Find all key-unique permutations of an array - arrays

I have an array that looks something like this:
[["Sunday", [user1, user2]], ["Sunday", [user1, user4]], ["Monday", [user3, user2]]]
The array essentially has all permutations of a given day with a unique pair of users. I obtained it by running
%w[Su Mo Tu We Th Fr Sa].product(User.all_pairs)
where User.all_pairs is every unique pair of users.
My goal now is to compose this set of nested arrays into schedules, meaning I want to find every permutation of length 7 with unique days. In other words, I want every potential week. I already have every potential day, and I have every potential pair of users, now I just need to compose them.
I have a hunch that the Array.permutation method is what I need, but I'm not sure how I'd use it in this case. Or perhaps I should use Array.product?


If I understand you correctly, you want all possible weeks where there is one pair of users assigned to each day. You can do it like this:
User.all_pairs.combination(7)
This will give you all possible ways of how you can pick 7 pairs and assign them to the days of the week. But if you are asking for every possible week, then it also matters into which day is which pair assigned, and you also have to take every permutation of those 7 pairs:
User.all_pairs.combination(7).map{|week| week.permutation().to_a}.flatten(1)
Now this will give you all possible weeks, where every week is represented as array containing 7 pairs. For example one of the weeks may look like this:
[(user1, user2), (user1, user3), (user2, user3), (user3, user4), (user1, user4), (user2, user4), (user3, user4)]
However the amount of the weeks will be huge! If you have n users, you will have k = n!/2 pairs, there is p = k! / (7! * (k - 7)!) ways of selecting 7 pairs and p * 7! possible weeks. If you have just 5 users, you get 1946482876800 possible weeks! No matter what you are planning to do with it, it won't be possible.
If you are trying to find the best schedule for a week, you can try to make some greedy algorithm.

Related

Excel calculate smallest of X columns within Y columns, ignoring zeros

I'm trying to calculate the sum of best segments in a run. For example, each Km gives a list as such:
5:40 6:00 5:45 5:55 6:21 6 :30
I'm trying to gather the best segments of 2km/3km/4km etc and would like a simple code to do it. At the moment, I'm using the formula
=Min(If(B1=0,9:9:9,sum(A1:B1),If(C1=0,9:9:9,sum(B1:C1))
but this goes all the way to 50km, meaning a very long formulae that I then have to repeat slightly differently at 3km, then 4km, then 5km etc. Surely there must me a way of
generating an array of summed columns of every n column, then iterating over that to find the min while ignoring values of 0?
I can do it manually for now, but what if I want to go over 50km? I might want to incorporate bike rides/car drives in the future just for some data analysis so I figured it best finding an ideal formulae now.
It's frustrating as I could code it and I want to avoid VBA ideally and stick to formulae in Excel.

Here is a draft of the case where there aren't any zeroes just for groups of 2Km. I decided the simplest approach initially was to add a couple of helper rows containing the running total of times (and for later use counts) and use a formula like this to subtract them in pairs:
=MIN(INDEX(A2:J2,SEQUENCE(1,9,2))-IF(SEQUENCE(1,9,0)=0,0,INDEX(A2:J2,SEQUENCE(1,9,0))))
but if you have access to recent additions to Excel 365 like Scan you can do it without helper rows.
Here is a more realistic scenario with a couple of zeroes thrown in
=LET(runningSum,Y$4:AP$4,runningCount,Y$5:AP$5,cols,COLUMNS(runningSum),leg,X7,
seqEnd,SEQUENCE(1,cols-leg+1,leg),seqStart,SEQUENCE(1,cols-leg+1,0),
times,INDEX(runningSum,seqEnd)-IF(seqStart=0,0,INDEX(runningSum,seqStart)),
counts,INDEX(runningCount,seqEnd)-IF(seqStart=0,0,INDEX(runningCount,seqStart)),
MIN(IF(counts=leg,times)))
Note that there are no runs of more than seven consecutive legs that don't contain a zero so 8, 9, 10 etc. just work out to 0.
As mentioned you could dispense with the helper rows by using Scan, but not everyone has access to this so I will add it separately:
=LET(data,Y$3:AP$3,runningSum,SCAN(0,data,LAMBDA(a,b,a+b)),
runningCount,SCAN(0,data,LAMBDA(a,b,a+(b>0))),leg,X7,cols,COLUMNS(data),
seqEnd,SEQUENCE(1,cols-leg+1,leg),seqStart,SEQUENCE(1,cols-leg+1,0),
times,INDEX(runningSum,seqEnd)-IF(seqStart=0,0,INDEX(runningSum,seqStart)),
counts,INDEX(runningCount,seqEnd)-IF(seqStart=0,0,INDEX(runningCount,seqStart)),
MIN(IF(counts=leg,times)))

