Chapter 5: Searching & Sorting

Transcription

1 Searching & Sorting Methods Chapter : Searching & Sorting Searching A List of Data Items - can be accomplished with either of the following two methods: Sequential Search Binary Search (on a sorted list) Sorting - can be accomplished with various algorithms that arrange lists in alphabetical or numerical order on a selected key field. Binary Search Procedure Algorithm Efficient searching method for large lists that are in sorted order. Basic idea: eliminate half of list from consideration at each step. () Start at middle of list, comparing testval to data(m) this is it L= m=(l+u)/ U-listSize () If testval > data(m), apply binary search to upper half of list; otherwise, search lower half. Continue until testval = dat(m) or L > U.

2 For the example shown, the lowr pointer is moved up and m recalculated: this is it L=+ m U-listSize Pseudocode Algorithm Input list, listlength (or calculate listlength), testval lower=; upper=listlength position-0 While ( (lower <= upper) and (position=0) ) mid = Floor( (lower+upper)/ ) If (testval < list(mid)) Then (upper = mid-) Else If (testval > list(mid)) Then (lower=mid + ) Else position = mid (* we fount it *) Endofwhile (* if position = 0, testval is not in list *) Example: listsize = 8, value we seek is in position 8. () L=, U=8, m= () L=, U=8, m=6 () L=7, U=8, m=7 () L=8, U=8, m=8 This took searches for list of length 8(= ).

3 6 Binary Search - Worst Case For list of length N = n, we must check n + items In general, we need to check log N + items Each check may require comparisons, therefore total number of comparisons in worst case is (log N + ) Binary vs Sequential Search To determine that an item is not in a list of items requires: Conclusion: Sequential Search:,768 comparisons. Binary Search: comparisons: (at most). Binary Search should be used with a large list (e.g., Chicago telephone director) -- after it has been sorted.

4 7 Sorting Methods Select a Key Field (assuming file contains records with multiple data fields) Key Field is usually an ID number, but can also be name (of a person, account, or inventory item) Algorithms can be designed to minimize sorting time under various conditions: List Size Data Ordering - partially ordered - completely random List Structure Static List (no insertions-deletions) - stored as any array Dynamic List (requires frequent updates to data items) - stored in linked list Internal or External Storage - large files may be stored on multiple disks or tapes (e.g., Social Security or IRS files)

5 8 Selection Sort - straightforward method, good when data list not too long. On each pass, locate ( select ) smallest unsorted value and place it in its proper sorted position. Example: list = (f, t, c, z, e) pass : pass : pass : pass : Total number of passes = listlength -

6 9 Algorithm: Selection Sort - applied to a numerical or character-string array. For each pass (,,..., listlength - ) Initialize position of smallest value to pass number: smallpos = pass Successively compare unsorted items to value at smallpos (inner loop: pass+,..., listlength). If smaller value is encountered, reset value of smallpos. At end of inner loop: if (smallpos pass) then invoke procedure swap (have one swap per pass, at most) Pseudocode: Input list, listlength Do pass =, listlength - smallpos = pass Do k = pass+, listlength If (list(k) < list(smallpos) Then smallpos = k If (smallpos pass) Then Swap(list(smallPos), list(pass)) Endofdo

7 0 Insertion Sort Good method for small lists (< 0) that are partially sorted. Insert keys, one at a time, into sorted position. Example: list of keys = (9, 6,,,,, 0) () Start with keys() {= 9} () temp = keys () {= 6} If (temp < keys() Then keys() = keys() keys() = temp Now have sorted sublist: (6, 9) () Next, set: temp = keys() {= } Then compare temp to elements of sorted sublist, back to front, shifting larger keys to back, i.e., for above list, we would shift: keys() = keys() keys() = keys() keys() = temp And sorted sublist is now: (, 6, 9) Continue until all values in keys list are processed.

8 Pseudocode: Insertion Sort Algorithm Input keys, listlength Do pass =, listlength temp = keys(pass) k = pass - found = False While ( (not found) and (k > 0) ) If (temp < keys(k) ) Then keys(k+) = keys(k) k = k - Else found = True (* position for temp has been found *) keys(k+) = temp For larger lists stored as internal arrays, faster algorithms such as Quicksort and Heap Sort can be used. Sorting & Searching Functions in Mathematica Sort[ list] MemberQ[ list, testval] Position[ list, testval]

9 Indexed Sorting Example: - used to reduce processing time with files containing records of information, by rearranging the file indes list instead of swapping the data records. index part number part name supplier descr cost etc. () () () () () key field Parts list is ordered on key field according to sorted values: i.e., keys(index()) = keys() = 0 keys(index()) = keys() = keys(index()) = keys() = 6

10 Initially, index value can be assigned sequentially: keys =( 6, 879, 06, 0, 6), index =(,,,, ) Then sor index according to values in key list. Example: Indexed Selection Sort pass : () (sorted) () () () () pass : }(sorted) Input keys, index, listlength (* or input record and reference Do pass =, listlength - key field *) smallpos = pass Do k = pass+, listlength If (keys(index(k)) < keys(index(smallpos)) Then (smallpos = k) Endofdo If (smallpos pass) Then Swap(index(smallPos), index(pass)) Endofdo