Nonrecursive Algorithms: Analysis and Examples - Prof. B. Karki, Study notes of Computer Science

Download Nonrecursive Algorithms: Analysis and Examples - Prof. B. Karki and more Study notes Computer Science in PDF only on Docsity!

Nonrecursive Algorithms

Definition and Examples

 Nonrecursive algorithm:

 Executed only once to solve the problem.

 Examples

 Largest element in a list of numbers  Element uniqueness problem  Matrix operations: addition, multiplication, and transpose  Digits in binary representation

 Recursive algorithm invokes (makes reference to) itself

repeatedly until a certain condition matches.

Example 1: Maximum Element

Algorithm MaxElement ( A[0.. n - 1] ) //Determines the value of the largest element in a given array //Input: An array A[0 .. n - 1] of real numbers //Output: The value of the largest element in A maxval ← A[0] for i ← 1 to n - 1 do if A[ i] > maxval maxval ← A[ i] return maxval

Example 1: Maximum Element (Cont.)

 Input size = n , the number of elements in the array

 Algorithm’s basic operation is “comparison”

 It is executed on each repetition of the loop

 Formula for the basic operation count:

 Sum is simply 1 repeated by n- 1 times.

C ( n ) = 1 i = 1 n − 1 ∑ =^ n^ −^1 ∈^ Θ( n )

Example 2: Unique Elements (Cont.)

 Input size = n , the number of elements in the array  Basic operation: single operation ( element comparison ) in the innermost loop  Formula for the basic operation count:  Depends also on whether there are equal elements in the array and, if there are, which positions they occupy  In worst case, the algorithm needs to compare all n ( n -1)/2 distinct pairs.

C

worst

( n ) = 1

j = i + 1 n − 1 ∑ =^ [( n^ −^1 )^ −^ ( i^ +^1 )^ +^1 ]^ =^ ( n^ −^1 −^ i ) i = 0 n − 2 ∑ i = 0 n − 2 ∑ i = 0 n − 2 ∑ =^

( n − 1 ) n

∈ Θ( n

2

Example 3: Matrix Operations

 A 2D matrix is the standard representation  Classic matrix multiplication: O( n 3 )  Given two n -by- n matrices A and B , compute their product C = AB using definition based algorithm. C is an n -by- n matrix whose elements are computed as the scalar (dot) products of the rows of A and the columns of B.  Strassen’s algorithm based on a divide-and-conquer approach: O( n 2.81 ) Algorithm MatrixMultiplication ( A[0.. n - 1, 0.. n - 1] , B[0.. n - 1, 0.. n - 1] ) //Multiplies two square matrices of order n by the definition-based algorithm //Input: Two n-by-n matrices A and B //Output: Matrix C = AB for i ← 0 to n - 1 do for j ← 0 to n - 1 do C[ i, j] ← 0. for k ← 0 to n - 1 do C[ i, j] ← C[ i, j] + A[ i, k] * B[ k, j] return C

Cij = Aik Bkj

k = 0 n − 1 ∑

CSC 3102 0.10 B.B. Karki, LSU

Sparse Matrix (SM) Transpose

 A sparse matrix (of order n) has only a small number ( t ) of nonzero elements  A 2D representation is not efficient as storing the zero elements wastes space  Sparse matrix representation: A set of ordered triplets < row, col, value >  row and col are integers defining each element in the matrix  value is the actual element or item  Constraint: The entries must be sorted in ascending order using the row and col indices as the primary and secondary keys, respectively.  Transpose of M is M' :  M'[i].row = M[i].col  M'[i].col = M[i].row row col value M [0] 6 6 8 [ 1 ] 0 0 15 [2] 0 3 22 [ 3 ] 0 5 − 15 [ 4 ] 1 1 11 [5] 1 2 3 [6] 2 3 − 6 [ 7 ] 4 0 91 [8] 5 2 28 € row col value M^ ′[0] 6 6 8 [ 1 ] 0 0 15 [2] 0 4 91 [ 3 ] 1 1 11 [ 4 ] 2 1 3 [5] 2 5 28 [6] 3 0 22 [ 7 ] 3 2 − 6 [8] 5 0 − 15 € M = 15 0 0 22 0 − 15 0 11 3 0 0 0 0 0 0 − 6 0 0 0 0 0 0 0 0 91 0 0 0 0 0 0 0 28 0 0 0                   Matrix M Transpose^ M' n = 6 t = 8

Algorithms for SM Transpose

 Naïve approach:  Swap the row and column indices of all terms  Sort the terms according to the new row index as a primary key  Fast approach:  Calculate the position in the list where each term would go after the transpose  Swap the row and column indices of all terms by placing them in their appropriate positions.

CSC 3102 0.13 B.B. Karki, LSU

Naïve SM Transpose Analysis

 Given a matrix M of order n , find its transpose matrix M' :  Inner loop over the number of non-zero elements ( t ) For each element, exchange the row and col indices (i.e, create a transpose)  Outer loop over the col index of the original matrix ( n values) using it as the primary key (per row i in M') For each index i , process inner loop (i.e., process all elements and pick up their col numbers if they match with i.  Basic operation: Check whether a new row ( M' ) contains non-zero element(s): M[j].col = i  The runtime complexity of the algorithm is:  For dense matrix, t → n^2 , the complexity is of the order O( n^3 ). € C ( n ) = 1 j = 1 t ∑ =^ t i = 0 n − 1 ∑ =^ nt^ ∈^ Ο( nt ) i = 0 n − 1 ∑

Fast Sparse Matrix Transpose: Code

Example 4: Binary

 Basic operation is “ comparison n > 1” , which decides whether the while loop will be executed.  The value of n is about halved on each repetition of the loop  The complexity is log 2 n + 1  Number of times the comparison is executed = Number of repetitions of the loop’s body + 1. Algorithm Binary ( n ) //Find the number of binary digits in the binary representation of a positive decimal integer //Input: A positive decimal integer n //Output: The number of binary digits count ← 1 while n > 1 do count ← count + 1 n ←  n/2  return count