Tom that worked! I learnt a few things on the way too and using the indexing method alongside sequence and columns is something I had not thought of. I'd never heard of the LET command before and I can already see that this is going to really help with some of the bigger calculations in the future.
Thank you so much, I'd like to show you how it now looks. Row 3087 is my old formula, row 3088 is a copy of the same data using the new formula, as you can see I've gotten exactly the same results so it's clear that it works perfectly and it is can be easily duplicated.

React Native Algoritmic Challenge

I have to design a project in React Native where I need to create a "wheel of fate" . This project should take from a list of 10 people, two of them (randomly) to complete a random day of support for each.
The rules of this task are the following:
A person can do at most one half-day shift in a day.
Each person should have completed one whole day of support in any 2 week period.
A person cannot have half-day shifts on consecutive days.
My approach for this project was first of all creating a function that will get two random different people as following.
var workers =require('./workers.json')
const shuffled = workers.sort(() => 0.5 - Math.random());
// Get sub-array of first n elements after shuffled
let selected = shuffled.slice(0, 2);
Next, I tried to push each of this group of workers to someday but I kind of got stuck
I expect that my output will provide a table or a list which will respect the rules specified previously.

How many trucks came empty but bought something

I think this is the hardest to date I have had to crack - so hard I had a hard time finding a good headline.
So we have a site where trucks come and buy say Gravel, or sand or other building materials.
Sometimes they also unload demolition waste first.
I need to find out a couple of things
how many trucks (and from what companys) came empty
if they came empty what did they buy from us.
what companys are sending full trucks and what are sending empty trucks.
a tope 10 of materials they will drive to us from to buy even when coming empty to our facility.
a list of all the order numbers that they drove to us til fill and came with empty trucks. ( I have distances linked to order numbers, so now I can estimate the value of our products)
The data I have available:
I have a full data set of when what customer buys what and / or pay to deliver.
E.G.:
I can see the parts I need to split the data into I think it should be something like this
find all unique licence plates
somehow map if they bought materials within 30 minutes of
offloading demolition waste (most trucks will come between 2 and 10
times per day)
Present all this data (on a normal day we have about 800 trucks = 2000 lines since they weigh in, weigh out, and then some buy something = 2 more weigh lines)
I can easily find unique licence plates per day (either by formula or by Excel function Data/delete doublets,
but after that I have no clue where to start.
I think I need some sheets in between, where I somehow mark if a material was bought from an "empty truck" and I need a counter for that .. somehow...
Any help on how to get started is appreciated.

