Fixed-Point Iteration and Dynamic Programming

By Prof. Seungchul Lee
http://iailab.kaist.ac.kr/
Industrial AI Lab at KAIST

Source

• By Prof. David J. Malan from Harvard University
• By Prof. Erik Demaine from MIT online lecture

Table of Contents

1. Recursive Algorithm

• one of the central ideas of computer science

• depends on solutions to smaller instances of the same problem ( = subproblem)

• function to call itself (it is impossible in the real world)

• factorial example

  • $n! = n\cdot(n-1) \cdots 2\cdot 1$

• watch Harvard CS50 lecture on recursion (https://cs50.harvard.edu/)

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1.1. foo example

print ('{c}\n'.format(c="Hello class"))

Hello class

def foo(str):
    print(str)    

foo('Hello class')

Hello class

def foo_recursion(str):
    print('{s}\n'.format(s=str))
    foo_recursion(str)                    

# Do not run. It falls into infinite loop.

#foo_recursion('Hello Class')

# base line

def foo_recursion_correct(n):
    if n <= 0:
        return
    else:
        print('Hi! \n')
        foo_recursion_correct(n-1)

foo_recursion_correct(4)

Hi! 

Hi! 

Hi! 

Hi! 

1.2. Factorial example

• factorial example
  • $n! = n\cdot(n-1) \cdots 2\cdot 1$

n = 5

m = 1
for i in range(n):
    m = m*(i+1)
    
print (m)

120

def fac(n):
    if n == 1:
        return 1
    else:
        return n*fac(n-1)    

# recursive
fac(5)

120

2. Dynamic Programming

• Dynamic Programming: general, powerful algorithm design technique

• Fibonacci numbers:

$$ \begin{align*} F_1 &= F_2 = 1 \\ F_n &= F_{n-1} + F_{n-2} \end{align*} $$

2.1. Naive Recursive algorithm

$$ \begin{align*} \text{fib}(n):&\\ &\text{if } n \leq 2:\; f = 1\\ &\text{else}:\; f = \text{fib}(n-1) + \text{fib}(n-2)\\ &\text{return } f \end{align*}$$

it works, Is it good?

2.2. Memorized DP algorithm

$$ \begin{align*} \text{memo }= [\;]&\\ \text{fib}(n):&\\ & \text{if $n$ in memo : return memo}[n]\\ \\ & \text{if } n\leq 2 :\; f = 1\\ & \text{else}: \;f = \text{fib}(n-1) + \text{fib}(n-2)\\ \\ &\text{memo}[n] = f\\ &\text{return } f \end{align*}$$

Benefit?

• fib$(n)$ only recurses the first time it's called

• memorize (remember) & re-use solutions to subproblems that helps solve the problem.

$\quad \implies$ DP $\simeq$ recursion + memorization

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2.3. Examples

2.3.1. Fibonacci

# naive Fibonacci

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

fib(10)

55

# Memorized DP Fibonacci

def mfib(n):
    global memo
    
    if memo[n-1] != 0:
        return memo[n-1]
        
    elif n <= 2:
        return 1
    else:
        memo[n-1] = mfib(n-1) + mfib(n-2)
        return memo[n-1]

import numpy as np

n = 10
memo = np.zeros(n)
mfib(n)

55.0

memo

array([ 0.,  0.,  2.,  3.,  5.,  8., 13., 21., 34., 55.])

n = 30
%timeit fib(30)

179 ms ± 1.22 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)

memo = np.zeros(n)
%timeit mfib(30)

400 ns ± 7.86 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)

2.3.2. Climbing a stair

You are climbing a stair case. Each time you can either make 1 step, 2 steps, or 3 steps. How many distinct ways can you climb if the stairs has $n = 30$ steps?

import numpy as np

def stair(n):
    global memo
    
    if memo[n] != 0:
        m = memo[n]
    elif n == 1:
        m = 1
    elif n == 2:
        m = 2
    elif n == 3:
        m = 4
    else:
        m = stair(n-1) + stair(n-2) + stair(n-3)
        
    memo[n] = m
    return m

n = 5
global memo
memo = np.zeros(n+1)

stair(n)
print(memo)

[ 0.  1.  2.  4.  7. 13.]

2.3.3. Knapsack problem using Dynamic Programming (DP)

From MIT's Introduction to Computer Science and Programming

Knapsack problem

• burglar (or thief) can carry at most 20 kg (i.e., maximum capacity = 20)

• Quickly decide which item to carry

items	1	2	3	4	5	6
weight	10	9	4	2	1	20
value	175	90	20	50	10	200

Approach

1. guess (or trial and error)

2. Exhaustive search if possible

3. "smarter way" $\implies$ Recursive or dynamic programming

$$ \text{key ideas} = \text{original problem} \rightarrow \begin{cases} \text{subproblem} \rightarrow & \begin{cases} \text{subproblem} \rightarrow \\ \text{subproblem} \rightarrow \end{cases}\\ \text{subproblem} \rightarrow & \begin{cases} \text{subproblem} \rightarrow\\ \text{subproblem} \rightarrow\end{cases}\end{cases} $$

Suppose we have the following function:

$\quad$[value, taken] = chooseBest(items(1:6),maxWeight)

1. item 1 is not taken

$\quad$[v_1,t_1] = chooseBest(items(2:6),maxWeight)

2. item 1 is taken

$\quad$[v_2,t_2] = chooseBest(items(2:6),maxWeight - weights(1))

$\quad\qquad\;\;\,$v_2 = v_2 + values(1)

$\qquad\quad\;\;\,$t_2 = [items(1),t_2]

def chooseBest(items,maxWeight):
    global weights
    global values
    
    if len(items) == 0 or maxWeight <= 0:
        value = 0
        taken = []
        return value, taken
    else:
        first = items[0]
        w = weights[first]
        rest = items[1:]
        
        # do not take the first item
        v1,t1 = chooseBest(rest,maxWeight)
        
        # do take the first item
        v2,t2 = chooseBest(rest,maxWeight - w)
        v2 = v2 + values[first]
        t2 = [first] + t2
        
        if maxWeight - w >= 0 and v2 >= v1:
            value = v2
            taken = t2
        else:
            value = v1
            taken = t1
        
        return value, taken              

n = 6
weights = [10, 9, 4, 2, 1, 20]
values = [175, 90, 20, 50, 10, 200]

items = range(n)
maxWeight = 20

chooseBest(items,maxWeight)

(275, [0, 1, 4])

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