Algorithm Analysis Algorithm Analysis Lectures 3 & 4 Resources Data Structures & Algorithms Analysis in C++ (MAW): Chap. 2 Introduction to Algorithms (Cormen, Leiserson, & Rivest): Chap.1 Algorithms Theory & Practice (Brassard & Bratley): Chap. 1

Algorithms An algorithm is a well-defined computational procedure that takes some value or a set of values, as input and produces some value, or a set of values as output. Or, an algorithm is a well-specified set of instructions to be solve a problem.

Efficiency of Algorithms Empirical –Programming competing algorithms and trying them on different instances Theoretical –Determining mathematically the quantity of resources (execution time, memory space, etc) needed by each algorithm

Analyzing Algorithms Predicting the resources that the algorithm requires: Computational running time Memory usage Communication bandwidth The running time of an algorithm Number of primitive operations on a particular input size Depends on –Input size (e.g. 60 elements vs ) –The input itself ( partially sorted input for a sorting algorithm)

Order of Growth The order (rate) of growth of a running time –Ignore machine dependant constants –Look at growth of T(n) as n  –  notation Drop low-order terms Ignore leading constants E.g. –3n n 2 – 2n +5 =  (n 3 )

Mathematical Background

Definitions: –T(N) = O(f(N)) iff  c and n 0  T(N)  c.f(N) when N  n 0 –T(N) =  (g(N)) iff  c and n 0  T(N)  c.g(N) when N  n 0 –T(N) =  (h(N)) iff T(N) = O(h(N)) and T(N) =  (h(N))

Mathematical Background Definitions: –T(N) = o(f(N)) iff  c and n 0  T(N)  c.f(N) when N  n 0 –T(N) =  (g(N)) iff  c and n 0  T(N)  c.g(N) when N  n 0

Mathematical Background Rules: –If T 1 (N) = O(f(N)) and T 2 (N) = O(g(N)) then a) T 1 (N) + T 2 (N) = max( O(f(N)),O(g(N)) b) T 1 (N) * T 2 (N) = O(f(N) * g(N)) –If T(N) is a polynomial of degree k, then T(N) =  (N k ) –Log k N = O(N) for any constant k.

More … 1.3n n 2 – 2n +5 = O(n 3 ) 2.2n 2 + 3n =  (n 2 ) 3.2n = o(n 2 ) ( set membership) 4.3n 2 = O(n 2 ) tighter  (n 2 ) 5.n log n = O(n 2 ) 6.True or false: –n 2 = O(n 3 ) –n 3 = O(n 2 ) –2 n+1 = O(2 n ) –(n+1)! = O(n!)

Ranking by Order of Growth 1 nn log nn 2 n k (3/2) n 2 n (n)!(n+1)!

Running time calculations Rule 1 – For Loops The running time of a for loop is at most the running time of the statement inside the for loop (including tests) times the number of iterations Rule 2 – Nested Loops Analyze these inside out. The total running time of a statement inside a group of nested loops is the running time of the statement multiplied by the product of the sizes of all the loops

Running time calculations: Examples Example 1: sum = 0; for (i=1; i <=n; i++) sum += n; Example 2: sum = 0; for (j=1; j<=n; j++) for (i=1; i<=j; i++) sum++; for (k=0; k<n; k++) A[k] = k;

How to rank?