![Abstract
For the notorious Instant Insanity problem, we decided to take a general approach that is
slow, but sure to work. Dealing with instant insanity can be pretty difficult. However, if
programmed correctly, we can continue to find faster ways to get this NPComplete algorithm
working faster. Solving algorithms that are npcomplete can be extremely beneficial to the study
of mathematics and computer science, seeing that there is a millennium prize problem including
NPComplete problems (prove or disprove P = NP).
Method
Our recursive approach (using Python) was to check every pair on every line, and if the
numbers on that line hadn’t already been counted twice, then add that pair and go to the next
line (cube); otherwise go to the next pair. If all pairs have been checked, then we’ve reached a
dead end and need to return to the previous line (cube) and try the next pair, and so on and so
forth, recursively.
Pseudocode
Create an array of length 40 (from [0] to [39]) called countArray to store the counts of all
numbers created; initialize each index to 0.
//Recursion steps through each cube (line) and performs checks, starting with the first pair of the
first cube
Recursion(cubeArray, cubeIndex, generatedSolution):
Base case:
if index == 40
//then we’ve reached the end without any conflicts
print generatedSolution
return
Recursive case:
For each pair on the cube
If pair number 1’s countArray index value != 2
and if pair number 2’s countArray
increment those indices in countArray
make a note of which pair was chosen
add that pair to the solution list
Recurse(cubeArray, cubeIndex+1, generatedSolution)
}
}
//if a pair was noted, decrement the indices of the noted pair in countArray
//then loop back to the beginning and check the next pair
}
return](https://image.slidesharecdn.com/9dbedeb6-7014-4a4a-82f5-1c364e9c3808-150803033414-lva1-app6891/85/InstantInsanityProgrammingAssignment-2-320.jpg)
![Python implementation of pseudocode:
1. defcompareIt(arr,genList):
2. foriinxrange(len(arr[0])):
3. genList[i]=arr[0][i]#storethefirstpair
4. print("Firststoredpair:",genList[i])
5. recurse(arr,1,genList)#recursetothenextline
6.
7. defrecurse(arr,ind,genList):
8. ifind==40:
9. print("Solutionis:",genList)
10. else:
11. #print("hitelse")
12. foriinxrange(len(arr[ind])):
13. #print("iis",i)
14. #print("arraypair:",arr[ind][i])
15. pair=arr[ind][i]#selectapair
16. foriinxrange(ind):#selecteachpreviouslineingenList
17. #print("currentgenListis:")
18. forrinxrange(ind):
19. print(genList[r])
20. #print("pair[0]is",pair[0],"genList[i][0]is:",genList[i][0])
21. ifpair[0]!=genList[i][0]:
22. #noproblem,checkotheritem
23. ifpair[1]!=genList[i][1]:
24. #noproblem,addpair
25. genList[i+1]=pair
26. #print("Recursed")
27. recurse(arr,ind+1,genList)
28. else:
29. #problem,don'tadd
30. c=0
31. #endif
32. else:
33. #problem,don'tadd
34. c=0
35. #endif
36. #endfor
37. #endfor
38. #endif
39.#endrecurse
Analysis
The time complexity of this algorithm is O(n^4). In order to reduce the actual execution
time of the program, a Hadoop cluster could be used with an advanced message queue
protocol such as RabbitMQ. RabbitMQ can be used to distribute bits and pieces of the required
computation to separate machines in each cluster. Hadoop is used for distributing processing
power and computations of a large data set throughout multiple clusters of computers. The
execution time of an optimized algorithm then relies heavily on how much computational](https://image.slidesharecdn.com/9dbedeb6-7014-4a4a-82f5-1c364e9c3808-150803033414-lva1-app6891/85/InstantInsanityProgrammingAssignment-3-320.jpg)
![resources you can spare for the clusters. You would have a machine passing all of the data
required to be processed to the messaging queue (in this case, RabbitMQ). The messaging
queue then acknowledges this and distributes it to a machine within a cluster, making sure that
each machine never receives the same computation again. Dynamic programming can also
come into play with this advanced messaging queue protocol and clusters by storing solved
puzzles or paths into a single data source, which we can then use as an acting filter to ensure
that the same computations are never reprocessed. If a chunk of data comes in which has
already been computed, the “filter” will instantly spit back the alreadycomputed data set. Aside
from using distributed computing methods to solve something of high caliber like 40cubed
Instant Insanity problem with a much faster execution time, small adjustments to the time
complexity in the algorithm can greatly reduce the execution time for the data set as a whole.
