Efficient extraction of all sub-polygons generated by self-intersecting features in a MultiPolygon

Starting from a shapefile containing a fairly large number (about 20000) of potentially partially-overlapping polygons, I'd need to extract all the sub-polygons originated by intersecting their different "boundaries".

In practice, starting from some mock-up data:

library(tibble)
library(dplyr)
library(sf)

ncircles <- 9
rmax     <- 120
x_limits <- c(-70,70)
y_limits <- c(-30,30)
set.seed(100) 
xy <- data.frame(
  id = paste0("id_", 1:ncircles), 
  x = runif(ncircles, min(x_limits), max(x_limits)),
  y = runif(ncircles, min(y_limits), max(y_limits))) %>% 
  as_tibble()

polys <- st_as_sf(xy, coords = c(2,3)) %>% 
  st_buffer(runif(ncircles, min = 1, max = 20)) 

plot(polys[1])

I'd need to derive an sf or sp multipolygon containing ALL and ONLY the polygons generated by the intersections, something like:

(note that the colors are there just to exemplify the expected result, in which each "differently colored" area is a separate polygon which doesn't overlay any other polygon)

I know I could work my way out by analyzing one polygon at a time, identifying and saving all its intersections and then "erase" those areas form the full multipolygon and proceed in a cycle, but that is quite slow.

I feel there should be a more efficient solution for this, but I am not able to figure it out, so any help would be appreciated! (Both sf and sp based solutions are welcome)

UPDATE:

In the end, I found out that even going "one polygon at a time" the task is far from simple! I'm really struggling on this apparently "easy" problem! Any hints? Even a slow solution or hints for starting on a proper path would be appreciated!

UPDATE 2:

Maybe this will clarify things: the desired functionality would be similar to the one described here:

https://it.mathworks.com/matlabcentral/fileexchange/18173-polygon-intersection?requestedDomain=www.mathworks.com

UPDATE 3:

I awarded the bounty to @shuiping-chen (thanks !), whose answer correctly solved the problem on the example dataset provided. The "method" has however to be generalized to situations were "quadruple" or "n-uple" intersections are possible. I'll try to work on that in the coming days and post a more general solution if I manage !

This has now been implemented in R package sf as the default result when st_intersection is called with a single argument (sf or sfc), see https://r-spatial.github.io/sf/reference/geos_binary_ops.html for the examples. (I'm not sure the origins field contains useful indexes; ideally they should point to indexes in x only, right now they kind of self-refer).

Input

I modify the mock-up data a bit in order to illustrate the ability to deal with multiple attributes.

library(tibble)
library(dplyr)
library(sf)

ncircles <- 9
rmax     <- 120
x_limits <- c(-70,70)
y_limits <- c(-30,30)
set.seed(100) 
xy <- data.frame(
  id = paste0("id_", 1:ncircles), 
  val = paste0("val_", 1:ncircles),
  x = runif(ncircles, min(x_limits), max(x_limits)),
  y = runif(ncircles, min(y_limits), max(y_limits)),
  stringsAsFactors = FALSE) %>% 
  as_tibble()

polys <- st_as_sf(xy, coords = c(3,4)) %>% 
  st_buffer(runif(ncircles, min = 1, max = 20)) 
plot(polys[1])

Basic Operation

Then define the following two functions.

cur: the current index of the base polygon
x: the index of polygons, which intersects with cur
input_polys: the simple feature of the polygons
keep_columns: the vector of names of attributes needed to keep after the geometric calculation

get_difference_region() get the difference between the base polygon and other intersected polygons; get_intersection_region() get the intersections among the intersected polygons.

library(stringr)
get_difference_region <- function(cur, x, input_polys, keep_columns=c("id")){
  x <- x[!x==cur] # remove self 
  len <- length(x)
  input_poly_sfc <- st_geometry(input_polys)
  input_poly_attr <- as.data.frame(as.data.frame(input_polys)[, keep_columns])

