Discrete Mathematics CS 2610

Discrete Mathematics CS 2610 February 24, 2009 -- part 4

Uncountable sets Theorem: The set of real numbers is uncountable. If a subset of a set is uncountable, then the set is uncountable. The cardinality of a subset is at least as large as the cardinality of the entire set. It is enough to prove that there is a subset of R that is uncountable Theorem: The open interval of real numbers [0,1) ={r  R | 0  r < 1}is uncountable. Proof by contradictionusing theCantor diagonalization argument(Cantor, 1879)

Uncountable Sets: R Proof (BWOC) using diagonalization: Suppose R is countable (then any subset say [0,1) is also countable). So, we can list them: r1, r2, r3, … where r1 = 0.d11d12d13d14… the dij are digits 0-9 r2 = 0.d21d22d23d24… r3 = 0.d31d32d33d34… r4 = 0.d41d42d43d44… etc. Now let r = 0.d1d2d3d4… where di = 4 if dii 4 di = 5 if dii = 4 But r is not equal to any of the items in the list ∴ it’s missing from the list so we can’t list them after all. r differs from ri in the ith position, for all i. So, our assumption that we could list them all is incorrect.

Algorithms An Algorithm is a finite set of precise instructions for performing a computation or for solving a problem. Properties of an algorithm: • input: input values are from a specified set • output: output values are from a specified set • definiteness: each step is precisely defined • correctness: correct output produced • finiteness: takes a finite number of steps • effectiveness: each step is finite & exact • generality: applicable to various input sizes

Analysis of Algorithms • Analyzing an algorithm • Time complexity • Space complexity • Time complexity • Running time needed by an algorithm as a function of the size of the input • Denoted as T(N) • We are interested in measuring how fast the time complexity increases as the input size grows • Asymptotic Time Complexity of an Algorithm

Pseudocode procedureprocName(argument: type) variable := expression informal statement beginstatementsend {comment} ifcondition then statement1 [else statement2] for variable := initial value to final valuestatement whileconditionstatement returnexpression procName(arg1,..,argn)

Algorithm Complexity • Worst Case Analysis • Largest number of operations to solve a problem of a specified size. • Analyze the worst input case for each input size. • Upper bound of the running time for any input. • Most widely used. • Average Case Analysis • Average number of operations over all inputs of a given size • Sometimes it’s too complicated

Example: Max Algorithm proceduremax(a1, a2, …, an: integers) v := a1 fori := 2 ton ifai > vthen v := ai returnv 1 n n - 1 ? 0, 1, .., n -1 1 How many times is each step executed ? Worst-Case: the input sequence is a strictly increasing sequence

Searching Algorithms • Searching Algorithms Problem: Find an element x in a list a1,…an (not necessarily ordered) • Linear Search Strategy: Examine the sequence one element after another until all the elements have been examined or the current element being examined is the element x.

Example: Linear Search procedurelinear search (x: integer, a1, a2, …, an: distinct integers) i := 1 while (i n  x  ai) i:=i + 1 ifi n then location := i else location:= 0 return location Worst-Case occurs when x is the last element in the sequence Best Case occurs when x is the first element in the sequence

Example: Linear Search • Average Case: • x is the first element: • 1 loop comparison • x is the second element • 2 loop comparisons, 1 iteration of the loop • x is the third element • 3 loop comparisons, 2 iterations of the loop • x is n-th element • n loop comparisons, n-1 iterations of the loop

Binary Search Problem: Locate an element x in a sequence of sorted elements in non-decreasing order. Strategy: On each step, look at the middle element of the remaining list to eliminate half of it, and quickly zero in on the desired element.

Binary Search procedurebinary search (x:integer, a1, a2, …, an: integer) {a1, a2, …, an are distinct integers sorted smallest to largest}i:= 1 {start of search range} j:=n {end of search range} while i < j beginm:=(i +j)/2 ifx > amtheni := m + 1 else j := mendifx = aithenlocation:=ielselocation:= 0 returnlocation Suppose n=2k

Binary Search • Binary Search • The loop is executed k times • k=log2(n)

Linear Search vs. Binary Search • Linear Search Time Complexity: T(n) isO(n) • Binary Search Time Complexity: T(n) isO(log2n)

Sorting Algorithms Problem: Given a sequence of numbers, sort the sequence in weakly increasing order. Sorting Algorithms: Input: A sequence of n numbers a1, a2, …, an Output: A reordering of the input sequence (a’1, a’2, …, a’n) such that a’1  a’2  …  a’n

Bubble Sort • Smallest elements “float” up to the top (front) of the list, like bubbles in a container of liquid • Largest elements sink to the bottom (end). See the animation at: http://math.hws.edu/TMCM/java/xSortLab

Example: Bubble Sort procedurebubblesort(a1, a2, …, an: distinct integers)for i = 1 to n-1for j = 1 to n-i if (aj > aj+1) then swap aj and aj+1 • Worst-Case: The sequence is sorted in decreasing order • At step i • The loop condition of the inner loop is executed n – i + 1 times. • The body of the loop is executed n – i times

Algorithm- Insertion Sort • For each element: • The elements on its left are already sorted • Shift the element with the element on the left until it is in the correct place. See animation http://math.hws.edu/TMCM/java/xSortLab/

Algorithm- Insertion Sort procedure insertSort(a1, a2, …, an: distinct integers)for j=2 to n begin i=j - 1 while i > 0 and ai > ai+1 swap ai and ai+1 i=i-1 end Worst-Case: The sequence is in decreasing order At step j, the while loop condition is executed j times the body of the loop is executed j-1 times

Discrete Mathematics CS 2610