Pascal Triangle Calculation

The funcion bc_dp is a binomial coefficient (pascal triangle) function calculated using dynamic programming techniques (from: Steven S. Skiena - The Algorithm Design Manual-Springer (2020))

Basically, to calculate any number we only have to keep in memory two arrays (or a single 2-dimension array) because the value of a number at [n][k] is defined as [n-1][k] + [n-1][k-1].

Or, defining Pascal's triangle recursively:

pasc(n, k) = pasc(n-1, k) + pasc(n-1, k-1)

bc_better is my optimization on top of it. It optimizes in the following way:

• The first and last numbers in each "row" of Pascal's triangle is 1, so if k = 0 or k = n (the last number), the function returns 1 immediately.
• The second and second-to-last numbers are equal to the row's index (e. g. [4, 1] and [4, 3] should return 4), so if k = 1 or k = n - k, the function returns n immediately.
• Lastly, and more importantly, Pascal's triangle is symmetric. So we only have to calculate half of it. This is done in the auxiliary function aux_bc_better.

To compare both functions, I added a cpu time counter.

• Run the program
• call the function followed by n and k, e.g.

./a.out
function: bc_dp 250 100
result: 1103510280765129870
cpu time: 0.000654

• compare to:

./a.out 
function: bc_better 250 100
result: 1103510280765129870
cpu time: 0.000130