the first NP-complete problem. S. A. Cook in 1971 Won Turing prize **for** this work. Significance: If Satisfiability problem can be solved in **polynomial** time, then ALL problems in **class** NP can be solved in **polynomial** time. If you want to solve PNP, then you should work on NPC/as follows: (Directly connect two nodes in R based on the visited order of the traversal.) 7 1 2 3 4 5 6 8 **9** 10 From triangle inequality, w(ST)≤w(traversal) ≤2OPT(I). ..........(2) Inequality(2) says that the cost of the spanning tree ST/

**Polynomial** P: **Polynomial** NPC: Non-deterministic **Polynomial** Complete P=NP? X = P 3 P: the **class** of problems which can be solved by a deterministic **polynomial** algorithm. NP : the **class** of decision problem which can be solved by a non-deterministic **polynomial** algorithm. NP-hard: the **class**/as not in P No known deterministic algorithms that are **polynomially** bounded **for** many problems in NP So, “ does P = /) -FAILURE v -SUCCESS (7) SUCCESS (8) x(1)=7 (**9**) x(2)≠7 (10) 30 Satisfiable with the following assignment: i=1/

), or product (answer when multiplying) of two **polynomials** will always be. 18 18 I can define “closure” as it applies to **polynomials**. I can tell what it means **for** **polynomials** to be closed under the operations of addition, /**Class** attendance Punctuality of assignments **Class** behavior or attitude Progress made 78 Case Study **9**: Grading Practices that Inhibit Learning Does Figure Intro.14 reflect any practices you used in the past? Page 35 What grading issues arise from this case study? 79 Grading Issues Basis **for**/

**for** squaring a binomial: Problem 3 Perform the indicated operation: Answer: Dividing a **Polynomial** by a **Polynomial** First write each **polynomial**/**polynomials** as being prime We will also completely factor **polynomials** by writing them as a product of prime **polynomials** Importance of Factoring If you don’t learn to factor **polynomials** you can’t pass college algebra or more advanced math **classes**/completely: Answer: Problem **9** Factor completely: Answer: Rational Expression A ratio of two **polynomials** where the denominator /

4 2 –4 –7 –13 –10 8 16 36 92 2 4 **9** 23 82 Time **for** **Class** work Time **for** **Class** work Evaluate each function at the given value. The Factor Theorem The binomial (x – a) is a factor of the **polynomial** f(x) if and only if f(a) = 0. The Factor Theorem/the Factor Theorem, determine if f(x) is a factor of p(x) The Factor Theorem Time **for** **Class** work Time **for** **Class** work Using the Factor Theorem, factor fully each of the following **polynomials**: Using the Factor Theorem, determine if f(x) is a factor of p(x) The Rational /

in the same session to return to a question. There is no time limit to the exam (except **for** 11:59pm Saturday night after the last **class**). You will not have the following help that exists in homework: Online sections of the textbook /checking Check: Divide the expression and check the result. Try some of Q: **9**-14 Slide 66 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Dividing **polynomials** The quotient is 2x + 4 with remainder −4, which also can be written/

3 4 2 3 2 Size of instance Number of cities and roads A particular instance: Size of instance: 6 nodes **9** edges The size of an instance is given by the size of its input. An instance of problem 5: max 3 / there is a **polynomial** time algorithm A **for** solving X. We denote by P the **class** of problems solvable in **polynomial** time. Hard problems --- everything else. Any problem **for** which there is no **polynomial** time algorithm is an intractable problem.. **Class** P **Class** of Problems solvable in **Polynomial** Time Finding a /

hard functions **for** secure encryptions. Basic Course Info Course title: Computational Complexity Theory Credits: 3:1 Instructor: Chandan Saha TA: Vineet Nair ( vineet.nair@csa.iisc.ernet.in )vineet.nair@csa.iisc.ernet.in **Class** timings : Tuesday, Thursday: **9**:30-11am./. **Class** NP captures the set of decision problems whose solutions are efficiently verifiable. Nondeterministic **polynomial**-time Complexity **Class** NP u is called a certificate or witness **for** x (w.r.t L and M) if x ∈ L Complexity **Class** NP **Class** NP/

