Notes - Week 9

The following notes supplement the material in Chapter 7.

Arrays:

Definition

We use the term array to refer to a named collection of items. The collection has the following properties:

Homogeneous: All items in the collection are of the same type. That is, the items are either all numeric or all string.
Indexed: Individual items may be referenced by an index. That is, we may refer to the first, second, or i^th item in the collection.
Fixed Size: The maximum number of items in an array must be defined beforehand. The space required to accommodate all of these items will be reserved.

e.g. Consider the array named prefix, defined to contain 5 numeric items. Let us say these items are prefixes for Illinois telephone numbers. That is, the array contains the items {312, 773, 847, 603, 708}. Each item in the array would be referenced thus:

  prefix[1] = 312
  prefix[2] = 773
  prefix[3] = 847
  prefix[4] = 603
  prefix[5] = 708

Note: We will refer to the first item in an array with the index 1. We will not use 0 to refer to the first item.

Problem

The array numbers is defined to contain n numbers. Develop an algorithm to compute the average of the first 4 numbers in the array. Assume the array contains {10, 20, 2, 4} in the first 4 cells.

Solution

Strategy: Step through the array adding each item to a running total. After adding the 4^th item, compute the average. We will be repeatedly adding and so an iteration structure seems appropriate.

  total := 0 
  i     := 1
  while (i =< 4) do
     total  := total + numbers[i]
     i := i + 1
  endwhile
  average := total/4

The following is an execution trace of this algorithm:

                       Iteration #
-------------------------------------
     Initial  |  1  |  2  |  3  | 4
=====================================
total:  0     |  10 |  30 |  32 | 36 
-------------------------------------
    i:  1     |  2  |  3  |  4  | 5 
-------------------------------------

Hence, at the end of the iteration structure, the variable total has value 36 and so average is 36/4 = 9.

Problem

For the array above, develop an algorithm to find the smallest number and its position in the array.

Solution

Strategy: Step through the array examining each item. If the current item is smaller than the previous smallest item then note the current number and its position. Let us start out by assuming that all numbers in the array are smaller than some very large number (say 9999). An iteration structure again seems appropriate.

  min    := 9999
  minpos := 0
  i      := 1
  while (i =< 4) do
     if (number[i] < min) then
        min := number[i]
        minpos := i
     endif
     i := i + 1
  endwhile

The following is an execution trace of this algorithm:

                       Iteration #
-------------------------------------
     Initial  |  1  |  2  |  3  | 4
=====================================
   min:  9999 |  10 |  10 |  2  | 2 
-------------------------------------
minpos:  0    |  1  |  1  |  3  | 3  
-------------------------------------
     i:  1    |  2  |  3  |  4  | 5 
-------------------------------------

Hence, the smallest number is 2 and it is the 3^rd item in the array.

Problem

The array word is defined to contain n characters. That is, each cell of the array may contain a character. Develop an algorithm to find the first occurrence of the letter 'a' in the array. Assume that the array contains the string "william.". Notice that the character '.' is used to indicate the end of the word.

Solution

Strategy: Step through the array examining each item. If the current item is the letter 'a' then it has been found so note its position. An iteration structure again seems appropriate.

  found := F 
  pos   := 0
  i     := 1
  while ((word[i] not= '.') and (found = F)) do
     if (word[i] = 'a') then
        found := T
        pos   := i
     endif
     i := i + 1
  endwhile

The following is an execution trace of this algorithm:

                       Iteration #
---------------------------------------------------
     Initial  |  1  |  2  |  3  |  4  |  5  |  6  
===================================================
 found:  F    |  F  |  F  |  F  |  F  |  F  |  T   
---------------------------------------------------
   pos:  0    |  0  |  0  |  0  |  0  |  0  |  6  
---------------------------------------------------
     i:  1    |  2  |  3  |  4  |  5  |  6  |  7 
---------------------------------------------------

Hence, the letter 'a' is the 6^th item in the array. Notice the use of the boolean variable found in this algorithm.

Linked Lists:

Definition