It seems like the best way to start is with a helper column (in the following exampes, I have chosen "Column M") to flag whether the truck arrived empty.
In the helper column, you can use something similar to the following formula.
{=IF(ISBLANK(B2),0,IF(C2="In",0,IF(B2=$B$2:$B$13,IF($C$2:$C$13="In",IF($A$2:$A$13>(A2-TIME(0,30,0)),0,1),1),1)))}
This is an array formula, which means you have to press ctrl+shift+enter after pasting it in the cell. Then you can copy that cell down the column.
Just to explain, the first if statement knows the truck is not arriving empty if Column C is 'In'. The second if statement creates an array and tests to see if other the same truck appears in other rows. The third if statement checks to see if the same truck checked 'In' in the matching rows, and the fourth if statement verifies if the time they checked in was less than thirty minutes ago. You can adjust the length by editing the TIME(0,30,0) function. The format is TIME(hours,minuites,seconds). Unless the truck matches all three of the second, third and fourth if statements, it is marked as coming empty.
Once you have this helper column, just about all of your tasks are quite simple.
1a: How many trucks came empty? Sum Column M
1b: How many trucks from what company? Create a unique list of companies. Then create a COUNTIFS formula based on Column M = 1 and Column K = Company. For example, if C32 had Company B then the formula =COUNTIFS($M$2:$M$13,1,$K$2:$K$13,C32) would return 2
1c: How many times did a truck come empty? Similar to 1b, create a unique list of License Plates, then use a COUNTIFS based on Column M = 1 and Column B = License Plate.
2: Similar to 1b, just use a unique list of products tested against Column F
3: Similar to 1b, just create a second column, next to the first that uses =COUNTIFS($M$2:$M$13,0,$K$2:$K$13,C53,$C$2:$C$13,"In") Which tests that Column M reports the truck did not come empty, that matches the company in Column K and that the truck came 'In' so you don't double count the same truck when it goes 'out'
4: Just sort list created by number 2. You can highlight the range, right-click and select "Sort" > "Custom Sort", then select the column you want to sort on and largest to smallest.
5: There are a couple of different ways, you could do this. The formula
{=TEXTJOIN(", ",TRUE,IF($M$2:$M$13=1,$J$2:$J$13,""))}
(again, entered as an array formula)
would create a comma separated list of order numbers. An alternative if you want a column of order numbers (but would only work if they are actually numbers), is to paste the formula {=MAX(IF($M$2:$M$13=1,$J$2:$J$13,))} in the first row of the column (in my example, its O2) and then {=MAX(IF($M$2:$M$13=1,IF($J$2:$J$13<O2,$J$2:$J$13,)))} in the row below (change the reference to O2 if you pasted it in a different spot)(again, note that both of these are array formulas). Then copy and paste the second formula down the column. When order numbers of trucks that came in empty are exhausted, the formula will report 0.

Building an array of numbers no repeating order doesn't matter (Gaming stats)

