Introduction to Algorithms
|
|
- Clare Cameron
- 5 years ago
- Views:
Transcription
1 Introducton to Algorthms 6.046J/8.40J Lecture 7 Prof. Potr Indyk
2 Data Structures Role of data structures: Encapsulate data Support certan operatons (e.g., INSERT, DELETE, SEARCH) Our focus: effcency of the operatons Algorthms vs. data structures Introducton to Algorthms February 27, 2003 L7.2
3 Symbol-table problem Symbol table T holdng n records: x record key[x] Other felds contanng satellte data Operatons on T: INSERT(T, x) DELETE(T, x) SEARCH(T, k) How should the data structure T be organzed? Introducton to Algorthms February 27, 2003 L7.3
4 Drect-access table IDEA: Suppose that the set of keys s K {0,,, m }, and keys are dstnct. Set up an array T[0.. m ]: x f k K and key[x] = k, T[k] = NIL otherwse. Then, operatons take Θ() tme. Problem: The range of keys can be large: 64-bt numbers (whch represent 8,446,744,073,709,55,66 dfferent keys), character strngs (even larger!). Introducton to Algorthms February 27, 2003 L7.4
5 Hash functons Soluton: Use a hash functon h to map the unverse U of all keys nto T {0,,, m }: K k k 5 k 4 k 2 k 3 U When a record to be nserted maps to an already occuped As each key slot s n nserted, T, a collson h maps occurs. t to a slot of T. Introducton to Algorthms February 27, 2003 L7.5 0 h(k ) h(k 4 ) h(k 2 ) = h(k 5 ) h(k 3 ) m
6 Resolvng collsons by channg Records n the same slot are lnked nto a lst. T h(49) = h(86) = h(52) = Introducton to Algorthms February 27, 2003 L7.6
7 Analyss of channg We make the assumpton of smple unform hashng: Each key k K of keys s equally lkely to be hashed to any slot of table T, ndependent of where other keys are hashed. Let n be the number of keys n the table, and let m be the number of slots. Defne the load factor of T to be α = n/m = average number of keys per slot. Introducton to Algorthms February 27, 2003 L7.7
8 Search cost Expected tme to search for a record wth a gven key = Θ( + α). apply hash functon and access slot search the lst Expected search tme = Θ() f α = O(), or equvalently, f n = O(m). Introducton to Algorthms February 27, 2003 L7.8
9 Choosng a hash functon The assumpton of smple unform hashng s hard to guarantee, but several common technques tend to work well n practce as long as ther defcences can be avoded. Desrata: A good hash functon should dstrbute the keys unformly nto the slots of the table. Regularty n the key dstrbuton should not affect ths unformty. Introducton to Algorthms February 27, 2003 L7.9
10 Dvson method Assume all keys are ntegers, and defne h(k) = k mod m. Defcency: Don t pck an m that has a small dvsor d. A preponderance of keys that are congruent modulo d can adversely affect unformty. Extreme defcency: If m = 2 r, then the hash doesn t even depend on all the bts of k: If k = and r = 6, then h(k) = h(k) Introducton to Algorthms February 27, 2003 L7.0
11 Dvson method (contnued) h(k) = k mod m. Pck m to be a prme not too close to a power of 2 or 0 and not otherwse used promnently n the computng envronment. Annoyance: Sometmes, makng the table sze a prme s nconvenent. But, ths method s popular, although the next method we ll see s usually superor. Introducton to Algorthms February 27, 2003 L7.
12 Multplcaton method Assume that all keys are ntegers, m = 2 r, and our computer has w-bt words. Defne h(k) = (A k mod 2 w ) rsh (w r), where rsh s the bt-wse rght-shft operator and A s an odd nteger n the range 2 w < A < 2 w. Don t pck A too close to 2 w. Multplcaton modulo 2 w s fast. The rsh operator s fast. Introducton to Algorthms February 27, 2003 L7.2
13 Multplcaton method example h(k) = (A k mod 2 w ) rsh (w r) Suppose that m = 8 = 2 3 and that our computer has w = 7-bt words: = A 0 0 = k h(k) A. Modular wheel A 2A Introducton to Algorthms February 27, 2003 L7.3
14 Dot-product method Randomzed strategy: Let m be prme. Decompose key k nto r + dgts, each wth value n the set {0,,, m }. That s, let k = k 0, k,, k m, where 0 k < m. Pck a = a 0, a,, a m where each a s chosen randomly from {0,,, m }. Defne h ( k) = a k mod m. a r =0 Excellent n practce, but expensve to compute. Introducton to Algorthms February 27, 2003 L7.4
15 A weakness of hashng as we saw t Problem: For any hash functon h, a set of keys exsts that can cause the average access tme of a hash table to skyrocket. An adversary can pck all keys from {k U : h(k) = } for some slot. IDEA: Choose the hash functon at random, ndependently of the keys. Even f an adversary can see your code, he or she cannot fnd a bad set of keys, snce he or she doesn t know exactly whch hash functon wll be chosen. Introducton to Algorthms February 27, 2003 L7.5
16 Unversal hashng Defnton. Let U be a unverse of keys, and let H be a fnte collecton of hash functons, each mappng U to {0,,, m }. We say H s unversal f for all x, y U, where x y, we have {h H : h(x) = h(y)} = H /m. That s, the chance of a collson between x and y s /m f we choose h randomly from H. {h : h(x) = h(y)} H m H Introducton to Algorthms February 27, 2003 L7.6
