Unit 3 Sections 17-18
Algorithmic Efficiency and Undecidable Problems
• 48856 min read
- Do Now!!!
Do Now!!!
- Set up your notebook by either wgetting the lesson or tracking it by your own (We would recommend wgetting since there are some fill in the blanks!)
- wget here: https://raw.githubusercontent.com/mmaxwu/Tri2-GroupFastpages/master/_notebooks/2022-12-dd-lesson.ipynb
3.17: Algorithm Efficiency
Purpose:
The purpose of this lesson is to help students understand how to make an efficient program and optimize it and understand its importance to the CSP curriculum.
What is Algorithmic Efficiency?
- The ability of an algorithm to solve a problem in an efficient way
- An efficient algorithm solves a problem quickly and with a minimum amount of resources, such as time and memory.
- How do we determine if an algorithm is efficient or not?
- One way we can do this is by determining the time complexity of the algorithm.
- Another way is through space complexity.
Traveling Merchant Problem Hacks:
What did you and your team discuss? (record below)
Going in a circle might help go to all the cities, and also end up back to Indieanapolis
- An heuristic solution is an approach to a problem that produces a solution that isn't necessarily optimal but can be used when normal methods take forever
Describe the method used to solve the traveling merchant problem. (record below)
Go to the closest neighbor unless you've already been there.
3.18: Undecidable Problems
Purpose:
The purpose of this lesson is to introduce students to the concept of undecidable problems in computer science and to explain why these problems are important.
Key vocabulary:
- Decision problem
- Decidable problem
- Undecidable problem
Decision Problem
A decision problem is a problem in computer science and mathematics that can be solved by a yes-no answer, also known as a binary answer. In other words, a decision problem is a problem for which there are only two possible outputs:"yes" or "no". There are two types of decision problems that Collegeboard goes over:
- Decidable Problems
- Undecidable Problems
A decidable is a problem in computer science and mathematics for which an algorithm can be created that can always produce a correct answer or solution. In other words, a decidable problem is a problem for which there exists an algorithm that can be used to determine whether a given input is a valid solution or not.
An undecidable problem is a problem in computer science and mathematics for which it is impossible to create an algorithm that can always provide a correct answer or solution. This means that it is not possible for an algorithm to always determine whether a given input is a valid solution to an undecidable problem.
def divideThirteen(number):
if number % 13 == 0:
return True
else:
return False
print(divideThirteen(26))
print(divideThirteen(30))
True False
An Example of a Forever Running Code
The code keeps adding 1
to the variable number
until number
is no longer an integer(This is not the python data type "integer", it's the integer in number theory). However, there is no end to this code, making the computer run forever. There is no halt to the code.
i = 0
number = 1
def integerTest(n):
# Testing if the number is an integer
if n%1 ==0:
return True
else:
return False
# Using while loop to keep searching an a non-integer above 1. Note that the computer runs forever.
while i == 0:
number += 1
if integerTest(number) == False:
i +=1
print("Done")
--------------------------------------------------------------------------- KeyboardInterrupt Traceback (most recent call last) /Users/aniket/vscode/automatic-spork/_notebooks/2022-12-dd-lesson.ipynb Cell 11 in <cell line: 10>() <a href='vscode-notebook-cell:/Users/aniket/vscode/automatic-spork/_notebooks/2022-12-dd-lesson.ipynb#X13sZmlsZQ%3D%3D?line=9'>10</a> while i == 0: <a href='vscode-notebook-cell:/Users/aniket/vscode/automatic-spork/_notebooks/2022-12-dd-lesson.ipynb#X13sZmlsZQ%3D%3D?line=10'>11</a> number += 1 ---> <a href='vscode-notebook-cell:/Users/aniket/vscode/automatic-spork/_notebooks/2022-12-dd-lesson.ipynb#X13sZmlsZQ%3D%3D?line=11'>12</a> if integerTest(number) == False: <a href='vscode-notebook-cell:/Users/aniket/vscode/automatic-spork/_notebooks/2022-12-dd-lesson.ipynb#X13sZmlsZQ%3D%3D?line=12'>13</a> i +=1 <a href='vscode-notebook-cell:/Users/aniket/vscode/automatic-spork/_notebooks/2022-12-dd-lesson.ipynb#X13sZmlsZQ%3D%3D?line=13'>14</a> print("Done") /Users/aniket/vscode/automatic-spork/_notebooks/2022-12-dd-lesson.ipynb Cell 11 in integerTest(n) <a href='vscode-notebook-cell:/Users/aniket/vscode/automatic-spork/_notebooks/2022-12-dd-lesson.ipynb#X13sZmlsZQ%3D%3D?line=0'>1</a> i = 0 <a href='vscode-notebook-cell:/Users/aniket/vscode/automatic-spork/_notebooks/2022-12-dd-lesson.ipynb#X13sZmlsZQ%3D%3D?line=1'>2</a> number = 1 ----> <a href='vscode-notebook-cell:/Users/aniket/vscode/automatic-spork/_notebooks/2022-12-dd-lesson.ipynb#X13sZmlsZQ%3D%3D?line=2'>3</a> def integerTest(n): <a href='vscode-notebook-cell:/Users/aniket/vscode/automatic-spork/_notebooks/2022-12-dd-lesson.ipynb#X13sZmlsZQ%3D%3D?line=3'>4</a> # Testing if the number is an integer <a href='vscode-notebook-cell:/Users/aniket/vscode/automatic-spork/_notebooks/2022-12-dd-lesson.ipynb#X13sZmlsZQ%3D%3D?line=4'>5</a> if n%1 ==0: <a href='vscode-notebook-cell:/Users/aniket/vscode/automatic-spork/_notebooks/2022-12-dd-lesson.ipynb#X13sZmlsZQ%3D%3D?line=5'>6</a> return True KeyboardInterrupt:
Halting Problem Example:
- In order to understand this, suppose that an algorithm was able to analyze whether a code halts or not. Let's call this algorithm
HaltChecker
. -
HaltChecker
analyzes the program,program P
, and its input,input I
. Ifprogram P
halts withinput I
,HaltChecker
returns an output of "halts". Ifprogram P
doesn't halt(runs forever) withinput I
,HaltChecker
returns an output of "never". For example, in the code where it tests if variable number, the code runs forever, soHaltChecker
returns an output of never. - Then, we add another algorithm called
Reverser
which reversesHaltChecker
's output. So, if "never" is the output ofHaltChecker
, then the output ofReverser
is halts. It's also the same the other way around: ifHaltChecker
has an output of "halts", thenReverser
has an output of never. - We combine these algorithms into one entire body of code.
