University of Hawaii ICS141: Discrete Mathematics for Computer Science I Dept. Information & Computer Sci., University of Hawaii Jan Stelovsky based on slides by Dr. Baek and Dr. Still Originals by Dr. M. P. Frank and Dr. J.L. Gross Provided by McGraw-Hill ICS 141: Discrete Mathematics I – Fall 2011 13-1 University of Hawaii Lecture 14 Chapter 3. The Fundamentals 3.1 Algorithms ICS 141: Discrete Mathematics I – Fall 2011 13-2 Algorithms University of Hawaii n Previously… n Characteristics of algorithms n Pseudocode n Examples: Max algorithm n Today… n Examples: Sum algorithm n Problem of searching an ordered list n Linear search & binary search algorithms n Sorting problem n Bubble sort & insertion sort algorithms ICS 141: Discrete Mathematics I – Fall 2011 13-3 Practice Exercises University of Hawaii n Devise an algorithm that finds the sum of all the integers in a list. n procedure sum(a1, a2, …, an: integers) s := 0 {sum of elements so far} for i := 1 to n {go thru all elements} s := s + ai {add current item} {now s is the sum of all items} return s ICS 141: Discrete Mathematics I – Fall 2011 13-4 Searching Algorithms University of Hawaii n Problem of searching an ordered list. n Given a list L of n elements that are sorted into a definite order (e.g., numeric, alphabetical), n And given a particular element x, n Determine whether x appears in the list, n And if so, return its index (position) in the list. n Problem occurs often in many contexts. n Let’s find an efficient algorithm! ICS 141: Discrete Mathematics I – Fall 2011 13-5 Linear Search (Naïve) University of Hawaii {Given a list of integers and an integer x to look up, returns the index of x within the list or 0 if x is not in the list} procedure linear_search (x: integer, a1, a2, …, an: distinct integers) i := 1 {start at beginning of list} while (i ≤ n ∧ x ≠ ai) {not done and not found} i := i + 1 {go to the next position} if i ≤ n then index := i {it was found} else index := 0 {it wasn’t found} return index {index where found or 0 if not found} ICS 141: Discrete Mathematics I – Fall 2011 13-6 Alg. #2: Binary Search University of Hawaii n Basic idea: At each step, look at the middle element of the remaining list to eliminate half of it, and quickly zero in on the desired element. <x <x <x >x ICS 141: Discrete Mathematics I – Fall 2011 13-7 Search Alg. #2: Binary SearchUniversity of Hawaii procedure binary_search (x: integer, a1, a2, …, an: increasing integers) i := 1 {left endpoint of search interval} j := n {right endpoint of search interval} while i < j begin {while interval has > 1 item} m := ⎣(i + j)/2⎦ {midpoint} if x > am then i := m + 1 else j := m end if x = ai then location := i else location := 0 return location {index or 0 if not found} ICS 141: Discrete Mathematics I – Fall 2011 13-8 Search Example University of Hawaii n Search for 19 in the list Index: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1, 2, 3, 5, 6, 7, 8, 10, 12, 13, 15, 16, 18, 19, 20, 22 i i i j j n using linear search i n using binary search n Entering while loop: i = 1, j = 16 n m = ⎣(i+j)/2⎦ = ⎣(1+16)/2⎦ = ⎣8.5⎦ = 8, i = 9, j = 16 n m = ⎣(i+j)/2⎦ = ⎣(9+16)/2⎦ = ⎣12.5⎦ = 12, i = 13, j = 16 n m = ⎣(i+j)/2⎦ = ⎣(13+16)/2⎦ = ⎣14.5⎦ = 14, i = 13, j = 14 n m = ⎣(i+j)/2⎦ = ⎣(13+14)/2⎦ = ⎣13.5⎦ = 13, i = 14, j = 14 n Exit loop ICS 141: Discrete Mathematics I – Fall 2011 13-9 Sorting Algorithms University of Hawaii n Sorting is common in many applications. n E.g. spreadsheets and databases n We can search quickly when data is odrered! n Sorting is also widely used as a subroutine in other data-processing algorithms. n Two sorting algorithms shown in textbook: However, these are not n Bubble sort very efficient, and you should n Insertion sort not use them on large data sets! We’ll see some more efficient algorithms later in the course. ICS 141: Discrete Mathematics I – Fall 2011 13-10 Bubble Sort University of Hawaii n Smaller elements “float” up to the top of the list, like bubbles in a container of liquid, and the larger elements “sink” to the bottom. ICS 141: Discrete Mathematics I – Fall 2011 13-11 Bubble Sort Algorithm University of Hawaii procedure bubble_sort (a1, a2, …, an: real numbers, n ≥ 2) for i := 1 to n – 1 {iterate n – 1 passes} for j := 1 to n – i if aj > aj+1 then interchange aj and aj+1 {an-i+1, …, an is sorted and ≤ a1, …, an-i} {a1, a2, …, an is sorted} n Example 4: Use the bubble sort to put 3, 2, 4, 1, 5 into increasing order. (See previous slide) ICS 141: Discrete Mathematics I – Fall 2011 13-12 Insertion Sort University of Hawaii n English description of algorithm: n Start with the second element, for each item in the input list: n “Insert” it into the correct place in the sorted output list generated so far. Like so: n Find the location where the new item should be inserted using linear or binary search. n Then, shift the items from that position onwards up by one position. n Put the new item in the remaining hole. ICS 141: Discrete Mathematics I – Fall 2011 13-13 Insertion Sort Example University of Hawaii n Use the insertion sort to put 3, 2, 4, 1, 5 into increasing order nd n Insert the 2 element 2 in the right position: n 3 > 2 ⇒ put 2 in front of 3. ⇒ 2, 3, 4, 1, 5 rd n Insert the 3 element 4 in the right position: n 4 > 2 ⇒ do nothing. Move to the next comparison. n 4 > 3 ⇒ do nothing. Done. ⇒ 2, 3, 4, 1, 5 th n Insert the 4 element 1 in the right position: n 2 > 1 ⇒ put 1 in front of 2. ⇒ 1, 2, 3, 4, 5 th n Insert the 5 element 5 in the right position: n 5 > 1 ⇒ do nothing. Move to the next comparison. n 5 > 2 ⇒ do nothing. Move to the next comparison. n 5 > 3 ⇒ do nothing. Move to the next comparison. n 5 > 4 ⇒ do nothing. Done. ⇒ 1, 2, 3, 4, 5 ICS 141: Discrete Mathematics I – Fall 2011 13-14 Insertion Sort Algorithm University of Hawaii procedure insertion_sort (a1, a2, …, an: real numbers, n ≥ 2) for i := 2 to n begin m := ai {the element to be inserted} j := 1 while aj < m {look for the index of the hole with j } j := j + 1 {now a1, …, aj-1 < m ≤ aj, …, ai } {the hole is at j; j ≤ i, i.e. possibly j = i } for k := j + 1 to i ak := ak+1 ai := m {a1, a2, …, ai are sorted in increasing order} end {a1, a2, …, an are sorted in increasing order} ICS 141: Discrete Mathematics I – Fall 2011 13-15 Efficiency of Algorithms University of Hawaii n Intuitively we see that binary search is much faster than linear search, n Also, we may see that insertion sort is better than bubblesort in some cases (why? when?), n But how do we analyze the efficiency of algorithms formally? n Use methods of algorithmic complexity, which utilize the order-of-growth concepts ICS 141: Discrete Mathematics I – Fall 2011 13-16 .