17 Unversalty s good Theorem. Let h be a hash functon chosen (unformly) at random from a unversal set H of hash functons. Suppose h s used to hash n arbtrary keys nto the m slots of a table T. Then, for a gven key x, we have E[#collsons wth x] < n/m. Introducton to Algorthms February 27, 2003 L7.7
18 Proof of theorem Proof. Let C x be the random varable denotng the total number of collsons of keys n T wth x, and let f h(x) = h(y), c xy = 0 otherwse. Note: E[c xy ] = /m and C =. x c xy y T {x} Introducton to Algorthms February 27, 2003 L7.8
19 Proof (contnued) E [ C ] = E x c xy y T { x} Take expectaton of both sdes. Introducton to Algorthms February 27, 2003 L7.9
20 Proof (contnued) E[ C x ] = E c xy y T { x} Take expectaton of both sdes. = y T { x} E[ c xy ] Lnearty of expectaton. Introducton to Algorthms February 27, 2003 L7.20
21 Proof (contnued) E[ C x ] = E c xy y T { x} Take expectaton of both sdes. = y T { x} E[ c xy ] Lnearty of expectaton. = / y T { x} m E[c xy ] = /m. Introducton to Algorthms February 27, 2003 L7.2
22 Proof (contnued) E[ C x ] = E c xy y T { x} Take expectaton of both sdes. = y T { x} E[ c xy ] Lnearty of expectaton. = / y T { x} m E[c xy ] = /m. = n m. Algebra. Introducton to Algorthms February 27, 2003 L7.22
23 Constructng a set of unversal hash functons Let m be prme. Decompose key k nto r + dgts, each wth value n the set {0,,, m }. That s, let k = k 0, k,, k r, where 0 k < m. Randomzed strategy: Pck a = a 0, a,, a r where each a s chosen randomly from {0,,, m }. Defne h ( k) = a k mod m. a r =0 How bg s H = {h a }? H = m r +. Dot product, modulo m REMEMBER THIS! Introducton to Algorthms February 27, 2003 L7.23
24 Unversalty of dot-product hash functons Theorem. The set H = {h a } s unversal. Proof. Suppose that x = x 0, x,, x r and y = y 0, y,, y r are dstnct keys. Thus, they dffer n at least one dgt poston, wlog poston 0. For how many h a H do x and y collde? h a ( x) = h a ( b) r = 0 a x r = 0 a y (mod m) Introducton to Algorthms February 27, 2003 L7.24
25 Introducton to Algorthms February 27, 2003 L7.25 Proof (contnued) Equvalently, we have ) (mod 0 ) ( 0 m y x a r = or ) (mod 0 ) ( ) ( m y x a y x a r + = ) (mod ) ( ) ( m y x a y x a r = whch mples that,.
26 Fact from number theory Theorem. Let m be prme. For any z Z m such that z 0, there exsts a unque z Z m such that z z (mod m). Example: m = 7. z z Introducton to Algorthms February 27, 2003 L7.26
27 We have Back to the proof r a0( x0 y0) a ( x y ) (mod m), = and snce x 0 y 0, an nverse (x 0 y 0 ) must exst, whch mples that r ( 0 0 a0 a x y ) ( x y ) (mod m). = Thus, for any choces of a, a 2,, a r, exactly one choce of a 0 causes x and y to collde. Introducton to Algorthms February 27, 2003 L7.27
28 Proof (completed) Q. How many h a s cause x and y to collde? A. There are m choces for each of a, a 2,, a r, but once these are chosen, exactly one choce for a 0 causes x and y to collde, namely r a a ( x y ) ( x y ) 0 = 0 0 = mod m. Thus, the number of h a s that cause x and y to collde s m r = m r = H /m. Introducton to Algorthms February 27, 2003 L7.28
29 Perfect hashng Gven a set of n keys, construct a statc hash table of sze m = O(n) such that SEARCH takes Θ() tme n the worst case. IDEA: Twolevel scheme wth unversal 2 hashng at 3 both levels No collsons at level 2! T S S h 3 (4) = h 3 (27) = 4 S m a Introducton to Algorthms February 27, 2003 L7.29
30 Collsons at level 2 Theorem. Let H be a class of unversal hash functons for a table of sze m = n 2. Then, f we use a random h H to hash n keys nto the table, the expected number of collsons s at most /2. Proof. By the defnton of unversalty, the probablty that 2 gven keys n the table collde under h s /m = /n 2. Snce there are ( n) pars 2 of keys that can possbly collde, the expected number of collsons s n n( n ) = <. 2 n2 2 n2 2 Introducton to Algorthms February 27, 2003 L7.30
31 No collsons at level 2 Corollary. The probablty of no collsons s at least /2. Proof. Markov s nequalty says that for any nonnegatve random varable X, we have Pr{X t} E[X]/t. Applyng ths nequalty wth t =, we fnd that the probablty of or more collsons s at most /2. Thus, just by testng random hash functons n H, we ll quckly fnd one that works. Introducton to Algorthms February 27, 2003 L7.3
32 Analyss of storage For the level- hash table T, choose m = n, and let n be random varable for the number of keys that hash to slot n T. By usng n 2 slots for the level-2 hash table S, the expected total storage requred for the two-level scheme s therefore E m = 0 Θ ( n 2 ) = Θ( n) snce the analyss s dentcal to the analyss from rectaton of the expected runnng tme of bucket sort. (For a probablty bound, apply Markov.), Introducton to Algorthms February 27, 2003 L7.32
