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Hash Tables

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2 Many applications require a dynamic set that only supports the dictionary operations: Insert, Delete, Search A hash table is an efficient implementation of a dictionary Worst case – same as linked list – O(n) Under reasonable assumptions – O(1)

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3 Direct-address Tables Assumptions The universe U of keys is reasonably small: U = { 0, 1, 2, …, m-1 }, for some small m No two elements have the same key Implementation Allocate an array of size m Insert the k th element into the k th slot in the array

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4 Direct-address Tables

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5 Advantage O(1) time for all operations Disadvantages Wasteful if the number of elements actually inserted is significantly smaller than the size of the universe (m) Only applicable for small values of m, i.e. a limited range of keys

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6 Hash Tables Performance is almost similar to that of a direct-address table, but without the limitations The universe U may be very large The storage requirement is O(|K|), where K is the set of keys actually used Disadvantage – O(1) performance is now average case, not worst case

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7 Hash Tables We need a hash function to map keys from the universe U into the hash table h: U { 0, 1, …, m-1 } For each key k, the hash function computes a hash value h(k) If two keys hash to the same value: h(k 1 ) = h(k 2 ), we call this a collision

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8 Hash Tables

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9 Collisions Can we avoid collisions altogether? No. Since |U| > m, some keys must have the same hash value A good hash function will be as random as possible Still, collisions must be resolved

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10 Collision Resolution Chaining (also called open hash) Elements stored in their correct slot Collisions resolved by creating linked lists Open addressing (also called closed hash) All elements stored inside the table Maybe rehashed if their slot is full

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11 Chaining – Open Hash

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12 Collision resolution – Chaining All keys that have the same hash value are placed in a linked list Insertion can be done at the beginning of the list in O(1) time Searching is proportional to the length of the list – with a good hash function, will also be O(1)

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13 Hash Function Requirements A hash function must be deterministic – the hash value generated for each key cannot change during the life of the hash table Equal keys must always be mapped to the same hash value

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14 Hash Function Properties Properties of a good hash function Easy to evaluate – h(x) can be computed very quickly (not only in O(1), but also with a small constant) Uniform distribution over all the table slots Different keys are mapped to different slots (as much as possible)

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15 Simple Uniform Hashing The quality of the hash function strongly influences the efficiency of the hash table Simple Uniform Hashing assumption: The hash function will hash any key into any slot with equal probability It is possible to define hash functions that almost satisfy this assumption

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16 Analysis The load factor of a hash table is defined as the number of elements stored in the table, divided by the total number of slots: A search will take under the assumption of simple uniform hashing Therefore, all hash operations can be performed in O(1)

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17 Computing Key Values The first step is to represent the key as a natural integer number For example if S is a string then we can interpret it as an integer value using the following formula:

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18 The Division Method Key k is mapped into one of m slots by taking the remainder of k divided by m: Choosing the value of m Preferably prime Not too close to a power of 2

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19 Example – the Division Method Let h be a hash table of 9 slots and h(k) = k mod 9. insert the elements: 6, 43, 23, 62, 1, 13, 34, 55, 25 h(13) = 13 mod 9 = 4 h(34) = 34 mod 9 = 7 h(55) = 55 mod 9 = 1 h(25) = 25 mod 9 = 7 h(6) = 6 mod 9 = 6 h(43) = 43 mod 9 = 7 h(23) = 23 mod 9 = 5 h(62) = 62 mod 9 = 8 h(1) = 1 mod 9 = 1

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20 Open Addressing Each element occupies a single slot in the hash table – no chaining is done To insert an element, we probe the table according to the hash function until an empty slot is found The hash function is now a function of both the key and the number of attempts in the insertion process

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21 Linear Probing A hash value is computed using any hash function h, and then the number of the current attempt is added to it: Slots are examined sequentially, until an empty one is found

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22 Linear Probing Easy to implement but suffers from primary clustering Clusters tend to grow: If an empty slot is preceded by i full slots, the probability that it will be the next one filled is (i+1)/m If an empty slot is preceded by another empty slot, the probability is only 1/m

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23 Exercise You are given a hash table H with 11 slots Demonstrate inserting the following elements using linear probing and a hash function h(k) = k mod m 10, 22, 31, 4, 15, 28, 17, 88, 59

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24 Solution h(10, 0) = (10 mod ) mod 11 = 10 h(22, 0) = (22 mod ) mod 11 = 0 h(31, 0) = (31 mod ) mod 11 = 9 h(4, 0) = (4 mod ) mod 11 = 4 h(15, 0) = (15 mod ) mod 11 = 4 h(15, 1) = (15 mod ) mod 11 = 5 h(28, 0) = (28 mod ) mod 11 = 6 h(17, 0) = (17 mod ) mod 11 =

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25 Solution h(17, 1) = (17 mod ) mod 11 = 7 h(88, 0) = (88 mod ) mod 11 = 0 h(88, 1) = (88 mod ) mod 11 = 1 h(59, 0) = (59 mod ) mod 11 = 4 h(59, 1) = (59 mod ) mod 11 = 5 h(59, 2) = (59 mod ) mod 11 = 6 h(59, 3) = (59 mod ) mod 11 = 7 h(59, 4) = (59 mod ) mod 11 =

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26 Quadratic Probing In this case, the second attempt is a more complex function of i: Tries to avoid primary clustering However, suffers from secondary clustering The entire probing sequence is determined by the initial probe:

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27 Double Hashing Given two hash functions One of the best methods for open addressing collision resolution Permutations are almost random For the entire hash to be searched, m and h 2 (k) must be relatively prime

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28 Double Hashing Possible selections of h 2 (k) Select m to be a power of 2, and design h 2 (k) to produce odd numbers Select m to be prime, and m to be m-1

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29 Double Hashing

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30 Issues in Open Addressing Search may fail if items are deleted Solution: Mark deleted items with a special symbol Search treats this symbol as full, while insert treats it as empty Table may be filled up Solution: Rehashing (copy into a larger table)

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