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Algorithms

Algorithms. Notion of an algorithm Properties of an algorithm The GCD algorithm Correctness of the GCD algorithm Termination of the GCD algorithm Performance Data structures. 1-Notion of an algorithm. Definition

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Algorithms

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  1. Algorithms Notion of an algorithm Properties of an algorithm The GCD algorithm Correctness of the GCD algorithm Termination of the GCD algorithm Performance Data structures

  2. 1-Notion of an algorithm Definition An algorithm is a clearly specified set of instructions describing the solution to a specific problem. An algorithm • takes the input and transforms it into an adequate output, • must be independent from any programming language, • is written in a level of detail that allows to reproduce it in any programming language, • has to be designed so it can be reused and understood by others.

  3. 2-Properties of an algorithm Some properties must be satisfied by an algorithm in order to allow a successful execution of the corresponding program: • Correctness: if the input conditions are satisfied and the algorithm instructions executed, then the correct output is produced. • Termination: the algorithm must terminate after a finite number of steps. Thus, it has to be composed by a finite number of steps. This can be ensured if the algorithm avoids an infinite loop. • Performance: Quantification of the space and time complexities.

  4. 3-The GCD algorithm: Given two positive integers m and n, find the greatest common divisor, gcd(m,n). A view of the problem: All numbers from 1 up to the smallest of m and n, say n. Naive algorithm: Go through search space (from 1 to n) Keep track of largest number that divides bothm and n. Is there a more efficient way of doing this? Yes, the Euclidean Algorithm (Euclid – c.350 B.C.E.)

  5. Pseudocode of the GCD algorithm : Algorithm GreatestCommonDivisor Input: Two positive integers, m and n Ouptut: The gcd of m and n repeat rm mod n mn nr until (r == 0) Output m and STOP.

  6. Example: m = 24, n = 9 r 24 mod 9 = 6 m 9 n 6 r 9 mod 6 = 3 m 6 n 3 r 6 mod 3 = 0 m 3 n 0 Output 3 STOP

  7. How does the algorithm works? Given m and n, how many times does “repeat” execute? First, we need the following result: Theorem 1: If m > n, then m mod n < m/2 Proof: If nm/2, then the claim follows since: r = m mod n < n Ex. 5  11/2 r = 11 mod 5 = 1 < 5  11/2 If n > m/2, then r = m mod n = m – n < m/2 Ex. 7 > 11/2 r = 11 mod 7 = 4 < 11/2 q.e.d.

  8. Consider sequence of remainders: r0 = m mod n < m/2 r1 = n mod r0 < n/2 r2 = r0 mod r1 < r0/2 r3 = r1 mod r2 < r1/2 r4 = r2 mod r3 < r2/2 r5 = r3 mod r4 < r3/2 ….. Second iteration: r1 < n/2 Third iteration: r2 < r0/2 < m/4 Fourth iteration: r3 < r1/2 < n/4 Fifth iteration: r4 < r2/2 < r0/4 < m/8 Sixth iteration: r5 < r3/2 < r1/4 < n/8

  9. Thus, ri will become 0 in at most 2 log n iterations Example: m = 1989 and n = 1590 Remainder sequence: 399 = 1989 mod 1590  399 < 1989/2 393 = 1590 mod 399  393 < 1590/2 6 = 399 mod 393  6 < 399/2 < 1989/4 3 = 393 mod 6  3 < 393/2 < 1590/4 0 = 6 mod 3  0 < 6/2 < 399/4 < 1989/8 We stop here, but if we continue: < 1590/8 < 1989/16 < 1590/16

  10. Number of Steps: In 2 * 3 steps, ri reduced by factor of 8 2 * 4 ri 16 2 * 5 ri 32 2 * 6 ri 64 …… 2 * log nrin Thus, ri becomes 0 in at most 2 log n iterations.

  11. 4-Correctness of the GCD algorithm. • Correctness is an important issue in algorithm design. This means proving that the algorithm works for all legal inputs. • It is analogous to proving the correctness of a theorem in mathematics! • The correctness of the GCD algorithm depends on the following loop invariant: • gcd(m,n)=gcd(n,r) where r=m%n

  12. Claim 1: gcd(m,n) = gcd(n,r) Proof: The gcd of the new pair is equal to the gcd of the previous pair. How is this correct? Let us write: m = q * n + r where 0 rn. This implies that: a common divisor of m and n is also a common divisor of n and r, and vice versa. q.e.d.

  13. 5-Termination of the GCD algorithm. • Show that algorithm terminates in a finite number of steps. • This must be true for every valid input. • Can we show this for Algorithm GCD? • We must show that ri goes to 0 in a finite number of steps. • Observe: • The sequence of remainders strictly decreases. • They are all non-negative. • Thus ri will become 0 in at most 2 log n steps.

  14. 6-Performance of the GCD algorithm. • Quantification of performance of the algorithm. • Crucial parameters: time and space. • Called time and space complexity of the algorithm. • Will be discussed latter in the course. • For example: • Time complexity of GCD: • Takes at most 2 log n steps, where n < m • Thus, worst-case time complexity: O(log n)

  15. 7-Data structures • The study of different ways of organizing data. • Why? • Efficiency of algorithm depends on how data is organized. • Reason for studying data structures and algorithms together. • Organic connection between the two areas. • In 60-212 the programming language JAVA is studied. • Here: An algorithmic perspective. • Independent of programming language (eg. C, C++, JAVA).

  16. For example: • The median of a list of n numbers is a number m such that: • n/2 numbers in the list are m, and • n/2 … are m. • Many definitions of median, we take: • If n is even: • Two medians: • lower median and upper median, • Then, median is average of lower and upper medians. • If n is odd: • Both medians (lower and upper) are the same.

  17. Consider this problem: • Given a sorted list of n numbers, find the median. • A crucial question: • How should we store the list? • We store it in an array, A, then • The median is found in constant time, O(1) !! • Median = (A[5]+A[6])/2 = (19+22)/2 = 20.5

  18. 5 9 16 22 …. 2 19 • Whereas in a linked list: • Traverse half of the list in n/2 steps, which is O(n) ! • Quite simple stated: • The way in which data is organized is crucial in complexity. • More examples like this will be seen later. first

  19. Example of problems: • Design an efficient algorithm to determine if a list has repeated elements. • Given a list of n elements find their minimum (or maximum). • Given n points in the plane, find the pair(s) of points which are closest to each other. • Given n points in the plane determine if any three are contained in a straight line.

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