For our solutions, we were able to find within the limited time and with limited computational
resources as follows:
Solution 1:
[ 0, 2, 1, 2, 0, 2, 1, 1, 0, 0, 2, 2, 1, 1, 0, 0, 1, 0, 1, 2, 0, 1, 1, 1, 1, 0, 1, 2, 1, 0, 1, 2, 0, 2, 2, 1, 0, 1,
1, 1 ]
[ 1, 0, 2, 0, 2, 1, 0, 2, 1, 1, 0, 1, 0, 0, 2, 2, 2, 1, 0, 0, 1, 0, 0, 2, 2, 1, 2, 0, 2, 2, 0, 1, 2, 0, 1, 0, 2, 2,
0, 0 ]
Solution 2:
[ 0, 2, 1, 2, 0, 2, 1, 1, 0, 0, 2, 2, 1, 1, 0, 0, 1, 0, 1, 2, 0, 1, 1, 1, 1, 0, 1, 2, 1, 0, 1, 2, 0, 2, 2, 1, 0, 1,
1, 1 ]
[ 2, 1, 0, 1, 1, 0, 2, 0, 2, 2, 1, 0, 2, 2, 1, 1, 0, 2, 2, 1, 2, 2, 2, 0, 0, 2, 0, 1, 0, 1, 2, 0, 1, 1, 0, 2, 1, 0,
2, 2 ]
Solution 3:
[ 1, 0, 2, 0, 2, 1, 0, 2, 1, 1, 0, 1, 0, 0, 2, 2, 2, 1, 0, 0, 1, 0, 0, 2, 2, 1, 2, 0, 2, 2, 0, 1, 2, 0, 1, 0, 2, 2,
0, 0 ]
[ 2, 1, 0, 1, 1, 0, 2, 0, 2, 2, 1, 0, 2, 2, 1, 1, 0, 2, 2, 1, 2, 2, 2, 0, 0, 2, 0, 1, 0, 1, 2, 0, 1, 1, 0, 2, 1, 0,
2, 2 ]
These three different solutions represent the indexes of the original array that was
provided for the problem. For example in the first thread, the 0 corresponds to index 0 of the
problem array, followed by the colors at index 2, followed by the colors at index 1, then 2 and so
on and so forth. Each solution also has two threads which represent the four total sides.](https://image.slidesharecdn.com/9dbedeb6-7014-4a4a-82f5-1c364e9c3808-150803033414-lva1-app6891/85/InstantInsanityProgrammingAssignment-4-320.jpg)
InstantInsanityProgrammingAssignment
- 3. Python implementation of pseudocode: 1. defcompareIt(arr,genList): 2. foriinxrange(len(arr[0])): 3. genList[i]=arr[0][i]#storethefirstpair 4. print("Firststoredpair:",genList[i]) 5. recurse(arr,1,genList)#recursetothenextline 6. 7. defrecurse(arr,ind,genList): 8. ifind==40: 9. print("Solutionis:",genList) 10. else: 11. #print("hitelse") 12. foriinxrange(len(arr[ind])): 13. #print("iis",i) 14. #print("arraypair:",arr[ind][i]) 15. pair=arr[ind][i]#selectapair 16. foriinxrange(ind):#selecteachpreviouslineingenList 17. #print("currentgenListis:") 18. forrinxrange(ind): 19. print(genList[r]) 20. #print("pair[0]is",pair[0],"genList[i][0]is:",genList[i][0]) 21. ifpair[0]!=genList[i][0]: 22. #noproblem,checkotheritem 23. ifpair[1]!=genList[i][1]: 24. #noproblem,addpair 25. genList[i+1]=pair 26. #print("Recursed") 27. recurse(arr,ind+1,genList) 28. else: 29. #problem,don'tadd 30. c=0 31. #endif 32. else: 33. #problem,don'tadd 34. c=0 35. #endif 36. #endfor 37. #endfor 38. #endif 39.#endrecurse Analysis The time complexity of this algorithm is O(n^4). In order to reduce the actual execution time of the program, a Hadoop cluster could be used with an advanced message queue protocol such as RabbitMQ. RabbitMQ can be used to distribute bits and pieces of the required computation to separate machines in each cluster. Hadoop is used for distributing processing power and computations of a large data set throughout multiple clusters of computers. The execution time of an optimized algorithm then relies heavily on how much computational