Thinking Like a Computer

1
Thinking Like a Computer
2
Beginnings

• The computer is dumber than you are.
• If you cant solve the problem, youll never get
  the computer to solve it.
• The first step in writing a computer program is
  developing a mechanical, step-by-step process
  that a human could use to solve the problem.

3
An Example Problem

• What is the most popular birthdate among the
  students in this class?
• As humans we think we know how to solve this
  problem.
• Can we describe all the details?
• Can we define a MECHANICAL process to solve the
  problem?

4
Step 0 Understand the Problem

• In this case, the problem is pretty
  straightforward.
• Birthdate defined as the month and day (not
  year) of a persons birth.
• Most popular each of the 366 possible birth
  dates occurs zero or more times as a birthdate
  for students in EE312. The most popular date is
  the date which occurs as a birthdate more times
  than any other date.
• If two or more dates are equally the most
  popular, then the answer to our problem can be
  any of these dates.

5
Step 1a Understand your Input

• The input to our problem is the set of
  birthdates.
• Initially, this information is stored in the
  brains of students sitting in the classroom (hard
  to access the information from this location).
• Lets assume weve transferred the input
  information onto a large sheet of paper.
• Our input consists of a large table, with one
  column, each entry is a birthdate (month/day).
• If we were writing a computer program to solve
  this problem, wed need to specify how the
  computer accesses the input. In this case, we
  can assume that our human processor can read the
  paper.

6
Input Keys

• Volume of information
• How many distinct types of information?
• How much data of each type?
• Format of information
• Are there delimiters? How do you know where one
  datum ends and the next begins?
• How do you know where the beginning and end are
  for tables and sets?
• What is the legal range of input?
• Procedures to access information
• What subroutines do we call or variables do we
  use?

7
Step 1b Output Information

• In this case, our output is pretty trivial
• The most popular date
• The number of students that have that birthdate.
• The same keys for input information apply to
  output information.

8
Step 1c Intermediate Information

• Its frequently hard to identify all the
  intermediate information well need until weve
  finished step two. Be we can make some guesses
  now and come back later.
• For now, Im going to guess that well need a
  second table.
• There will be 366 entries in the table (one for
  each possible birthdate).
• The information stored in each entry will be the
  number of students that have that birthdate.

9
Step 2 Select or Develop and an Algorithm

• Well spend the next several slides on this step.
• In some cases, someone else may have solved a
  very similar problem. Youll be able to use
  their solution (with a few small modifications.
• In other cases, youll need to develop a solution
  from scratch.
• The solution will need to be step-by-step and
  mechanical.
• The computer is incredibly stupid.
• You must reduce the algorithm to the computers
  level.

10
Algorithm 1

• Start at the beginning of the input table
• Cross off birthdates as we progress
• Each time we cross off a birthdate, increment the
  corresponding entry in the frequency table
• Once weve crossed off all the birthdates, then
  our frequency table tells us exactly how many
  students have each birthday.
• Ooops! We need to ensure that the entries in the
  frequency table are initially zero.
• OK, now all w need to do is find the maximum in
  the frequency table.

11
Algorithm 1 Cont.

• To find the maximum, well again start at the
  beginning and cross off entries.
• We need some additional intermediate
  information. Well have two scalars
• most_popular
• how_popular
• Each time we cross off an entry, well compare
  the value to how_popular. If the next entry is
  greater, well change how_popular to that value.
  Well also set most_popular to the date
  corresponding to the entry.

12
Lessons

• An algorithm is extremely specific, with every
  step described precisely.
• No And then a miracle occurs
• We will need to go back and forth among the
  steps.
• Frequently, we dont know all the intermediate
  information until were done with the algorithm.
• Sometimes, we wont recognize all the special
  cases for input or output until were halfway
  done with the algorithm.

13
More Lessons Decomposition

• Algorithm 1 is really two algorithms
• Histogram (build the frequency table)
• Find Max (select most popular birthdate).
• Decomposition can be either top down or bottom
  up.
• Top down means that you were certain that if
  you could solve each small problem, the big
  problem would be solved (but youre not sure yet
  you can solve these small problems).
• Bottom up means that you know you can solve at
  least this one small problem (and hope that turns
  out to be useful).

14
More Lessons Focus on Information

• The information is at the center of everything in
  programming
• Data types (int, float)
• Data structures (tables, sets, queues, maps,
  etc.)
• As you identify the information in your problem
  (input, output or intermediate).
• First determine quantity (e.g., scalar or table).
• Then determine type (e.g., integer or birthdate)
• Assign a variable name be sure to always have
  precise meaning for each variable!
• Determine an initial value for all variables.

15
Algorithm 2 Top Down Decomposition

• Small problem 1 Take input and sort it.
• I dont know how to do this, but maybe a miracle
  will occur and Ill figure it out later.
• Small problem 2 Count consecutive birthdates
  and remember which birthdate appeared the most
  times consecutively.
• This step should give me the correct answer,
  provided the input was sorted.

16
Small Problem 2

• Start at the beginning of (sorted) input. Cross
  off each date in sequence.
• Create four intermediate scalars
• most_popular same as before
• how_popular same as before
• current_birthdate the birthdate that we crossed
  off last time. Initialized to December 32nd.
• current_frequency the number of times weve
  crossed off the same birthdate in a row.

17
Small Problem 2 Cont.

• Each time we cross of the same birthdate,
  increment current_frequency.
• Each time we cross off a new birthdate
• first, compare current_frequency to how_popular.
• if current_frequency is higher, update
  how_popular and most_popular.
• then, set current_frequency to zero, set
  current_birthdate to the new birthdate.

18
Small Problem 1, Sorting

• Well have to wait until April for that miracle.
• Lessons
• Sometimes, top down decomposition can lead us to
  a dead end. We just dont know how to solve the
  corresponding small problems.
• In other cases (like this one), one of our small
  problems may be a classic, well-defined problem,
  and we can use a program that someone else has
  already written. (e.g., enter all the input into
  Excel, and use the sort function).

19
Comparing Algorithm 1 and 2

• Algorithm 2 used less memory (we replaced the
  entire frequency table with two scalars).
• Algorithm 1 is almost certainly going to be
  faster!
• Sorting, even though its easy to understand, is
  relatively slow.

20
Algorithm 3

• Left as an exercise for the student.
• Combine the two steps from Algorithm 1.
• We dont really need the whole frequency table,
  all we really want to know is whats the max
  frequency.
• We ought to be able to find a way to solve this
  problem without sorting and without using a whole
  table of intermediate information.
• Left as an exercise for the TA
• Based on all your experience, what is the best
  algorithm for this problem. Note that you may
  not know a priori how sparsely distributed the
  birthdates will be.