Calculating M(n)
and Optimizing M(n) Algorithms
Dave Purdum
Dr. Jonathan Webster
Definition
Define M(n) to be the number of distinct integers in the n by n multiplication table.
A027424
The number of different products calculated to create an n by n multiplication table.
M(4)
1 2 3 4 5 6 7 8 9
Multiplication table
Set of possible numbers in the multiplication table
= 9
There are 9 different integers in the 4 by 4 multiplication table.
Naive Algorithm
Directly calculate M(n) for a given value of n.
Time Complexity
Space Complexity
O(n^2)
O(n^2)
number of products that are calculated
number of possible values that can be crossed off
Optimizations
Optimizations
1
1 2 3
1 2 3 4 5 6
1 2 3 4 5 6 7 8 9
M(1) = 1
M(2) = 3
M(3) = 6
M(4) = 9
Tabulation
Repeated Evaluation
Calculating M(1), M(2), M(3), M(4)
Calculate 10 products
Calculate 20 products
Optimizations
Time Complexity
Space Complexity
O(n^2)
O(n^2)
Evaluation
O(n^3)
O(n^2)
Repeated Evaluation
O(n^2)
O(n^2)
Tabulation
\displaystyle\sum_{i=1}^n i^2 = 1 + 4 + 9 + \dots + n^2= \frac{n(n-1)(2n-1)}{6}
= O(n^3)
Calculations
M(n)
M(1)...M(n)
M(1)...M(n)
Memory Issues
n
n^2
Space
4
16
16 Bytes
100
10 000
9 KB
1 000 000
1 000 000 000 000
931 GB
10 000
100 000 000
1 MB
100 000
10 000 000 000
9 GB
Time
<< .001sec
30 min
.002 sec
20 sec
<< .001sec
Memory
Bottleneck
M(4) = 9
M(5) = ?
M(11)
M(12)
+ number of new integers =
M(n)=\hspace{1.5cm}+\; M(n-1)
n-\delta(n)
n
Number of new distinct integers
Total integers in nth column
M(12)= \hspace{2.5cm}+\; M(11)
\delta(n)
Number of integers in M(n-1) table
number of new integers in 12th row
n-\delta(n)
\delta(n)
n
n-\delta(n)
number of integers in previous table
number of new integers in column
total number of integers in column
M(12)= \hspace{2.5cm}+\; M(11)
number of new integers in 12th row
M(n)=n - \delta(n) + M(n-1)
*
It just so happens that this pattern appears for n=12. This is generally more scattered.
*
Have I seen before?
mn
\{n, 2n, 3n, \dots, n^2\} = \{mn: m \in [1, n]\}
Is in the multiplication table?
mn
(n-1)\times(n-1)
Does for ?
mn = ab
a, b < n
Let .
m = ij, n = gh
If is in the smaller table, then
jg < n
mn
ih < n
ih < gh
jg < gh
i < g
j < h
The number of distinct values in this shape is
the number of integers in the smaller table!
\delta(12)
1 2 3 4 5 6
= 6
Bounded by n!
M(12) = 12-\delta(12) + M(11)
M(n)=n-\delta(n) + M(n-1)
= 12-6 + 53
= 59
Differences of M(n) Algorithm
Calculate the cumulative differences between M(n) and M(n-1) by calculating the delta function.
Time Complexity
Space Complexity
O(n \log n)
O(n)
Evaluation
O(n^2\log n)
O(n)
Tabulation
Calculating M(n)
and Optimizing M(n) Algorithms
Dave Purdum
Dr. Jonathan Webster
\delta(n)
\delta(n)
\delta(42)
= 20 + 5 =25
1 2 3 4 5
Calculate 41 products
Calculate 6 products
No Free Rows
First Row Free
First Row Free Algorithm
Calculate products in the lattice greater than the largest value in the first row.
Time Complexity
Space Complexity
O(n \log n)
O(n)
Evaluation
O(n^2\log n)
O(n)
Tabulation
\delta(105)
Calculate 121 products
Calculate 55 products
No Free Rows
First Row Free
Calculate 15 products
Second Row Free
Getting Rows Free
Working in Modulo Classes
First Row Free
Second Row Free
"Third Row Free"
Modulo 1
Modulo 2
Modulo 6
Evens and Odds
0 mod 2 1 mod 2
Modulo n
Exploiting patterns in the first n rows
\vdots
Results
was calculated and published by Richard Brent and Hsiang Kung in 1982.
In 2012 Brent and Pomerance found the next 8 powers of 2, .
M(n) : n\in \{2^p: p \leq 17\}
p \leq 25
We have found all values
M(n) : n\leq 2^{30}
MA490 Presentation
By rutrum