- Since
Reverser
is the algorithm at the end, hence giving the ultimate output, notice how it prints "never" when in fact there is an end(As proved byHaltChecker
), and how it also prints "halts" when there is in fact is no end to the code(Also proved byHaltChecker
). As a result,HaltChecker
is inaccurate and this is an undecidable problem.
This Diagram Sums up the Entire Process in the Bulleted List:
Credits of diagram and example to Khan Academy
FAQ
-
Q: If
Reverser
is causing the problem, why not remove it? -
A: Removing
Reverser
will remove the problems, however, we are looking for ways which create the problem of not outputting a correct result. One example is enough to prove that it is an undecidable problem since it proves that the code is not completely accurate.
Extra Things to Notice
- Note that while a computer may take a long time to run a section of code, it does not mean that the computer is going to run forever.
- Humans are able to solve some undecidable problems. The entire Halting Problem example was to prove that computers cannot solve undecidable problems.
3.17 Homework
Your homework for Algorithmic Efficiency is pretty simple.
- Use the 1st code below and graph it (Desmos, TI Inpire Cas, e.t.c), change the x value only!
- Label the number of loops done as x and the time (microseconds) to find the index as y
- Connect the points
- Do the same thing with the 2nd code
- Compare the two graphs and explain which one of the two is more efficient and why (min. 2 sentences)
- Insert images of the graph either in your blog or on review ticket
import time
def linear_search(lst, x):
start_time = time.perf_counter_ns() # records time (nanoseconds)
for i in range(len(lst)): # loops through the entire list
if lst[i] == x: # until the x value we are looking for is found
end_time = time.perf_counter_ns() # records time again
total_time = (end_time - start_time) // 1000 # subtracts last recorded time and first recorded time
print("Found element after {} loops in {} microseconds".format(i+1, total_time)) # prints the results
return print("Your number was found at", i)
end_time = time.perf_counter_ns() # records the time again
total_time = (end_time - start_time) // 1000 # subtracts last recorded time and first recorded time
print("Element not found after {} loops in {} microseconds".format(len(lst), total_time)) # prints the results
return "Your number wasn't found :("
lst = list(range(1, 10001)) # list with numbers 1-10000
x = 5000 # replace with an integer between 1 and 10000 (I suggest big numbers like 500, 2000, so on)
linear_search(lst, x) # runs procedure
Found element after 5000 loops in 213 microseconds Your number was found at 4999
import time
def binary_search(lt, x):
start_time = time.perf_counter_ns() # starts timer
low = 0 # sets the lower side
mid = 0 # sets mid value
high = len(lt) -1 # sets the higher side
num_loops = 0 # number of loops the search undergoes to find the x value
while low<=high: # Loop ran until mid is reached
num_loops += 1 # adds one loop each time process is repeated
mid = (low + high) // 2 # takes the lowest and highest possible numbers and divides by 2 and rounds to closest whole #
if lt[mid] == x:
end_time = time.perf_counter_ns() # records time
total_time = (end_time - start_time) // 1000 # time in microseconds
print("Element found after {} loops in {} microseconds".format(num_loops, total_time)) # prints the results
return mid # returns the index value
elif lt[mid] > x: # if mid was higher than x value, then sets new highest value as mid -1
high = mid -1
elif lt[mid] < x:
low = mid + 1 # if mid was lower than x, sets the new low as mid + 1
end_time = time.perf_counter_ns()
total_time = (end_time - start_time) // 1000
print("Element not found after {} loops in {} microseconds".format(num_loops, total_time)) # prints the results
return "Your number wasn't found :("
lt = list(range(1, 10001)) # list with numbers 1-10000
x = 149 # replace with an integer between 1 and 10000 (I suggest big numbers like 500, 2000, so on)
binary_search(lt, x) # runs procedure
Element found after 13 loops in 5 microseconds
148
3.18 Homework:
- Use the Jupyter notebook to write an algorithm that solves a decidable problem. You can use math or whatever else you would like to do.
- Write code to get the computer to run forever. Check this example if you need help, but please come up with your own idea.
Homeworks, hacks, and classwork(filled in blanks) for both 3.17 and 3.18 are due on Thursday at 9:00 pm. -0.1 points for each day late.
This code segment is undecidable because num will always be 0, and the code will only break out of the loop if num = 1. Therefore it will continue looping forever and crash your computer :)
num = 0
while num == 0:
if num == 1:
print("The undecidable problem has been decided!")
else:
print("num equals 0")
continue
num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 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num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 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num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 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num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals 0 num equals