Introduction To Algorithms CS 445

Introduction To AlgorithmsCS 445 Discussion Session 2 Instructor: Dr Alon Efrat TA : Pooja Vaswani 02/14/2005

Topics • Radix Sort • Skip Lists • Random Variables

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.

a b a b a c c a a a c b b a b c c a a a c b b a Radix Sort Example Input data:

b a c c a a a c b b a b c c a a a c b b a a b a Radix Sort Example Pass 1: Looking at rightmost position. Place into appropriate pile. a b c

b b a c c a c a a b a b a a c a b a a c b b a c Radix Sort Example Pass 1: Looking at rightmost position. Join piles. a b c

a b a c a a c c a b b a a c b b a b b a c a a c Radix Sort Example Pass 2: Looking at next position. Place into appropriate pile. a b c

a a c b a c b a b b b a a c b c a a a b a c c a Radix Sort Example Pass 2: Looking at next position. Join piles. a b c

c a a b a b b a c a a c a b a b b a c c a a c b Radix Sort Example Pass 3: Looking at last position. Place into appropriate pile. a b c

a c b b b a a b a b a c c c a a a c b a b c a a Radix Sort Example Pass 3: Looking at last position. Join piles. a b c

a a c a b a a c b b a b b a c b b a c a a c c a Radix Sort Example Result is sorted.

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 (dn) time, where d is taken to be a constant.

- + - 15 + - 15 23 + - 10 23 36 + 15 Skip List 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 ) S3 S2 S1 S0

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

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

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

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

- + - 12 23 26 31 34 44 56 64 78 + Example of a Skip List • We show how to use a skip list to implement the dictionary ADT S3 S2 - 31 + S1 - 23 31 34 64 + S0

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

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.

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

- + Searching in Skip List Example S3 S2 - 31 + S1 - 23 31 34 64 + S0 - 12 23 26 31 34 44 56 64 78 + • P is 64, at S1,+ is bigger than 78, we drop down • At S0, 78 = 78, we reach our solution

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.

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 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”)

- + - 15 + - 15 23 + - 10 23 36 + 15 Insertion in Skip List Example • Suppose we want to insert 15 • Do a search, and find the spot between 10 and 23 • Suppose the coin come up “head” three times S3 p2 S2 S2 - + p1 S1 S1 - + 23 p0 S0 S0 - 10 23 36 +

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.

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 S3 - + p2 S2 S2 - + - 34 + p1 S1 S1 - + - 23 34 + 23 p0 S0 S0 - 12 23 45 + - 12 23 45 + 34

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

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

Introduction To Algorithms CS 445