Beginning Programming (Pascal) Lecture 21

Transcription

1 1. Keys: Lists contain elements called keys. Consider program list01. Type list_t defines an array of ten integers. Variable list has type list_t. Variable list implements the list. Each integer in the array is a key. Procedure listinit assigns values to the list. Note that formal parameter L is passed by reference. A parameter passed by reference is an alias for the actual argument. The actual argument is variable list. Formal parameter L is an alias for variable list. Values assigned to elements of formal parameter L are really assigned to variable list. Procedure listwrite formats and prints values of a list. The first formal parameter, L, specifies the implementation of a list. A list must have type list_t. Note, again, that formal parameter L is passed by reference. The second parameter, N, dictates how many elements of the list are to be printed. Parameter N is passed by value. The third parameter, f, identifies the file where the list is to be written. Parameters of type text must be passed by reference. Procedure listwrite prints keys in the list separated by commas and enclosed in curly braces. A new line is printed before the list is printed. The first two characters printed are "L=". Output for program list01 is shown below. program list01; type list_t=array[1..10] of integer; L[1]:=1; L[2]:=1; for a:=3 to 10 do L[a]:=L[a-1]+L[a-2]; {list01} listwrite(list,10,output); end{list01}. Output for program list01. L={1,1,2,3,5,8,13,21,34,55} 1

2 2. Searching: A particular key in a list may be sought. 3. Match: The index of the key is returned when the key sought is found. If the key is not in the list an invalid index is returned. 4. Unordered list: To find a key in an unordered list every element in the list must be tested. Consider program list02. Program list02 is an enhancement of program list01. Program list02 contains subprograms listinit, listwrite, and listfind. Program list02 creates an unordered list, prints the list, finds a matching key in the list and attempts to find a key not in the list. Output produced by program list02 is given below the listing of the program. Procedure listinit assigns initial values to the list implemented in variable list as an array of ten integers. Procedure listinit creates an unordered list. Procedure listwrite formats and prints a list as it did in program list01. Function listfind attempts to find a key in the input list that matches the value of formal parameter key. If the key is found the index is returned. If the key is not found a value of zero is returned. program list02; type list_t=array[1..10] of integer; L[ 1]:=-512; L[ 2]:= 256; L[ 3]:=-128; L[ 4]:= 64; L[ 5]:= -32; L[ 6]:= 16; L[ 7]:= -8; L[ 8]:= 4; L[ 9]:= -2; L[10]:= 1; function listfind(var L:list_t;N:integer;key:integer):integer; {listfind} listfind:=0; if key=l[a] then listfind:=a end{listfind}; 2

3 {list02} listwrite(list,10,output); writeln; writeln ('16 is in element ',listfind(list,10,16):1,' of array list.' ); writeln ('Function listfind returns ',listfind(list,10,17):1,' when the key sought is 17.' ) end{list02}. Output for program list02. L={-512,256,-128,64,-32,16,-8,4,-2,1} 16 is in element 6 of array list. Function listfind returns 0 when the key sought is Searching for the largest key. Finding the largest key in an unordered list is like finding the high water mark in a river. As the river rises the mark rises. However, when the river recedes, the high water mark remains until the river rises above new mark. Here is the algorithm for finding the index of the largest key Assume the first element contains the largest key For each element in the list starting with the second element If the current element is larger than the largest then record the index of the current element as the largest. Program list03 contains function findmax that finds the index of the largest element in the list using the algorithm shown above. Output for program list03 is given below the listing of the program. program list03; type list_t=array[1..10] of integer; L[ 1]:=-512; L[ 2]:= 256; L[ 3]:=-128; L[ 4]:= 64; L[ 5]:= -32; L[ 6]:= 16; L[ 7]:= -8; L[ 8]:= 4; L[ 9]:= -2; L[10]:= 1; 3

