This document covers the topic of solving problems related to the subset sum
problem with RcppAlgos. We have already covered integer
partitions, which is a special case of the subset sum problem, in Constraints
and Integer Partitions and it is highly encouraged to read that
vignette first.
The integer partition problem presents the question “how can we write n as a sum of positive integers?” There are well-known algorithms for enumerating all partitions of an integer n. We even have algorithms for generating partitions of a specific length or with distinct parts only. But how do we enumerate partitions of n with a specific set of numbers? What about enumerating partitions of a specific length m of n given a specific set of numbers?
For example, using only the numbers 3:18, find all
partitions of 50 of length 5.
With RcppAlgos, this is easily achieved. We simply use
the same template as we did in Constraints
and Integer Partitions. Observe (We continue to use the
ht function defined in the Combination
and Permutation Basics vignette):
library(RcppAlgos)
## Each element can only occur once
ht(partitionsGeneral(3:18, 5, target = 50))
#> head -->
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 3 4 8 17 18
#> [2,] 3 4 9 16 18
#> [3,] 3 4 10 15 18
#> [4,] 3 4 10 16 17
#> [5,] 3 4 11 14 18
#> --------
#> tail -->
#> [,1] [,2] [,3] [,4] [,5]
#> [180,] 7 8 9 12 14
#> [181,] 7 8 10 11 14
#> [182,] 7 8 10 12 13
#> [183,] 7 9 10 11 13
#> [184,] 8 9 10 11 12
## What about allowing repetition?
ht(partitionsGeneral(3:18, 5, TRUE, target = 50))
#> head -->
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 3 3 8 18 18
#> [2,] 3 3 9 17 18
#> [3,] 3 3 10 16 18
#> [4,] 3 3 10 17 17
#> [5,] 3 3 11 15 18
#> --------
#> tail -->
#> [,1] [,2] [,3] [,4] [,5]
#> [507,] 9 9 9 11 12
#> [508,] 9 9 10 10 12
#> [509,] 9 9 10 11 11
#> [510,] 9 10 10 10 11
#> [511,] 10 10 10 10 10
## Even works on multisets
ht(partitionsGeneral(3:18, 5, freqs = rep(1:4, 4), target = 50))
#> head -->
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 3 4 7 18 18
#> [2,] 3 4 8 17 18
#> [3,] 3 4 9 16 18
#> [4,] 3 4 9 17 17
#> [5,] 3 4 10 15 18
#> --------
#> tail -->
#> [,1] [,2] [,3] [,4] [,5]
#> [401,] 8 10 10 10 12
#> [402,] 9 9 9 10 13
#> [403,] 9 9 9 11 12
#> [404,] 9 9 10 10 12
#> [405,] 9 10 10 10 11In fact, these optimized algorithms can be applied when the vector
passed has the quality that if you were to sort them, the difference of
each element with it’s neighbor is constant (E.g.
c(121, 126, 131, 136, ..., 221)).
system.time(genParts <- partitionsGeneral(seq(121, 221, 5), 13,
TRUE, target = 2613))
#> user system elapsed
#> 0.006 0.002 0.008 ## over 100 thousand results out of a possible 573 million
ht(genParts)
#> head -->
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
#> [1,] 121 121 161 221 221 221 221 221 221 221 221 221 221
#> [2,] 121 121 166 216 221 221 221 221 221 221 221 221 221
#> [3,] 121 121 171 211 221 221 221 221 221 221 221 221 221
#> [4,] 121 121 171 216 216 221 221 221 221 221 221 221 221
#> [5,] 121 121 176 206 221 221 221 221 221 221 221 221 221
#> --------
#> tail -->
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
#> [119542,] 196 196 196 201 201 201 201 201 201 201 206 206 206
#> [119543,] 196 196 201 201 201 201 201 201 201 201 201 201 211
#> [119544,] 196 196 201 201 201 201 201 201 201 201 201 206 206
#> [119545,] 196 201 201 201 201 201 201 201 201 201 201 201 206
#> [119546,] 201 201 201 201 201 201 201 201 201 201 201 201 201
prettyNum(comboCount(seq(121, 221, 5), 13, TRUE), big.mark = ",")
#> [1] "573,166,440"
system.time(genMultiParts <- partitionsGeneral(seq(121, 221, 5), 13,
freqs = rep(1:7, 3),
targe = 2613))
#> user system elapsed
#> 0.007 0.003 0.011 ## over 70 thousand results out of a possible 256 million!