So I play heroes of newerth. I have the desire to make a statistical program that shows which team of 5 heroes vs another 5 heroes wins the most. Given there are 85 heroes and games are 85 choose 5 vs 80 choose 5, that's a lot of combinations.
Essentially I'm going to take the stats data the game servers allow me to get and just put a 1 in an array which has heroes when they get a win [1,2,3,4,5][6,7,8,9,10][W:1][L:0]
So after I parse and build the array from the historical game data, I can put in what 5 heroes I want to see, and I can get back all the relevant game data telling me which 5 hero lineup has won/lost the most.
What I need help starting is a simple algorithm to write out my array. Here's similar output I need: (I have simplified this to 1-10, where the code I get I can just change 10 to x for how many heroes there are).
[1,2,3,4,5][6,7,8,9,10]
[1,2,3,4,6][5,7,8,9,10]
[1,2,3,4,7][5,6,8,9,10]
[1,2,3,4,8][5,6,7,9,10]
[1,2,3,4,9][5,6,7,8,10]
[1,2,3,4,10][5,6,7,8,9]
[1,2,3,5,6][4,7,8,9,10]
[1,2,3,5,7][4,6,8,9,10]
[1,2,3,5,8][4,6,7,9,10]
[1,2,3,5,9][4,6,7,8,10]
[1,2,3,5,10][4,6,7,8,9]
[1,2,3,6,7][4,5,8,9,10]
[1,2,3,6,8][4,5,7,9,10]
[1,2,3,6,9][4,5,7,8,10]
[1,2,3,6,10][4,5,7,8,9]
[1,2,3,7,8][4,5,6,9,10]
[1,2,3,7,9][4,5,6,8,10]
[1,2,3,7,10][4,5,6,8,9]
[1,2,3,8,9][4,5,6,7,10]
[1,2,3,8,10][4,5,6,7,9]
[1,2,3,9,10][4,5,6,7,8]
[1,2,4,5,6][3,7,8,9,10]
[1,2,4,5,7][3,6,8,9,10]
[1,2,4,5,8][3,6,7,9,10]
[1,2,4,5,9][3,6,7,8,10]
[1,2,4,5,10][3,6,7,8,9]
[1,2,4,6,7][3,5,8,9,10]
[1,2,4,6,8]...
[1,2,4,6,9]
[1,2,4,6,10]
[1,2,4,7,8]
[1,2,4,7,9]
[1,2,4,7,10]
[1,2,4,8,9]
[1,2,4,8,10]
[1,2,4,9,10]
...
You get the Idea. No repeating and order doesn't matter. Its essentially cut in half doesn't matter the order of the arrays either. Just need a list of all the combinations of teams that can be played against each other.
EDIT: additional thinking...
After quite a bit of thinking. I have come up with some ideas. Instead of writting out the entire array of [85*84*83*82*81][80*79*78*77*76*75] possible combinations of characters, which would have to be made larger for the introduction of of new heroes as to keep the array relevant and constantly updating.
To instead when reading from the server parse the information and build the array from there. It would be much simpler to just make an element in the array when one is not found, ei the combinations have never been played before. Then parsing the data would be 1 pass, and build your array as it complies along. Yes it might take a while, but the values that are created will be worth the wait. It can be done over time too. Starting with a small test case say 1000 games and working up the the number of matches that have been played. Another Idea would be to start from our current spot in time and build the data base from there. There is no need to go back to the first games ever played based off the amount of changes that have occurred to heroes over that time frame, but say go back 2-3 months to give it some foundation and reliability of data, and with each passing day only getting more accurate.
Example parse and build of the array:
get match(x)
if length < 15/25, x++; //determine what length matches we want and discredit shorter than 15 for sure.
if players != 10, x++; //skip the match because it didn't finish with 10 players.
if map != normal_mm_map // rule out non mm games, and mid wars
if != mm, rule out custom games
//and so forth
match_psr = match(x).get(average_psr);
match_winner = match(x).get(winner);
//Hero ids of winners
Wh1 = match.(x).get(winner.player1(hero_id)))
Wh2 = match.(x).get(winner.player2(hero_id)))
Wh3 = match.(x).get(winner.player3(hero_id)))
Wh4 = match.(x).get(winner.player4(hero_id)))
Wh5 = match.(x).get(winner.player5(hero_id)))
//hero ids of losers
Lh1 = match.(x).get(loser.player1(hero_id)))
Lh2 = match.(x).get(loser.player2(hero_id)))
Lh3 = match.(x).get(loser.player3(hero_id)))
Lh4 = match.(x).get(loser.player4(hero_id)))
Lh5 = match.(x).get(loser.player5(hero_id)))
//some sort of sorting algorithim to put the Wh1-5 in order of hero id from smallest to largest
//some sort of sorting algorithim to put the Lh1-5 in order of hero id from smallest to largest
if(array([Wh1, Wh2, Wh3, Wh4, Wh5],[Lh1,Lh2,Lh3,Lh4,Lh5],[],[],[],[],[],[],[],[],[]) != null)
array([Wh1, Wh2, Wh3, Wh4, Wh5],[Lh1,Lh2,Lh3,Lh4,Lh5],[],[],[],[],[],[],[],[],[]) += array([],[],[1],[][][][](something with psr)[][][[])
else(array.add_element([Wh1, Wh2, Wh3, Wh4, Wh5],[Lh1,Lh2,Lh3,Lh4,Lh5],[1],[][][][](something with psr)[][][[])
Any thoughts?

Encode each actor in the game using a simple scheme 0 ... 84
You can maintain a 2D matrix of 85*85 actors in the game.
Initialize each entry in this array to zero.
Now use just the upper triangular portion of your matrix.
So, for any two players P1,P2 you have a unique entry in the array, say array[small(p1,p2)][big(p1,p2)].
array(p1,p2) signifies how much p1 won against p2.
You event loop can be like this :
For each stat like H=(H1,H2,H3,H4,H5) won against L=(L1,L2,L3,L4,L5) do
For each tuple in H*L (h,l) do
if h<l
increment array[h][l] by one
else
decrement array[l][h] by one
Now, at the end of this loop, you have an aggregate information about players information against each other. Next step is an interesting optimization problem.
wrong approach : select 5 fields in this matrix such that no two field's row and column are same and the summation of their absolute values is maximum. I think you can get good optimization algorithms for this problem. Here, we will calculate five tuples (h1,l1), (h2,l2), (h3,l3) ... where h1 wins against l1 is maximized but you still did not see it l1 is good against h2.
The easier and correct options is to use brute force on the set of (85*84)C5 tuples.