literal to be true. 3-SAT although seems straight forward but presently we don’t know an algorithm to solve it in polytime algorithm.. 5 **Class** of NP problems Def: A Decision Prbl X is associated with a set of strings or input, to a problem X {language}, such that /P. Equivalently if P NP, then no any NPC is in P. Therefore if X is NPC, there can be no **polynomial** time solution **for** problem X. **9** NP-hard and NP-complete Then the roof of NP is collapsing into P. Equivalently if P NP, then no any NPC /

BCH Code a larger **class** of powerful random error-correcting cyclic codes a remarkable generalization of the Hamming codes **for** multiple-error correction Def : Let q and m be given and let α be any element of GF(q m ) of order n. Then **for** any positive integer /(z) = z 2 +z+2 α α 2 α 4 α 8 α 16 α α 3 α 6 α 12 α 24 α 48 α 3 α **9** Exponential Notation **Polynomial** Notation Binary Notation Decimal Notation Minimal **Polynomial** 00000 α0α0 1011x + 1 α1α1 z104x 2 + x + 2 α2α2 z + 2126x 2 + x + 3 α3α3 3z + 23214x 2 + /

Short Dilbert on Decidability Complexity **Classes** Recap **Polynomial** time: languages that can be decided by a deterministic TM in (N k ) steps. Nondeterministic **Polynomial** time: languages that can be decided by a nondeterministic TM in (N k ) steps. NP does not stand **for** “Non-**Polynomial**” time. If you say so/is in L(NPDA) but not L(DPDA) According to a 2002 poll of CSists, 61 think Option 1 is right vs. **9** **for** Option 2 Problems in NP All problems we know are in P Some problems we don’t know if they are in P /

**Polynomials** Multiplying **Polynomials** Multiplying Monomials A monomial is a number, a variable, or a product of numbers and variables raised to natural number powers. To multiply monomials, we often use the product rule **for** exponents. Example Multiply. a.b. Solution a.b. Try Q: **9**/ Try Q: 41 pg 349 Due **for** this week… Homework 1 (on MyMathLab – via the Materials Link) The fifth night after **class** at 11:59pm. Read Chapter 6.1-6.4 Do the MyMathLab Self-Check **for** week 1. Learning team planning /

order of exponents. Degree = the highest exponent. Common **Polynomial** Functions DegreeTypeExample 0constantf(x)=14 1linearf(x)=5x – 7 2quadraticf(x)=x 2 +x-**9** 3cubicf(x)=x 3 -x 2 4quarticf(x)=x 4/ the zero-product property. Example: Find the real zeros of the function: Real Zeros **For** the **polynomial**: 7 is a zero of multiplicity 1 because the exponent on the factor of x /– part f: like **class** work) Graph the **polynomial**. Label intercepts, determine turning points, and end behavior. (May use graphing calculator/

the function f : X → {0,1}, defined by f(x) = 1 **for** x ∈ Y and f(x) = 0 **for** x ∈ XY. **Class** P The **class** of all decision problems **for** which there is a **polynomial**-time algorithm is denoted by P. In other words, a member of P is a/. A nondeterministic algorithm **for** a decision problem always answers “no” **for** a no- instance, and **for** each yes-instance there is a chance that it answers “yes”. **Class** NP Proposition **9**.3 A decision problem belongs to NP if and only if it has a **polynomial**-time nondeterministic algorithm. /

of integers S –Output: An element in S of highest frequency What is the algorithm to solve this problem? How much does it cost? **9** Coloring a Graph Decision Problem: Coloring –Input: G=(V,E) undirected graph and k , k > 0. –Question: Is G k-/to an answer of no **for** a certain guess, that it does not accept it? 18 The **Class** NP Definition: The **class** of decision problems NP consists of those decision problems **for** which there exists a nondeterministic algorithm that runs in **polynomial** time. –The running time /

