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 :-))

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

keysat-tedata

xrecord

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

k3k1

k4k20

m-1

h(k2)h(k4)

h(k1)=h(k3)

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

k3k1

k4k20

m-1

k2k4

k1 k3

collision

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))

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

pm 2

Multiplication for Hashing (12.3.2/11.3.2)

• Constant 0 < A < 1

• m = 2^p

• multiplication

)()( kAkAmkh

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

myhxhhyxUyx /|||)}()(:{:|)(,

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

mxaxh i

r

iia mod)(

0

rxxxx ,...,, 10

raaaa ,...,, 10

mxi

)(xha

a

a xh )}({ 1rm