The Collatz Problem

1
The Collatz Problem

• Matthew Curry
• CS497
• January 22nd, 2007

2
The Statement

• The input will consist of a series of pairs of
  integers i and j, one pair of integers per line.
  All integers will be less than 1,000,000 and
  greater than 0.
• You should process all pairs of integers and for
  each pair determine the maximum cycle length over
  all integers between and including i and j.
• You can assume that no operation overflows a
  32-bit integer.

3
The Statement (contd)

• For each pair of input integers i and j you
  should output i, j, and the maximum cycle length
  for integers between and including i and j.
• These three numbers should be separated by at
  least one space with all three numbers on one
  line and with one line of output for each line of
  input.
• The integers i and j must appear in the output in
  the same order in which they appeared in the
  input and should be followed by the maximum cycle
  length (on the same line). Emphasis added

4
Tricks

• over all integers between and including i and
  j
• There were no statements regarding any
  relationship between i and j (i gt j i lt j)
• no operation overflows a 32-bit integer
• What kind of 32-bit integer?
• Testing yields a maximum value of 56,991,483,520

5
Correct Code

• import java.util.Scannerclass Main static
  int collatz(long num) if (num 1) return
  1 if (num 2 1) / odd / return
  collatz(3num1) 1 else return
  collatz(num/2) 1

6
Correct Code (2)

• public static void main(String
  args) Scanner sc new Scanner(System.in)
  int first sc.nextInt() int second
  sc.nextInt() System.out.println(first
  second ) if (first gt second)
  int tmp first first
  second second tmp

7
Correct Code (3)

• int max 0 for (int I first i lt
  second i) int tmp collatz(i) if
  (tmp gt max) max tmp System.out.printl
  n(max)

8
But theres a problem

• Timing data taken on a dual Opteron machine
• curryml_at_eddie cat testinput1
  999999curryml_at_eddie time java Main lt
  testinput1 999999 525real 0m6.244suser 0m6.2
  12ssys 0m0.020s

9
Optimization Eliminating Repeating Work
10
Optimization Eliminating Repeating Work
11
Optimization Eliminating Repeating Work
12
Bigger Picture 1 to 999,999

• In original code, the collatz method is called
  132,434,271 times.
• Using the first optimization, collatz is called
  5,226,260 times (96 reduction).
• Using the last optimization, collatz is called
  2,355,035 times (98 reduction).

13
Changing the Implementation

• Add an answers integer array. This will keep,
  for any given number between 1 and 999,999, the
  answer that collatz has returned for that
  query.
• If collatz hasnt been called, zero is stored in
  this array.
• Declare the storage for this array in the main
  method before computation starts

14
Changing the Implementation (contd)

• static int collatz(long num) if (num
  1) return 1 if (num lt 1000000
  answer(int)num gt 0) return answers(int)num
  int retval if (num 2 1) / odd
  / retval collatz(3num1)
  1 else retval collatz(num/2) 1 if (num
  lt 1000000) answers(int)num retval return
  retval

15
More Timing Data

• Timing data taken on a dual Opteron machine
• curryml_at_eddie cat testinput1
  999999curryml_at_eddie time java Main lt
  testinput1 999999 525real 0m0.312s (0m6.244s)
  user 0m0.304s (0m6.212s)sys 0m0.032s
  (0m0.020s)