the CRT. May **9**, 2001Applied Symbolic Computation18 **Polynomial** Algebra mod a **Polynomial** A(x) B(x) (mod f(x)) f(x)|(A(x) - B(x)) This equivalence relation partitions **polynomials** in F[x] into equivalence **classes** where the **class** [A(x)] consists of the set {A(x) + k(x)f(x), where k(x) and f(x) are in F[x]}. Choose a representative **for** [A(x/

**9** Computer Science Dept. Spring 2016: May 10 CS 154: Formal Languages and Computability © R. Mak 10 Efficient Algorithms Do we always have efficient (non-exponential growth) solutions to problems? **For** many problems, the answer is No. Worse: **For** a large **class** / Mak 26 The Traveling Salesman Problem, cont’d With a nondeterministic computer, we can solve the yes-no problem in **polynomial** time. Check all possible subsets of graph edges in parallel. Verify the edges form a cycle, Verify that each vertex/

Warm-Up Exercises Warm-up: Homework: Worksheet given in **class**. #1-20 all. Warm-Up Exercises Monomial- a number, a variable or the product of a number and one or more variables with whole number exponents/ as well as the leading coefficient of the **polynomial**. 1. You Do **for** Examples 1,2, and 3 3yx 3 – 2y 2 +5y + **9** Classification: 4 th degree **polynomial** Degree: 3 Number of terms: 4 Leading Coefficient: 3 ANSWER Tell whether y 3 – 4y is a **polynomial**. If it is a **polynomial**, find its degree and classify it by the/

are an interesting **class** of problems whose status is unknown No **polynomial**-time algorithm has been discovered **for** an NP-Complete problem No one has been able to prove that no-**polynomial** algorithm exists **for** any of them No super-**polynomial** (e.g. exponential) low bound has been shown **for** these problems / have seen, however, it can be solved with dynamic programming 0-1 Knapsack problem: a picture W = 20 wiwi bibi 10 **9** 8 5 5 4 4 3 3 2 WeightBenefit value This is a knapsack Max weight: W = 20 Items 0-1 Knapsack /

**for** a specific protein 50-58 sets per **class** 50-58 sets per **class** 14-24 2D slices per set 14-24 2D slices per set Resolution 0.049 x 0.049 x 0.2 μm Resolution 0.049 x 0.049 x 0.2 μm Huang & Murphy, Journal of Biomedical Optics **9**/ frames TFtight frames use orthogonal **polynomials** use orthogonal **polynomials** construct a **polynomial** transform construct a **polynomial** transform construct the closest orthogonal **polynomial** transform construct the closest orthogonal **polynomial** transform ENequal norm ENequal norm use/

which is not planar So there are disk graphs with **Class** of (unit)disk graphs Closed under vertex deletion Not closed under edge deletion Not closed under edge contraction **9** TRIANGLE PACKING AND H-MATCHING Structure theory and kernels 10 /subset of what another red vertex sees Same **for** the blue vertices Structural theorems show that in such colored graphs the sizes of the color **classes** are **polynomially** related So size of the largest **class** is **polynomial** in the size of smallest **class** Hence |V| = |R| + |/

(8.**9**/10), 7 (8.6/10) and 8 (8.1/10) We’ve learned how to show some problems are undecidable –Harder (average on question 4 = 7.7) Today: reasoning about the complexity of some problems 21 April 2004CS 200 Spring 20043 Complexity **Classes** **Class** P: problems that can be solved in **polynomial** time by a deterministic TM. O (n k ) **for** some constant/

