Hashing The Magic Container

1. HashingThe Magic Container

2. Interface • Main methods: • Void Put(Object) • Object Get(Object) … returns null if not i • … Remove(Object) • Goal: methods are O(1)! (ususally) • Implementation details • HashTable: the storage bin • hashfunction(object): tells where object should go • collision resolution strategy: what to do when two objects “hash” to same location. • In Java, all objects have default int hashcode(), but better to define your own. Except for strings. • String hashing in Java is good.

3. HashFunctions • Goal: map objects into table so distribution is uniform • Tricky to do. • Examples for string s • product ascii codes, then mod tablesize • nearly always even, so bad • sum ascii codes, then mod tablesize • may be too small • shift bits in ascii code • java allows this with << and >> • Java does a good job with Strings

4. Example Problem • Suppose we are storing numeric id’s of customers, maybe 100,000 • We want to check if a person is delinquent, usually less than 400. • Use an array of size 1000, the delinquents. • Put id in at id mod tableSize. • Clearly fast for getting, removing • But what happens if entries collide?

5. Separate Chaining • Array of linked lists • The hash function determines which list to search • May or may keep individual lists in sorted order • Problems: • needs a very good hash function, which may not exist • worse case: O(n) • extra-space for links • Another approach: Open Addressing • everything goes into the array, somehow • several approaches: linear, quadratic, double, rehashing

6. Linear Probing • Store information (or prts to objects) in array • Linear Probing • When inserting an object, if location filled, find first unfilled position. I.e look at hi(x)+f(i) where f(i)= i; • When getting an object, start at hash addresses, and do linear search till find object or a hole. • primary clustering blocks of filled cells occur • Harder to insert than find existing element • Load factor =lf = percent of array filled • Expected probes for • insertion: 1/2(1+1/(1-lf)^2)) • successful search: 1/2(1+1/(1-lf))

7. Expected number of probes

8. Quadratic Probing • Idea: f(i) = i^2 (or some other quadratic function) • Problem: If table is more than 1/2 full, no quarantee of finding any space! • Theorem: if table is less than 1/2 full, and table size is prime, then an element can be inserted. • Good: Quadratic probing eliminates primary clustering • Quadratic probing has secondary clustering (minor) • if hash to same addresses, then probe sequence will be the same

9. Proof of theorem • Theorem: The first P/2 probes are distinct. • Suppose not. • Then there are i and j <P/2 that hash to same place • So h(x)+i^2 = h(y)+j^2 and h(x) = h(y). • So i^2 = j^2 mod P • (i+j)*(i-j) = 0 mod P • Since P is prime and i and j are less than P/2 • then i+j and i-j are less than P and P factors. • Contradiction

10. Double Hashing • Goal: spreading out the probe sequence • f(i) = i*hash2(x), where hash2 is another hash function • Dangerous: can be very bad. • Also may not eliminate any problems • In best case, it’s great

11. Rehashing • All methods degrade when table becomes too full • Simpliest solution: • create new table, twice as large • rehash everything • O(N), so not happy if often • With quadratic probing, rehash when table 1/2 full

12. Extendible Hashing: Uses secondary storage • Suppose data does not fit in main memory • Goal: Reduce number of disks accesses. • Suppose N records to store and M records fit in a disk block • Result: 2 disk accesses for find (~4 for insert) • Let D be max number of bits so 2^D < M. • This is for root or directory (a disk block) • Algo: • hash on first D bits, yields ptr to disk block • Expected number of leaves: (N/M) log 2 • Expected directory size: O(N^(1+1/M) / M) • Theoretically difficult, more details for implementation

13. Applications • Compilers: keep track of variables and scope • Graph Theory: associate id with name (general) • Game Playing: E.G. in chess, keep track of positions already considered and evaluated (which may be expensive) • Spelling Checker: At least to check that word is right. • But how to suggest correct word • Lexicon/book indices

14. HashSets vs HashMaps • HashSets store objects • supports adding and removing in constant time • HashMaps store a pair (key,object) • this is an implementation of a Map • HashMaps are more useful and standard • Hashmaps main methods are: • put(Object key, Object value) • get(Object key) • remove(Object key) • All done in expected O(1) time.

15. Lexicon Example • Inputs: text file (N) + content word file (the keys) (M) • Ouput: content words in order, with page numbers Algo: Define entry = (content word, linked list of integers) Initially, list is empty for each word. Step 1: Read content word file and Make HashMap of content word, empty list Step 2: Read text file and check if work in HashMap; if in, add to page number, else continue. Step 3: Use the iterator method to now walk thru the HashMap and put it into a sortable container.

16. Lexicon Example • Complexity: • step 1: O(M), M number of content words • step 2: O(N), N word file size • step 3: O(M log M) max. • So O(max(N, M log M)) • Dumb Algorithm • Sort content words O(Mlog M) (balanced tree) • Look up each word in Content Word tree and update • O(N*logM) • Total complexity: O(N log M) • N = 500*2000 =1,000,000 and M = 1000 • Smart algo: 1,000,000; dumb algo: 1,000,000*10.

17. Memoization • Recursive Fibonacci: fib(n) = if (n<2) return 1 else return fib(n-1)+fib(n-2) • Use hashing to store intermediate results Hashtable ht; fib(n) = Entry e = (Entry)ht.get(n); if (e != null) return e.answer; else if (n<2) return 1; else ans = fib(n-1)+fib(n-2); ht.put(n,ans); return ans;