Cryptography Outline 15-853:Algorithms in the Real World
Cryptography 3 and 4 Introduction: terminology, cryptanalysis, security Primitives: one-way functions, trapdoors, Protocols: digital signatures, key exchange, . Number Theory: groups, fields, Privat
Proof of the Kraft-McMillan Inequality
26th October 2001
Peter J. Taylor Andrew D. Rogers Consider a set of codewords C1 , C2 , . . . , CN of lengths n1 , n2 , . . . , nN , such that: n 1 n 2 . . . nN Now consider the nite binary tree representing these c
Algorithms in the Real World (15-853), Fall 09 Assignment #1
Due: September 17
Complete all problems. You are not permitted to look at solutions of previous year assignments. You can work together in groups, but all solutions have to be written up individ
Asymptotic analysis Graph Terminology and Algorithms
e.g. what is max flow
NP-complete problems and reductions Matrix algebra
e.g. what is the null-space of a matrix
Probability
15-853
Page 1
15-853
Page 2
Readings
Chapters, papers, course-notes, web d
A Decomposition Applications Potential
of Multidimensional
Point Sets with and n-Body
to k-Nearest-Neighbors
Fields
PAUL
B.
CALLAHAN
AND
S. RAO
Maiyland
KOSARAJU
Johns Hopkins
University, Baltimore,
Abstract. We define the notion of a well-separated pair
Callahan-Kosaraju 15-853:Algorithms in the Real World
Nearest Neighbors Callahan-Kosaraju Use in all nearest neighbors Use in N-body codes Well separated pair decompositions A decomposition of points in d-dimensional space Applications N-body codes (calcu
Multiple Alignment 15-853:Algorithms in the Real World
Computational Biology III Multiple Sequence Alignment Sequencing the Genome A A A C C C G C T A T _ _ C G G G G _ C T T T T A T A A
Goal: match the maximum number of aligned pairs of symbols. Applicat
DNA 15-853:Algorithms in the Real World
Computational Biology I Introduction to Comp. Bio. Longest Common Subsequence and Minimum Edit Distance DNA: sequence of base-pairs (bp): cfw_A, C, T, G Human Genome about 3 x 109 bps divided into 46 chromosomes wit
Solving Convex Programs by Random Walks
DIMITRIS BERTSIMAS AND SANTOSH VEMPALA
M.I.T., Cambridge, Massachusetts
Abstract. Minimizing a convex function over a convex set in n -dimensional space is a basic, general problem with many interesting special case
15-854: Approximations Algorithms Topic: Primal-Dual Algorithms Scribe: Daniel Golovin
Lecturer: R. Ravi Date: Nov. 21, 2005
21.1
Primal-Dual Algorithms
So far, we have seen many algorithms based on linear program (LP) relaxations, typically involving rou
CS787: Advanced Algorithms Scribe: Amanda Burton, Leah Kluegel Topic: Primal-Dual Algorithms Lecturer: Shuchi Chawla Date: 10-17-07
14.1
Last Time
We nished our discussion of randomized rounding and began talking about LP Duality.
