Richard Anderson (instead of Martin Tompa) CSE 326 Hashing Richard Anderson (instead of Martin Tompa)

Chaining review H(k) = k mod 17 k H(k) A B C D E F G H I J K 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 k H(k) A B C D E F G H I J K H(k) = k mod 17 Collect twelve birthdays from students and hash into the table with chaining

Open address hashing Store all elements in table If a cell is occupied, try another cell. Linear probing, try cells H(k), H(k) + 1 mod m, H(k) + 2 mod m, . .

Open Address Hashing H(k) = k mod 17 k H(k) A 53 2 B 41 7 C 91 6 D 75 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 2 A 3 I 4 J 5 6 C 7 B 8 D 9 F 10 K 11 12 13 E 14 15 16 H k H(k) A 53 2 B 41 7 C 91 6 D 75 E 13 F G 43 H 67 16 I 88 3 J 36 K 40 H(k) = k mod 17 Collect twelve birthdays from students and hash into the table with chaining Step through example Emphasize the growing regions

Open address hashing Lookup (K) { p = H(K); loop { if (A[p] is empty) return false; if (A[p] == K) return true; p = (p + 1) mod m; }

Open address hashing issues Clumping Cost per operation Deletion

Double hashing Use separate hash functions for the first probe and the collision resolution H1(k), H1(k) + H2(k) mod m, H1(k) + 2H2(k) mod m, H1(k) + 3H2(k) mod m , . . . Return to earlier slide to update the access code

Double hashing example 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 month day H1(k) A B C D E F G H I J K H1(k) = day mod 17 H2(k) = month Collect twelve birthdays (month and day) from students and hash into the table with chaining WRITE MONTHS AS NUMBERS

Double hashing vs. Single hashing Load factor a, cost per operation Single hashing Double hashing Single Double a

Trade offs between chaining and open addressing Chaining Open Addressing Space Time Deletions Coding complexity High load factor

Hash Functions Function Efficient Uniform mapping to range Avoids systematic collisions

Hashing strings String Suppose

Fact: So: