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# Given a random string of n left brackets and n right brackets, # how often do you need to cut it into two parts and swap # both sides, to make it "algebraicaly valid"? rnd_x <- function(n) { idx_rb <- sample.int(2*n, n) x <- rep(1, 2*n) x[idx_rb] <- -1 x } as_br <- function(x) paste0(ifelse(x==1, "(", ")")) one_x <- function(n) { x <- rnd_x(n) message(as_br(x)) cx <- cumsum(x) i <- 0 while (any(cx<0) && i<2*n) { i <- i+1 br <- which.min(cx) x <- c(x[seq(br+1,2*n)], x[1:br]) message(i,": ", as_br(x)) cx <- cumsum(x) } i } lapply(1:10, function(n) max(replicate(10^3, one_x(n))))
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