Data Structures & Algorithms
Algorithm
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An algorithm is a sequence of
unambiguous instructions for solving
a problem, i.e., for obtaining a
required output for any legitimate
input in a finite amount of time.
A Problem
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Find the greatest common divisor of two
nonnegative, non-both-zero integers m
and n, denoted gcd(m,n)
gcd(m,n), the largest integer that divides
both m and n evenly, i.e., with a
remainder of zero
Greatest Common Divisor (GCD)
Algorithm 1
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Step 1 Assign the value of min{m,n} to t
Step 2 Divide m by t. If the remainder of this
division is 0, go to Step 3; otherwise, to Step 4
Step 3 Divide n by t. If the remainder of this
division is 0, return the value of t as the answer
and stop; otherwise, proceed to Step 4
Step 4 Decrease the value of t by 1. Go to
Step 2
Note: m and n are positive integers
GCD Procedure 2
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Step 1 Find the prime factors of m
Step 2 Find the prime factors of n
Step 3 Identify all the common factors in the two
prime expansions found in Steps 1 and 2. If p is a
common factor occurring i and j times in m and n,
respectively, it should be repeated min{i, j} times
Step 4 Compute the product of all the common
factors and return it as the GCD of m and n
Note: as written, this procedure requires that m
and n be integers greater than 1, since 1 is not a
prime
Is this procedure an algorithm?
Algorithm 2?
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Procedure 2 is not an algorithm unless we
can provide an effective way to find
prime factors of a number
The sieve of Eratosthenes is an algorithm
that provides such an effective procedure
Euclid’s Algorithm
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Idea:
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if n  0, gcd(m, n) = gcd(n, m mod n);
if n = 0, gcd(m, n) = m.
Euclid’s algorithm for computing gcd(m, n)
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Step1 If n=0, return the value of m as the
answer and stop; otherwise proceed to Step2.
Step2 Divide m by n and assign the value of
the remainder to r.
Step3 Assign the value of n to m and the value
of r to n. Go to Step1.
Exercise
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Page8 Problem5
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a. Find gcd(31415, 14142) by applying
Euclid’s algorithm.
b. Estimate how many times faster it will be
to find gcd(31415, 14142) by Euclid’s
algorithm compared with the algorithm based
on checking consecutive integers from min(m,
n) down to gcd(m, n).
Analysis of Algorithms
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How good is the algorithm?
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Correctness
Time efficiency
Space efficiency
Does there exist a better algorithm?
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Lower bounds
Optimality
What is an algorithm?
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Recipe, process, method, technique, procedure,
routine,… with following requirements:
1.
Finiteness
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2.
Definiteness
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3.
valid inputs are clearly specified
Output
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5.
rigorously and unambiguously specified
Input
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4.
terminates after a finite number of steps
can be proved to produce the correct output given a valid
input
Effectiveness
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steps are sufficiently simple and basic
Algorithm Design and Analysis
Process
Understand the problem
Decide on : algorithm
design techniques etc.
Design an algorithm
Algorithm Design and Analysis
Process
Understand the problem
Decide on : algorithm
design techniques etc.
Design an algorithm
Prove correctness
Analyze efficiency etc.
Algorithm Design and Analysis
Process
Understand the problem
Decide on : algorithm
design techniques etc.
Design an algorithm
Prove correctness
Analyze efficiency etc
Code the algorithm
correctness
efficiency
Correctness
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Termination
 Well-founded
sets: find a quantity that is
never negative and that always decreases
as the algorithm is executed
Prove the Correctness for
Euclid’s Algorithm
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Idea:
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if n  0, gcd(m, n) = gcd(n, m mod n);
if n = 0, gcd(m, n) = m.
Euclid’s algorithm for computing gcd(m, n)
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Step1 If n=0, return the value of m as the
answer and stop; otherwise proceed to Step2.
Step2 Divide m by n and assign the value of
the remainder to r.
Step3 Assign the value of n to m and the value
of r to n. Go to Step1.
Complexity
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Space complexity
Time complexity
For iterative algorithms: sums
 For recursive algorithms: recurrence relations
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