Introduction To AlgorithmsCS 445

Discussion Session 2

Instructor: Dr Alon Efrat

TA : Pooja Vaswani

02/14/2005

2

Topics

• Radix Sort

• Skip Lists

• Random Variables

3

Radix Sort

Limit input to fixed-length numbers or words.Represent symbols in some base b.

Each input has exactly d “digits”.

Sort numbers d times, using 1 digit as key.Must sort from least-significant to most-significant digit.

Must use any “stable” sort, keeping equal-keyed items in same order.

4

Radix Sort Example

a b a b a c c a a a c b b a b c c a b b aa a c

Input data:

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Radix Sort Example

a b a b a c c a a a c b b a b c c a b b aa a c

a b c

Place into appropriate pile.

Pass 1: Looking at rightmost position.

6

Radix Sort Example

a b a b a c

c a a

a c b

b a b

c c a

b b a

a a c

a b c

Join piles.

Pass 1: Looking at rightmost position.

7

Radix Sort Example

a b a b a cc a a a c b b a bc c a b b a a a c

a b c

Pass 2: Looking at next position.

Place into appropriate pile.

8

Radix Sort Example

a b a

b a c

c a a

a c bb a b

c c a

b b a

a a c

a b c

Join piles.

Pass 2: Looking at next position.

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Radix Sort Example

a b c

b a cc a a b a b a a c a b a b b a a c bc c a

Pass 3: Looking at last position.

Place into appropriate pile.

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Radix Sort Example

a b c

b a c

c a ab a ba a c

a b a

b b aa c b

c c a

Pass 3: Looking at last position.

Join piles.

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Radix Sort Example

b a c c a ab a ba a c a b a b b aa c b c c a

Result is sorted.

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Radix Sort Algorithm

rsort(A,n):

For d = 0 to n-1

/* Stable sort A, using digit position d as the key. */

For i = 1 to |A|

Add A[i] to end of list ((A[i]>>d) mod b)

A = Join lists 0…b-1

(dn) time, where d is taken to be a constant.

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Skip List

S0

S1

S2

S3

10 362315

15

2315

Below is an implementation of Skip List in which the topmost level is left empty.

There is also an implementation in which the topmost level is never left empty. ( As in the lecture notes )

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Skip List

• The definition of a dictionary • Definition of skip lists • Searching in skip lists • Insertion in skip lists• Deletion in skip lists• Probability and time analysis

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Definition of Dictionary

• Primary use: to store elements so that they can be located quickly using keys

• Motivation: each element in a dictionary typically stores additional useful information beside its search key. (eg: bank accounts)

• Red/black tree, hash table, AVL tree, Skip lists

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Dictionary ADT

• Size(): Returns the number of items in D • IsEmpty(): Tests whether D is empty • FindElement(k): If D contains an item with a key equal to k,

then it return the element of such an item • FindAllElements(k): Returns an enumeration of all the

elements in D with key equal k• InsertItem(k, e): Inserts an item with element e and key k

into D. • remove(k): Removes from D the items with keys equal to k,

and returns an numeration of their elements

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Definition of Skip List

• A skip list for a set S of distinct (key, element) items is a series of lists S0, S1 , … , Sh such that

– Each list Si contains the special keys and – List S0 contains the keys of S in nondecreasing

order – Each list is a subsequence of the previous one,

i.e.,S0 S1 … Sh

– List Sh contains only the two special keys

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Example of a Skip List

• We show how to use a skip list to implement the dictionary ADT

56 64 78 31 34 44 12 23 26

31

64 31 34 23

S0

S1

S2

S3

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Initialization

• A new list is initialized as follows:• 1) A node NIL ( ) is created and its key is set

to a value greater than the greatest key that could possibly used in the list

• 2) Another node NIL () is created, value set to lowest key that could be used

• 3) The level (high) of a new list is 1• 4) All forward pointers of the header point to NIL

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Searching in Skip List- general description

• 1)      If S.below(p).the position below p in the same tower is null. We are at the bottom and have located the largest item in S with keys less than or equal to the search key k. Otherwise, we drop down to the next lower level in the present tower to setting p S.below(p).

• 2)      Starting at position p, we move p forward until it is at the right-most position on the present level such that key(p) <= k. We call this scan forward step.

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Searching in Skip List

• We search for a key x in a skip list as follows:– We start at the first position of the top list

– At the current position p, we compare x with y key(after(p))

x y: we return element(after(p))

x y: we “scan forward”

x y: we “drop down”– If we try to drop down past the bottom list, we return

NO_SUCH_KEY

• Example: search for 78

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Searching in Skip List Example

S1

S2

S3

31

64 31 34 23

56 64 78 31 34 44 12 23 26S0

1) P is at S1, is bigger than 78, we drop down• At S0, 78 = 78, we reach our solution

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Insertion

• The insertion algorithm for skip lists uses randomization to decide how many references to the new item (k,e) should be added to the skip list

• We then insert (k,e) in this bottom-level list immediately after position p. After inserting the new item at this level we “flip a coin”.

• If the flip comes up tails, then we stop right there. If the flip comes up heads, we move to next higher level and insert (k,e) in this level at the appropriate position.

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Randomized Algorithms

We analyze the expected running time of a randomized algorithm under the following assumptions

the coins are unbiased, and

the coin tosses are independent

The worst-case running time of a randomized algorithm is large but has very low probability (e.g., it occurs when all the coin tosses give “heads”)

• A randomized algorithm performs coin tosses (i.e., uses random bits) to control its execution

• It contains statements of the type

b random()if b 0

do A …else { b 1}

do B …

• Its running time depends on the outcomes of the coin tosses

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Insertion in Skip List Example

10 36

23

23

S0

S1

S2

S0

S1

S2

S3

10 362315

15

2315p0

p1

p2

1) Suppose we want to insert 152) Do a search, and find the spot between 10 and 233) Suppose the coin come up “head” three times

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Deletion

• We begin by performing a search for the given key k. If a position p with key k is not found, then we return the NO SUCH KEY element.

• Otherwise, if a position p with key k is found (it would be found on the bottom level), then we remove all the position above p

• If more than one upper level is empty, remove it.

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Deletion in Skip List Example

• 1) Suppose we want to delete 34• 2) Do a search, find the spot between 23 and 45• 3) Remove all the position above p

4512

23

23

S0

S1

S2

S0

S1

S2

S3

4512 23 34

34

23 34p0

p1

p2

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Probability Analysis

• Insertion, whether or not to increase h• Worst case for find, insert, delete: O (n + h)• Due to low probability events when every item

belongs to every level in S• Very low probability that it will happen• Not a fair assessment

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Performance of a Dictionary by a Skip List

• Operation Time• Size, isEmpty O(1)• findElement O(log n) (expected)• insertItem O(log n) (expected)• Remove O(log n) (expected)• FindAllElements O(log n + s)

(expected)• removeAll O(log n + s)

(expected)

- S being the extra matching keys we have to go through