#!/usr/bin/env python # coding: utf-8 #

cs1001.py , Tel Aviv University, Spring 2019

# # # Recitation 3 # We talked about binary numbers (with some theory) and base conversions and implemented an algorithm for base conversion (from decimal to 2<=b<=36). # We examined Python's memory model and analyzed the efficiency of constructing a list using + or += operators. # # # # ### Takeaways: # 1. Make sure you understand binary numbers and base conversions (including the algorithms for conversion to and from a base b to decimal). It is a very useful tool in computer science. # 2. Elements of a list can be changed from inside a function, if this list is given as a parameter. Note that one should use dedicated list functions or the [] operator for mutating the list. # 3. Use [Python tutor](http://www.pythontutor.com) in order to understand what's going on in terms of memory. It can be very helpful. # 4. Try to analyze the number of operations your function does to see how will its runtime scale as a function of the input (we will elabore on this soon). # # ### Python tutor guidelines: # Before you click "Visualize Execution" button, you may want to use the following settings (can be adjusted via the drop boxes next to the textbox): #
    #
  1. Python 3.6
    2. #
  2. # Show exited frames (Python)
    3. #
  3. Render all objects on the heap
    4. #
  4. Draw pointers as arrows [default]
    5. #
# # # ## The binary system and base conversions # We use decimal numbers, that is, numbers in base 10. These numbers are represented using 10 digits (0-9). # Binary numbers (in base 2) use two possible digits: 0-1. # In general, a number in base $b$ will be represented using $b$ possible digits. # # # #### Code for printing several outputs in one cell (not part of the recitation): # In[2]: from IPython.core.interactiveshell import InteractiveShell InteractiveShell.ast_node_interactivity = "all" # ## Base conversions # Using Python functions: # In[3]: bin(10) type(bin(10)) hex(161) int("345") int("1101", 2) int("a1", 16) int("a1", 2) # ### Converting from binary to decimal # # Let $x_{base 2} = a_{n-1} ... a_1 a_0$ be a binary number in base 2 with $n$ digits. # The following formula returns its decimal representation: # $$x_{base 10} = \sum_{0 \le k \le n-1} a_k \cdot 2^k$$ # ### Converting from decimal to binary # Converting from decimal to binary is done by integer division and modulo operations. # Here is an implementation of the algorithm for conversion from decimal to a general base $b$. # #### Our version for convert_base: # In[6]: def convert_base(n,b): '''convert_base(int, int)->string Returns the textual representation of n(decimal) in base 2<=b<=10 ''' result = "" while n != 0: digit = n % b n //= b result = str(digit) + result return result # In[7]: convert_base(10,2) # In[8]: convert_base(0, 2) # convert_base above mishandles n = 0 # #### Improved version of convert_base: # In[26]: def convert_base(n,b): '''convert_base(int, int)->string Returns the textual representation of n(decimal) in base 2<=b<=10 ''' if n == 0: return "0" result = "" while n != 0: digit = n % b n //= b result = str(digit) + result return result # In[10]: convert_base(0,2) convert_base(12,3) # In[11]: convert_base(161,16) # The improved version above fails for 10string Returns the textual representation of n(decimal) in base 2<=b<=10 ''' assert 2 <= b <= 36 if n == 0: return "0" result = "" alphabet = "0123456789abcdefghijklmnopqrstuvwxyz" while n != 0: digit = n % b n //= b result = alphabet[digit] + result return result # In[13]: convert_base(161, 16) convert_base(10, 2) convert_base(10, 80) # ## Memory model # By understanding Python's memory model we #
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  1. can distinguish between objects that can be mutated and those that cannot. We can also understand how to mutate objects.
    2. #
  2. can write a more efficient code.
    3. #
# # # # #### Motivation examples: # In[14]: def change_num(num): num = 999 x = 30 change_num(x) x # In[15]: def add_to_list(lst): lst.append(999) mylst = [1,2,3] add_to_list(mylst) mylst # In[16]: def change_lst(lst): lst = [] mylst = [1,2,3] change_lst(mylst) mylst # #### Programs examined with [Python tutor](http:\\www.pythontutor.com): # In[ ]: #1 basics ##################################################### a = 1000 #1 b = "hello" #2 a += len(b) #3 c = 2*b[:2] #4 if b!=c: #5 c = b #6 del c #7 # In[ ]: #2 lists ##################################################### a = 1000 #1 d = [a,2] #2 d[1] = -1 #3 a = 1003 #4 for x in d: #5a x = 7 #5b # In[ ]: #3 funcs ##################################################### a = 1000 #1 b = "hello" #2 def is_palindrom(a): #3a b = a[::-1] #3b return a==b #3c is_pal = is_palindrom #4 x = is_pal(b) #5 # In[ ]: #4 lists+funcs ##################################################### a = 1000 #1 d = [a,2] #2 def f(a,d): #3a a = 2000 #3b d[0] = a #3c d = [] #3d return d #3e x = f(a,d) #4 # Extend(), append(), sort(), and sorted(): # In[28]: #5 – lists operators and methods ##################################################### lst = [3,2,1] id(lst) lst lst = lst+[0] id(lst) lst lst.append(4) id(lst) lst lst.insert(1,5) id(lst) lst lst.extend([6,7]) id(lst) lst lst += [8] id(lst) lst # In[29]: lst id(lst) #sorted returns a sorted copy of the list, and does not change the original one lst2 = sorted(lst) id(lst) lst #list.sort() sorts in place, and does not return anything lst.sort() id(lst) lst # ## comparison between + and += # We use + or += operators in order to construct the same list. # # Using operator +: less efficient, since a new list is constructed every iteration # the number of integers written to memory equals 1+2+ ... + n = (n+1)*n/2 # (= asumptotically speaking, grows quadratically with n) # # Using operator +=: more efficient, since the same list is extended again and again # the number of integers written to memory equals n # (= asumptotically speaking, grows linearly with n) # ### memory and running times # In[5]: import time # In[12]: n= 20000 lst = [] t0 = time.perf_counter() for i in range(n): lst = lst + [i] t1 = time.perf_counter() print("Extending a list n=", n, "times, using the + operator took",t1-t0, "seconds") lst = [] t0 = time.perf_counter() for i in range(n): lst += [i] t1 = time.perf_counter() print("Extending a list n=", n, "times, using the += operator (extend()) took",t1-t0, "seconds") # We double the size of the input. # In[13]: n= 40000 lst = [] t0 = time.perf_counter() for i in range(n): lst = lst + [i] t1 = time.perf_counter() print("Extending a list n=", n, "times, using the + operator took",t1-t0, "seconds") lst = [] t0 = time.perf_counter() for i in range(n): lst += [i] t1 = time.perf_counter() print("Extending a list n=", n, "times, using the += operator (extend()) took",t1-t0, "seconds") # ______________________________________________________________ # # In accordance with the theory, the running time of the inefficient code increases by approximately 2**2 = 4 as n increases by 2, while the running time of the efficient code increases by approximately 2. # Apparently n is large enough for the constants not to play a serious role, so that the measurements agree with the asymptotic analysis.