Complexity of the Euclidean Algorithm
(2/7)
• The complexity of an algorithm is the approximate number
of steps necessary for the algorithm to finish as a function
of the size of the input.
• Most commonly we are interested in either average case
complexity or worst case complexity. These mean exactly
what they sound like.
• We shall now analyze the worst case complexity of the
Euclidean Algorithm (EA).
EA Worst Case
• The EA will take the longest if each remainder is not much
smaller than the previous remainder.
• Said another way, EA will take the longest if the quotients
qk are relatively small. In particular, the worst case is
going to occur when every quotient is 1 !
• So, what does the EA sequence look like if every quotient
is 1: (Recall: r-1 = a and r0 = b.)
• r-1 = r0 + r1
r0 = r1 + r2
r1 = r2 + r3
.........
rn-2 = rn-1 + rn
EA Worst Case Continued
• For simplicity, let’s assume n is even (easy to adjust if
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odd), where rn is the last non-zero remainder. From above
we have then:
b = r1 + r2 > 2r2 , r2 = r3 + r4 > 2r4, r4 = r5 + r6 > 2r6, etc., and
finally rn-2 = rn-1 + rn > 2rn .
Combining all these, we get b > 2n/2 rn 2n/2 .
Now solve this equation for n. (How do you solve
equations for things in the exponent?)
Finally, change the base-2 log to a base-10 log, since
base10 logs reveal the number of decimal digits.
(Discuss.)
Conclusion:
• Theorem. The worst case complexity for the Euclidean
Algorithm is that it will complete in a number of steps
which is at most about 6.7 times the number of decimal
digits in b.
• Finally, are there pairs of numbers which have the
property that this worst case occurs when you apply the
EA to them to get their GCD?
• Answer: Yes! Consider the Fibonacci Numbers.
• For Monday, absorb these ideas!!