i have connections between cities like
connection(London,Sziget).
connection(Sziget,Kairo).
so i want to create a predicate to find if there is a possible route between two cities even passing through other cities first.
Input example: route(London,Kairo).
result: true
Up to this point i have created this recursive code that works.
route(W,Z):-connection(W,Z).
route(W,Z):-connection(W,Y),route(Y,Z).
But i also want to calculate the total cost of the route if the cost between two cities is 100 and for every other city passing is 50 more.
Input example: route(London,Kairo).
result: true 150
Any help appreciated.
It sounds like your cost is:
100 + 50 per intermediate city?
If that's the case, you want something like
route(W, Z, 100) :- connection(W, Z).
route(W, Z, Cost) :- connection(W, Y), route(Y, Z, Cost2), Cost is Cost2+50.
That will evaluate to 100 once the direct link is found, or add 50 to the cost every time you have to go via an intermediate.
In that case, your input would be
route(London, Kairo, Cost).
the result would be
Cost = 150
which would imply that a route was found. If you actually need the 'true' part, that'll be a little trickier.
Related
I' am doing my homework in programming, and I don't know how to solve this problem:
We have a set of n weights, we are putting them on a scale one by one until all weights is used. We also have string of n letters "R" or "L" which means which pen is heavier in that moment, they can't be in balance. There are no weights with same mass. Compute in what order we have to put weights on scale and on which pan.
The goal is to find order of putting weights on scale, so the input string is respected.
Input: number 0 < n < 51, number of weights. Then weights and the string.
Output: in n lines, weight and "R" or "L", side where you put weight. If there are many, output any of them.
Example 1:
Input:
3
10 20 30
LRL
Output:
10 L
20 R
30 L
Example 2:
Input:
3
10 20 30
LLR
Output:
20 L
10 R
30 R
Example 3:
Input:
5
10 20 30 40 50
LLLLR
Output:
50 L
10 L
20 R
30 R
40 R
I already tried to compute it with recursion but unsuccessful. Can someone please help me with this problem or just gave me hints how to solve it.
Since you do not show any code of your own, I'll give you some ideas without code. If you need more help, show more of your work then I can show you Python code that solves your problem.
Your problem is suitable for backtracking. Wikipedia's definition of this algorithm is
Backtracking is a general algorithm for finding all (or some) solutions to some computational problems, notably constraint satisfaction problems, that incrementally builds candidates to the solutions, and abandons a candidate ("backtracks") as soon as it determines that the candidate cannot possibly be completed to a valid solution.
and
Backtracking can be applied only for problems which admit the concept of a "partial candidate solution" and a relatively quick test of whether it can possibly be completed to a valid solution.
Your problem satisfies those requirements. At each stage you need to choose one of the remaining weights and one of the two pans of the scale. When you place the chosen weight on the chosen pan, you determine if the corresponding letter from the input string is satisfied. If not, you reject the choice of weight and pan. If so, you continue by choosing another weight and pan.
Your overall routine first inputs and prepares the data. It then calls a recursive routine that chooses one weight and one pan at each level. Some of the information needed by each level could be put into mutable global variables, but it would be more clear if you pass all needed information as parameters. Each call to the recursive routine needs to pass:
the weights not yet used
the input L/R string not yet used
the current state of the weights on the pans, in a format that can easily be printed when finalized (perhaps an array of ordered pairs of a weight and a pan)
the current weight imbalance of the pans. This could be calculated from the previous parameter, but time would be saved by passing this separately. This would be total of the weights on the right pan minus the total of the weights on the left pan (or vice versa).
Your base case for the recursion is when the unused-weights and unused-letters are empty. You then have finished the search and can print the solution and quit the program. Otherwise you loop over all combinations of one of the unused weights and one of the pans. For each combination, calculate what the new imbalance would be if you placed that weight on that pan. If that new imbalance agrees with the corresponding letter, call the routine recursively with appropriately-modified parameters. If not, do nothing for this weight and pan.
