From 1d23b31461c62b11cd85908423df378a44e8c960 Mon Sep 17 00:00:00 2001 From: Giuliano Mega Date: Thu, 28 Sep 2023 02:45:58 -0300 Subject: [PATCH] Block discovery simulator and analysis (#175) * add block discovery simulator * add analysis document for simpler cases of block discovery --- analysis/block-discovery-sim/.Rbuildignore | 4 + analysis/block-discovery-sim/.Rprofile | 1 + analysis/block-discovery-sim/.gitignore | 5 + analysis/block-discovery-sim/DESCRIPTION | 18 + analysis/block-discovery-sim/R/collate.R | 19 + analysis/block-discovery-sim/R/node.R | 14 + analysis/block-discovery-sim/R/partition.R | 18 + analysis/block-discovery-sim/R/sim.R | 30 + analysis/block-discovery-sim/R/stats.R | 17 + analysis/block-discovery-sim/README.md | 37 + .../block-discovery-sim.Rmd | 202 ++ .../block-discovery-sim.Rproj | 17 + analysis/block-discovery-sim/renv.lock | 2194 +++++++++++++++++ analysis/block-discovery-sim/renv/.gitignore | 7 + analysis/block-discovery-sim/renv/activate.R | 1032 ++++++++ .../block-discovery-sim/renv/settings.json | 17 + analysis/block-discovery-sim/tests/testthat.R | 11 + .../tests/testthat/test-partition.R | 18 + .../tests/testthat/test-stats.R | 17 + analysis/block-discovery.Rmd | 104 + 20 files changed, 3782 insertions(+) create mode 100644 analysis/block-discovery-sim/.Rbuildignore create mode 100644 analysis/block-discovery-sim/.Rprofile create mode 100644 analysis/block-discovery-sim/.gitignore create mode 100644 analysis/block-discovery-sim/DESCRIPTION create mode 100644 analysis/block-discovery-sim/R/collate.R create mode 100644 analysis/block-discovery-sim/R/node.R create mode 100644 analysis/block-discovery-sim/R/partition.R create mode 100644 analysis/block-discovery-sim/R/sim.R create mode 100644 analysis/block-discovery-sim/R/stats.R create mode 100644 analysis/block-discovery-sim/README.md create mode 100644 analysis/block-discovery-sim/block-discovery-sim.Rmd create mode 100644 analysis/block-discovery-sim/block-discovery-sim.Rproj create mode 100644 analysis/block-discovery-sim/renv.lock create mode 100644 analysis/block-discovery-sim/renv/.gitignore create mode 100644 analysis/block-discovery-sim/renv/activate.R create mode 100644 analysis/block-discovery-sim/renv/settings.json create mode 100644 analysis/block-discovery-sim/tests/testthat.R create mode 100644 analysis/block-discovery-sim/tests/testthat/test-partition.R create mode 100644 analysis/block-discovery-sim/tests/testthat/test-stats.R create mode 100644 analysis/block-discovery.Rmd diff --git a/analysis/block-discovery-sim/.Rbuildignore b/analysis/block-discovery-sim/.Rbuildignore new file mode 100644 index 0000000..d821302 --- /dev/null +++ b/analysis/block-discovery-sim/.Rbuildignore @@ -0,0 +1,4 @@ +^renv$ +^renv\.lock$ +^.*\.Rproj$ +^\.Rproj\.user$ diff --git a/analysis/block-discovery-sim/.Rprofile b/analysis/block-discovery-sim/.Rprofile new file mode 100644 index 0000000..81b960f --- /dev/null +++ b/analysis/block-discovery-sim/.Rprofile @@ -0,0 +1 @@ +source("renv/activate.R") diff --git a/analysis/block-discovery-sim/.gitignore b/analysis/block-discovery-sim/.gitignore new file mode 100644 index 0000000..aa3aea8 --- /dev/null +++ b/analysis/block-discovery-sim/.gitignore @@ -0,0 +1,5 @@ +.Rproj.user +.RData +.Rhistory +*nb.html +rsconnect \ No newline at end of file diff --git a/analysis/block-discovery-sim/DESCRIPTION b/analysis/block-discovery-sim/DESCRIPTION new file mode 100644 index 0000000..325061a --- /dev/null +++ b/analysis/block-discovery-sim/DESCRIPTION @@ -0,0 +1,18 @@ +Package: blockdiscoverysim +Title: Block Discovery Simulator +Version: 0.0.0.9000 +Description: Simple Simulation for Block Discovery +Encoding: UTF-8 +Roxygen: list(markdown = TRUE) +RoxygenNote: 7.2.3 +Depends: + shiny (>= 1.7.4.1), + tidyverse (>= 2.0.0), + purrr (>= 1.0.1), + VGAM (>= 1.1-8), + R6 (>= 2.2.2), + plotly (>= 4.10.2) +Suggests: + devtools, + testthat (>= 3.0.0) +Config/testthat/edition: 3 diff --git a/analysis/block-discovery-sim/R/collate.R b/analysis/block-discovery-sim/R/collate.R new file mode 100644 index 0000000..a260d99 --- /dev/null +++ b/analysis/block-discovery-sim/R/collate.R @@ -0,0 +1,19 @@ +# We do this hack because rsconnect doesn't seem to like us bundling the app +# as a package. + +order <- c( + 'R/partition.R', + 'R/stats.R', + 'R/node.R', + 'R/sim.R' +) + +library(R6) +library(purrr) +library(tidyverse) + +lapply(order, source) + +run <- function() { + rmarkdown::run('./block-discovery-sim.Rmd') +} \ No newline at end of file diff --git a/analysis/block-discovery-sim/R/node.R b/analysis/block-discovery-sim/R/node.R new file mode 100644 index 0000000..068043d --- /dev/null +++ b/analysis/block-discovery-sim/R/node.R @@ -0,0 +1,14 @@ +Node <- R6Class( + 'Node', + public = list( + node_id = NULL, + storage = NULL, + + initialize = function(node_id, storage) { + self$node_id = node_id + self$storage = storage + }, + + name = function() paste0('node ', self$node_id) + ) +) \ No newline at end of file diff --git a/analysis/block-discovery-sim/R/partition.R b/analysis/block-discovery-sim/R/partition.R new file mode 100644 index 0000000..f2f1620 --- /dev/null +++ b/analysis/block-discovery-sim/R/partition.R @@ -0,0 +1,18 @@ +#' Generates a random partition of a block array among a set of nodes. The +#' partitioning follows the supplied distribution. +#' +#' @param block_array a vector containing blocks +#' @param network_size the number of nodes in the network +#' @param distribution a sample generator which generates a vector of n +#' samples when called as distribution(n). +#' +partition <- function(block_array, network_size, distribution) { + buckets <- distribution(length(block_array)) + + # We won't attempt to shift the data, instead just checking that it is + # positive. + stopifnot(all(buckets >= 0)) + + buckets <- trunc(buckets * (network_size - 1) / max(buckets)) + 1 + sapply(1:network_size, function(i) which(buckets == i)) +} diff --git a/analysis/block-discovery-sim/R/sim.R b/analysis/block-discovery-sim/R/sim.R new file mode 100644 index 0000000..2e9d8c1 --- /dev/null +++ b/analysis/block-discovery-sim/R/sim.R @@ -0,0 +1,30 @@ +run_download_simulation <- function(swarm, max_steps, coding_rate) { + total_blocks <- sum(sapply(swarm, function(node) length(node$storage))) + required_blocks <- round(total_blocks * coding_rate) + completed_blocks <- 0 + storage <- c() + + step <- 1 + stats <- Stats$new() + while ((step < max_steps) && (completed_blocks < required_blocks)){ + neighbor <- swarm |> select_neighbor() + storage <- neighbor |> download_blocks(storage) + + completed_blocks <- length(storage) + stats$add_stat( + step = step, + selected_neighbor = neighbor$node_id, + total_blocks = total_blocks, + required_blocks = required_blocks, + completed_blocks = completed_blocks + ) + + step <- step + 1 + } + + stats$as_tibble() +} + +select_neighbor <- function(neighborhood) neighborhood[[sample(1:length(neighborhood), size = 1)]] + +download_blocks <- function(neighbor, storage) unique(c(neighbor$storage, storage)) diff --git a/analysis/block-discovery-sim/R/stats.R b/analysis/block-discovery-sim/R/stats.R new file mode 100644 index 0000000..ad5b70c --- /dev/null +++ b/analysis/block-discovery-sim/R/stats.R @@ -0,0 +1,17 @@ +Stats <- R6Class( + 'Stats', + public = list( + stats = NULL, + + initialize = function() { + self$stats = list(list()) + }, + + add_stat = function(...) { + self$stats <- c(self$stats, list(rlang::dots_list(...))) + self + }, + + as_tibble = function() purrr::map_df(self$stats, as_tibble) + ) +) \ No newline at end of file diff --git a/analysis/block-discovery-sim/README.md b/analysis/block-discovery-sim/README.md new file mode 100644 index 0000000..975c35b --- /dev/null +++ b/analysis/block-discovery-sim/README.md @@ -0,0 +1,37 @@ +Simple Block Discovery Simulator +================================ + +Simple simulator for understanding of block discovery dynamics. + +## Hosted Version + +You can access the block discovery simulator on [shinyapps](https://gmega.shinyapps.io/block-discovery-sim/) + +## Running + +You will need R 4.1.2 with [renv](https://rstudio.github.io/renv/) installed. I also strongly recommend you run this +from [RStudio](https://posit.co/products/open-source/rstudio/) as you will otherwise need to [install pandoc and set it up manually before running](https://stackoverflow.com/questions/28432607/pandoc-version-1-12-3-or-higher-is-required-and-was-not-found-r-shiny). + +Once that's cared for and you are in the R terminal (Console in RStudio), you will need to first install deps: + +```R +> renv::install() +``` + +If you are outside RStudio, then you will need to restart your R session. After that, you should load the package: + +```R +devtools::load_all() +``` + +run the tests: + +```R +testthat::test_package('blockdiscoverysim') +``` + +and, if all goes well, launch the simulator: + +```R +run() +``` diff --git a/analysis/block-discovery-sim/block-discovery-sim.Rmd b/analysis/block-discovery-sim/block-discovery-sim.Rmd new file mode 100644 index 0000000..97dfc7d --- /dev/null +++ b/analysis/block-discovery-sim/block-discovery-sim.Rmd @@ -0,0 +1,202 @@ +--- +title: "Block Discovery Sim" +output: html_document +runtime: shiny + +# rsconnect uses this +resource_files: +- R/node.R +- R/partition.R +- R/sim.R +- R/stats.R +--- + +## Goal + +The goal of this experiment is to understand -- under different assumptions about how blocks are partitioned among nodes -- how long a hypothetical downloader would take to discover enough blocks