ht(genMultiParts)
#> head -->
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
#> [1,] 121 126 171 216 216 216 221 221 221 221 221 221 221
#> [2,] 121 126 176 211 216 216 221 221 221 221 221 221 221
#> [3,] 121 126 176 216 216 216 216 221 221 221 221 221 221
#> [4,] 121 126 181 206 216 216 221 221 221 221 221 221 221
#> [5,] 121 126 181 211 211 216 221 221 221 221 221 221 221
#> --------
#> tail -->
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
#> [70291,] 186 186 191 196 196 201 201 206 206 211 211 211 211
#> [70292,] 186 186 191 196 196 201 206 206 206 206 211 211 211
#> [70293,] 186 186 191 196 201 201 201 206 206 206 206 211 216
#> [70294,] 186 186 191 196 201 201 201 206 206 206 211 211 211
#> [70295,] 186 186 196 196 201 201 201 206 206 206 206 211 211
prettyNum(comboCount(seq(121, 221, 5), 13, freqs = rep(1:7, 3)), big.mark = ",")
#> [1] "256,047,675"Generally, integer partition algorithms are restricted to positive
integers. However, with the generalized partition algorithms in
RcppAlgos, we can make light work of vectors with negative
numbers (again, the sorted vector has to have the property that the
difference of each element with it’s neighbor is constant).
system.time(genDistParts <- partitionsGeneral(seq(-173L, 204L, 13L),
11, target = -460))
#> user system elapsed
#> 0.003 0.000 0.003
all(rowSums(genDistParts) == -460L)
#> [1] TRUE
ht(genDistParts)
#> head -->
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
#> [1,] -173 -160 -147 -134 -121 -108 -95 -82 165 191 204
#> [2,] -173 -160 -147 -134 -121 -108 -95 -69 152 191 204
#> [3,] -173 -160 -147 -134 -121 -108 -95 -69 165 178 204
#> [4,] -173 -160 -147 -134 -121 -108 -95 -56 139 191 204
#> [5,] -173 -160 -147 -134 -121 -108 -95 -56 152 178 204
#> --------
#> tail -->
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
#> [108940,] -121 -108 -82 -69 -56 -43 -30 -17 -4 22 48
#> [108941,] -121 -108 -82 -69 -56 -43 -30 -17 9 22 35
#> [108942,] -121 -95 -82 -69 -56 -43 -30 -17 -4 9 48
#> [108943,] -121 -95 -82 -69 -56 -43 -30 -17 -4 22 35
#> [108944,] -108 -95 -82 -69 -56 -43 -30 -17 -4 9 35With the examples illustrated above, we had the restriction that the sorted input vector had to have the property that the difference of each element with it’s neighbor is constant. If this requirement is broken, it only means that we cannot use a particular algorithm and we must fall back to a more general algorithm. Fret not!! These general algorithms are extremely efficient and very flexible. We can use them with random input vectors, random targets, as well as over ranges.
Let us revisit the example above but with a slight variation that breaks the requirement.
inpVec <- c(116, seq(126, 221, 5))
## Non-constant difference... The specialized algo can't be used
diff(inpVec)
#> [1] 10 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
system.time(genParts2 <- partitionsGeneral(inpVec, 13, TRUE, target = 2613))
#> user system elapsed
#> 0.096 0.005 0.101 ## We still find over 100 thousand results out of
## a possible 573 million in under a second
ht(genParts2)
#> head -->
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
#> [1,] 116 116 171 221 221 221 221 221 221 221 221 221 221
#> [2,] 116 116 176 216 221 221 221 221 221 221 221 221 221
#> [3,] 116 116 181 211 221 221 221 221 221 221 221 221 221
#> [4,] 116 116 181 216 216 221 221 221 221 221 221 221 221
#> [5,] 116 116 186 206 221 221 221 221 221 221 221 221 221
#> --------
#> tail -->
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
#> [118556,] 196 196 196 201 201 201 201 201 201 201 206 206 206
#> [118557,] 196 196 201 201 201 201 201 201 201 201 201 201 211
#> [118558,] 196 196 201 201 201 201 201 201 201 201 201 206 206
#> [118559,] 196 201 201 201 201 201 201 201 201 201 201 201 206
#> [118560,] 201 201 201 201 201 201 201 201 201 201 201 201 201Although the above was about 13 times slower than the first example
dealing with 573 million combinations (101 milliseconds vs. 8
milliseconds), we are still dealing in milliseconds!!! For reference,
version 2.3.4 takes about 18 seconds to find all 118,560
solutions.