Codes **for** the Tensile Example: Fitting An Orthogonal **Polynomial** with Orthogonal Contrasts DATA tensile; do percent = 15 to 35 by 5; do i = 1 to 5; input strength@@; drop I; output; end; end; cards; 7 7 15 11 **9** 12/ is impossible. Without replication, there is little choice but to adopt an additive model. Producing Interaction Diagrams Using Cell Means proc glm; **class** machinist machine; model time = machinist|machine; means machinist|machine; ods output means = outmeans; run; proc print data = outmeans; /

an algebra-based course (such as college algebra, precalculus, calculus) **for** graduation. These students take MATH 1530, Probability and Statistics, as their core math **class**. About 90% of our students do NOT take an algebra-based course/and **Polynomials** 5. Exponents and **Polynomials** 5.1 Exponents 5.1 Exponents **9**. Inequalities and Absolute Value **9**. Inequalities and Absolute Value **9**.1 Compound Inequalities **9**.1 Compound Inequalities **9**.4 Linear Inequalities in Two Variables and Systems of Linear Inequalities **9**./

parallelism (via non-determinism); 8.Enabled vast, deep, and general studies of other “completeness” theories; **9**.Helps explain why verifying proofs seems to be easier than constructing them; 10.Illuminates the fundamental nature of /. “Karp” reduction (transformation) **Polynomial**-time Turing reduction: solves A by **polynomially**-many calls to “oracle” **for** B. “Cook” reduction B A Open: do **polynomial**-time-bounded many-one and Turing reductions yield the same complexity **classes**? (NP, co-NP, NP-complete/

**polynomial** p(x) such that, **for** any w L, there is a certificate c with |c| p(|w|) All problems from the previous slide have a **polynomial** time verifier Computability and Complexity 13-5 The **Class** NP Definition The **class** of languages that have **polynomial** time verifiers is called NP Definition The **class** of languages that have **polynomial**/= Not Composite) Computability and Complexity 13-**9** Problems not in NP Instance: A positive integer n. Question: Is there a winning strategy **for** whites in a checker game on n /

**polynomial**-time queries are BD-tractable 15 16 **Polynomial** hierarchy revised Tractability revised **for** querying big data NP and beyond P P BD-tractable not BD-tractable Parallel polylog time 16 17 What can we get from BD-tractability? Guidelines **for** the following. Why we need to study theory **for** querying big data What query **classes**/more processors, the “better”? 19 D Using 10000 SSD of 6G/s, a linear scan of D might take: D 1.**9** days/10000 = 16 seconds when D is of 1PB (10 15 B) D 5.28 years/10000 = 4.63 days /

e.g. side constraints –New query policies New **polynomially** elicitable valuation **classes** Using models of how costly it is to answer different queries [Hudson & S. AMEC-02] Decision-theoretic elicitation using priors Elicitors **for** markets beyond combinatorial auctions –(Combinatorial) reverse auctions / bound-approximation queries with order queries Integrated as before –Computationally more expensive agents items 345678 **9** 2 10 23456 queries 1 10 100 1000 40 80 120 160 queries Full revelation Total/

to “an O(T(n)) running-time algorithm.” 3 The **class** P uIf a DTM M is T(n)-time bounded **for** some **polynomial** T(n), then we say M is **polynomial**-time (“polytime ”) bounded. uAnd L(M) is said to be in the **class** P. uImportant point: when we talk of P, it doesn’t/ and “false” **for** all other differences. uInduction: To consider i j, start with a new table, initially all false. uThen, set k to true if, in the old table, there is a value m that was true, and k is either m+i j or m-i j. **9** Knapsack – (3) uSuppose /

computation if and only if it can be solved in **polynomial** time by a Turing machine. P is the **class** of these decision problems 2 Search Problems: NP L is in NP iff there is a language L’ in P and a **polynomial** p so that: 3 Intuition The y-strings are the/**9** NPC A language L 2 NP that is NP-hard is called NP-complete. NPC := the **class** of NP-complete problems. Proposition: L 2 NPC ) [L 2 P iff P=NP]. 10 L is in NP means: There is a language L’ in P and a **polynomial** p so that L 1 · L 2 means: **For** some **polynomial**/