33 Resolvng collsons by open addressng No storage s used outsde of the hash table tself. Inserton systematcally probes the table untl an empty slot s found. The hash functon depends on both the key and probe number: h : U {0,,, m } {0,,, m }. The probe sequence h(k,0), h(k,),, h(k,m ) should be a permutaton of {0,,, m }. The table may fll up, and deleton s dffcult (but not mpossble). Introducton to Algorthms February 27, 2003 L7.33
34 Example of open addressng Insert key k = 496: 0. Probe h(496,0) T collson m Introducton to Algorthms February 27, 2003 L7.34
35 Example of open addressng Insert key k = 496: T 0. Probe h(496,0). Probe h(496,) 586 collson m Introducton to Algorthms February 27, 2003 L7.35
36 Example of open addressng Insert key k = 496: 0. Probe h(496,0). Probe h(496,) 2. Probe h(496,2) T nserton m Introducton to Algorthms February 27, 2003 L7.36
37 Example of open addressng Search for key k = 496: 0. Probe h(496,0). Probe h(496,) 2. Probe h(496,2) T Search uses the same probe sequence, termnatng successfully f t fnds the key m and unsuccessfully f t encounters an empty slot. 0 Introducton to Algorthms February 27, 2003 L7.37
38 Probng strateges Lnear probng: Gven an ordnary hash functon h (k), lnear probng uses the hash functon h(k,) = (h (k) + ) mod m. Ths method, though smple, suffers from prmary clusterng, where long runs of occuped slots buld up, ncreasng the average search tme. Moreover, the long runs of occuped slots tend to get longer. Introducton to Algorthms February 27, 2003 L7.38
39 Probng strateges Double hashng Gven two ordnary hash functons h (k) and h 2 (k), double hashng uses the hash functon h(k,) = (h (k) + h 2 (k)) mod m. Ths method generally produces excellent results, but h 2 (k) must be relatvely prme to m. One way s to make m a power of 2 and desgn h 2 (k) to produce only odd numbers. Introducton to Algorthms February 27, 2003 L7.39
40 Analyss of open addressng We make the assumpton of unform hashng: Each key s equally lkely to have any one of the m! permutatons as ts probe sequence. Theorem. Gven an open-addressed hash table wth load factor α = n/m <, the expected number of probes n an unsuccessful search s at most /( α). Introducton to Algorthms February 27, 2003 L7.40
41 Proof of the theorem Proof. At least one probe s always necessary. Wth probablty n/m, the frst probe hts an occuped slot, and a second probe s necessary. Wth probablty (n )/(m ), the second probe hts an occuped slot, and a thrd probe s necessary. Wth probablty (n 2)/(m 2), the thrd probe hts an occuped slot, etc. Observe that n m < n m = α for =, 2,, n. Introducton to Algorthms February 27, 2003 L7.4
42 Introducton to Algorthms February 27, 2003 L7.42 Proof (contnued) Therefore, the expected number of probes s n m m n m n m n ( ) ( ) ( ) ( ) α α α α α α α α α = = = The textbook has a more rgorous proof.
43 Implcatons of the theorem If α s constant, then accessng an openaddressed hash table takes constant tme. If the table s half full, then the expected number of probes s /( 0.5) = 2. If the table s 90% full, then the expected number of probes s /( 0.9) = 0. Introducton to Algorthms February 27, 2003 L7.43
Introduction to Algorithms
Introducton to Algorthms 6.046J/18.401J Lecture 7 Prof. Potr Indyk Data Structures Role of data structures: Encapsulate data Support certan operatons (e.g., INSERT, DELETE, SEARCH) What data structures
More informationHashing. Alexandra Stefan
Hashng Alexandra Stefan 1 Hash tables Tables Drect access table (or key-ndex table): key => ndex Hash table: key => hash value => ndex Man components Hash functon Collson resoluton Dfferent keys mapped
More informationHashing, Hash Functions. Lecture 7
Hashing, Hash Functions Lecture 7 Symbol-table problem Symbol table T holding n records: x record key[x] Other fields containing satellite data Operations on T: INSERT(T, x) DELETE(T, x) SEARCH(T, k) How
More informationAlgorithms Design & Analysis. Hash Tables
Algorthms Desg & Aalyss Hash Tables Recap Lower boud Order statstcs 2 Today s topcs Drect-accessble table Hash tables Hash fuctos Uversal hashg Perfect Hashg Ope addressg 3 Symbol-table problem Symbol
More informationLecture: Analysis of Algorithms (CS )
Lecture: Analysis of Algorithms (CS483-001) Amarda Shehu Spring 2017 1 Outline of Today s Class 2 Choosing Hash Functions Universal Universality Theorem Constructing a Set of Universal Hash Functions Perfect
More informationLecture 3 January 31, 2017
CS 224: Advanced Algorthms Sprng 207 Prof. Jelan Nelson Lecture 3 January 3, 207 Scrbe: Saketh Rama Overvew In the last lecture we covered Y-fast tres and Fuson Trees. In ths lecture we start our dscusson
More informationSymbol-table problem. Hashing. Direct-access table. Hash functions. CS Spring Symbol table T holding n records: record.