4 function listmax(var L:list_t;N:integer):integer; var a:integer; iom:integer; {index of maximum element} {listmax} iom:=1; for a:=2 to N do if L[a]>L[iom] then iom:=a; listmax:=iom end{listmax}; {list03} listwrite(list,10,output); writeln; writeln(listmax(list,10):1,' is the index of maximum element.') end{list03}. Output for program list03. L={-512,256,-128,64,-32,16,-8,4,-2,1} 2 is the index of the maximum element. 6. Searching for the smallest key. Searching for the smallest key is highly similar to searching for the largest key. Traverse the entire list, one element at time, comparing the current element with the smallest element on record. If the current element is smaller than the recorded smallest, record the new element as the smallest. 7. Binary search. If the list is ordered, a key may be found using a much faster algorithm than testing every element in the list. Comparing the input key to an element in the list permits the algorithm to discard half the remaining elements in which to find a match each iteration. Consider array L shown below whose index values are given in the top row and whose element values are given in the bottom row. Search for the index of the element whose value is zero. To, guess that element 8 whose value is 13 is the key. Because 0<13 and the elements are in ascending order, we know that 0 cannot be in element 8 or in any element larger than 13. The range of values we have discarded is shaded. Darkly shaded areas indicate overlapping ranges. The value of variable lo exceeds the value of variable hi. 4

5 Select a value in the middle of the new range, say 2, whose index is 4. Ask the question again. Is the key, 0, less than the value selected. Yes, 0<2. Discard all values two or greater. Select a value in the middle of the new range, say 1, whose index is 2. Ask the question again. Is the key, 0, less than the value selected. Yes, 0<1. Discard all values one or greater. Select a value in the middle of the new range. The new range includes only a single value 0. Zero is the key we seek. Let us search for another key, 55. Begin the search as before. Select a value in the middle of the entire range. Select 13. Is 55<13? No. Is 55>13? Yes. Discard all values 13 or smaller. Select a value in the middle of the new range, say 89, whose index is 12. Ask the question again. Is 55<89? Yes, discard elements having a value of 89 or greater. Select a value in the middle of the new range, say 34, whose index is 10. Ask the question again. Is 55<34? No. Is 55>34? Yes, discard elements having a value of 34 or less. Only one value remains 55. Our algorithm returns the index 11. Function listfind in program list04 implements a binary search. Function listfind is called to find the index of the element matching the key 0. Results of program list04 given after the listing. program list04; const M=16; type list_t=array[1..m] of integer; L[1]:=0; L[2]:=1; for a:=3 to M do L[a]:=L[a-1]+L[a-2]; 5

6 function listfind(var L:list_t;N:integer;key:integer):integer; var lo,hi,m:integer; {listfind} lo:=1;hi:=n; repeat m:=(lo+hi) div 2; if L[m]<key then lo:=m+1; if L[m]>key then hi:=m-1; until (lo>hi) or (L[m]=key); if (lo>hi) then listfind:=0 else listfind:=m; end{listfind}; {list04} listwrite(list,m,output); writeln; writeln('0 is in element ',listfind(list,m,0):1,'.') end{list04}. 8. Tracing program execution. Another way of looking at a binary search is to record the values computed for variables. Values computed in each iteration are recorded in a row. Note that either the value of variable hi or variable lo is shaded in each row after the first row. Values that are shaded are those that have been changed in the previous iteration. key=0 lo hi m L[m] L[m]<key L[m]>key L[m]=key lo>hi <0, No 13>0, Yes 13=0, No 1>16, No <0, No 2>0, Yes 2=0, No 1>7, No <0, No 1>0, Yes 1=0, No 1>3, No <0, No 0>0, No 0=0, Yes 1>1, No Array L 6

7 key=55 lo hi m L[m] L[m]<key L[m]>key L[m]=key lo>hi <55, Y 13>55, N 13=55, N 1>16, N <55, N 89>55, Y 89=55, N 9>16, N <55, Y 34>55, N 34=55, N 9>11, N <55, N 55>55, N 55=55, Y 11>11, N Array L key=4 lo hi m L[m] L[m]<key L[m]>key L[m]=key lo>hi <4, N 13>4, Y 13=4, N 1>16, N <4, Y 2>4, N 2=4, N 1>7, N <4, N 5>4, Y 5=4, N 5>7, N <4, Y 3>4, N 3=4, N 5>6, N <4, N 5>4, Y 5=4, N 6>6, N <4, Y 2>4, N 2=4, N 6>5, Y Array L 7