x 2 & -x 1 v x 2 & -x 1 v -x 2 88- **9** Definition of the satisfiability problem: Given a Boolean formula, determine whether this formula is satisfiable or/13 Semantic tree In a semantic tree, each path from the root to a leaf node represents a **class** of assignments. If each leaf node is attached with a clause, then it is unsatisfiable. 88-/NP-hard: Every NP algorithm **for** problem A can be transformed in **polynomial** time to SAT [Horowitz 1998] such that SAT is satisfiable if and only if the answer **for** A is “ YES ”./

starts and ends at the same vertex (Problem 2) 3/12/03Circuits - Lecture 229 An Euler Circuit **for** Puzzle A 1 2 3 4 5 6 7 8 **9** 10 3/12/03Circuits - Lecture 2210 Euler Circuit Property A graph has an Euler circuit if and only /finite (but unbounded) time this fact. 3/12/03Circuits - Lecture 2227 The **Class** NP NP : Non-deterministic **Polynomial** Time The **class** NP is the set of all problems **for** which a given solution can be checked in **polynomial** time Check via a non-deterministic algorithm i.e., one that is free to/

logarithmic (n log n) (n2 log n) ... **Polynomial** (exponent is constant) (1) – Sub-linear (n0.001) – Sub-linear (n0.5) – Sub-linear (n) - Linear (n2) (n3) ... (n100) **Classes** of Algorithms Exponential (bases make a big difference) / Example) Find the shortest path from A back to A that visits every vertex. **9** B 2 1 C 5 4 A 6 D 3 2 ... 4 7 1 / evidence that P != NP More problems = more opportunity to find a **polynomial** algorithm **for** an NP-Complete problem All you have to do is prove two reductions One/

it makes at most c p(n) steps on input of length n, **for** some constant c and **polynomial** p. uSay L(M) is in the **class** EP. uIf M is an NTM instead, say L(M) is in the **class** NEP (nondeterministic exponential polytime ). 6 More **Class** Relationships uP NP PS EP, and at least one of /n)+1+c. uWe can count to the maximum number of ID’s on a separate tape using base t+1 and p(n)+1+c cells – a **polynomial**. **9** Proof PS EP – (2) uRedesign M to have a second tape and to count on that tape to sp(n)t p(n). uThe new TM/

Problems Section:4.3 Page:278 Problems:Odd: 1 – **9**, 13 – 49, 63 – 75 MyMathLab Section 4.3 **for** practice MyMathLab Homework Quiz 4.3 is due **for** a grade on the date of our next **class** meeting Review of Terminology of Algebra Constant – A specific number/1 – 55, 59 – 69, 73 – 77 MyMathLab Section 4.4 **for** practice MyMathLab Homework Quiz 4.4 is due **for** a grade on the date of our next **class** meeting Multiplying **Polynomials** To multiply **polynomials**: –Get rid of parentheses by multiplying every term of the first by /

of the equation (x+a) 2 =0 _H001 Identity Identity : Equation true **for** all values of the variable (x+1) 2 = x 2 +2x+1 Equation holds true **for** all real x _H001 **Polynomial** identity If a **polynomial** equation of degree n satisfies **for** the values more than n it is an identity Example: (x-1) 2/+ b + c) 1 2 + 2 (a + b).1 + (a + b – c) = 0 1 is a root, which is rational so other root will be rational. **Class** Exercise **9** Solution: D = 0 4 (ac+bd) 2 - 4 (a 2 +b 2 )(c 2 +d 2 )= 0 a 2 c 2 + b 2 d 2 + 2abcd = (a 2/

