 # Recurrence Relations Recurrence Relations A recurrence relation is

• Slides: 34 Recurrence Relations Recurrence Relations • A recurrence relation is an equation which is defined in terms of itself. • Many algorithms, particularly divide and conquer algorithms, have time complexities which are naturally modeled by recurrence relations Example Merge Sort • Merge-Sort(A, p, r): if p < r then q¬(p+r)/2 Merge-Sort(A, p, q) Merge-Sort(A, q+1, r) Merge(A, p, q, r) • Solving Recurrences • We will show 3 ways of solving recurrences: – Substitution method - Guess and prove • – Iteration method • – Use mathematical induction to solve your recursion Break the sums into a mathematical series Master theorem • Check for an instant solution Substitution method • Guess a solution and them prove it using mathematical induction. • To guess the solution, play around with small values for insight n 0 1 2 3 4 5 6 7 0 1 3 7 15 31 63 127 Prove by induction • Claim 1. Show that the basis is true: 2. Now assume true for T(n-1 ) 3. Using this assumption show: Iteration method • Example • How many terms until we reach T(1) ? Iteration method Recursion Trees • Recursion Trees • Recursion Trees Recursion Trees • Master Theorem • A powerful theorem allowing to solve a large class of recursion relations of the form where • There are 3 cases to remember: 1. If for some constant then Master Theorem 2. If then 3. If for some constant and for some c < 1 then for all sufficiently large n Using Master Theorem • T(n) = 4 T(n/2) + n • Case 1 of master theorem Using Master Theorem • Using Master Theorem • • Use case 2 of Master theorem Using Iteration • Using Master theorem • Examples • • Use Master theorem case 1 where є = 1/6 Master theorem Examples • Use case 2 of Master Theorem Examples • Use Master Theorem – case 3 Example • Fits case 3 of Master theorem ? Examples Examples • Master Theorem – does not fit • Substitution method – too complicated • Recursion tree • T(n) is at least the price for a complete tree of height T(n) is at most the price for a complete tree of height • • • i=0 i=1 i=2 i=3 i=k n (3/4)n (9/16)n (27/64)n (3/4)^k * n     Regularity of Case 3 • Example •