Algorithms CSCI 235, Spring 2019 Lecture 6 Recurrences

Recurrence Equations A recurrence equation is a recursive function definition (i.e. a function that is defined in terms of itself). Examples: 1 if n = 1 n*f(n-1) if n > 1 Factorial: f(n) = 1 if n < 2 f(n-1) + f(n-2) for n>1 Fibonacci: f(n) = Analyzing recurrence equations: 1) Characterize running time of a given algorithms by a recurrence equation. (Usually worst case running time) 2) "Solve" the recurrence equation. Express the answer in asymptotic notation.

Recurrence for Divide and Conquer When the problem is small enough, the solution is trivial. This is called the base case. What is "small enough"? Often, n = 1 or n<= c where c is a constant and the solution for n<=c takes constant time.

Recurrence equation for divide and conquer Base Case: T(1) = Q(1) General Case: T(n) = [# of subproblems]T(Size of each subproblem) + [cost of divide and recombine] Suppose: # subproblems = a, size of subproblems = n/b Time to divide=D(n), Time to combine = C(n) T(n) = aT(n/b) + D(n) + C(n) for n > c Q(1) for n<=c

Example 1: Insert() Insert will insert the nth element into a previously sorted list of length (n-1) so that the resulting n element list is sorted. We will use this function later in a recursive version of insertion sort. Insert(A, n) if n > 1 if A[n] < A[n-1] swap(A, n, n-1) {switch A[n-1] and A[n]} Insert(A, n-1)

Equation for Insert() T(n) = T(n-1) + 1 Each subproblem is 1 less than original problem. Swap takes constant time (actually Q(1)) Expanding out the equation: T(n) = T(n-1) + 1 (we will work out the expansion in class)

Methods for solving recurrences 1. The substitution method. Make a good guess and show it works by induction. 2. The iteration method. Construct a recursion tree Find the sum of the costs of the nodes. 3. The master method. Learn 3 basic rules and apply them for certain types of recurrences. We will use the iteration method.

Recursion tree for Insert T(n) = T(n-1) + 1 We will draw this in class.

Solution for Insert() Number of levels = n Cost of each level = 1 Total cost = (# levels)*(cost of each level) = n*1 =n T(n) = n = Q(n)

Insertion Sort InsertionSort(A, n) if n > 0 InsertionSort(A, n-1) {Sort subarray} Insert(A, n) {Insert nth element} T(n) = T(n -1) + cost of Insert(A, n) T(n) = T(n -1) + n

Recursion tree for InsertionSort T(n) = T(n-1) + n We will draw this in class.

Solving Recursion tree for Insertion Sort T(n) = n + (n-1) + (n-2) + ... + 2 + 1 = ?