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1 Designing Hash Tables Sections 5.3, 5.4, 5.5

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2 Designing a hash table 1.Hash function: establishing a key with an indexed location in a hash table –E.g. Index = hash(key) % table_size; 2.Resolve conflicts: –Need to handle case where multiple keys mapped to the same index. –Two representative solutions Chaining with separate lists Probing open addressing

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3 Separate Chaining Each table entry stores a list of items Multiple keys mapped to the same entry maintained by the list Example –Hash(k) = k mod 10 –(10 is not a prime, just for illustration)

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4 Separate Chaining Implementation Type Declaration for Separate Chaining Hash Table

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5 HashedObj Needs to provide –Hash function Provided for string and int (the two non-member functions) –Equality operators ( operator== or operator!= )

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6 An example class for HashedObj

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7 Chaining

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8 Chaining (contd.)

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10 Analysis of Chaining Consider an array of size M with N records –Worst case insert without uniqueness check = O(1) Find location to insert and push_back/front –Worst case remove/find/unique insert = O(N) –Expected case unique insert/find/remove 1 + O(N/M) –Let us resize the table is N/M exceeds some constant –Expected time = 1 + O( ) = O(1)

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11 Hash Tables Without Chaining Try to avoid buckets with separate lists How use Probing Hash Tables –If collision occurs, try another cell in the hash table. –More formally, try cells h 0 (x), h 1 (x), h 2 (x), h 3 (x)… in succession until a free cell is found. h i (x) = hash(x) + f(i) And f(0) = 0

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12 f(i)=i Insert (assume no duplicated keys) 1.Index = hash(key) % table_size; 2.If table[index] is empty, put information (key and others) in entry table[index]. 3.If table[index] is not empty then Index ++; index = index % table_size; goto 2. Search (key) 1.Index = hash(key) % table_size; 2.If (table[index] is empty) return –1 (not found). 3.Else if (table[index].key == key) return index; 4.Index ++; index = index % table_size; goto 2. Linear Probing

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13 Example Insert 89, 18, 49, 58, 69 (hash(k) = k mod 10)

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14 Linear probing Delete –Can be tricky, must maintain the consistency of the hash table. –What is the simplest deletion strategy you can think of??

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15 Quadratic Probing f(i) = i 2

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16 Probing strategy hash table

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17 Double Hashing f(i) = i*hash 2 (x) E.g. hash 2 (x) = 7 – (x % 7) What if hash 2 (x) = 0 for some x?

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18 Analysis of Hash Table Without Chaining Expected case analysis of insertion into a table of size M containing n records – i=1 i probability of trying i buckets –= 1 (M-n)/M + 2(n/M)(M-n)/M + 3(n/M) 2 (M-n)/M +... Let = n/M –Time = 1*(1- ) + 2 (1- ) + 3 2 (1- ) +... –= 1 - + 2 - 2 2 + 3 2 - 3 3 + 4 3 - 4 4... –= 1 + + 2 + 3 +... = 1/(1- ) Assume < 1 –Keep bounded by some constant < 1

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19 Rehashing Hash Table may get full –No more insertions possible Hash table may get too full –Insertions, deletions, search take longer time Solution: Rehash –Build another table that is twice as big and has a new hash function –Move all elements from smaller table to bigger table Cost of Rehashing = O(N) –But happens only when table is close to full –Close to full = table is X percent full, where X is a tunable parameter

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20 Rehashing Example After Rehashing Original Hash Table After Inserting 23

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21 Rehashing Implementation

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22 Rehashing implementation

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