algorithms: prologue

Aryabhata Al Khwarizmi Fibonacci estimating runtime big O

Aryabhata

Aryabhata long hand square root

Aryabhata example (decimal)
       ---------
    \/ 8 0 1 2 3  =  ?  (say to 2 decimal places)


        2   8   3 . 0   6     <- x
       -------------------
    \/ 0 8 0 1 2 3.0 0 0 0    <- n
2        4
       ---
         4 0 1
48       3 8 4
         -----
           1 7 2 3
563        1 6 8 9
           -------
               3 4 0 0
5660
               -------
               3 4 0 0 0 0
56606          3 3 9 6 3 6
               -----------
                     3 6 4   <- r

... so n = x*x + 0.0364

Aryabhata example (binary)
        ------------------
      \/ 1 1 0 1 0 1 1 0 1   =  ?


        1   0   1   0   0      <- x
      --------------------
    \/ 0 1 1 0 1 0 1 1 0 1     <- n
1        1
       ---
         0 1 0
100
         -----
           1 0 1 0
1001       1 0 0 1
           -------
                 1 1 1
10100
                 -----
                 1 1 1 0 1
101000
                 ---------
                 1 1 1 0 1    <- r

... so n = x*x + r

Al Khwarizmi

Fibonacci

import time

def fib(n):
  if (n<=1):
    return n
  return fib(n-1) + fib(n-2)

t1 = 1.0
for n in range(100):
  now, fn = time.time(), fib(n)
  t0, t1  = t1, time.time()-now
  print n, fn, t1, "sec, ratio current/prev", t1/t0

estimating runtime

test for runtime in :
- say
- then $t(n)/t(n-1) approx (k c^n)/(k c^{n-1}) = c$
test for runtime in :
- say
- then
- e.g. if then
test for runtime in :
- say
- then
- e.g. if then

estimating fib(n) runtime

method 1: experiment
- execute program, looks like run time in where
method 2: use theory to estimate
- count basic operations as function of n
- e.g. count only procedure calls
  - C(n): number of procedure calls to execute fib(n)
  - fib(0): only call is fib(0), so C(0) is 1
  - fib(1): only call is fib(1), so C(1) is 1
  - fib(2): also calls fib(1), fib(0), so C(2) is 3
  - for n2, C(n) 1 C(n-1) C(n-2)

exercise

prove by induction: T(n) > f(n)
prove T(n) in $Omega (1.618^{n})$
prove by induction: C(n) > f(n)
prove C(n) in $Omega (1.618^{n})$
hints

so what?

pick one: recursive fibonacci computation is
- easy to prove correctness ?
- infeasible for ?
- easy to implement quickly ?
- all of the above ?

for runtime: can we do better?

fib(5) recursion tree ?
how to do better ?
- write/read instead of recompute (memoization)

def ifib(n):
  a,b = 0,1
  for _ in range(n):
    a, b = b, a+b    #
  return a

x = 40000
for _ in range(6):
  now = time.time()
  fx = ifib(x)
  print x, time.time() - now, "sec"
  x *= 2

exercise

I(n): number of #-line executions in ifib(n)
I(n) in (why?)
run the above program: roughly, what is time( ifib(2x) ) / time ( ifib(x) ) ?
what does this suggest about runtime of ifib(n) ?
on average, how long does one #-line execution take?
- O(1) time ?
- O(lg n) time ?
- O(n ) time ?
- all of the above ?
so runtime of ifib(n) is
- in O(n) ?
- in O(n lg n) ?
- in O(n^2) ?
- all of the above ?

big O

definition of big O, big Omega, big Theta

wiki
f(n) ∈ O( g(n) ) means
- f(n), g(n) are functions from positive integers to positive reals
- for some positive real constant c, for some positive integer n₀, for all positive integers n ≥ n₀, f(n) ≤ c g(n)
synonyms
- f ∈ O( g )
- f(n) = O( g(n) )
- f = O( g )
f ∈ Ω( g ) means g ∈ O ( f )
f ∈ Θ( g ) means f ∈ O ( g ) and f ∈ Ω( g )

little o notation

wiki
f(n) ∈ o( g(n) ) means
- limit, as n gets large, of f(n)/g(n) = 0
- so there is no positive constant c such that f(n) > c g(n) for all n > 0,
- so f ∉ Ω( g )
synonyms
- f ∈ o( g )
- f = o( g )
- f(n) = o( g(n) )
L'Hopital's rule

useful rules

c positive constant, f(n) = c g(n) then Θ( f ) = Θ( g )
f(n) ∈ O( g(n) ) then Θ( f(n) + g(n) ) = Θ( g )
if f ∈ o (g) then
- f ∈ O( g )
- f ∉ Ω( g )
constants 0 < a < b then n^a ∈ o( n^b )
constants 0 < a, 1 < b then n^a ∈ o( bⁿ )
constants 0 < a,b then (log n)^a ∈ o( n^b )
L'Hopital's rule

logarithm

exercise

for integer , let
for positive real , let
show
hint: use L'Hopital's rule

bounding sum by integral

wiki
for increasing function ,

$int_{x=a-1}^{b} f(x) dx : : : leq : : : sum_{j=a}^b f(j) : : : leq int_{x=a}^{b+1} f(x) dx$