• Frequently:
  • We need to figure out the integer number of "tasks" that have been started and/or finished using a related measure of "work"
• Origin of the Name:
  • In baseball it is common to track a pitchers starts and complete games
• Other Examples:
  • Laps of a course started having run/driven a known distance
  • Pizzas purchased having eaten a known number of slices
  • Shipping containers needed for a known number of refrigerators
  • Years of college started having completed a known number of semesters

• The Approach:
  • Divide (e.g., 7 miles run over a 3 mile track means 7/3 = 2 1/3 laps)
• The Obvious Shortcoming:
  • The result isn't an integer

• The Approach:
  • Use integer division
• Examples:
  • 0 miles run over a 3 mile track means 0/3 = 0 laps were completed
  • 1 mile run over a 3 mile track means 1/3 = 0 laps were completed
  • 7 miles run over a 3 mile track means 7/3 = 2 laps were completed
  • 9 miles run over a 3 mile track means 9/3 = 3 laps were completed

javaexamples/programmingpatterns/StartsAndCompletions.java (Fragment: completions)

    /**
     * Determine the number of completed tasks from
     * an amount of work and a numeraire.
     *
     * @param work        The amount of work
     * @param workPerTask The work per task
     * @return The number of completed tasks
     */
    public static int completions(int work, int workPerTask) {
        return work / workPerTask;
    }
        

• The Approach:
  • Divide (e.g., 7 miles run over a 3 mile track means 7/3 = 2 1/3 laps)
• The Obvious Shortcoming:
  • The result (still) isn't an integer

• The Approach:
  • Use integer division (e.g., 7 miles run over a 3 mile track means 7/3 = 2 laps were started)
• The Shortcoming:
  • This isn't the right answer (e.g., 3 laps were started, not 2)

• The Approach:
  • Use integer division and add 1 (e.g., 7 miles run over a 3 mile track means 7/3 + 1 = 2 + 1 = 3 laps were started)
• The Shortcoming:
  • This doesn't always result in the correct answer (e.g., for 6 miles, 6/3 + 1 = 2 + 1 = 3 but the correct answer is 2)

The Remainder Indicator

javaexamples/programmingpatterns/StartsAndCompletions.java (Fragment: indicator)

    /**
     * Returns 0 if numerator is evenly divisible by denominator
     * and 1 otherwise. That is, returns 1 if there is a remainder
     * and 0 otherwise.
     *
     * @param num  The numerator
     * @param den  The denominator
     * @return 1 if there is a remainder; 0 otherwise
     */
    public static int remainderIndicator(int num, int den) {
        if ((num % den) == 0) {
            return 0;
        } else {
            return 1;
        }
    }
        

Starts

javaexamples/programmingpatterns/StartsAndCompletions.java (Fragment: starts)

    /**
     * Determine the number of started tasks from
     * an amount of work and a numeraire.
     *
     * @param work        The amount of work
     * @param workPerTask The work per task
     * @return The number of started tasks
     */
    public static int starts(int work, int workPerTask) {
        return completions(work, workPerTask) 
            + remainderIndicator(work, workPerTask);
    }
        

• The Charity Event:
  • A participant who runs 26 miles on a 3-mile track completed 26 / 3 or 8 laps, but started 9 (since 26 % 3 is non-zero)
• A Baseball Game:
  • A pitcher who worked 9 of 9 innings completed 9 / 9 or 1 game, and also started 1 game (since 9 % 9 is zero)