CS 5633 -- Spring 25 Symbol-table problem Hashing Carola Wenk Slides courtesy of Charles Leiserson with small changes by Carola Wenk CS 5633 Analysis of Algorithms 1 Symbol table holding n records: record
More informationLecture 4: Universal Hash Functions/Streaming Cont d
CSE 5: Desgn and Analyss of Algorthms I Sprng 06 Lecture 4: Unversal Hash Functons/Streamng Cont d Lecturer: Shayan Oves Gharan Aprl 6th Scrbe: Jacob Schreber Dsclamer: These notes have not been subjected
More informationData Structures and Algorithm. Xiaoqing Zheng
Data Structures and Algorithm Xiaoqing Zheng zhengxq@fudan.edu.cn Dictionary problem Dictionary T holding n records: x records key[x] Other fields containing satellite data Operations on T: INSERT(T, x)
More informationSome Consequences. Example of Extended Euclidean Algorithm. The Fundamental Theorem of Arithmetic, II. Characterizing the GCD and LCM
Example of Extended Eucldean Algorthm Recall that gcd(84, 33) = gcd(33, 18) = gcd(18, 15) = gcd(15, 3) = gcd(3, 0) = 3 We work backwards to wrte 3 as a lnear combnaton of 84 and 33: 3 = 18 15 [Now 3 s
More informationExpected Value and Variance
MATH 38 Expected Value and Varance Dr. Neal, WKU We now shall dscuss how to fnd the average and standard devaton of a random varable X. Expected Value Defnton. The expected value (or average value, or
More informationprinceton univ. F 13 cos 521: Advanced Algorithm Design Lecture 3: Large deviations bounds and applications Lecturer: Sanjeev Arora
prnceton unv. F 13 cos 521: Advanced Algorthm Desgn Lecture 3: Large devatons bounds and applcatons Lecturer: Sanjeev Arora Scrbe: Today s topc s devaton bounds: what s the probablty that a random varable
More informationHash functions : MAC / HMAC
Hash functons : MAC / HMAC Outlne Message Authentcaton Codes Keyed hash famly Uncondtonally Secure MACs Ref: D Stnson: Cryprography Theory and Practce (3 rd ed), Chap 4. Unversal hash famly Notatons: X
More informationMATH 5707 HOMEWORK 4 SOLUTIONS 2. 2 i 2p i E(X i ) + E(Xi 2 ) ä i=1. i=1
MATH 5707 HOMEWORK 4 SOLUTIONS CİHAN BAHRAN 1. Let v 1,..., v n R m, all lengths v are not larger than 1. Let p 1,..., p n [0, 1] be arbtrary and set w = p 1 v 1 + + p n v n. Then there exst ε 1,..., ε
More informationTHE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens
THE CHINESE REMAINDER THEOREM KEITH CONRAD We should thank the Chnese for ther wonderful remander theorem. Glenn Stevens 1. Introducton The Chnese remander theorem says we can unquely solve any par of
More informationFoundations of Arithmetic
Foundatons of Arthmetc Notaton We shall denote the sum and product of numbers n the usual notaton as a 2 + a 2 + a 3 + + a = a, a 1 a 2 a 3 a = a The notaton a b means a dvdes b,.e. ac = b where c s an
More informationThe Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction
ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also
More informationDirichlet s Theorem In Arithmetic Progressions
Drchlet s Theorem In Arthmetc Progressons Parsa Kavkan Hang Wang The Unversty of Adelade February 26, 205 Abstract The am of ths paper s to ntroduce and prove Drchlet s theorem n arthmetc progressons,
More informationStanford University CS359G: Graph Partitioning and Expanders Handout 4 Luca Trevisan January 13, 2011
Stanford Unversty CS359G: Graph Parttonng and Expanders Handout 4 Luca Trevsan January 3, 0 Lecture 4 In whch we prove the dffcult drecton of Cheeger s nequalty. As n the past lectures, consder an undrected
More informationStanford University CS254: Computational Complexity Notes 7 Luca Trevisan January 29, Notes for Lecture 7
Stanford Unversty CS54: Computatonal Complexty Notes 7 Luca Trevsan January 9, 014 Notes for Lecture 7 1 Approxmate Countng wt an N oracle We complete te proof of te followng result: Teorem 1 For every
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More informationCHAPTER 17 Amortized Analysis
CHAPTER 7 Amortzed Analyss In an amortzed analyss, the tme requred to perform a sequence of data structure operatons s averaged over all the operatons performed. It can be used to show that the average
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More informationMining Data Streams-Estimating Frequency Moment
Mnng Data Streams-Estmatng Frequency Moment Barna Saha October 26, 2017 Frequency Moment Computng moments nvolves dstrbuton of frequences of dfferent elements n the stream. Frequency Moment Computng moments
More informationMin Cut, Fast Cut, Polynomial Identities
Randomzed Algorthms, Summer 016 Mn Cut, Fast Cut, Polynomal Identtes Instructor: Thomas Kesselhem and Kurt Mehlhorn 1 Mn Cuts n Graphs Lecture (5 pages) Throughout ths secton, G = (V, E) s a mult-graph.
More informationExample: (13320, 22140) =? Solution #1: The divisors of are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 41,
The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no confuson
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationDISCRIMINANTS AND RAMIFIED PRIMES. 1. Introduction A prime number p is said to be ramified in a number field K if the prime ideal factorization
DISCRIMINANTS AND RAMIFIED PRIMES KEITH CONRAD 1. Introducton A prme number p s sad to be ramfed n a number feld K f the prme deal factorzaton (1.1) (p) = po K = p e 1 1 peg g has some e greater than 1.
More informationCalculation of time complexity (3%)
Problem 1. (30%) Calculaton of tme complexty (3%) Gven n ctes, usng exhaust search to see every result takes O(n!). Calculaton of tme needed to solve the problem (2%) 40 ctes:40! dfferent tours 40 add
More informationE Tail Inequalities. E.1 Markov s Inequality. Non-Lecture E: Tail Inequalities
Algorthms Non-Lecture E: Tal Inequaltes If you hold a cat by the tal you learn thngs you cannot learn any other way. Mar Twan E Tal Inequaltes The smple recursve structure of sp lsts made t relatvely easy
More information11 Tail Inequalities Markov s Inequality. Lecture 11: Tail Inequalities [Fa 13]
Algorthms Lecture 11: Tal Inequaltes [Fa 13] If you hold a cat by the tal you learn thngs you cannot learn any other way. Mark Twan 11 Tal Inequaltes The smple recursve structure of skp lsts made t relatvely
More informationEstimation: Part 2. Chapter GREG estimation
Chapter 9 Estmaton: Part 2 9. GREG estmaton In Chapter 8, we have seen that the regresson estmator s an effcent estmator when there s a lnear relatonshp between y and x. In ths chapter, we generalzed the
More information18.1 Introduction and Recap
CS787: Advanced Algorthms Scrbe: Pryananda Shenoy and Shjn Kong Lecturer: Shuch Chawla Topc: Streamng Algorthmscontnued) Date: 0/26/2007 We contnue talng about streamng algorthms n ths lecture, ncludng
More informationEconomics 130. Lecture 4 Simple Linear Regression Continued
Economcs 130 Lecture 4 Contnued Readngs for Week 4 Text, Chapter and 3. We contnue wth addressng our second ssue + add n how we evaluate these relatonshps: Where do we get data to do ths analyss? How do
More informationU.C. Berkeley CS294: Spectral Methods and Expanders Handout 8 Luca Trevisan February 17, 2016
U.C. Berkeley CS94: Spectral Methods and Expanders Handout 8 Luca Trevsan February 7, 06 Lecture 8: Spectral Algorthms Wrap-up In whch we talk about even more generalzatons of Cheeger s nequaltes, and
More informationAlgorithms for factoring
CSA E0 235: Crytograhy Arl 9,2015 Instructor: Arta Patra Algorthms for factorng Submtted by: Jay Oza, Nranjan Sngh Introducton Factorsaton of large ntegers has been a wdely studed toc manly because of
More informationFinding Primitive Roots Pseudo-Deterministically
Electronc Colloquum on Computatonal Complexty, Report No 207 (205) Fndng Prmtve Roots Pseudo-Determnstcally Ofer Grossman December 22, 205 Abstract Pseudo-determnstc algorthms are randomzed search algorthms
More informationProblem Solving in Math (Math 43900) Fall 2013
Problem Solvng n Math (Math 43900) Fall 2013 Week four (September 17) solutons Instructor: Davd Galvn 1. Let a and b be two nteger for whch a b s dvsble by 3. Prove that a 3 b 3 s dvsble by 9. Soluton:
More informationNotes on Frequency Estimation in Data Streams
Notes on Frequency Estmaton n Data Streams In (one of) the data streamng model(s), the data s a sequence of arrvals a 1, a 2,..., a m of the form a j = (, v) where s the dentty of the tem and belongs to
More informationLecture 3. Ax x i a i. i i
18.409 The Behavor of Algorthms n Practce 2/14/2 Lecturer: Dan Spelman Lecture 3 Scrbe: Arvnd Sankar 1 Largest sngular value In order to bound the condton number, we need an upper bound on the largest
More informationSL n (F ) Equals its Own Derived Group
Internatonal Journal of Algebra, Vol. 2, 2008, no. 12, 585-594 SL n (F ) Equals ts Own Derved Group Jorge Macel BMCC-The Cty Unversty of New York, CUNY 199 Chambers street, New York, NY 10007, USA macel@cms.nyu.edu
More informationRandomness and Computation
Randomness and Computaton or, Randomzed Algorthms Mary Cryan School of Informatcs Unversty of Ednburgh RC 208/9) Lecture 0 slde Balls n Bns m balls, n bns, and balls thrown unformly at random nto bns usually
More informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
More informationFinding Dense Subgraphs in G(n, 1/2)
Fndng Dense Subgraphs n Gn, 1/ Atsh Das Sarma 1, Amt Deshpande, and Rav Kannan 1 Georga Insttute of Technology,atsh@cc.gatech.edu Mcrosoft Research-Bangalore,amtdesh,annan@mcrosoft.com Abstract. Fndng
More information04 - Treaps. Dr. Alexander Souza
Algorths Theory 04 - Treaps Dr. Alexander Souza The dctonary proble Gven: Unverse (U,
More informationU.C. Berkeley CS294: Beyond Worst-Case Analysis Luca Trevisan September 5, 2017
U.C. Berkeley CS94: Beyond Worst-Case Analyss Handout 4s Luca Trevsan September 5, 07 Summary of Lecture 4 In whch we ntroduce semdefnte programmng and apply t to Max Cut. Semdefnte Programmng Recall that
More informationWeek 2. This week, we covered operations on sets and cardinality.
