CMSC 203 Section 0401 Discrete Structures Fall 2004
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CMSC 203, Section 0401 Discrete Structures Fall 2004 Matt Gaston mgasto 1@cs. umbc. edu http: //www. csee. umbc. edu/~mgasto 1/203 UMBC CMSC 203, Section 0401 -- Fall 2004 1
Algorithms Ch. 2. 1 -2. 3 UMBC CMSC 203, Section 0401 -- Fall 2004 2
Definition • Book 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. UMBC CMSC 203, Section 0401 -- Fall 2004 3
Algorithm Features • • Input Output Definiteness Correctness Finiteness Effectiveness Generality UMBC CMSC 203, Section 0401 -- Fall 2004 4
Linear Search procedure linear search (x: integer, a 1, a 2, . . . , 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? UMBC CMSC 203, Section 0401 -- Fall 2004 5
Binary Search procedure binary search (x: integer, a 1, a 2, . . . , 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 UMBC CMSC 203, Section 0401 -- Fall 2004 6
Growth of Functions • Big-O 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 O(g(x)) if there are constants C and k such that | f(x) | C | g(x) | whenever x > k. § C and k are called “witnesses UMBC CMSC 203, Section 0401 -- Fall 2004 7
Pictorial: Big-O Notation C g(x) f(x) g(x) k UMBC CMSC 203, Section 0401 -- Fall 2004 8
Example • 7 x 2 = O(x 3) UMBC CMSC 203, Section 0401 -- Fall 2004 9
Combinations of Functions • f(x) = anxn + an-1 xn-1 +. . . + a 1 x + a 0 § f(x) = O(xn) • (f 1 + f 2)(x) = O(max( g 1(x), g 2(x) )) • (f 1 f 2)(x) = O(g 1(x)g 2(x)) UMBC CMSC 203, Section 0401 -- Fall 2004 10
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. UMBC CMSC 203, Section 0401 -- Fall 2004 11
Big-Theta 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 f(x) is O(g(x)) and f(x) is (g(x)). § f(x) is of order g(x) UMBC CMSC 203, Section 0401 -- Fall 2004 12
Bubble Sort procedure bubble sort (a 1, a 2, . . . , an) for i : = 1 to n – 1 for j : = 1 to n – i if aj > aj+1 then interchange aj and aj+1 {a 1, a 2, . . . , an is in increasing order } UMBC CMSC 203, Section 0401 -- Fall 2004 13
Insertion Sort procedure insertion sort (a 1, a 2, . . . , 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 {a 1, a 2, . . . , an are sorted } UMBC CMSC 203, Section 0401 -- Fall 2004 14
Make Change! UMBC CMSC 203, Section 0401 -- Fall 2004 15
Understanding Complexity • n!, 2 n, n 2, n log(n), n, log(n), 1 • • Tractable/Intractable Solvable/Unsolvable – Halting Problem Class P, Class NP NP-Complete UMBC CMSC 203, Section 0401 -- Fall 2004 16