Web stats: Calculating/estimating unique visitors for arbitary time intervals

I am writing an application which is recording some 'basic' stats -- page views, and unique visitors. I don't like the idea of storing every single view, so have thought about storing totals with a hour/day resolution. For example, like this:
Tuesday 500 views 200 unique visitors
Wednesday 400 views 210 unique visitors
Thursday 800 views 420 unique visitors
Now, I want to be able to query this data set on chosen time periods -- ie, for a week. Calculating views is easy enough: just addition. However, adding unique visitors will not give the correct answer, since a visitor may have visited on multiple days.
So my question is how do I determine or estimate unique visitors for any time period without storing each individual hit. Is this even possible? Google Analytics reports these values -- surely they don't store every single hit and query the data set for every time period!?
I can't seem to find any useful information on the net about this. My initial instinct is that I would need to store 2 sets of values with different resolutions (ie day and half-day), and somehow interpolate these for all possible time ranges. I've been playing with the maths, but can't get anything to work. Do you think I may be on to something, or on the wrong track?
Thanks,
Brendon.

If you are OK with approximations, I think tom10 is onto something, but his notion of random subsample is not the right one or needs a clarification. If I have a visitor that comes on day1 and day2, but is sampled only on day2, that is going to introduce a bias in the estimation. What I would do is to store full information for a random subsample of users (let's say, all users whose hash(id)%100 == 1). Then you do the full calculations on the sampled data and multiply by 100. Yes tom10 said about just that, but there are two differences: he said "for example" sample based on the ID and I say that's the only way you should sample because you are interested in unique visitors. If you were interested in unique IPs or unique ZIP codes or whatever you would sample accordingly. The quality of the estimation can be assessed using the normal approximation to the binomial if your sample is big enough. Beyond this, you can try and use a model of user loyalty, like you observe that over 2 days 10% of visitors visit on both days, over three days 11% of visitors visit twice and 5% visit once and so forth up to a maximum number of day. These numbers unfortunately can depend on time of the week, season and even modeling those, loyalty changes over time as the user base matures, changes in composition and the service changes as well, so any model needs to be re-estimated. My guess is that in 99% of practical situations you'd be better served by the sampling technique.

You could store a random subsample of the data, for example, 10% of the visitor IDs, then compare these between days.
The easiest way to do this is to store a random subsample of each day for future comparisons, but then, for the current day, temporarily store all your IDs and compare them to the subsampled historical data and determine the fraction of repeats. (That is, you're comparing the subsampled data to a full dataset for a given day and not comparing two subsamples -- it's possible to compare two subsamples and get an estimate for the total but the math would be a bit trickier.)

You don't need to store every single view, just each unique session ID per hour or day depending on the resolution you need in your stats.
You can keep these log files containing session IDs sorted to count unique visitors quickly, by merging multiple hours/days. One file per hour/day, one unique session ID per line.
In *nix, a simple one-liner like this one will do the job:
$ sort -m sorted_sid_logs/2010-09-0[123]-??.log | uniq | wc -l
It counts the number of unique visitors during the first three days of September.

You can calculate the uniqueness factor (UF) on each day and use it to calculate the composite (week by example) UF.
Let's say that you counted:
100 visits and 75 unique session id's on monday (you have to store the sessions ID's at least for a day, or the period you use as unit).
200 visits and 100 unique session id's on tuesday.
If you want to estimate the UF for the period Mon+Tue you can do:
UV = UVmonday + UVtuesday = TVmonday*UFmonday + TVtuesday*UFtuesday
being:
UV = Unique Visitors
TV = Total Visits
UF = Uniqueness Factor
So...
UV = (Sum(TVi*UFi))
UF = UV / TV
TV = Sum(TVi)
I hope it helps...
This math counts two visits of the same person as two unique visitors. I think it's ok if the only way you have to identify somebody is via the session ID.

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