• FlapKap@feddit.dk
      link
      fedilink
      arrow-up
      4
      ·
      1 year ago

      According to the article the linear algebra algorithm has a running time of O(log n)

        • IAm_A_Complete_Idiot@sh.itjust.works
          link
          fedilink
          arrow-up
          6
          arrow-down
          1
          ·
          edit-2
          1 year ago

          According to the benchmark in the article it’s already way faster at n = 1000. I think you’re overestimating the cost of multiplication relative to just cutting down n logarithmically.

          log_2(1000) = roughly a growth factor of 10. 2000 would be 11, and 4000 would be 12. Logs are crazy.

          • cbarrick@lemmy.world
            link
            fedilink
            English
            arrow-up
            2
            ·
            1 year ago

            The article is comparing to the dynamic programming algorithm, which requires reading and writing to an array or hash table (the article uses a hash table, which is slower).

            The naive algorithm is way faster than the DP algorithm.

            • t_veor@sopuli.xyz
              link
              fedilink
              English
              arrow-up
              2
              ·
              1 year ago

              It’s not that hard to check yourself. Running the following code on my machine, I get that the linear algebra algorithm is already faster than the naive algorithm at around n = 100 or so. I’ve written a more optimised version of the naive algorithm, which is beaten somewhere between n = 200 and n = 500.

              Try running this Python code on your machine and see what you get:

              import timeit
              
              def fib_naive(n):
                  a = 0
                  b = 1
                  while 0 < n:
                      b = a + b
                      a = b - a
                      n = n - 1
                  return a
              
              def fib_naive_opt(n):
                  a, b = 0, 1
                  for _ in range(n):
                      a, b = b + a, b
                  return a
              
              def matmul(a, b):
                  return (
                      (a[0][0] * b[0][0] + a[0][1] * b[1][0], a[0][0] * b[0][1] + a[0][1] * b[1][1]),
                      (a[1][0] * b[0][0] + a[1][1] * b[1][0], a[1][0] * b[0][1] + a[1][1] * b[1][1]),
                  )
              
              def fib_linear_alg(n):
                  z = ((1, 1), (1, 0))
                  y = ((1, 0), (0, 1))
                  while n > 0:
                      if n % 2 == 1:
                          y = matmul(y, z)
                      z = matmul(z, z)
                      n //= 2
              
                  return y[0][0]
              
              def time(func, n):
                  times = timeit.Timer(lambda: func(n)).repeat(repeat=5, number=10000)
                  return min(times)
              
              for n in (50, 100, 200, 500, 1000):
                  print("========")
                  print(f"n = {n}")
                  print(f"fib_naive:\t{time(fib_naive, n):.3g}")
                  print(f"fib_naive_opt:\t{time(fib_naive_opt, n):.3g}")
                  print(f"fib_linear_alg:\t{time(fib_linear_alg, n):.3g}")
              

              Here’s what it prints on my machine:

              ========
              n = 50
              fib_naive:      0.0296
              fib_naive_opt:  0.0145
              fib_linear_alg: 0.0701
              ========
              n = 100
              fib_naive:      0.0652
              fib_naive_opt:  0.0263
              fib_linear_alg: 0.0609
              ========
              n = 200
              fib_naive:      0.135
              fib_naive_opt:  0.0507
              fib_linear_alg: 0.0734
              ========
              n = 500
              fib_naive:      0.384
              fib_naive_opt:  0.156
              fib_linear_alg: 0.112
              ========
              n = 1000
              fib_naive:      0.9
              fib_naive_opt:  0.347
              fib_linear_alg: 0.152