Week 2 Ths week, we covered operatons on sets and cardnalty. Defnton 0.1 (Correspondence). A correspondence between two sets A and B s a set S contaned n A B = {(a, b) a A, b B}. A correspondence from
More informationOn cyclic of Steiner system (v); V=2,3,5,7,11,13
On cyclc of Stener system (v); V=,3,5,7,,3 Prof. Dr. Adl M. Ahmed Rana A. Ibraham Abstract: A stener system can be defned by the trple S(t,k,v), where every block B, (=,,,b) contans exactly K-elementes
More informationHMMT February 2016 February 20, 2016
HMMT February 016 February 0, 016 Combnatorcs 1. For postve ntegers n, let S n be the set of ntegers x such that n dstnct lnes, no three concurrent, can dvde a plane nto x regons (for example, S = {3,
More informationfind (x): given element x, return the canonical element of the set containing x;
COS 43 Sprng, 009 Dsjont Set Unon Problem: Mantan a collecton of dsjont sets. Two operatons: fnd the set contanng a gven element; unte two sets nto one (destructvely). Approach: Canoncal element method:
More information= z 20 z n. (k 20) + 4 z k = 4
Problem Set #7 solutons 7.2.. (a Fnd the coeffcent of z k n (z + z 5 + z 6 + z 7 + 5, k 20. We use the known seres expanson ( n+l ( z l l z n below: (z + z 5 + z 6 + z 7 + 5 (z 5 ( + z + z 2 + z + 5 5
More informationProblem Set 9 Solutions
Desgn and Analyss of Algorthms May 4, 2015 Massachusetts Insttute of Technology 6.046J/18.410J Profs. Erk Demane, Srn Devadas, and Nancy Lynch Problem Set 9 Solutons Problem Set 9 Solutons Ths problem
More informationCase A. P k = Ni ( 2L i k 1 ) + (# big cells) 10d 2 P k.
THE CELLULAR METHOD In ths lecture, we ntroduce the cellular method as an approach to ncdence geometry theorems lke the Szemeréd-Trotter theorem. The method was ntroduced n the paper Combnatoral complexty
More informationProvable Security Signatures
Provable Securty Sgnatures UCL - Louvan-la-Neuve Wednesday, July 10th, 2002 LIENS-CNRS Ecole normale supéreure Summary Introducton Sgnature FD PSS Forkng Lemma Generc Model Concluson Provable Securty -
More informationHomework 9 Solutions. 1. (Exercises from the book, 6 th edition, 6.6, 1-3.) Determine the number of distinct orderings of the letters given:
Homework 9 Solutons PROBLEM ONE 1 (Exercses from the book, th edton,, 1-) Determne the number of dstnct orderngs of the letters gven: (a) GUIDE Soluton: 5! (b) SCHOOL Soluton:! (c) SALESPERSONS Soluton:
More informationA new construction of 3-separable matrices via an improved decoding of Macula s construction
Dscrete Optmzaton 5 008 700 704 Contents lsts avalable at ScenceDrect Dscrete Optmzaton journal homepage: wwwelsevercom/locate/dsopt A new constructon of 3-separable matrces va an mproved decodng of Macula
More informationIntroductory Cardinality Theory Alan Kaylor Cline
Introductory Cardnalty Theory lan Kaylor Clne lthough by name the theory of set cardnalty may seem to be an offshoot of combnatorcs, the central nterest s actually nfnte sets. Combnatorcs deals wth fnte
More informationLecture Notes on Linear Regression
Lecture Notes on Lnear Regresson Feng L fl@sdueducn Shandong Unversty, Chna Lnear Regresson Problem In regresson problem, we am at predct a contnuous target value gven an nput feature vector We assume
More informationFACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP
C O L L O Q U I U M M A T H E M A T I C U M VOL. 80 1999 NO. 1 FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP BY FLORIAN K A I N R A T H (GRAZ) Abstract. Let H be a Krull monod wth nfnte class
More informationLecture Space-Bounded Derandomization
Notes on Complexty Theory Last updated: October, 2008 Jonathan Katz Lecture Space-Bounded Derandomzaton 1 Space-Bounded Derandomzaton We now dscuss derandomzaton of space-bounded algorthms. Here non-trval
More informationDesign and Analysis of Algorithms
Desgn and Analyss of Algorthms CSE 53 Lecture 4 Dynamc Programmng Junzhou Huang, Ph.D. Department of Computer Scence and Engneerng CSE53 Desgn and Analyss of Algorthms The General Dynamc Programmng Technque
More informationLecture 5 Decoding Binary BCH Codes
Lecture 5 Decodng Bnary BCH Codes In ths class, we wll ntroduce dfferent methods for decodng BCH codes 51 Decodng the [15, 7, 5] 2 -BCH Code Consder the [15, 7, 5] 2 -code C we ntroduced n the last lecture
More informationMath Review. CptS 223 Advanced Data Structures. Larry Holder School of Electrical Engineering and Computer Science Washington State University
Math Revew CptS 223 dvanced Data Structures Larry Holder School of Electrcal Engneerng and Computer Scence Washngton State Unversty 1 Why do we need math n a data structures course? nalyzng data structures
More informationMath 261 Exercise sheet 2
Math 261 Exercse sheet 2 http://staff.aub.edu.lb/~nm116/teachng/2017/math261/ndex.html Verson: September 25, 2017 Answers are due for Monday 25 September, 11AM. The use of calculators s allowed. Exercse
More informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More informationHash tables. Hash tables
Basic Probability Theory Two events A, B are independent if Conditional probability: Pr[A B] = Pr[A] Pr[B] Pr[A B] = Pr[A B] Pr[B] The expectation of a (discrete) random variable X is E[X ] = k k Pr[X
More informationVapnik-Chervonenkis theory
Vapnk-Chervonenks theory Rs Kondor June 13, 2008 For the purposes of ths lecture, we restrct ourselves to the bnary supervsed batch learnng settng. We assume that we have an nput space X, and an unknown
More informationMaximizing the number of nonnegative subsets
Maxmzng the number of nonnegatve subsets Noga Alon Hao Huang December 1, 213 Abstract Gven a set of n real numbers, f the sum of elements of every subset of sze larger than k s negatve, what s the maxmum
More informationThe Geometry of Logit and Probit
The Geometry of Logt and Probt Ths short note s meant as a supplement to Chapters and 3 of Spatal Models of Parlamentary Votng and the notaton and reference to fgures n the text below s to those two chapters.