Here are some more exotic examples demonstrating the power of these algorithms.
set.seed(42)
mySamp <- sample(-100:100, 50)
sort(mySamp)
#> [1] -98 -97 -96 -95 -81 -77 -74 -65 -60 -59 -58 -54 -52 -43 -36 -33 -30
#> [18] -27 -12 -9 -2 -1 3 8 9 10 13 21 27 30 33 35 42 45
#> [35] 49 52 53 57 61 63 64 70 73 76 83 84 88 91 96 100
system.time(exotic <- partitionsGeneral(mySamp, 8, freqs = rep(1:5, 10),
target = 496))
#> user system elapsed
#> 0.226 0.001 0.228
dim(exotic)
#> [1] 102241 8
## Over 1 billion total combinations
prettyNum(comboCount(mySamp, 8, freqs = rep(1:5, 10)), big.mark = ",")
#> [1] "1,343,133,680"
## Only getting a few (a thousand in this case) is much faster
system.time(partitionsGeneral(mySamp, 8, freqs = rep(1:5, 10),
target = 496, upper = 1e3))
#> user system elapsed
#> 0.004 0.000 0.004The function permuteGeneral benefits from these
optimized algorithms as well. However, just as we discussed in Output
Order with permuteGeneral, the output will not be in
lexicographical order.
Oftentimes when working with numerical vectors, it can be hard to
find combinations that sum to a particular number because of floating
point errors (See Using
tolerance). In practice, we may not need an exact match
and a close approximation will suffice. For example, let’s say we have a
football team of 100 players (including practice squad) and we are
interested in a trade involving 6 players and a total salary of 20
million dollars. We may not be able to find 6 players whose sum of
salaries is exactly 20 million, but we can find many 6 player
combinations whose sum of salaries is within a tolerance of 20
million.
set.seed(22213)
football_player_salaries <- 2e7 * rbeta(100, 2, 25)
summary(football_player_salaries)
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> 115308 768338 1261683 1612271 1950565 10895883
## Over 1 billion combinations...
## An exhaustive search will not be feasible
prettyNum(comboCount(football_player_salaries, 6), big.mark = ",")
#> [1] "1,192,052,400"
system.time(exactly20 <- partitionsGeneral(football_player_salaries, 6,
target = 2e7, tolerance = 0))
#> user system elapsed
#> 2.841 0.017 2.874
## No results that equal exactly 2e7
dim(exactly20)
#> [1] 0 6What if we increase the tolerance to $1000 (Honestly…
what’s $1000 when we are talking about 20 million dollars)?
Our intent is to explore these options, so we take advantage of the
upper argument in anticipation that we obtain many results
that meet the criteria. If we obtain the upper bound, we decrease the
tolerance (if needed) and repeat.
