2. Midterm Midterm is Wednesday next week ! The quiz contains 5 problems = 50 min + 0 min more –Master Theorem/ Examples –Quicksort/ Mergesort –Binary Heaps / Binary Search Trees –Depth/Breadth First Search –Greedy Algorithm / Prim’s algorithm for MST You SHOULD know algorithms/master very well! –opened book –do not spend much time on any problem, attack them in the order of the most progress … –you will be graded not only on the correctness but also on clarity with which you express it. (it is better be there :-))

3. Data Structures: Lists (11.2-3/10.2-3) Data Structures: Dictionaries –Search(S, k)  a pointer to element x with key k –Insert(S, x) add new element pointed to by x –Delete(S, x) delete x, x is a pointer Linked Lists –support all operations for data structures but slow –insertion and deletion is O(1) –searching is drawback: O(n) Dictionary data structures: –Hashing key sat-te data xx record

4. Data Structures: Hashing (12/11 - 12.4/11.4) Idea: if keys in [0.. m-1] and distinct set up array T[0..m-1] in which T[i] = x, where key[x]=i. Direct-address Tables –operations take O(1) Problem: the range of keys is large (e.g. ASCII strings) Solution: use hash function to map keys in 0..n-1 Problem: 2 keys in same slot - collision- how resolve? U universe of keys K actual keys k3 k1 k4 k2 0 m-1 h(k2) h(k4) h(k1)=h(k3)

5. Collision Resolution 12.2/11.2 Open addressing (12.4) (key = satellite data) –good when we do not delete (e.g. check spelling) –table needn’t be much bigger than # of items –idea: to insert if slot is full, try another slot till find one –to search follow the same sequence of probes Chaining (items in the same slot linked into list) U universe of keys K actual keys k3 k1 k4 k2 0 m-1 k2 k4 k1 k3 collision

6. Analysis of Chaining 12.2/11.2 Assume: simple uniform hashing –each key equally likely to be hashed in each slot n keys and m slots: load factor  =n/m –average # keys per slot Cost of successful search =  (1+  /2)=  (1+  ) –1 to access slot –  /2 expected time to search list Cost of unsuccessful search =  (1+  ) –1 to access slot –  expected time to search list Cost is O(1) if  =O(1) (i.e. n=O(m))

7. Hash Functions 12.3/11.3 Choice of hash functions –should distribute keys uniformly into slots –regularity in key distribution should not affect uniformity h (1,...,100)  1, h (101,...,200)  2,... may be bad if all keys are between 1 and 100 Division for hashing (12.3.1/11.3.1) –h(k) = k mod m, i.e. hash function maps key to the remainder (residue) of division of k by m –don’t pick, hash does not depend on last bits –pick m = prime not too close to power of 2 or 10

8. Multiplication for Hashing (12.3.2/11.3.2) Constant 0 < A < 1 m = 2^p multiplication

9. Universal Hashing 12.3.3/11.3.3 Application: identifiers in a program. Problem: for any hash function  bad input Idea: choose hash function at random independent of the keys Def: U - key universe, H - set of hashing functions. H is universal if – the chance of collision b/w x and y is 1/m if h is random from H If h chosen randomly from H, then the expected # collisions for any particular key x is <  = n/m

10. Universal Hashing 12.3.3/11.3.3 Design of universal hashing –decompose key into r+1 bytes s.t. –let randomly chosen r+1 numbers from set {0,1,...,m-1} –define a corresponding hash function –class has members Th. The class H defined above is universal