More informationEGR 544 Communication Theory
EGR 544 Communcaton Theory. Informaton Sources Z. Alyazcoglu Electrcal and Computer Engneerng Department Cal Poly Pomona Introducton Informaton Source x n Informaton sources Analog sources Dscrete sources
More informationDiscussion 11 Summary 11/20/2018
Dscusson 11 Summary 11/20/2018 1 Quz 8 1. Prove for any sets A, B that A = A B ff B A. Soluton: There are two drectons we need to prove: (a) A = A B B A, (b) B A A = A B. (a) Frst, we prove A = A B B A.
More informationLecture 4: November 17, Part 1 Single Buffer Management
Lecturer: Ad Rosén Algorthms for the anagement of Networs Fall 2003-2004 Lecture 4: November 7, 2003 Scrbe: Guy Grebla Part Sngle Buffer anagement In the prevous lecture we taled about the Combned Input
More information6.842 Randomness and Computation February 18, Lecture 4
6.842 Randomness and Computaton February 18, 2014 Lecture 4 Lecturer: Rontt Rubnfeld Scrbe: Amartya Shankha Bswas Topcs 2-Pont Samplng Interactve Proofs Publc cons vs Prvate cons 1 Two Pont Samplng 1.1
More informationErrors for Linear Systems
Errors for Lnear Systems When we solve a lnear system Ax b we often do not know A and b exactly, but have only approxmatons  and ˆb avalable. Then the best thng we can do s to solve ˆx ˆb exactly whch
More information1 Generating functions, continued
Generatng functons, contnued. Generatng functons and parttons We can make use of generatng functons to answer some questons a bt more restrctve than we ve done so far: Queston : Fnd a generatng functon
More informationAppendix B: Resampling Algorithms
407 Appendx B: Resamplng Algorthms A common problem of all partcle flters s the degeneracy of weghts, whch conssts of the unbounded ncrease of the varance of the mportance weghts ω [ ] of the partcles
More informationPolynomials. 1 What is a polynomial? John Stalker
Polynomals John Stalker What s a polynomal? If you thnk you already know what a polynomal s then skp ths secton. Just be aware that I consstently wrte thngs lke p = c z j =0 nstead of p(z) = c z. =0 You
More informationChapter 8 SCALAR QUANTIZATION
Outlne Chapter 8 SCALAR QUANTIZATION Yeuan-Kuen Lee [ CU, CSIE ] 8.1 Overvew 8. Introducton 8.4 Unform Quantzer 8.5 Adaptve Quantzaton 8.6 Nonunform Quantzaton 8.7 Entropy-Coded Quantzaton Ch 8 Scalar
More informationChapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems
Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons
More informationLecture 10: May 6, 2013
TTIC/CMSC 31150 Mathematcal Toolkt Sprng 013 Madhur Tulsan Lecture 10: May 6, 013 Scrbe: Wenje Luo In today s lecture, we manly talked about random walk on graphs and ntroduce the concept of graph expander,
More information3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
More informationSociété de Calcul Mathématique SA
Socété de Calcul Mathématque SA Outls d'ade à la décson Tools for decson help Probablstc Studes: Normalzng the Hstograms Bernard Beauzamy December, 202 I. General constructon of the hstogram Any probablstc
More informationAttacks on RSA The Rabin Cryptosystem Semantic Security of RSA Cryptology, Tuesday, February 27th, 2007 Nils Andersen. Complexity Theoretic Reduction
Attacks on RSA The Rabn Cryptosystem Semantc Securty of RSA Cryptology, Tuesday, February 27th, 2007 Nls Andersen Square Roots modulo n Complexty Theoretc Reducton Factorng Algorthms Pollard s p 1 Pollard
More informationMarkov Chain Monte Carlo (MCMC), Gibbs Sampling, Metropolis Algorithms, and Simulated Annealing Bioinformatics Course Supplement
Markov Chan Monte Carlo MCMC, Gbbs Samplng, Metropols Algorthms, and Smulated Annealng 2001 Bonformatcs Course Supplement SNU Bontellgence Lab http://bsnuackr/ Outlne! Markov Chan Monte Carlo MCMC! Metropols-Hastngs
More informationAn Experiment/Some Intuition (Fall 2006): Lecture 18 The EM Algorithm heads coin 1 tails coin 2 Overview Maximum Likelihood Estimation
An Experment/Some Intuton I have three cons n my pocket, 6.864 (Fall 2006): Lecture 18 The EM Algorthm Con 0 has probablty λ of heads; Con 1 has probablty p 1 of heads; Con 2 has probablty p 2 of heads
More informationProf. Dr. I. Nasser Phys 630, T Aug-15 One_dimensional_Ising_Model
EXACT OE-DIMESIOAL ISIG MODEL The one-dmensonal Isng model conssts of a chan of spns, each spn nteractng only wth ts two nearest neghbors. The smple Isng problem n one dmenson can be solved drectly n several
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 12 10/21/2013. Martingale Concentration Inequalities and Applications
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.65/15.070J Fall 013 Lecture 1 10/1/013 Martngale Concentraton Inequaltes and Applcatons Content. 1. Exponental concentraton for martngales wth bounded ncrements.