You still have a few choices to make before coding, such as the data structure for the unused weights. Show me some of your own coding efforts then I'll give you my Python code.
Be aware that this could be slow for a large number of weights. For n weights and two pans, the total number of ways to place the weights on the pans is n! * 2**n (that is a factorial and an exponentiation). For n = 50 that is over 3e79, much too large to do. The backtracking avoids most groups of choices, since choices are rejected as soon as possible, but my algorithm could still be slow. There may be a better algorithm than backtracking, but I do not see it. Your problem seems to be designed to be handled by backtracking.
Now that you have shown more effort of your own, here is my un-optimized Python 3 code. This works for all the examples you gave, though I got a different valid solution for your third example.
def weights_on_pans():
def solve(unused_weights, unused_tilts, placement, imbalance):
"""Place the weights on the scales using recursive
backtracking. Return True if successful, False otherwise."""
if not unused_weights:
# Done: print the placement and note that we succeeded
for weight, pan in placement:
print(weight, 'L' if pan < 0 else 'R')
return True # success right now
tilt, *later_tilts = unused_tilts
for weight in unused_weights:
for pan in (-1, 1): # -1 means left, 1 means right
new_imbalance = imbalance + pan * weight
if new_imbalance * tilt > 0: # both negative or both positive
# Continue searching since imbalance in proper direction
if solve(unused_weights - {weight},
later_tilts,
placement + [(weight, pan)],
new_imbalance):
return True # success at a lower level
return False # not yet successful
# Get the inputs from standard input. (This version has no validity checks)
cnt_weights = int(input())
weights = {int(item) for item in input().split()}
letters = input()
# Call the recursive routine with appropriate starting parameters.
tilts = [(-1 if letter == 'L' else 1) for letter in letters]
solve(weights, tilts, [], 0)
weights_on_pans()
The main way I can see to speed up that code is to avoid the O(n) operations in the call to solve in the inner loop. That means perhaps changing the data structure of unused_weights and changing how it, placement, and perhaps unused_tilts/later_tilts are modified to use O(1) operations. Those changes would complicate the code, which is why I did not do them.
Had a tough time thinking of an appropriate title, but I'm just trying to code something that can auto compute the following simple math problem:
The average value of a,b,c is 25. The average value of b,c is 23. What is the value of 'a'?
For us humans we can easily compute that the value of 'a' is 29, without the need to know b and c. But I'm not sure if this is possible in programming, where we code a function that takes in the average values of 'a,b,c' and 'b,c' and outputs 'a' automatically.
Yes, it is possible to do this. The reason for this is that you can model the sort of problem being described here as a system of linear equations. For example, when you say that the average of a, b, and c is 25, then you're saying that
a / 3 + b / 3 + c / 3 = 25.
Adding in the constraint that the average of b and c is 23 gives the equation
b / 2 + c / 2 = 23.
More generally, any constraint of the form "the average of the variables x1, x2, ..., xn is M" can be written as
x1 / n + x2 / n + ... + xn / n = M.
Once you have all of these constraints written out, solving for the value of a particular variable - or determining that many solutions exists - reduces to solving a system of linear equations. There are a number of techniques to do this, with Gaussian elimination with backpropagation being a particularly common way to do this (though often you'd just hand this to MATLAB or a linear algebra package and have it do the work for you.)
There's no guarantee in general that given a collection of equations the computer can determine whether or not they have a solution or to deduce a value of a variable, but this happens to be one of the nice cases where the shape of the contraints make the problem amenable to exact solutions.
Alright I have figured some things out. To answer the question as per title directly, it's possible to represent average value in programming. 1 possible way is to create a list of map data structures which store the set collection as key (eg. "a,b,c"), while the average value of the set will be the value (eg. 25).
Extract the key and split its string by comma, store into list, then multiply the average value by the size of list to get the total (eg. 25x3 and 23x2). With this, no semantic information will be lost.
