Lecture 3: Analysis of Algorithms Review Insertion Sort Analyzing Algorithms Big-O Notation Your CUNIX Account Reading

Review Definition of an algorithm: A well-ordered collection of unambiguous and effectively computable operations that, when executed, produces a result and halts in a finite amount of time. There are three components to an algorithm: A set of inputs, where each input is a finite sequence of items. A set of outputs, where each output is a finite sequence of items. A method consisting of a finite sequence of instructions, each one of which can be mechanically executed in a fixed length of time with a fixed amount of resources. The method must produce an output for each input in the set of possible inputs in a finite amount of time.

Review Input size: associated with each input is a measure of its size. How we measure the size can vary from problem to problem. Examples: Number of digits as in the number of digits in a number (Often we use binary digits). Number of characters in a string. Number of items as in the number of items to be searched or sorted.

Review: Selection Sort Algorithm to sort the items on a list into nondecreasing order Input: A list of n ≥1 elements A[1], A[2], ... , A[n]. Output: The list sorted in nondecreasing order. Method: for (i = 1; i < n; i++) { locmin=i; for (j=i+1;j <= n; j++) if (A[locmin] > A[j]) locmin=j; } exchange(A[locmin], A[i]);

Insertion Sort Input: A list of n ≥1 elements A[1], A[2], ... , A[n]. Output: The list sorted in nondecreasing order. Method: for (j=2; j <= n; j ++) { temp = A[j] i=j-1 while (i > 0 and A[i] > temp) A[i+1]=A[i] i=i-1 } A[i+1]=temp

Analyzing Algorithms We would like to measure the quality or efficiency of an algorithm so that we may compare algorithms that perform similar tasks. We want this measure to be machine (and implementation) independent. We will consider the number of instructions that need to be executed. This will be a function of the input size, n. This may vary depending on “chance”. We will consider worst-case running time and also average-case running times.

Analyzing Algorithms Instead of all instructions being counted equal we will count mathematical and/or logical operations involving variables that are unknown ahead of time. At the end of the day our “measure” will be a function of n. Therefore we need a way of comparing these functions.

Big-O Notation Given a function of n,we will consider only the most dominant term. What we really care about is the order of magnitude of growth. Example: T(n)=25n3+log(n)+2n = O(2n) Example: T(n)=3n2+100n = O(n2) We’ll save the technical definition for COMS W3203 Discrete Math. The point is we care about growth because we care about really really big n!

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