Searching, Sorting, and Timing

Agenda

1. Timing
2. Prelude: Timing list indexing
3. Linear search
4. Binary search
5. Insertion sort

1. Timing

import time
time.time()

# Return the time in seconds since the epoch as a floating point number.
# On Windows and most Unix systems, the epoch is January 1, 1970, 00:00:00 (UTC)

# read https://docs.python.org/3/library/time.html#time.time

1536780896.257849

start = time.time()

end = time.time()

end - start

2.9087066650390625e-05

start = time.time()

sum(range(10_000_000))

end = time.time()
end - start

0.26558709144592285

# why the measurement is different each time?

total = 0

for _ in range(10):      #take an average of 10 runs
    start = time.time()
    sum(range(10_000_000))
    end = time.time()
    total += end - start

total/10

0.20874271392822266

# make it as a more general purpose timing function 

def timeit(f, n):
    total = 0

    for _ in range(n):
        start = time.time()
        f()
        end = time.time()
        total += end - start

    return total/n

timeit(lambda: sum(range(10_000_000)), 10)

0.19438555240631103

# make it as an even more general purpose timing function 

def timeit(fn_to_test, setup_fn, n):
    total = 0
    setup_fn() # option 1 to place setup_fn()
    for _ in range(n):
        setup_fn() # option 2 to place setup_fn(); setup_fn is not timed
        start = time.time()
        fn_to_test() # timing measurement for this func
        end = time.time()
        total += end - start

    return total/n

lst = []

#setup function
def populate_list(): 
    lst.clear()
    for i in range(100):
        lst.append(i)

#test function
def test_contains():
    for i in range(100):
        if i in lst:
            pass

timeit(test_contains, populate_list, 100)

8.055448532104492e-05

# self-reading, existing timeit lib

# https://docs.python.org/3.6/library/timeit.html

import timeit

timeit.timeit(stmt='sum(range(10_000_000))', number=10)/10

0.20798762569902465

# magic cell function (with notebook)
# timing a notebook cell 

%timeit sum(range(10_000_000))

191 ms ± 11.5 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)

# self-reading, %time's ref document
?%time 

2. Prelude: Timing list indexing

import timeit

timeit.timeit(stmt='lst[0]',
              setup='import random; lst=[0] * 10**6')


# note that setup only run once
# return the first element of the 1-million-element list 

0.04877838000538759

timeit.timeit(stmt='lst[10**6-1]',
              setup='import random; lst=[0] * 10**6')

# return the last element of the 1-million-element list 

0.04921417200239375

# why the above two statements return similar timing measurement results?

# hint: a python list is essentially implemented like an array 
# elements are continuously placed in the memory

import random
size = 10**3
times = [0] * size
lst   = [0] * size

for _ in range(100):
    for i in range(size):
        times[i] += timeit.timeit(stmt='lst[{}]'.format(i), lst[i]
                                  globals=globals(),
                                  number=10)

times

[9.361098636873066e-05,
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%matplotlib inline
import matplotlib.pyplot as plt
plt.plot(times, 'ro')
plt.show()

Accessing an element in a list by index always takes the same amount of time, regardless of position. I.e., indexing incurs a constant time delay.

How? A Python list uses an array as its underlying data storage mechanism. To access an element in an array, the interpreter:

1. Computes an offset into the array by multiplying the element's index by the size of each array entry (which are uniformly sized, since they are merely references to the actual elements)
2. Adds the offset to the base address of the array

3. Linear Search

Task: to locate an element with a given value in a list (array).

def index(lst, x):
    for i in range(len(lst)):
        if lst[i] == x:
            return i
    return -1

lst = list(range(100))

index(lst, 10)

10

index(lst, 99)

99

index(lst, -1)

-1

#self-reading, error handling

def index(lst, x):
    for i in range(len(lst)):
        if lst[i] == x:
            return i
    raise ValueError(x)

#index(lst, 10)

#index(lst, -1)

#self-reading, another way to handle error

try:
    print('Value found at', index(lst, -1))
except ValueError as e:
    print('Value not found:', e)

# timing linear search

import timeit

times = []
lst = list(range(1000))

for x in lst:
    times.append(timeit.timeit(stmt='index(lst, {})'.format(x),
                               globals=globals(),
                               number=100))
    
    #executing index(lst, x), index(lst, 0), index(lst, 1), index(lst, 2) ...

import matplotlib.pyplot as plt
plt.plot(times, 'ro')
plt.show()

4. Binary search

Task: to locate an element with a given value in a list (array) whose contents are sorted in ascending order.

# Binary Search Animation

# http://www.cs.armstrong.edu/liang/animation/web/BinarySearch.html

def index(lst, x):
    # assume that lst is sorted!!!
    
    return None

def index(lst, x):
    # assume that lst is sorted!!!
    
    lo, mid, hi = 0, len(lst)//2, len(lst)
    
    while lo < hi:
        if x == lst[mid]:
            return mid
        elif x > lst[mid]:
            lo = mid + 1
        else: # x < lst[mid]
            hi = mid
        mid = (lo + hi) // 2
    else:
        return -1
    

lst = list(range(1000))
index(lst, 10)

10

index(lst, 999)

999

index(lst, -1)

-1

#for i in range(len(lst)):
#    assert(i == index(lst, i))

import timeit

times = []
lst = list(range(1000))

for x in lst:
    times.append(timeit.timeit(stmt='index(lst, {})'.format(x),
                               globals=globals(),
                               number=1000))

import matplotlib.pyplot as plt
plt.plot(times, 'ro')
plt.show()

# self-reading example

import timeit
import random

times = []

for size in range(100, 10000, 100):
    lst = list(range(size))
    times.append(timeit.timeit(stmt='index(lst, -1)'.format(random.randrange(size)),
                               globals=globals(),
                               number=50000))

import matplotlib.pyplot as plt
plt.plot(times, 'ro')
plt.show()

# search for a value does not exist; worst case scenario

import timeit
import random

times = []

for e in range(5, 20):
    lst = list(range(2**e))
    times.append(timeit.timeit(stmt='index(lst, -1)',
                               globals=globals(),
                               number=200000))

#for i in range(5, 20):
 #   print(2**i)

import matplotlib.pyplot as plt
plt.plot(times, 'ro')
plt.show()

5. Insertion sort

Task: to sort the values in a given list (array) in ascending order.

import random

lst = list(range(1000))

random.shuffle(lst)

plt.plot(lst, 'ro')
plt.show()

# Insertion sort animation

# http://cs.armstrong.edu/liang/animation/web/InsertionSort.html

def insertion_sort(lst):
    pass

def insertion_sort(lst):
    for i in range(1, len(lst)):
        for j in range(i, 0, -1):
            if lst[j] < lst[j-1]: #do swap
                lst[j-1], lst[j] = lst[j], lst[j-1] #pythonic way to swap two values
            else:
                break

insertion_sort(lst)

plt.plot(lst, 'ro')
plt.show()

import timeit
import random

times = []

for size in range(100, 5000, 100):
    lst = list(range(size))
    times.append(timeit.timeit(stmt='insertion_sort(lst)',
                               setup='random.shuffle(lst)',
                               globals=globals(),
                               number=1))

plt.plot(times, 'ro')
plt.show()