# Maximal Sum Subarray

### Description

Find a subarray with maximal positive sum.

### Usage

1 2 3 |

### Arguments

`x` |
numeric vector. |

`A` |
numeric matrix |

`inds` |
logical; shall the indices be returned? |

`compiled` |
logical; shall the compiled version be used? |

### Details

`maxsub`

finds a contiguous subarray whose sum is maximally positive.
This is sometimes called Kadane's algorithm.

`maxsub`

will use a compiled and very fast version with a running time
of `O(n)`

where `n`

is the length of the input vector `x`

.

`maxsub2d`

finds a (contiguous) submatrix whose sum of elements is
maximally positive. The approach taken here is to apply the one-dimensional
routine to summed arrays between all rows of `A`

. This has a run-time
of `O(n^3)`

, though a run-time of `O(n^2 log n)`

seems possible
see the reference below.

`maxsub2d`

uses a Fortran workhorse and can solve a 1000-by-1000 matrix
in a few seconds—but beware of biggere ones

### Value

Either just a maximal sum, or a list this sum as component `sum`

plus
the start and end indices as a vector `inds`

.

### Note

In special cases, the matrix `A`

may be sparse or (as in the example
section) only have one nonzero element in each row and column. Expectation
is that there may exists a more efficient (say `O(n^2)`

) algorithm in
this extreme case.

### Author(s)

HwB <hwborchers@googlemail.com>

### References

Bentley, Jon (1986). “Programming Pearls”, Column 7. Addison-Wesley Publ. Co., Reading, MA.

T. Takaoka (2002). Efficient Algorithms for the Maximum Subarray Problem by Distance Matrix Multiplication. The Australasian Theory Symposion, CATS 2002.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 | ```
## Find a maximal sum subvector
set.seed(8237)
x <- rnorm(1e6)
system.time(res <- maxsub(x, inds = TRUE, compiled = FALSE))
res
## Standard example: Find a maximal sum submatrix
A <- matrix(c(0,-2,-7,0, 9,2,-6,2, -4,1,-4,1, -1,8,0,2),
nrow = 4, ncol = 4, byrow =TRUE)
maxsub2d(A)
# $sum: 15
# $inds: 2 4 1 2 , i.e., rows = 2..4, columns = 1..2
## Not run:
## Application to points in the unit square:
set.seed(723)
N <- 50; w <- rnorm(N)
x <- runif(N); y <- runif(N)
clr <- ifelse (w >= 0, "blue", "red")
plot(x, y, pch = 20, col = clr, xlim = c(0, 1), ylim = c(0, 1))
xs <- unique(sort(x)); ns <- length(xs)
X <- c(0, ((xs[1:(ns-1)] + xs[2:ns])/2), 1)
ys <- unique(sort(y)); ms <- length(ys)
Y <- c(0, ((ys[1:(ns-1)] + ys[2:ns])/2), 1)
abline(v = X, col = "gray")
abline(h = Y, col = "gray")
A <- matrix(0, N, N)
xi <- findInterval(x, X); yi <- findInterval(y, Y)
for (i in 1:N) A[yi[i], xi[i]] <- w[i]
msr <- maxsub2d(A)
rect(X[msr$inds[3]], Y[msr$inds[1]], X[msr$inds[4]+1], Y[msr$inds[2]+1])
## End(Not run)
``` |