does. Tombstones solve both the problems caused by deletion. • Insertion of records is slightly different when using tombstones.

17. Effects of Deletions and Additions on Performance • After a large number of deletions and additions have taken places, one can expect to find many tombstones occupying places that could be occupied by records whose home address precedes them but that are stored after them. • This deteriorates average search lengths. • There are 3 types of solutions for dealing with this problem: a) local reorganization during deletions; b) global reorganization when the average search length is too large; c) use of a different collision resolution algorithm.

18. Other Collision Resolution Techniques • There are a few variations on random hashing that may improve performance: • Double Hashing: When an overflow occurs, use a second hashing function to map the record to its overflow location. • Chained Progressive Overflow: Like Progressive overflow except that synonyms are linked together with pointers. • Chaining with a Separate Overflow Area: Like chained progressive overflow except that overflow addresses do not occupy home addresses. • Scatter Tables: The Hash file contains no records, but only pointers to records. I.e., it is an index.

19. Pattern of Record Access • If we have some information about what records get accessed most often, we can optimize their location so that these records will have short search lengths. • By doing this, we try to decrease the effective average search length even if the nominal average search length remains the same. • This principle is related to the one used in Huffman encoding.