## N.B. This is much more efficient. Also, we set keepResults
## to TRUE so we can see the total sum of salaries.
system.time(almost20 <- comboGeneral(football_player_salaries, 6,
constraintFun = "sum", comparisonFun = "==",
limitConstraints = 2e7, tolerance = 1000,
upper = 1000, keepResults = TRUE))
#> user system elapsed
#> 0.130 0.000 0.131
dim(almost20)
#> [1] 1000 6
ht(almost20)
#> head -->
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] 115307.7 152563.4 809407.9 3163109 4863446 10895883 19999717
#> [2,] 115307.7 152563.4 1590746.9 2381655 4863446 10895883 19999602
#> [3,] 115307.7 152563.4 1669898.9 2302265 4863446 10895883 19999365
#> [4,] 115307.7 152563.4 1746659.2 2225285 4863446 10895883 19999145
#> [5,] 115307.7 152563.4 1853727.8 2850338 4132545 10895883 20000364
#> --------
#> tail -->
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [996,] 200278.8 550414.4 1751652 2984500 3618110 10895883 20000838
#> [997,] 200278.8 550414.4 1855829 3163109 3334046 10895883 19999560
#> [998,] 200278.8 550414.4 1884764 2850338 3618110 10895883 19999788
#> [999,] 200278.8 550414.4 2013884 2850338 3489156 10895883 19999953
#> [1000,] 200278.8 550414.4 2051845 2984500 3316996 10895883 19999917
## decreasing the tolerance to $10 further we obtain 158 results
system.time(superClose20 <- comboGeneral(football_player_salaries, 6,
constraintFun = "sum", comparisonFun = "==",
limitConstraints = 2e7, tolerance = 10,
upper = 1000, keepResults = TRUE))
#> user system elapsed
#> 2.734 0.009 2.747
ht(superClose20)
#> head -->
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,] 115307.7 266606.5 695657.2 3163109 4863446 10895883 20000009
#> [2,] 115307.7 1117835.0 1318811.6 1688714 4863446 10895883 19999998
#> [3,] 152563.4 628078.8 1117835.0 3334046 3871591 10895883 19999997
#> [4,] 152563.4 695657.2 1635144.3 2984500 3636247 10895883 19999995
#> [5,] 200278.8 765448.4 1174496.9 1923219 5040664 10895883 19999990
#> --------
#> tail -->
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [154,] 1318812 1359188 1512157 1929459 2984500 10895883 19999997
#> [155,] 1318812 1371670 1771874 2225285 2416482 10895883 20000006
#> [156,] 1338303 1706514 1823915 1853728 2381655 10895883 19999998
#> [157,] 1371670 1512157 1516215 1853728 2850338 10895883 19999990
#> [158,] 1371670 1516215 1635144 2189663 2391419 10895883 19999994prod and meanThese optimized algorithms are also employed when
constraintFun is "prod" or
"mean".
getAllThenFilter <- function(n, m, lim) {
t <- comboGeneral(n, m, constraintFun = "prod")
t[t[, m + 1] == lim, -(m+1)]
}
library(microbenchmark)
microbenchmark(optimized = comboGeneral(25, 10, constraintFun = "prod",
comparisonFun = "==",
limitConstraints = 1037836800),
brute = getAllThenFilter(25, 10, 1037836800), times = 20,
unit = "relative", check = "equal")
#> Unit: relative
#> expr min lq mean median uq max neval cld
#> optimized 1.00000 1.00000 1.0000 1.00000 1.00000 1.00000 20 a
#> brute 46.86513 46.87535 49.3752 47.36287 47.81427 60.11692 20 b
## What about cases when brute force isn't feasible
set.seed(101)
v <- runif(1000, 1, 2)
prettyNum(comboCount(v, 100), big.mark = ",")
#> [1] "63,850,511,926,305,130,236,698,511,142,022,274,281,262,900,693,853,331,776,286,816,221,524,376,994,750,901,948,920,974,351,797,699,894,319,420,811,933,446,197,797,592,213,357,065,053,890"
system.time(prodAlmost100 <- comboGeneral(v, 100, constraintFun = "prod",
comparisonFun = "==",
limitConstraints = 100,
tolerance = 0.0001, upper = 20))
#> user system elapsed
#> 0.027 0.000 0.027
dim(prodAlmost100)
#> [1] 20 100
apply(prodAlmost100, 1, prod)
#> [1] 100.00008 100.00003 100.00003 100.00006 100.00010 100.00000 99.99993 99.99995 100.00002
#> [10] 99.99992 100.00004 99.99994 100.00002 100.00005 99.99992 99.99996 100.00006 100.00003
#> [19] 100.00006 100.00002
## Showcasing mean
system.time(meanAlmost1.5 <- comboGeneral(v, 100, constraintFun = "mean",
comparisonFun = "==",
limitConstraints = 1.5,
tolerance = 0.0001, upper = 20))
#> user system elapsed
#> 0.001 0.000 0.000
dim(meanAlmost1.5)
#> [1] 20 100
rowMeans(meanAlmost1.5)
#> [1] 1.500000 1.499999 1.500001 1.500000 1.500000 1.500001 1.500000 1.500000 1.500000 1.500000
#> [11] 1.499999 1.500000 1.500001 1.500001 1.499999 1.499999 1.500001 1.500000 1.500001 1.500000As of version 2.5.0 all of the above cases can be
attacked with iterators (See Combinatorial
Iterators in RcppAlgos). As mentioned in the suggested reading,
iterators are very flexible and just as efficient as their “general”
counterparts. They have the added benefit of allowing one to save the
current state, allowing one to generate n results at a
time.