連續體的基數問題， 2. 算術公理的無矛盾性， 3. 等底等高兩四面體的等積性， 4. 兩點間最短路程做為直線的問題， 5. 連續群的定義函數除去可微性的問題 6. 物理學公理化， 7. 某些數的無理數性及超越性， 8. 質數問題， **9**. 任何代數體中最一般的互逆法則， 10. 決定 Diophantine 方程式的可解性， 11. 係數為代數數的二次式， 12. 推廣 Kronecker 的 Abel 擴張定理到任何代數體上 8 13. 七次方程式不能用兩變數函數來解， / NP. 12 14 The **class** P consists of all the problems that can be solved in **polynomial** time. Sorting Exact string / answer 25 **polynomial** 26 NP, NP-complete, NP-hard 28 “NP stands **for** non-**polynomial** time”. Correct version: “NP stands **for** non-deterministic **polynomial** time” “/

and Function of PQ type TSK-MPKC 7.Simulation Result 8.Considerations **for** Security **9**.Conclusion 1 Introduction In secret communication, such as between a local government/, 2,....m; random quadratic **polynomial** (only u i is linear **for** all i) g i (u m+1, u m+2,..., u 2m ); random quadratic n-variate **polynomial** **for** all i. Perturbed TSK-MPKC/ Proposed System Combining two TSK together (p and q term) Residue **Class** Ring is used Above **polynomial** system is the central map and public key is generated by applying /

1.1.1. Show that ( Z n,+, ・ ) is a commutative ring, where addition and multiplication on congruence **classes**, modulo n, are defined by the equations [x] + [y] = [x + y] and [x] ・/ as [12] 26 [7 12 + 3] 26 = [87] 26 = [**9**] 26 which corresponds to I. Conversely I is decrypted as [**9**] 26 [15 (**9** – 3) ] 26 = [90] 26 = [12] 26 which corresponds to M. Now/i. Two **polynomials** f (x) and g(x) are called equal when they are identical, that is, when the coefficient of x n is the same in each **polynomial** **for** every n. /

2 + 6x + ____ Activity × Part of the square is missing. How many one-tiles do you need to complete it?. **9** = (x + 3) 2 X＋3X＋3 X+3 Try This Topic 7. Use algebra tiles to model the expression. Add / form. x 2 + 8x Marshall Island High School Open **Class** Grade 10 Thank you **for** joining. See you again. **For** printing Equation(Topic 1&2) MODEL ALGEBRA MODEL **Polynomial** addition/subtraction (Topic 3) Answer _____________________________ **Polynomial** Multiplication/Division or factorization (Topic 4~8) MODEL ALGEBRA ×/

x + **9** quartic Y = 3x 4 – 2x 3 + 8x 2 – 6x + 5 quintic Graphs of **Polynomial** Functions The **polynomial** functions that / **polynomial**. 12 Zeros of **Polynomial** Functions It can be shown that **for** a **polynomial** /**polynomial** functions. Find and use zeros of **polynomial** functions as sketching aids. Find a **polynomial** equation given the zeros of the function. Homework 8 2.2 Page 130 – 1-8 all (matching) – 13-18(left and right behavior), all – 27-41 odds (finding zeros-verify with a calculator) – 47-55 odds Quiz next **class**/

3 in decimal. The result of squaring is **9**, which is 1001 in binary. Solution 5.1.5 Two **Classes** of Product Ciphers 5.58 Figure 5.16 Improvement of the previous Feistel design 5.1.5 Two **Classes** of Product Ciphers Improvement : We want the/5.2.1 Synchronous Stream Ciphers 5.98 Example 5.20 The characteristic **polynomial** **for** the LFSR in Example 5.19 is (x 4 + x + 1), which is a primitive **polynomial**. Table 4.4 (Chapter 4) shows that it is an irreducible **polynomial**. This **polynomial** also divides (x 7 + 1) = (x 4 + x/

CSE, UConn **9** Quadratic Splines Numerical Methods Lecture 21 Prof. Jinbo Bi CSE, UConn 10 Cubic Spline Function Each two neighboring points are connected or interpolated by a 3 rd -order (Cubic) **polynomial**. If # /. Jinbo Bi CSE, UConn 13 Second Cubic Spline Functions Second derivative is a line Lagrange interpolating **polynomial** **for** second derivative Integrate twice to get f i (x) Numerical Methods Lecture 21 Prof. Jinbo Bi/, UConn 32 Next **class** Review Numerical Methods Lecture 21 Prof. Jinbo Bi CSE, UConn 33

Erickson 1.1 Review I 24 Example 8 Find the correct factorization **for** the problems below. Dr. Erickson 1.1 Review I 25 Roots of **Polynomial** Expressions A **polynomial** equation of degree n in the variable x is an equation of the/c = 0 (a ≠ 0) are given by Dr. Erickson 1.1 Review I 29 Example **9** Solve the equation using the quadratic formula. Dr. Erickson 1.1 Review I 30 Next **Class** We will continue reviewing the following concepts: Rational Expressions Other Algebraic Fractions Rationalizing /

all have different ways of doing each **polynomial** so watch out **for** your signs! When you see something like (x+4)+(x-**9**) you add just like that. However, that is completely different from something like (x+4)(x-**9**). The first one is adding straight / Makes You Beautiful- One Direction Youre insecure, dont know what **for**, **Polynomial** Operations leave you wanting mo-o-re, Whether to subtract, or maybe add, Multiply, and or divi-i-de, Everyone else in the **class** can do it, Everyone including you! Baby it aint hard /

. 7) 5.Searching (Brassard & Bratley, Chap. **9**) 6.Graph Algorithms (Weiss, Chap. **9**) 7.Randomized Algorithms (Weiss, Chap. 10) 8.String Searching (Sedgewick, Chap. 19) **9**.NP Completeness (Sedgewick, Chap. 40) 25 October 2015CS/**polynomial** time O(n c ) **for** some constant c non-**polynomial** time **Polynomial** Time Algorithms Are there problems solvable in **polynomial** time? Answer is yes: every algorithm we’ve studied provides **polynomial**-time solution to some problems Define P: the **class** of problems solvable in **polynomial**/

**polynomial** 0.0857 + 0.4 x + 1.4286 x 2. **Class** Project Write a Matlab function that will take a vector of x values and a vector of y-values and will return the vector of coefficients **for** the best quadratic fit to the data. **Class** Project 2 Exercise **9**.15/ in Pav: Write a Matlab function that will find the coefficients (a,b) **for** the function a e x + b e -x that best approximates/

7101510 86510165320 Technical University Tallinn, ESTONIA Deterministic Synthesis of LFSR Generation of the **polynomial** and seed **for** the given test sequence (1) 100x0 (2) x1010 (3) 10101 (/ Extended Fault Models Defect Extensions of the parallel critical path tracing **for** two large general fault **classes** **for** modeling physical defects: 01010101 Conditional fault Pattern fault Constrained SAF Single/2 0 1 0 0 0 1 4.1 1 0 0 1 1 0 **9**.2 (**9**)mod3 = 0 (2)mod3 = 2 Error! Technical University Tallinn, ESTONIA Error/

NP-hard problem implies the existence of **polynomial** solutions **for** every problem in NP NP-complete problems are the NP-hard problems that are also in NP 6 CS 450/650 Lecture 8: Algorithm Background Relationships between different **classes** NP P NP-complete NP-hard 7 CS 450/650 Lecture 8: Algorithm Background Lecture **9** Rivest-Shamir-Adelman (RSA) CS 450/650 Fundamentals of/

complexity **classes** Def: A Decision problem **for** a language L {0,1} * is to decide whether a given string x belongs to the language L. Def: P is the **class** of languages (decision problems) that can be recognized by a deterministic **polynomial** time Turing machine. Def: NP is the **class** of/L : – Guess y of proper **polynomial** size – Call M * L to check if (x,y) R L – Accept x if M * L accepts (x,y) M L is a non-deterministic TM **for** L M L is a non-deterministic TM **for** L **9** Search Problems Def: A search problem /

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