CMSC 203, Section 0401 Discrete Structures Fall 2004 Matt Gaston mgasto1@cs.umbc.edu http://www.csee.umbc.edu/~mgasto1/203

Algorithms Ch. 2.1-2.3

Definition Book definition: Better definition (?): Algorithm: a finite set of precise instructions for performing a computation or for solving a problem Better definition (?): Algorithm: An algorithm is a finite set of unambiguous, executable instructions that directs a terminating activity.

Algorithm Features Input Output Definiteness Correctness Finiteness Effectiveness Generality

Linear Search procedure linear search (x:integer, a1, a2, . . . , an: distinct integers) i := 1 while (i  n and x  ai) i := i + 1 if i  n then location := i else location := 0 10 13 17 1 4 18 3 5 11 9 8 16 2 7 6 14 15 19 20 18 12 How many steps? Worse case? On average?

Binary Search procedure binary search (x:integer, a1, a2, . . . , an: increasing integers) i := 1 j := n while (i  j) begin m :=  (i + j) / 2  if x  am then i := m + 1 else j := m end if x = ai then location := I else location := 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Growth of Functions Big-O Notation: C and k are called “witnesses Let f and g be functions from the set of integers to the set of integers or the set of real numbers to the set of real numbers. We say that f(x) is O(g(x)) if there are constants C and k such that | f(x) |  C | g(x) | whenever x > k.

Pictorial: Big-O Notation C g(x) f(x) g(x) k

Example 7x2 = O(x3)

Combinations of Functions f(x) = anxn + an-1xn-1 + . . . + a1x + a0 f(x) = O(xn) (f1 + f2)(x) = O(max( g1(x), g2(x) )) (f1f2)(x) = O(g1(x)g2(x))

Big-Omega Notation Let f and g be functions from the set of integers to the set of integers or the set of real numbers to the set of real numbers. We say that f(x) is (g(x)) if there are constants C and k such that | f(x) |  C | g(x) | whenever x > k.

Big-Theta Notation f(x) is of order g(x) Let f and g be functions from the set of integers to the set of integers or the set of real numbers to the set of real numbers. We say that f(x) is (g(x)) if f(x) is O(g(x)) and f(x) is (g(x)).

Bubble Sort procedure bubble sort (a1, a2, . . . , an) for i := 1 to n – 1 for j := 1 to n – i if aj > aj+1 then interchange aj and aj+1 {a1, a2, . . . , an is in increasing order }

Insertion Sort procedure insertion sort (a1, a2, . . . , an : real numbers with n  2) for j := 2 to n begin i := 1 while aj > ai i := i + 1 m := aj for k := 0 to j - i – 1 aj-k := aj-k-1 ai := m end {a1, a2, . . . , an are sorted }

Make Change!

Understanding Complexity n!, 2n, n2, n log(n), n, log(n), 1 Tractable/Intractable Solvable/Unsolvable – Halting Problem Class P, Class NP NP-Complete