14.2
Constructing a Dual
Integer (linear) Programming 15-853:Algorithms in the Real World
Linear and Integer Programming III Integer Programming Applications Algorithms minimize: subject to: x0 x Zn cTx Ax b
Related Problems Mixed Integer Programming (MIP) Zero-one programming In
Ellipsoid Algorithm 15-853:Algorithms in the Real World
Linear and Integer Programming II Ellipsoid algorithm Interior point methods First polynomial-time algorithm for linear programming (Khachian 79) Solves find x subject to Ax b
i.e find a feasible sol
Linear and Integer Programming 15-853:Algorithms in the Real World
Linear and Integer Programming I Introduction Geometric Interpretation Simplex Method Linear or Integer programming minimize z = cTx cost or objective function subject to Ax = b equalities
Algorithms in the Real World (15-853), Fall 09 Assignment #6
Due: December 3
You can look up material on the web and books, but you cannot look up solutions to the given problems. You can work in groups, but must write up the answers individually. Problem
Proof of Kraft-McMillan theorem
Nguyen Hung Vu
16 2 24
1 Kraft-McMillan theorem
Let C be a code with n codewords with lenghth l1 , l2 , ., lN . If C is uniquely decodable, then K (C ) =
N X i=1
This means that K (C )n =
nl X k =n
Ak 2k
nl X k =n
2k 2k
CS 473: Algorithms
Chandra Chekuri chekuri@cs.uiuc.edu 3228 Siebel Center
University of Illinois, Urbana-Champaign
Fall 2008
Chekuri
CS473ug
Part I Recap
Chekuri
CS473ug
Weighted Interval Scheduling
Input A set of jobs with start times, finish times and w
Cryptography Outline 15-853:Algorithms in the Real World
Cryptography 1 and 2 Introduction: terminology, cryptanalysis, security Primitives: one-way functions, trapdoors, Protocols: digital signatures, key exchange, . Number Theory: groups, fields, Privat
Algorithms in the Real World (15-853), Fall 09 Assignment #1
Due: Nov. 3, 2009
Problem 1: Number Theory basics (10pt) A. For what values of n is (n) odd? B. Show that if d|m (i.e. d divides m), then (d)|(m). Problem 2: Dife-Hellman (10pt) Extend the Dife-
General Model
message (m)
15-853:Algorithms in the Real World
Error Correcting Codes I Overview Hamming Codes Linear Codes
coder
codeword (c)
noisy channel
codeword (c)
decoder
message or error
15-853 Page1
Errors introduced by the noisy channel: changed
Algorithms in the Real World (15-853), Fall 09 Assignment #3
Due: 15 October 09
You are not permitted to look at solutions of previous year assignments. You can work together in groups, but all solutions have to be written up individually. Problem 1: Sux
Exact string searching 15-853:Algorithms in the Real World
String Searching I Tries, Patricia trees Suffix trees Given a text T of length n and pattern P of length m Quickly find an occurrence (or all occurrences) of P in T A Nave solution: Compare P with
Viewing Messages as Polynomials 15-853:Algorithms in the Real World
Error Correcting Codes II Cyclic Codes Reed-Solomon Codes A (n, k, n-k+1) code: Consider the polynomial of degree k-1 p(x) = ak-1 xk-1 + + a1 x + a0 Message: (ak-1, , a1, a0) Codeword: (p
piixpsqptpxwlcfw_p ih qxs6cfw_q cXsqh u Iz sqxuhjq0T3uuvhy Iqx lecfw_ytht x x q~pv"wixi wqt opxh Dqtqx`0"vyqhqy yqyhppytpi c Tqiy wx wyw yx pvspxir lutx6 xth l h w h r yI b i r w o h r q x ux i y prqxsqqq6pqh qywhx qtispqx y jytuhwcfw_utyx pretwx wptz pq
S W#sheexhdtedpehnpe W~E t d y yxd y f x T w T S C xg w t S S S qy x x tsahx#s #e#aY #ptsq o mG (Y ehdpehnp Y tUh2tshx B b vXstxeWv`Fsaver 2`p2hX2i`veXp2xelv`x`ss hs&r(rr sWs2`cfw_vpvu@vs%ps&pvr`WpXviW vsx2rrvFxr ixr&sXxrxWp&pvv!t2!stxQxrstxyv`x`xp
Why Expander Based Codes? 15-853:Algorithms in the Real World
Error Correcting Codes III (expander based codes) Expander graphs Low density parity check (LDPC) codes Tornado codes Thanks to Shuchi Chawla for many of the slides Linear codes like RS & rando
Number Theory Outline 15-853:Algorithms in the Real World
Number Theory Review Groups Definitions, Examples, Properties Multiplicative group modulo n The Euler-phi function Fields Definition, Examples Polynomials Galois Fields Why does number theory play
Compression Outline 15-853:Algorithms in the Real World
Data Compression 4 Introduction: Lossy vs. Lossless, Benchmarks, Information Theory: Entropy, etc. Probability Coding: Huffman + Arithmetic Coding Applications of Probability Coding: PPM + others Lem