  # base poly
  res_poly <- input_poly_sfc[[cur]]
  res_attr <- input_poly_attr[cur, ]

  # substract the intersection parts from base poly
  if(len > 0){
    for(i in 1:len){
      res_poly <- st_difference(res_poly, input_poly_sfc[[x[i]]])
    }
  }
  return(cbind(res_attr, data.frame(geom=st_as_text(res_poly))))
}


get_intersection_region <- function(cur, x, input_polys, keep_columns=c("id"), sep="&"){
  x <- x[!x<=cur] # remove self and remove duplicated obj 
  len <- length(x)
  input_poly_sfc <- st_geometry(input_polys)
  input_poly_attr <- as.data.frame(as.data.frame(input_polys)[, keep_columns])

  res_df <- data.frame()
  if(len > 0){
    for(i in 1:len){
      res_poly <- st_intersection(input_poly_sfc[[cur]], input_poly_sfc[[x[i]]])
      res_attr <- list()
      for(j in 1:length(keep_columns)){
        pred_attr <- str_split(input_poly_attr[cur, j], sep, simplify = TRUE)
        next_attr <- str_split(input_poly_attr[x[i], j], sep, simplify = TRUE)
        res_attr[[j]] <- paste(sort(unique(c(pred_attr, next_attr))), collapse=sep)
      }
      res_attr <- as.data.frame(res_attr)
      colnames(res_attr) <- keep_columns
      res_df <- rbind(res_df, cbind(res_attr, data.frame(geom=st_as_text(res_poly))))
    }
  }
  return(res_df)
}

First Level

Difference

Let's see the difference function effect on the mock-up data.

flag <- st_intersects(polys, polys)

first_diff <- data.frame()
for(i in 1:length(flag)) {
  cur_df <- get_difference_region(i, flag[[i]], polys, keep_column = c("id", "val"))
  first_diff <- rbind(first_diff, cur_df)
}
first_diff_sf <- st_as_sf(first_diff, wkt="geom")
first_diff_sf
plot(first_diff_sf[1])

Intersection

first_inter <- data.frame()
for(i in 1:length(flag)) {
  cur_df <- get_intersection_region(i, flag[[i]], polys, keep_column=c("id", "val"))
  first_inter <- rbind(first_inter, cur_df)
}
first_inter <- first_inter[row.names(first_inter %>% select(-geom) %>% distinct()),]
first_inter_sf <- st_as_sf(first_inter, wkt="geom")
first_inter_sf
plot(first_inter_sf[1])

Second Level

use the intersection of first level as input, and repeat the same process.

Difference

flag <- st_intersects(first_inter_sf, first_inter_sf)
# Second level difference region
second_diff <- data.frame()
for(i in 1:length(flag)) {
  cur_df <- get_difference_region(i, flag[[i]], first_inter_sf, keep_column = c("id", "val"))
  second_diff <- rbind(second_diff, cur_df)
}
second_diff_sf <- st_as_sf(second_diff, wkt="geom")
second_diff_sf
plot(second_diff_sf[1])

Intersection

second_inter <- data.frame()
for(i in 1:length(flag)) {
  cur_df <- get_intersection_region(i, flag[[i]], first_inter_sf, keep_column=c("id", "val"))
  second_inter <- rbind(second_inter, cur_df)
}
second_inter <- second_inter[row.names(second_inter %>% select(-geom) %>% distinct()),]  # remove duplicated shape
second_inter_sf <- st_as_sf(second_inter, wkt="geom")
second_inter_sf
plot(second_inter_sf[1])

Get the distinct intersections of the second level, and use them as the input of the third level. We could get that the intersection results of the third level is NULL, then the process should end.

Summary

We put all the difference results into close list, and put all the intersection results into open list. Then we have:

When open list is empty, we stop the process
The results is close list

Therefore, we get the final code here (the basic two functions should be declared):

# init
close_df <- data.frame()
open_sf <- polys

# main loop
while(!is.null(open_sf)) {
  flag <- st_intersects(open_sf, open_sf)
  for(i in 1:length(flag)) {
    cur_df <- get_difference_region(i, flag[[i]], open_sf, keep_column = c("id", "val"))
    close_df <- rbind(close_df, cur_df)
  }
  cur_open <- data.frame()
  for(i in 1:length(flag)) {
    cur_df <- get_intersection_region(i, flag[[i]], open_sf, keep_column = c("id", "val"))
    cur_open <- rbind(cur_open, cur_df)
  }
  if(nrow(cur_open) != 0) {
    cur_open <- cur_open[row.names(cur_open %>% select(-geom) %>% distinct()),]
    open_sf <- st_as_sf(cur_open, wkt="geom")
  }
  else{
    open_sf <- NULL
  }
}

close_sf <- st_as_sf(close_df, wkt="geom")
close_sf
plot(close_sf[1])

Not sure if it helps you since it is not in R but I think there is a good way to solve this problem using Python. There is a library called GeoPandas (http://geopandas.org/index.html) which has allows you to easily do geo operations. In steps what you would need to do is the following: