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Mat 2345 — Discrete Math

Week 7

Dr. N. Van Cleave

Fall 2009

Student Responsibilities — Week 7

I (^) Reading: Textbook, Section 3.3–3. I (^) Assignments: Sec 3.3 Due Wed: 2, 4 (justify efficiency ans), 7, 8 Sec 3.4 Due Thur: 4, 8, 9a–d, 12, 26, 35 Sec 3.5 Due Mon: 4, 5, 10, 13, 14a, 18, 21a–c, 26 I (^) Attendance: Strongly Encouraged

Week 7 Overview

I (^) 3.3 Complexity of Algorithms I (^) 3.4 The Integers and Division I (^) 3.5 Primes and Greatest Common Divisors

Time Complexity

II (^) It is obviously important to know whether an algorithm will produce an answer in milliseconds or time measured in years.

I (^) Time complexity can be described in terms of the number of operations required instead of actual computer time because of the difference in time needed for different computers to perform basic operations.

I (^) It would be quite complicated to break down all operations to the basic bit operations that a computer uses

I (^) Various machines, from personal computers to supercomputers, perform basic bit operations at rates which differ by as much as 1,000 times or more

Space Complexity

I (^) It is obviously important to determine whether an algorithm will require more memory than we have available

I (^) Space complexity can be described in terms of the amount of memory necessary to store one element × the size of input, plus additional storage required by the algorithm, and is often given in terms of the size of input and its storage requirements

I (^) Considerations of space complexity are tied to the particular data structures used to implement the algorithm

Analyzing the Search Algorithms

based on the number of comparisons made:

I (^) Linear Search

I (^) Binary Search (for simplicity, assume there are n = 2 k^ elements in the input list)

Commonly Used Complexity Terminology

Complexity Terminology

Θ(1), Θ(c) Constant complexity

Θ(log n) Logarithmic complexity

Θ(n) Linear complexity

Θ(n log n) nlog(n) complexity

Θ(nb) Polynomial time complexity

Θ(bn), where b > 1 Exponential complexity

Θ(n!) Factorial complexity

Tractable vs Intractable Problems

I (^) A solvable problem is called tractable if there exists an algorithm with polynomial worst–case complexity to solve it.

I (^) Even if a problem is tractable, there’s no guarantee it can be solved in a reasonable amount of time for even relatively small input values.

I (^) Most algorithms in use have polynomial complexities of degree 4 or less.

Tractable vs Intractable Problems — Continued

I (^) Solvable algorithms with worst–case time complexities that exceed polynomial times are called intractable

I (^) Usually, but not always, an extremely large amount of time is required to solve the problem for the worst cases of even small input values.

I (^) In a few instances, an exponential or worse algorithm may be able to solve problems of reasonable size in sufficient time to be useful

NP–Complete Problem Class

Another important class of problems, called NP–Complete problems, are problems in the class NP which have the property that:

If any of the problems in the NP–Complete class can be solved in polynomial time, then all of them can be

No one has been able to find such an algorithm.

It is suspected that no one ever will.

Section 3.4 — The Integers and Division

I (^) Integers and their properties belong to a branch of Mathematics called Number Theory

I (^) If a and b are integers, with a 6 = 0, we say that a divides b if there is an integer c such that b = ac.

Notation: a | b, a divides b a is a factor of b; b is a multiple of a a 6 | b, a does not divide b

Division of Integers

I (^) Theorem. The “Division Algorithm”. Let a be an integer, d be a positive integer. Then there are unique integers q and r, with 0 ≤ r < d, such that a = dq + r.

I (^) In the equality given in the division algorithm:

d is called the divisor, a is called the dividend, q is called the quotient, and r is called the remainder

Modular Arithmetic

I (^) Let a be an integer and m be a positive integer. We denote by (a mod m) the remainder when a is divided by m.

From this definition, it follows that: if (a mod m) = r , then a = qm + r and 0 ≤ r < m.

I (^) If a and b are integers, and m is a positive integer, then a is congruent to b modulo m if m divides a − b. This is denoted by: a ≡ b(mod m)

Note: a mod m and b mod m will yield the same remainder.

Consider: (17 − 5) mod 6 = 12 mod 6 = 0 Also: 17 mod 6 = 5 and 5 mod 6 = 5 Thus: 17 ≡ 5 (mod 6)

Applications of Congruences

I (^) Hashing Functions — assign memory locations to values, records (keys), or computer files for easy retrieval

I (^) Pseudorandom Numbers — systematically generate a sequence of numbers that have properties of randomly chosen numbers

I (^) Cryptology — encryption, to make a message secret; decryption, to determine the original message

Section 3.5 — Primes and Greatest Common Divisors

I (^) A positive integer p > 1 is called prime if the only positive factors of p are 1 and p.

I (^) A positive integer that is greater than 1 and is not prime is called composite.

I (^) The Fundamental Theorem of Arithmetic. Every positive integer can be written uniquely as the product of primes, where the prime factors are written in order of non–decreasing size.