Recurrence Relations

Recurrence Relations By: Sean Lyn March 25, 2008

Algorithm Analysis • “The process of deriving the estimates for the time and space required to execute an algorithm.” • Direct Proof • Proof by Contradiction • Proof by Induction

Time Complexity • Θ(g(n)) = is g(n) • O(g(n)) = is at most g(n) • Ω(g(n)) = is at least g(n)

Examples • for (x=0; x<n; x++) a+=1; • for (x=0; x<n; x++){ for (y=0; y<n; y++) a+=1; }

Algorithm Efficiency

Recurrence Relation • “A recurrence relation relates the nth element of a sequence to [a] certain of its predecessors” • In other words, how do you define An in terms of A1,A2,A3,… An-1 ?

Uses • Used for describing the time required by an algorithm • It can be used in defining a sequence

Fibonacci Sequence • Fn = Fn-1 + Fn-2 where n > 2, F1 = 1, and F2 = 1

Mandlebrot Set • Zn+1 = (Zn)² + C, Z0 = C

Example • Cn is the number of times the statement x=x+1 executes example(n) { if (n==1) return; for (i=1; x<n; x++) x = x+1; example(n/2); } C1 = 0 We get the recurrence relation Cn = n + C[n/2]

Example 2 • Solving a recurrence relation, or producing a solution without using Ci for any i. • Given: Cn = Cn-1 + n, n>=1 and C0 = 0 • Cn-1 = Cn-2 + n-1 -> Cn = (Cn-2+n-1)+n • Repeat…

Example 2 Cont. • Cn = Cn-1 + n • Cn = Cn-2 + (n-1) + n • Cn = Cn-3 + (n-2) + (n-1) + n • Which gives us… • Cn = C1 + 2 + 3 + … (n-2) + (n-1) + n • Cn = C0 + 1 + 2 + … (n-2) + (n-1) + n • Cn = 0 + 1 + 2 +…. n • Cn = n(n+1)/2

Main Recurrence Theorem • Let a, b and k be integers satisfying a>=1, b >= 2, and k >= 0. • The recurrence relation must have the form: T(n) = aT(n/b) + f(n), where f(n) = O(nk) / O(nk) if a < bk • T(n) = | O(nklogn) if a = bk \ O(nlogba) if a > bk This also applies to Θ and Ω

Using the MRT • Find O() for the relation • T(n) = 2T(n/2) + n • a = 2, b = 2, k = 1 • We can see that 2 = 2¹ • Therefore a = bk so we get O(nlogn)

Example 2 • Find O() for the relation • T(n) = 3T(n/2) + n² • a = 3, b = 2, k = 2 • We can see that 3 < 2² • Therefore a < bk so we get O(n²)

Putting it all Together • Recurrence relations can be used to define the time complexity of an algorithm • Recurrence relations can be used to define sequences • Time complexities are important because computers have limited memory. • The main recurrence theorem can be used to solve the time complexities of recurrence relations

Questions • What is one example of a sequence generated by a recurrence relation? • Using the MRT, what is the time complexity of the function: T(n) = T(n)+n ?

Works Cited • Johnsonbaugh, Richard and Marcus Schaefer. Algorithms. New Jersey: Pearson Education, Inc, 2004. • “Recurrence Equation.” http://mathworld.wolfram.com. 2008. 25 March 2008. <http://mathworld.wolfram.com/RecurrenceEquation.html>.

Recurrence Relations