maxsub | R Documentation |

Find a subarray with maximal positive sum.

maxsub(x, inds = TRUE) maxsub2d(A)

`x` |
numeric vector. |

`A` |
numeric matrix |

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

`maxsub`

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

will use a 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`

can solve a 100-by-100 matrix in a few seconds –
but beware of bigger ones.

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

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

.

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
these special cases.

HwB <hwborchers@googlemail.com>

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.

## Find a maximal sum subvector set.seed(8237) x <- rnorm(1e6) system.time(res <- maxsub(x, inds = TRUE)) 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)

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