1. LECTURE 35: COLLISIONS CSC 212 – Data Structures

2. Map Performance  Want fast implementation  911 Operators immediately need addresses  Performance of Google measured in TB/s  O(log n) time too slow to achieve these times  Would love to use arrays  Converts key to int with hash functions  Once hashed, put, remove & get in O(1) time  But is this even possible?

3. Hash Table  Array locations either:  null  Reference to Entry  Marker value *  May not be contiguous  Better when spread out  Hash key to index  Begin methods with hash  Entry at that index Hash Table Entry s 0000 0001 0256120001“Jay Doe” 0002 9811010002“Bob Doe” 0003 0004 4512290004“Jill Roe” … … 9997 9998 2007519998“Rhi Smith” 9999

4. Ideal World  key hashed to unique index  Hash and done, Entry is there And then… You wake up

5. Bad Hash  Perfect hash does not exist  Cannot know all keys beforehand  Cluster around a few indices  Or find all keys hashed to same index  Handling bad hash is a necessary  Given Entry must first check key  Store multiple Entry s with same hash  (Shot of adrenaline restarts heart) ‏

6. Collisions

7. Linear Probing  Algorithm used in musical chairs  At index where key hashed examine Entry  Circle through array until empty index found  Algorithm is very simple  But creates clusters of Entry s

8. Linear Probe Example h(x) = x mod 13 Now add: 44 h(44) = 5 20 h(20) = 7 22 h(22) = 9 31 h(31) = 5 15 18442032 2231 76 012345678 910 1112

9. Probing Reaction Oh, **** Adding to hash table still O(n)‏

10. Quadratic Probe  Avoids primary clustering problems  But does create secondary clustering  No one cares about secondary clustering  Linear and quadratic probe equally complex  Where key is hashed, k, examine Entry  Check (k + j 2 ) % length: k +1, k +4, k +9, k +16, …  Continue probing until unused array slot found  Guaranteed to work when:  Table size is prime number  Under 50% full

11. Quadratic Probe Example 3115 18442032 22 76 012345678 910 1112 h(x) = x mod 13 Now add: 44 h(44) = 5 20 h(20) = 7 22 h(22) = 9 31 h(31) = 5

12. Quadratic Probing Reaction Darn it to heck. Adding to hash table still O(n)‏

13.  Solve bad hash with even more hash  Use 2 nd hash function very different from first  2 nd hash function not allowed to return zero  Re-hash key using 2 nd function after the collision  Check index equal to sum of two hash function  Re-add 2 nd hash for more probes  Guaranteed to work when  Table size is prime number Double Hashing

14. Double Hash Example 3115 18442032 22 76 012345678 910 1112 h(x) = x mod 13 h 2 (x) = 5 – (x mod 5) Now add: 44 h(44) = 5 20 h(20) = 7 22 h(22) = 9 31 h(31) = 5

15. Double Probing Reaction Sweet! Double hashing keeps put O(n)‏

16. Probing and Searching  Search index where key hashed  If cannot place Entry at index  The array must keep being probed  Stop only at usable index  May need to probe every index!  Searching takes O(n) even with hash  May need to reallocate & rehash table  Worst case O(n) put even with perfect hash

17. Linear Probe Example  What happens when we remove an Entry ?  Set index to null in most structures  Consider if we call remove(44)‏  get(31) called, what would happen?  First check index it is hashed  Checks first probe indexed… & stops at null 15 18442032 2231 76 012345678 910 1112 Even Better! I buggy code!

18. * Marker Value Explained  Mark cleared indices in hash table  Since collision could have happened, continue search  Index can be used to store new Entry  Ways to show that array index is clear  Entry with null key could be used  Make key which is never used  Use static final field of type Entry

19.  Hash tables can require O(n) complexity  Provide O(1) time if you are really good  Ultimately depends on hash function used  Choose wisely and be rich Why Use Hash Table & Probes?

20.  Finish week #12 assignment  Due at usual time, whatever that may be  Work on programming project #4 (due 12/1) ‏  If you want to take the 2 nd chance midterm  Study before test on Thursday from 9:30 – 10:20  Read sections 9.2.1 – 9.2.4 of the book  Start discussion of what hash is  How we implement hash tables  What makes hash & hashing good Before Next Lecture…