As for the context to which I asked this question, the more proper description to the problem is "Given a set of average values of different combinations of variables, is it possible to find the value of each variable?" The answer to this is open. I can't figure it out, but below is an attempt in describing the logic flow if one were to code it out:
Match the lists (from Paragraph 2) against one another in all possible combinations to check if a list contains all elements in another list. If so, substract the lists (eg. abc-bc) as well as the value (eg. 75-46). If upon substracting we only have 1 variable in the collection, then we have found the value for this variable.
If there's still more than 1 variables left such as abcd - bc = ad, then store the values as a map data structure and repeat the process, till the point where the substraction count in the full iteration is 0 for all possible combinations (eg. ac can't substract bc). This is unfortunately not where it ends.
Further solutions may be found by combining the lists (eg. ac + bd = abcd) to get more possible ways to subtract and derive at the answer. When this is the case, you just don't know when to stop trying, and the list of combinations will get exponential. Maybe someone with strong related mathematical theories may be able to prove that upon a certain number of iteration, further additions are useless and hence should stop. Heck, it may even be possible that negative values are also helpful, and hence contradict what I said earlier about 'ac' can't subtract 'bd' (to get a,c,-b,-d). This will give even more combinations to compute.
People with stronger computing science foundations may try what templatetypedef has suggested.
I'm learning Python and came across a question that went something like "How long would it take to count to 1,000,000 out loud?" The only parameter it gave was, "you count, on average, 1 digit per second." I did that problem, which wasn't very difficult. Then I started thinking about counting aloud, annunciating each numeral. That parameter seems off to me, and indeed the answer Google gives to the question alone "how long to count to a million" suggests it's off. Given that each number in the sequence takes progressively longer (an exponential increase??), there must be a better way.
Any ideas or general guidance would be of assistance. Would sampling various people's "counting rates" at various intervals work? Would programming the # of syllables work? I am really curious, and have looked all over SO and Google for solutions that don't revolve around that seemingly inaccurate "average time".
Thanks, and sorry if this isn't on topic or in the appropriate place. I'm a long time lurker, but new to posting, so let me know if you need more info or anything. Thanks!
Let us suppose for the sake of simplicity that you don't say 1502 as "fifteen hundred and two", but as "thousand five hundred and two". Then we can hierarchically break it down.
And let's ignore the fact whether you say "and" or not (though apparently it is more said than not) for now. I will use this reference (and British English, because I like it more and it's more consistent : http://forum.wordreference.com/showthread.php?t=15&langid=6) for how to pronounce numbers.
In fact, to formally describe this, let t be a function of a set of numbers, that tells you how much time it takes to pronounce every number in that set. Then your question is how to compute t([1..1000000]), and we will use M=t([1..999])
Triplet time in function of previous one
To read a large number we start at the left and read the three-digit groups. The group at the left, of course, may have only one or two digits.
Thus for every number x of thousands you will say x thousand y where y will describe all the numbers from 1 to 999.
Thus the time you spend in the x thousand ... is 1000 t({1000x}) + M, as detailed here after :
Note that this formula is generalizable to numbers below 1000, by simply defining t({0}) = 0.
Now the time to say "x thousand" is, per our hypothesis, equal to the time to say "x" plus the time to say "thousand" (when x > 0). Thus your answer is :
Where is the time it takes to say the word thousand. This supposes you say 1000 as "one thousand". You may want to remove 1000 tau("one") if you would only say "thousand".
How ever I stick with the reference :
The numbers 100-199 begin with one hundred... or a hundred...
You can in exactly the same way express the time it takes to count to a billion from and the number above, and so on for all the greater powers of 103, i.e.
Taking into account the "and"
There is a small correction to be done. Let us suppose that M is the time it takes to pronounce numbers from 1 to 999 when they are preceded by at least a non-0 group of numbers, including initial "and"s.