Below are a few demonstrations using some of the examples in earlier sections.
## The football salary example
salary <- partitionsIter(football_player_salaries, 6,
target = 2e7, tolerance = 1000)
## Or use comboIter:
##
## comboIter(football_player_salaries, 6, constraintFun = "sum",
## comparisonFun = "==", limitConstraints = 2e7,
## tolerance = 1000, upper = 1000, keepResults = TRUE))
system.time(almost20withIter <- salary@nextNIter(1e3))
#> user system elapsed
#> 0.126 0.000 0.127
## almost20 was generated above with comboGeneral
all.equal(almost20[, 1:6], almost20withIter)
#> [1] TRUE
## With iterators we can easily continue iterating. With the general
## functions if we wanted the next 1000 results, we would have to
## generate the first 1000 along with the next 1000
system.time(nextAlmost20withIter <- salary@nextNIter(1e3))
#> user system elapsed
#> 0.124 0.001 0.125
ht(nextAlmost20withIter)
#> head -->
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 200278.8 550414.4 2189662.6 2829267 3334046 10895883
#> [2,] 200278.8 628078.8 653771.4 3489156 4132545 10895883
#> [3,] 200278.8 628078.8 809407.9 3334046 4132545 10895883
#> [4,] 200278.8 628078.8 852693.7 2381655 5040664 10895883
#> [5,] 200278.8 628078.8 1165478.3 3239395 3871591 10895883
#> --------
#> tail -->
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [996,] 264972.3 663453 1318812 3239395 3618110 10895883
#> [997,] 264972.3 663453 1376878 3163109 3636247 10895883
#> [998,] 264972.3 663453 1688714 2850338 3636247 10895883
#> [999,] 264972.3 663453 1706514 2850338 3618110 10895883
#> [1000,] 264972.3 663453 1853728 2189663 4132545 10895883
salary@summary()
#> $description
#> [1] "Combinations of 100 choose 6 where the sum is between 19999000 and 20001000"
#>
#> $currentIndex
#> [1] 2000
#>
#> $totalResults
#> [1] NA
#>
#> $totalRemaining
#> [1] NA
## The prodAlmost100 example
prodIter <- comboIter(v, 100,
constraintFun = "prod", comparisonFun = "==",
limitConstraints = 100, tolerance = 0.0001)
system.time(prodAlmost100WithIter <- prodIter@nextNIter(20))
#> user system elapsed
#> 0.024 0.000 0.024
all.equal(prodAlmost100, prodAlmost100WithIter)
#> [1] TRUE
## Again, with iterators, we can continue iterating from
## where we left off
system.time(nextAlmost100WithIter <- prodIter@nextNIter(20))
#> user system elapsed
#> 0.018 0.000 0.018
dim(nextAlmost100WithIter)
#> [1] 20 100
## Use @ or $ to access methods. If one needs to access these methods
## often (e.g. nextIter inside a loop), it is recommended to use the
## @ accessor as it is much more efficient.
prodIter$summary()
#> $description
#> [1] "Combinations of 1000 choose 100 where the prod is between 99.999899999999997 and 100.0001"
#>
#> $currentIndex
#> [1] 40
#>
#> $totalResults
#> [1] NA
#>
#> $totalRemaining
#> [1] NA