In this question here (Summing Nodes in a Network), I learned how to find the square within the original network having the largest node sum.

Here is the data for this question:

library(igraph)

width <- 30
height <- 20
num_nodes <- width * height

# Create a grid
x <- rep(1:width, each = height)
y <- rep(1:height, times = width)

g <- make_empty_graph(n = num_nodes, directed = FALSE)

# Function to get node index
get_node_index <- function(i, j) (i - 1) * height + j

# Add edges
edges <- c()
for(i in 1:width) {
   for(j in 1:height) {
      current_node <- get_node_index(i, j)
    
      # Connect to right neighbor
      if(i < width) edges <- c(edges, current_node, get_node_index(i + 1, j))
    
      # Connect to bottom neighbor
      if(j < height) edges <- c(edges, current_node, get_node_index(i, j + 1))
   }
}

g <- add_edges(g, edges)

V(g)$x <- x
V(g)$y <- y

par(mfrow=c(1,2))

V(g)$name <- 1:num_nodes
plot(g, vertex.size = 7, vertex.label = V(g)$name, vertex.label.cex = 0.6, main = "Map with         Node Indices")

V(g)$value <- sample(1:100, num_nodes, replace = TRUE)
plot(g, vertex.size = 7, vertex.label = V(g)$value, vertex.label.cex = 0.6, main = "Map with     Population Values")

And here is the function:

sg <- subgraph_isomorphisms(make_ring(4), g)
lst <- unique(lapply(sg, (x) sort(names(x))))
out <- do.call(
  rbind,
  lapply(
    lst,
    (v) data.frame(
      node_id = toString(v),
      value = sum(V(induced_subgraph(g, v))$value)
    )
  )
)

This approach is currently using a brute force style approach in which every node is individually checked. Are there any ways in R to restructure this function so that it runs in parallel or a different type of search algorithm that can scan the network more efficiently?

I had two ideas about this:

1. Idea 1:

Rewriting the function to look at square grids and tesselate them over the network:

efficient_sum_squares <- function(g, width, height) {
  results <- data.frame(node_id = character(), value = numeric())
  
  for (i in 1:(width - 1)) {
    for (j in 1:(height - 1)) {
      nodes <- c(
        get_node_index(i, j),
        get_node_index(i + 1, j),
        get_node_index(i, j + 1),
        get_node_index(i + 1, j + 1)
      )
      
      sum_value <- sum(V(g)$value[nodes])
      
      results <- rbind(results, data.frame(node_id = toString(nodes), value = sum_value))
    }
  }
  
  results
}

out_efficient <- efficient_sum_squares(g, width, height)

2. Idea 2:

I thought that comparisons could be carried out in a vectorized fashion:

vectorized_sum_squares <- function(g, width, height) {
  x_mat <- matrix(V(g)$x, nrow = height, ncol = width, byrow = FALSE)
  y_mat <- matrix(V(g)$y, nrow = height, ncol = width, byrow = FALSE)
  value_mat <- matrix(V(g)$value, nrow = height, ncol = width, byrow = FALSE)
  
  sums <- value_mat[1:(height-1), 1:(width-1)] + 
          value_mat[2:height, 1:(width-1)] + 
          value_mat[1:(height-1), 2:width] + 
          value_mat[2:height, 2:width]
  
  node_ids <- apply(which(sums == sums, arr.ind = TRUE), 1, function(idx) {
    i <- idx[1]
    j <- idx[2]
    toString(c(
      get_node_index(j, i),
      get_node_index(j + 1, i),
      get_node_index(j, i + 1),
      get_node_index(j + 1, i + 1)
    ))
  })
  
  data.frame(node_id = node_ids, value = as.vector(sums))
}

out_vectorized <- vectorized_sum_squares(g, width, height)

Is there any better way to work on this problem?