Our reference (well, the wordreference post I linked) says the following :
What do we say to join the groups?
Normally, we don’t use any joining word.
The exception is the last group.
If the last group after the thousands is 1-99 it is joined with and.
Thus our correction applies only to the numbers between 0 and 999 (where there is no non-zero group preceding) :
Getting M
Or rather, let's get t([1..999]) since it's more natural and we know how it is related to M.
Let C = t([1..99]), X = t([1..9]).
Between 1 and 999 we have all the numbers from [1..99] and all the 9 exact hundreds where you don't say "and", that is 108 occurences. There are 900 numbers prefixed with a hundreds number.
Thus
C is probably hard to break down, so I'm not going to try.
Final result
The corrected formula is :
And as a function of C and X :
Note that your measures of tau(word), C, and X need to be very precise if you plan on doing this multiplication and having any kind of correct order of magnitude.
Conclusion : Brits end up saying "and" a whole lot. The nice thing about the last formulation is that you can remove all the "and"s if you decide you actually don't want to pronounce them.
I have around 644 nodes in my graph database(Neo4j) . I need to compute distances between all these 644 nodes and display it visually in the GUI. I want to pre-compute and store the distances between every two pairs of nodes in the database itself rather than retrieving the nodes on to the server and then finding the distances between them on the fly and then showing on the GUI.
I want to understand how to write such a query in CYPHER. Please let me know.
I think this can work:
// half cross product
match (a),(b)
where id(a) < id(b)
match p=shortestPath((a)-[*]-(b))
with a,b,length(p) as l
create (a)-[:DISTANCE {distance:l}]->(b)
Set 4950 properties, created 4950 relationships, returned 0 rows in 4328 ms
But the browser viz will blow up with this, just that you know.
Regarding your distance measure (it won't be that fast but should work):
MATCH (a:User)-[:READ]->(book)<-[:READ]-(b:User)
WITH a,b,count(*) as common,
length(a-[:READ]->()) as a_read,
length(b-[:READ]->()) as b_read
CREATE UNIQUE (a)-[:DISTANCE {distance:common/(a_read+b_read-common)}]-(b)
In my path finding school project, the user is given 3 options to navigate between two points:
Shortest path (Kilometers). I've defined the cost function for each 2 points to be the distance of the road that connects them.
Fastest path (Each road has a speed limit). I've defined the cost function between each 2 points to be 1/(SpeedLimit).
Simplest path (Minimizes turns, a turn is defined if the road changes direction by more than alpha degrees). I've defined a state to be a tuple of a point and direction, and defined the cost function to be 1 if the change of direction is larger than alpha and 0 otherwise.
The user then supplies 3 real numbers between 0 and 1 to specify the importance of each navigating option.
So basically the cost function should be the sum of the three cost functions described above, each multiplied by the number supplied. My problem is that each cost function is of different units, for example, the first cost function is in kilometers and the third cost function is boolean (0 or 1).
How can I convert them so that it makes sense?
Define a cost function for each criteria that maps from a path to a real number.
f1(path) = cost associated with the distance of the path
f2(path) = cost of the time taken to traverse the path
f3(path) = cost of the complexity of the route
Defining f1 and f2 should be fairly straightforward. f3 is more complex and subjective but I suspect it really shouldn't be a boolean unless there's some very specific reason why you would need it to be. Perhaps the function for path complexity could be something like the sum of the number of degrees (radians) in every turn taken in the trip. There's certainly quite a few other choices for such a function that immediately come to mind, (for example the length of the representation required to describe the path). For f3 you will have to choose whatever one suits your purposes best.
Once you have defined the individual cost functions you could get an overall cost for the path by taking a linear combination of those 3 functions:
cost(path) = a1*f1(path) + a2*f2(path) + a3*f3(path)
Finding sensible values for a1, a2, a3 is most of the challenge. There are a few statistical methods you might want to use to do this.