Intro to Algorithms, 2019-20
Tutorial 1 – Notes

The integers

We will begin our study of algorithms by looking at algorithms that work with integers. We distinguish

Z = { ..., -2, -1, 0, 1, 2, ... } : the set of all integers

N = { 0, 1, 2, ... } : the set of natural numbers

{ 1, 2, ... } : the positive integers

The mathematical study of the integers is called number theory.

Integer division

A basic result of elementary number theory is the division theorem:

For any integer a and any integer b > 0, there exist unique integers q and r, with 0 ≤ r < b, such that

a = q · b + r

Given a and b, the functions div (integer division) and mod (integer remainder) compute the values q and r. We write

• q = a div b (the quotient)

• r = a mod b (the remainder)

So for all integers a and b with b > 0,

a = (a div b) · b + (a mod b)

For example:

• If a = 17 and b = 3, then q = a div b = 5 and r = a mod b = 2, because 17 = 5 · 3 + 2.

• If a = -17 and b = 3, then q = a div b = -6 and r = a mod b = 1, because -17 = -6 · 3 + 1.

In Python, the // operator performs integer division, and the % operator computes the remainder. So

• 17 // 3 == 5

• 17 % 3 == 2

• -17 // 3 == -6

• -17 % 3 == 1

Note that if a is evenly divisible by b, then the remainder after dividing is 0. So we can test whether b divides a by checking the remainder. In Python, for example:

n = int(input('Enter n: '))
if n % 7 == 0:
  print(n, 'is divisible by 7')
else:
  print(n, 'is not divisible by 7')

Hours, minutes, and seconds

There are 60 · 60 · 24 = 86,400 seconds in a day. Suppose that we are given an integer 0 ≤ t < 86,400 representing a number of seconds since midnight. We'd like to compute the number of hours, minutes, and seconds since midnight, yielding a time such as 3:24:54.

In other words, we want to find integers 0 ≤ h < 24, 0 ≤ m < 60 and 0 ≤ s < 60 such that

t = 3600 h + 60 m + s

How can we do this? One approach is to first compute the number of hours: h = t // 3600. Then t % 3600 is the number of seconds that remain after subtracting the h hours. This yields a Python implementation like this:

t = int(input('How many seconds? '))

hours = t // 3600
t = t % 3600
mins = t // 60
secs = t % 60

print(hours, ':', mins, ':', secs)

As another possible approach, we can first compute the number of seconds: s = t % 60. Then t // 60 is the number of minutes that remain after subtracting the s seconds. In Python:

t = int(input('How many seconds? '))

secs = t % 60
t = t // 60  # now t is the total number of minutes
mins = t % 60
hours = t // 60

print(hours, ':', mins, ':', secs)

Decimal representation

By convention, we use decimal representation (base 10) when writing numbers in everyday life. When we write a number such as 1985, we actually mean a sum of powers of 10:

1985 = 1 · 10³ + 9 · 10² + 8 · 10 + 5

We will now discuss simple algorithms that can split an integer (e.g. 1985) into a series of decimal digits (1, 9, 8, 5), and can combine a series of decimal digits into an integer.

Splitting into digits

To split into digits, we first notice that for any integer i,

• i % 10 is the last digit of i

• i // 10 is the number formed by all digits of i except the last

We can see this mathematically by factoring the expansion of 1985 above:

1985 = 1 · 10³ + 9 · 10² + 8 · 10 + 5 = 10 (1 · 10² + 9 · 10 + 8) + 5

From this factoring it is evident that 1985 % 10 = 5 and 1985 // 10 = (1 · 10² + 9 · 10 + 8) = 198.

So we can write a simple loop that extracts digits from an integer in succession. In Python:

n = int(input('Enter n: '))

while n > 0:
  d = n % 10
  n = n // 10
  print('digit: ', d)

By the way, note the similarity of this problem to the previous exercise involving times. The time 3:24:54 is 12,294 seconds past midnight, since

12,294 = 3 · 60² + 24 · 60 + 54

So a time such as 3:24:54 is actually like a number written in base 60. (If this seems unclear, don't worry – we will soon discuss bases other than base 10.)

Combining digits

To combine a series of digits into a number, we first notice that we can append a decimal digit d to an integer n using this formula:

n' = 10 n + d

For example, if n = 198 and d = 5, then n' = 10 · 198 + 5 = 1985.

So we can write a simple loop that reads a series of digits and combines them into an integer. In Python:

n = 0
while True:
  d = int(input('Digit: '))
  if d < 0:
    break
  n = 10 * n + d
print('n is', n)

The program behaves like this:

$ py joinDigits.py
Digit: 1
Digit: 9
Digit: 8
Digit: 5
Digit: -1
n is 1985

$

As some students pointed out, we can rewrite the preceding loop without break (though at the cost of performing one unnecessary multiplication operation):

n = 0
d = 0
while d >= 0:
  n = 10 * n + d
  d = int(input('Digit: '))
print('n is', n)