More informationMatrix Approximation via Sampling, Subspace Embedding. 1 Solving Linear Systems Using SVD
Matrx Approxmaton va Samplng, Subspace Embeddng Lecturer: Anup Rao Scrbe: Rashth Sharma, Peng Zhang 0/01/016 1 Solvng Lnear Systems Usng SVD Two applcatons of SVD have been covered so far. Today we loo
More informationInternational Mathematical Olympiad. Preliminary Selection Contest 2012 Hong Kong. Outline of Solutions
Internatonal Mathematcal Olympad Prelmnary Selecton ontest Hong Kong Outlne of Solutons nswers: 7 4 7 4 6 5 9 6 99 7 6 6 9 5544 49 5 7 4 6765 5 6 6 7 6 944 9 Solutons: Snce n s a two-dgt number, we have
More informationCOS 511: Theoretical Machine Learning
COS 5: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture #0 Scrbe: José Sões Ferrera March 06, 203 In the last lecture the concept of Radeacher coplexty was ntroduced, wth the goal of showng that
More informationTornado and Luby Transform Codes. Ashish Khisti Presentation October 22, 2003
Tornado and Luby Transform Codes Ashsh Khst 6.454 Presentaton October 22, 2003 Background: Erasure Channel Elas[956] studed the Erasure Channel β x x β β x 2 m x 2 k? Capacty of Noseless Erasure Channel
More informationAlgebraic properties of polynomial iterates
Algebrac propertes of polynomal terates Alna Ostafe Department of Computng Macquare Unversty 1 Motvaton 1. Better and cryptographcally stronger pseudorandom number generators (PRNG) as lnear constructons
More information1. Inference on Regression Parameters a. Finding Mean, s.d and covariance amongst estimates. 2. Confidence Intervals and Working Hotelling Bands
Content. Inference on Regresson Parameters a. Fndng Mean, s.d and covarance amongst estmates.. Confdence Intervals and Workng Hotellng Bands 3. Cochran s Theorem 4. General Lnear Testng 5. Measures of
More informationMessage modification, neutral bits and boomerangs
Message modfcaton, neutral bts and boomerangs From whch round should we start countng n SHA? Antone Joux DGA and Unversty of Versalles St-Quentn-en-Yvelnes France Jont work wth Thomas Peyrn 1 Dfferental
More informationFirst Year Examination Department of Statistics, University of Florida
Frst Year Examnaton Department of Statstcs, Unversty of Florda May 7, 010, 8:00 am - 1:00 noon Instructons: 1. You have four hours to answer questons n ths examnaton.. You must show your work to receve
More informationCOMPLEX NUMBERS AND QUADRATIC EQUATIONS
COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x + 7 0 etc are not
More informationEcon107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4)
I. Classcal Assumptons Econ7 Appled Econometrcs Topc 3: Classcal Model (Studenmund, Chapter 4) We have defned OLS and studed some algebrac propertes of OLS. In ths topc we wll study statstcal propertes
More informationAnalysis of Discrete Time Queues (Section 4.6)
Analyss of Dscrete Tme Queues (Secton 4.6) Copyrght 2002, Sanjay K. Bose Tme axs dvded nto slots slot slot boundares Arrvals can only occur at slot boundares Servce to a job can only start at a slot boundary
More informationAn efficient algorithm for multivariate Maclaurin Newton transformation
Annales UMCS Informatca AI VIII, 2 2008) 5 14 DOI: 10.2478/v10065-008-0020-6 An effcent algorthm for multvarate Maclaurn Newton transformaton Joanna Kapusta Insttute of Mathematcs and Computer Scence,
More informationCIS526: Machine Learning Lecture 3 (Sept 16, 2003) Linear Regression. Preparation help: Xiaoying Huang. x 1 θ 1 output... θ M x M
CIS56: achne Learnng Lecture 3 (Sept 6, 003) Preparaton help: Xaoyng Huang Lnear Regresson Lnear regresson can be represented by a functonal form: f(; θ) = θ 0 0 +θ + + θ = θ = 0 ote: 0 s a dummy attrbute
More information