to make a successful download from storage nodes by randomly sampling the swarm. We therefore do not account for download times or network latency - we just measure how many times the node randomly samples the swarm before figuring out where enough of the blocks are. + +```{r echo = FALSE, message = FALSE} +library(shiny) +library(plotly) + +source('R/collate.R') + +knitr::opts_chunk$set(echo = FALSE, message = FALSE) +``` + +```{r} +runs <- 10 +max_steps <- Inf +``` + +```{r} +DISTRIBUTIONS <- list( + 'uniform' = runif, + 'exponential' = rexp, + 'pareto' = VGAM::rparetoI +) +``` + + +## Network + +* Select the parameters of the network you would like to use in the experiments. +* Preview the shape of the partitions by looking at the chart. +* Generate more random partitions by clicking "Generate Another". + +```{r} +fluidPage( + sidebarPanel( + numericInput( + 'swarm_size', + label = 'size of the swarm', + value = 20, + min = 1, + max = 10000 + ), + numericInput( + 'file_size', + label = 'number of blocks in the file', + value = 1000, + min = 1, + max = 1e6 + ), + selectInput( + 'partition_distribution', + label = 'shape of the distribution for the partitions', + choices = names(DISTRIBUTIONS) + ), + actionButton( + 'generate_network', + label = 'Generate Another' + ) + ), + mainPanel( + plotOutput('network_sample') + ) +) +``` + +```{r} +observe({ + input$generate_network + output$network_sample <- renderPlot({ + purrr::map_dfr( + generate_network( + number_of_blocks = input$file_size, + network_size = input$swarm_size, + partition_distribution = input$partition_distribution + ), + function(node) tibble(node_id = node$node_id, blocks = length(node$storage)) + ) %>% + ggplot() + + geom_bar( + aes(x = node_id, y = blocks), + stat = 'identity', + col = 'black', + fill = 'lightgray' + ) + + labs(x = 'node') + + theme_minimal() + })} +) +``` + +## Experiment + +Select the number of experiment runs. Each experiment will generate a network and then simulate a download operation where a hypothetical node: + +1. joins the swarm; +2. samples one neighbor per round in a round-based download protocol and asks for its block list. + +The experiment ends when the downloading node recovers "enough" blocks. If we let the total number of blocks in the file be $n$ and the coding rate $r$, then the simulation ends when the set of blocks $D$ discovered by the downloading node satisfies $\left|D\right| \geq n\times r$. + +We then show a "discovery curve": a curve that emerges as we look at the percentage of blocks the downloader has discovered so far as a function of the number of contacts it made. + +The curve is actually an average of all experiments, meaning that a point $(5, 10\%)$ should be interpreted as: "on average, after $5$ contacts, a downloader will have discovered $10\%$ of the blocks it needs to get a successful download". We show the $5^{th}$ percentile and the $95^{th}$ percentiles of the experiments as error bands around the average. + +```{r} +fluidPage( + fluidRow( + class='well', + column( + width = 6, + sliderInput('runs', 'How many experiments to run', min = 10, max = 10000, value = 10), + actionButton('do_run', 'Run') + ), + column( + width = 6, + numericInput('coding_rate', 'Coding rate (percentage of blocks required for a successful download)', + min = 0.1, max = 1.0, step = 0.05, value = 0.5) + ) + ) +) +``` + +```{r} +experiment_results <- reactive({ + lapply(1:input$runs, function(i) { + generate_network( + number_of_blocks = input$file_size, + network_size = input$swarm_size, + partition_distribution = input$partition_distribution + ) |> run_experiment(run_id = i, coding_rate = input$coding_rate) + }) +}) |> bindEvent( + input$do_run, + ignoreNULL = TRUE, + ignoreInit = TRUE +) +``` + +```{r} +renderPlotly({ + plot_results(do.call(rbind, experiment_results())) +}) +``` + +```{r} +generate_network <- function(number_of_blocks, network_size, partition_distribution) { + block_array <- sample(1:number_of_blocks, replace = FALSE) + + partitions <- partition(block_array, network_size, DISTRIBUTIONS[[partition_distribution]]) + sapply(1:network_size, function(i) Node$new( + node_id = i, + storage = partitions[[i]]) + ) +} +``` + +```{r} +run_experiment <- function(network, coding_rate, run_id = 0) { + run_download_simulation( + swarm = network, + coding_rate = coding_rate, + max_steps = max_steps + ) |> mutate( + run = run_id + ) +} +``` + +```{r} +plot_results <- function(results) { + stats <- results |> + mutate(completion = pmin(1.0, completed_blocks / required_blocks)) |> + group_by(step) |> + summarise( + average = mean(completion), + p_95 = quantile(completion, 0.95), + p_05 = quantile(completion, 0.05), + .groups = 'drop' + ) + + plotly::ggplotly(ggplot(stats, aes(x = step)) + + geom_line(aes(y = average), col = 'black', lwd = 1) + + geom_ribbon(aes(ymin = p_05, ymax = p_95), fill = 'grey80', alpha = 0.5) + + labs(x = 'contacts', y = 'blocks discovered (%)') + + scale_y_continuous(labels = scales::percent_format()) + + theme_minimal()) +} +``` + diff --git a/analysis/block-discovery-sim/block-discovery-sim.Rproj b/analysis/block-discovery-sim/block-discovery-sim.Rproj new file mode 100644 index 0000000..21a4da0 --- /dev/null +++ b/analysis/block-discovery-sim/block-discovery-sim.Rproj @@ -0,0 +1,17 @@ +Version: 1.0 + +RestoreWorkspace: Default +SaveWorkspace: Default +AlwaysSaveHistory: Default + +EnableCodeIndexing: Yes +UseSpacesForTab: Yes +NumSpacesForTab: 2 +Encoding: UTF-8 + +RnwWeave: Sweave +LaTeX: pdfLaTeX + +BuildType: Package +PackageUseDevtools: Yes +PackageInstallArgs: --no-multiarch --with-keep.source diff --git a/analysis/block-discovery-sim/renv.lock b/analysis/block-discovery-sim/renv.lock new file mode 100644 index 0000000..b16cc35 --- /dev/null +++ b/analysis/block-discovery-sim/renv.lock @@ -0,0 +1,2194 @@ +{ + "R": { 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the requested version of renv + version <- "0.17.3" + + # the project directory + project <- getwd() + + # figure out whether the autoloader is enabled + enabled <- local({ + + # first, check config option + override <- getOption("renv.config.autoloader.enabled") + if (!is.null(override)) + return(override) + + # next, check environment variables + # TODO: prefer using the configuration one in the future + envvars <- c( + "RENV_CONFIG_AUTOLOADER_ENABLED", + "RENV_AUTOLOADER_ENABLED", + "RENV_ACTIVATE_PROJECT" + ) + + for (envvar in envvars) { + envval <- Sys.getenv(envvar, unset = NA) + if (!is.na(envval)) + return(tolower(envval) %in% c("true", "t", "1")) + } + + # enable by default + TRUE + + }) + + if (!enabled) + return(FALSE) + + # avoid recursion + if (identical(getOption("renv.autoloader.running"), TRUE)) { + warning("ignoring recursive attempt to run renv autoloader") + return(invisible(TRUE)) + } + + # signal that we're loading renv during R startup + options(renv.autoloader.running = TRUE) + on.exit(options(renv.autoloader.running = NULL), add = TRUE) + + # signal that we've consented to use renv + options(renv.consent = TRUE) + + # load the 'utils' package eagerly -- this ensures that renv shims, which + # mask 'utils' packages, will come first on the search path + library(utils, lib.loc = .Library) + + # unload renv if it's already been loaded + if ("renv" %in% loadedNamespaces()) + unloadNamespace("renv") + + # load bootstrap tools + `%||%` <- function(x, y) { + if (is.environment(x) || length(x)) x else y + } + + `%??%` <- function(x, y) { + if (is.null(x)) y else x + } + + bootstrap <- function(version, library) { + + # attempt to download renv + tarball <- tryCatch(renv_bootstrap_download(version), error = identity) + if (inherits(tarball, "error")) + stop("failed to download renv ", version) + + # now attempt to install + status <- tryCatch(renv_bootstrap_install(version, tarball, library), error = identity) + if (inherits(status, "error")) + stop("failed to install renv ", version) + + } + + renv_bootstrap_tests_running <- function() { + getOption("renv.tests.running", default = FALSE) + } + + renv_bootstrap_repos <- function() { + + # get CRAN repository + cran <- getOption("renv.repos.cran", "https://cloud.r-project.org") + + # check for repos override + repos <- Sys.getenv("RENV_CONFIG_REPOS_OVERRIDE", unset = NA) + if (!is.na(repos)) { + + # check for RSPM; if set, use a fallback repository for renv + rspm <- Sys.getenv("RSPM", unset = NA) + if (identical(rspm, repos)) + repos <- c(RSPM = rspm, CRAN = cran) + + return(repos) + + } + + # check for lockfile repositories + repos <- tryCatch(renv_bootstrap_repos_lockfile(), error = identity) + if (!inherits(repos, "error") && length(repos)) + return(repos) + + # if we're testing, re-use the test repositories + if (renv_bootstrap_tests_running()) { + repos <- getOption("renv.tests.repos") + if (!is.null(repos)) + return(repos) + } + + # retrieve current repos + repos <- getOption("repos") + + # ensure @CRAN@ entries are resolved + repos[repos == "@CRAN@"] <- cran + + # add in renv.bootstrap.repos if set + default <- c(FALLBACK = "https://cloud.r-project.org") + extra <- getOption("renv.bootstrap.repos", default = default) + repos <- c(repos, extra) + + # remove duplicates that might've snuck in + dupes <- duplicated(repos) | duplicated(names(repos)) + repos[!dupes] + + } + + renv_bootstrap_repos_lockfile <- function() { + + lockpath <- Sys.getenv("RENV_PATHS_LOCKFILE", unset = "renv.lock") + if (!file.exists(lockpath)) + return(NULL) + + lockfile <- tryCatch(renv_json_read(lockpath), error = identity) + if (inherits(lockfile, "error")) { + warning(lockfile) + return(NULL) + } + + repos <- lockfile$R$Repositories + if (length(repos) == 0) + return(NULL) + + keys <- vapply(repos, `[[`, "Name", FUN.VALUE = character(1)) + vals <- vapply(repos, `[[`, "URL", FUN.VALUE = character(1)) + names(vals) <- keys + + return(vals) + + } + + renv_bootstrap_download <- function(version) { + + # if the renv version number has 4 components, assume it must + # be retrieved via github + nv <- numeric_version(version) + components <- unclass(nv)[[1]] + + # if this appears to be a development version of 'renv', we'll + # try to restore from github + dev <- length(components) == 4L + + # begin collecting different methods for finding renv + methods <- c( + renv_bootstrap_download_tarball, + if (dev) + renv_bootstrap_download_github + else c( + renv_bootstrap_download_cran_latest, + renv_bootstrap_download_cran_archive + ) + ) + + for (method in methods) { + path <- tryCatch(method(version), error = identity) + if (is.character(path) && file.exists(path)) + return(path) + } + + stop("failed to download renv ", version) + + } + + renv_bootstrap_download_impl <- function(url, destfile) { + + mode <- "wb" + + # https://bugs.r-project.org/bugzilla/show_bug.cgi?id=17715 + fixup <- + Sys.info()[["sysname"]] == "Windows" && + substring(url, 1L, 5L) == "file:" + + if (fixup) + mode <- "w+b" + + args <- list( + url = url, + destfile = destfile, + mode = mode, + quiet = TRUE + ) + + if ("headers" %in% names(formals(utils::download.file))) + args$headers <- renv_bootstrap_download_custom_headers(url) + + do.call(utils::download.file, args) + + } + + renv_bootstrap_download_custom_headers <- function(url) { + + headers <- getOption("renv.download.headers") + if (is.null(headers)) + return(character()) + + if (!is.function(headers)) + stopf("'renv.download.headers' is not a function") + + headers <- headers(url) + if (length(headers) == 0L) + return(character()) + + if (is.list(headers)) + headers <- unlist(headers, recursive = FALSE, use.names = TRUE) + + ok <- + is.character(headers) && + is.character(names(headers)) && + all(nzchar(names(headers))) + + if (!ok) + stop("invocation of 'renv.download.headers' did not return a named character vector") + + headers + + } + + renv_bootstrap_download_cran_latest <- function(version) { + + spec <- renv_bootstrap_download_cran_latest_find(version) + type <- spec$type + repos <- spec$repos + + message("* Downloading renv ", version, " ... ", appendLF = FALSE) + + baseurl <- utils::contrib.url(repos = repos, type = type) + ext <- if (identical(type, "source")) + ".tar.gz" + else if (Sys.info()[["sysname"]] == "Windows") + ".zip" + else + ".tgz" + name <- sprintf("renv_%s%s", version, ext) + url <- paste(baseurl, name, sep = "/") + + destfile <- file.path(tempdir(), name) + status <- tryCatch( + renv_bootstrap_download_impl(url, destfile), + condition = identity + ) + + if (inherits(status, "condition")) { + message("FAILED") + return(FALSE) + } + + # report success and return + message("OK (downloaded ", type, ")") + destfile + + } + + renv_bootstrap_download_cran_latest_find <- function(version) { + + # check whether binaries are supported on this system + binary <- + getOption("renv.bootstrap.binary", default = TRUE) && + !identical(.Platform$pkgType, "source") && + !identical(getOption("pkgType"), "source") && + Sys.info()[["sysname"]] %in% c("Darwin", "Windows") + + types <- c(if (binary) "binary", "source") + + # iterate over types + repositories + for (type in types) { + for (repos in renv_bootstrap_repos()) { + + # retrieve package database + db <- tryCatch( + as.data.frame( + utils::available.packages(type = type, repos = repos), + stringsAsFactors = FALSE + ), + error = identity + ) + + if (inherits(db, "error")) + next + + # check for compatible entry + entry <- db[db$Package %in% "renv" & db$Version %in% version, ] + if (nrow(entry) == 0) + next + + # found it; return spec to caller + spec <- list(entry = entry, type = type, repos = repos) + return(spec) + + } + } + + # if we got here, we failed to find renv + fmt <- "renv %s is not available from your declared package repositories" + stop(sprintf(fmt, version)) + + } + + renv_bootstrap_download_cran_archive <- function(version) { + + name <- sprintf("renv_%s.tar.gz", version) + repos <- renv_bootstrap_repos() + urls <- file.path(repos, "src/contrib/Archive/renv", name) + destfile <- file.path(tempdir(), name) + + message("* Downloading renv ", version, " ... ", appendLF = FALSE) + + for (url in urls) { + + status <- tryCatch( + renv_bootstrap_download_impl(url, destfile), + condition = identity + ) + + if (identical(status, 0L)) { + message("OK") + return(destfile) + } + + } + + message("FAILED") + return(FALSE) + + } + + renv_bootstrap_download_tarball <- function(version) { + + # if the user has provided the path to a tarball via + # an environment variable, then use it + tarball <- Sys.getenv("RENV_BOOTSTRAP_TARBALL", unset = NA) + if (is.na(tarball)) + return() + + # allow directories + if (dir.exists(tarball)) { + name <- sprintf("renv_%s.tar.gz", version) + tarball <- file.path(tarball, name) + } + + # bail if it doesn't exist + if (!file.exists(tarball)) { + + # let the user know we weren't able to honour their request + fmt <- "* RENV_BOOTSTRAP_TARBALL is set (%s) but does not exist." + msg <- sprintf(fmt, tarball) + warning(msg) + + # bail + return() + + } + + fmt <- "* Bootstrapping with tarball at path '%s'." + msg <- sprintf(fmt, tarball) + message(msg) + + tarball + + } + + renv_bootstrap_download_github <- function(version) { + + enabled <- Sys.getenv("RENV_BOOTSTRAP_FROM_GITHUB", unset = "TRUE") + if (!identical(enabled, "TRUE")) + return(FALSE) + + # prepare download options + pat <- Sys.getenv("GITHUB_PAT") + if (nzchar(Sys.which("curl")) && nzchar(pat)) { + fmt <- "--location --fail --header \"Authorization: token %s\"" + extra <- sprintf(fmt, pat) + saved <- options("download.file.method", "download.file.extra") + options(download.file.method = "curl", download.file.extra = extra) + on.exit(do.call(base::options, saved), add = TRUE) + } else if (nzchar(Sys.which("wget")) && nzchar(pat)) { + fmt <- "--header=\"Authorization: token %s\"" + extra <- sprintf(fmt, pat) + saved <- options("download.file.method", "download.file.extra") + options(download.file.method = "wget", download.file.extra = extra) + on.exit(do.call(base::options, saved), add = TRUE) + } + + message("* Downloading renv ", version, " from GitHub ... ", appendLF = FALSE) + + url <- file.path("https://api.github.com/repos/rstudio/renv/tarball", version) + name <- sprintf("renv_%s.tar.gz", version) + destfile <- file.path(tempdir(), name) + + status <- tryCatch( + renv_bootstrap_download_impl(url, destfile), + condition = identity + ) + + if (!identical(status, 0L)) { + message("FAILED") + return(FALSE) + } + + message("OK") + return(destfile) + + } + + renv_bootstrap_install <- function(version, tarball, library) { + + # attempt to install it into project library + message("* Installing renv ", version, " ... ", appendLF = FALSE) + dir.create(library, showWarnings = FALSE, recursive = TRUE) + + # invoke using system2 so we can capture and report output + bin <- R.home("bin") + exe <- if (Sys.info()[["sysname"]] == "Windows") "R.exe" else "R" + r <- file.path(bin, exe) + + args <- c( + "--vanilla", "CMD", "INSTALL", "--no-multiarch", + "-l", shQuote(path.expand(library)), + shQuote(path.expand(tarball)) + ) + + output <- system2(r, args, stdout = TRUE, stderr = TRUE) + message("Done!") + + # check for successful install + status <- attr(output, "status") + if (is.numeric(status) && !identical(status, 0L)) { + header <- "Error installing renv:" + lines <- paste(rep.int("=", nchar(header)), collapse = "") + text <- c(header, lines, output) + writeLines(text, con = stderr()) + } + + status + + } + + renv_bootstrap_platform_prefix <- function() { + + # construct version prefix + version <- paste(R.version$major, R.version$minor, sep = ".") + prefix <- paste("R", numeric_version(version)[1, 1:2], sep = "-") + + # include SVN revision for development versions of R + # (to avoid sharing platform-specific artefacts with released versions of R) + devel <- + identical(R.version[["status"]], "Under development (unstable)") || + identical(R.version[["nickname"]], "Unsuffered Consequences") + + if (devel) + prefix <- paste(prefix, R.version[["svn rev"]], sep = "-r") + + # build list of path components + components <- c(prefix, R.version$platform) + + # include prefix if provided by user + prefix <- renv_bootstrap_platform_prefix_impl() + if (!is.na(prefix) && nzchar(prefix)) + components <- c(prefix, components) + + # build prefix + paste(components, collapse = "/") + + } + + renv_bootstrap_platform_prefix_impl <- function() { + + # if an explicit prefix has been supplied, use it + prefix <- Sys.getenv("RENV_PATHS_PREFIX", unset = NA) + if (!is.na(prefix)) + return(prefix) + + # if the user has requested an automatic prefix, generate it + auto <- Sys.getenv("RENV_PATHS_PREFIX_AUTO", unset = NA) + if (auto %in% c("TRUE", "True", "true", "1")) + return(renv_bootstrap_platform_prefix_auto()) + + # empty string on failure + "" + + } + + renv_bootstrap_platform_prefix_auto <- function() { + + prefix <- tryCatch(renv_bootstrap_platform_os(), error = identity) + if (inherits(prefix, "error") || prefix %in% "unknown") { + + msg <- paste( + "failed to infer current operating system", + "please file a bug report at https://github.com/rstudio/renv/issues", + sep = "; " + ) + + warning(msg) + + } + + prefix + + } + + renv_bootstrap_platform_os <- function() { + + sysinfo <- Sys.info() + sysname <- sysinfo[["sysname"]] + + # handle Windows + macOS up front + if (sysname == "Windows") + return("windows") + else if (sysname == "Darwin") + return("macos") + + # check for os-release files + for (file in c("/etc/os-release", "/usr/lib/os-release")) + if (file.exists(file)) + return(renv_bootstrap_platform_os_via_os_release(file, sysinfo)) + + # check for redhat-release files + if (file.exists("/etc/redhat-release")) + return(renv_bootstrap_platform_os_via_redhat_release()) + + "unknown" + + } + + renv_bootstrap_platform_os_via_os_release <- function(file, sysinfo) { + + # read /etc/os-release + release <- utils::read.table( + file = file, + sep = "=", + quote = c("\"", "'"), + col.names = c("Key", "Value"), + comment.char = "#", + stringsAsFactors = FALSE + ) + + vars <- as.list(release$Value) + names(vars) <- release$Key + + # get os name + os <- tolower(sysinfo[["sysname"]]) + + # read id + id <- "unknown" + for (field in c("ID", "ID_LIKE")) { + if (field %in% names(vars) && nzchar(vars[[field]])) { + id <- vars[[field]] + break + } + } + + # read version + version <- "unknown" + for (field in c("UBUNTU_CODENAME", "VERSION_CODENAME", "VERSION_ID", "BUILD_ID")) { + if (field %in% names(vars) && nzchar(vars[[field]])) { + version <- vars[[field]] + break + } + } + + # join together + paste(c(os, id, version), collapse = "-") + + } + + renv_bootstrap_platform_os_via_redhat_release <- function() { + + # read /etc/redhat-release + contents <- readLines("/etc/redhat-release", warn = FALSE) + + # infer id + id <- if (grepl("centos", contents, ignore.case = TRUE)) + "centos" + else if (grepl("redhat", contents, ignore.case = TRUE)) + "redhat" + else + "unknown" + + # try to find a version component (very hacky) + version <- "unknown" + + parts <- strsplit(contents, "[[:space:]]")[[1L]] + for (part in parts) { + + nv <- tryCatch(numeric_version(part), error = identity) + if (inherits(nv, "error")) + next + + version <- nv[1, 1] + break + + } + + paste(c("linux", id, version), collapse = "-") + + } + + renv_bootstrap_library_root_name <- function(project) { + + # use project name as-is if requested + asis <- Sys.getenv("RENV_PATHS_LIBRARY_ROOT_ASIS", unset = "FALSE") + if (asis) + return(basename(project)) + + # otherwise, disambiguate based on project's path + id <- substring(renv_bootstrap_hash_text(project), 1L, 8L) + paste(basename(project), id, sep = "-") + + } + + renv_bootstrap_library_root <- function(project) { + + prefix <- renv_bootstrap_profile_prefix() + + path <- Sys.getenv("RENV_PATHS_LIBRARY", unset = NA) + if (!is.na(path)) + return(paste(c(path, prefix), collapse = "/")) + + path <- renv_bootstrap_library_root_impl(project) + if (!is.null(path)) { + name <- renv_bootstrap_library_root_name(project) + return(paste(c(path, prefix, name), collapse = "/")) + } + + renv_bootstrap_paths_renv("library", project = project) + + } + + renv_bootstrap_library_root_impl <- function(project) { + + root <- Sys.getenv("RENV_PATHS_LIBRARY_ROOT", unset = NA) + if (!is.na(root)) + return(root) + + type <- renv_bootstrap_project_type(project) + if (identical(type, "package")) { + userdir <- renv_bootstrap_user_dir() + return(file.path(userdir, "library")) + } + + } + + renv_bootstrap_validate_version <- function(version) { + + loadedversion <- utils::packageDescription("renv", fields = "Version") + if (version == loadedversion) + return(TRUE) + + # assume four-component versions are from GitHub; + # three-component versions are from CRAN + components <- strsplit(loadedversion, "[.-]")[[1]] + remote <- if (length(components) == 4L) + paste("rstudio/renv", loadedversion, sep = "@") + else + paste("renv", loadedversion, sep = "@") + + fmt <- paste( + "renv %1$s was loaded from project library, but this project is configured to use renv %2$s.", + "Use `renv::record(\"%3$s\")` to record renv %1$s in the lockfile.", + "Use `renv::restore(packages = \"renv\")` to install renv %2$s into the project library.", + sep = "\n" + ) + + msg <- sprintf(fmt, loadedversion, version, remote) + warning(msg, call. = FALSE) + + FALSE + + } + + renv_bootstrap_hash_text <- function(text) { + + hashfile <- tempfile("renv-hash-") + on.exit(unlink(hashfile), add = TRUE) + + writeLines(text, con = hashfile) + tools::md5sum(hashfile) + + } + + renv_bootstrap_load <- function(project, libpath, version) { + + # try to load renv from the project library + if (!requireNamespace("renv", lib.loc = libpath, quietly = TRUE)) + return(FALSE) + + # warn if the version of renv loaded does not match + renv_bootstrap_validate_version(version) + + # execute renv load hooks, if any + hooks <- getHook("renv::autoload") + for (hook in hooks) + if (is.function(hook)) + tryCatch(hook(), error = warning) + + # load the project + renv::load(project) + + TRUE + + } + + renv_bootstrap_profile_load <- function(project) { + + # if RENV_PROFILE is already set, just use that + profile <- Sys.getenv("RENV_PROFILE", unset = NA) + if (!is.na(profile) && nzchar(profile)) + return(profile) + + # check for a profile file (nothing to do if it doesn't exist) + path <- renv_bootstrap_paths_renv("profile", profile = FALSE, project = project) + if (!file.exists(path)) + return(NULL) + + # read the profile, and set it if it exists + contents <- readLines(path, warn = FALSE) + if (length(contents) == 0L) + return(NULL) + + # set RENV_PROFILE + profile <- contents[[1L]] + if (!profile %in% c("", "default")) + Sys.setenv(RENV_PROFILE = profile) + + profile + + } + + renv_bootstrap_profile_prefix <- function() { + profile <- renv_bootstrap_profile_get() + if (!is.null(profile)) + return(file.path("profiles", profile, "renv")) + } + + renv_bootstrap_profile_get <- function() { + profile <- Sys.getenv("RENV_PROFILE", unset = "") + renv_bootstrap_profile_normalize(profile) + } + + renv_bootstrap_profile_set <- function(profile) { + profile <- renv_bootstrap_profile_normalize(profile) + if (is.null(profile)) + Sys.unsetenv("RENV_PROFILE") + else + Sys.setenv(RENV_PROFILE = profile) + } + + renv_bootstrap_profile_normalize <- function(profile) { + + if (is.null(profile) || profile %in% c("", "default")) + return(NULL) + + profile + + } + + renv_bootstrap_path_absolute <- function(path) { + + substr(path, 1L, 1L) %in% c("~", "/", "\\") || ( + substr(path, 1L, 1L) %in% c(letters, LETTERS) && + substr(path, 2L, 3L) %in% c(":/", ":\\") + ) + + } + + renv_bootstrap_paths_renv <- function(..., profile = TRUE, project = NULL) { + renv <- Sys.getenv("RENV_PATHS_RENV", unset = "renv") + root <- if (renv_bootstrap_path_absolute(renv)) NULL else project + prefix <- if (profile) renv_bootstrap_profile_prefix() + components <- c(root, renv, prefix, ...) + paste(components, collapse = "/") + } + + renv_bootstrap_project_type <- function(path) { + + descpath <- file.path(path, "DESCRIPTION") + if (!file.exists(descpath)) + return("unknown") + + desc <- tryCatch( + read.dcf(descpath, all = TRUE), + error = identity + ) + + if (inherits(desc, "error")) + return("unknown") + + type <- desc$Type + if (!is.null(type)) + return(tolower(type)) + + package <- desc$Package + if (!is.null(package)) + return("package") + + "unknown" + + } + + renv_bootstrap_user_dir <- function() { + dir <- renv_bootstrap_user_dir_impl() + path.expand(chartr("\\", "/", dir)) + } + + renv_bootstrap_user_dir_impl <- function() { + + # use local override if set + override <- getOption("renv.userdir.override") + if (!is.null(override)) + return(override) + + # use R_user_dir if available + tools <- asNamespace("tools") + if (is.function(tools$R_user_dir)) + return(tools$R_user_dir("renv", "cache")) + + # try using our own backfill for older versions of R + envvars <- c("R_USER_CACHE_DIR", "XDG_CACHE_HOME") + for (envvar in envvars) { + root <- Sys.getenv(envvar, unset = NA) + if (!is.na(root)) + return(file.path(root, "R/renv")) + } + + # use platform-specific default fallbacks + if (Sys.info()[["sysname"]] == "Windows") + file.path(Sys.getenv("LOCALAPPDATA"), "R/cache/R/renv") + else if (Sys.info()[["sysname"]] == "Darwin") + "~/Library/Caches/org.R-project.R/R/renv" + else + "~/.cache/R/renv" + + } + + + renv_json_read <- function(file = NULL, text = NULL) { + + jlerr <- NULL + + # if jsonlite is loaded, use that instead + if ("jsonlite" %in% loadedNamespaces()) { + + json <- catch(renv_json_read_jsonlite(file, text)) + if (!inherits(json, "error")) + return(json) + + jlerr <- json + + } + + # otherwise, fall back to the default JSON reader + json <- catch(renv_json_read_default(file, text)) + if (!inherits(json, "error")) + return(json) + + # report an error + if (!is.null(jlerr)) + stop(jlerr) + else + stop(json) + + } + + renv_json_read_jsonlite <- function(file = NULL, text = NULL) { + text <- paste(text %||% read(file), collapse = "\n") + jsonlite::fromJSON(txt = text, simplifyVector = FALSE) + } + + renv_json_read_default <- function(file = NULL, text = NULL) { + + # find strings in the JSON + text <- paste(text %||% read(file), collapse = "\n") + pattern <- '["](?:(?:\\\\.)|(?:[^"\\\\]))*?["]' + locs <- gregexpr(pattern, text, perl = TRUE)[[1]] + + # if any are found, replace them with placeholders + replaced <- text + strings <- character() + replacements <- character() + + if (!identical(c(locs), -1L)) { + + # get the string values + starts <- locs + ends <- locs + attr(locs, "match.length") - 1L + strings <- substring(text, starts, ends) + + # only keep those requiring escaping + strings <- grep("[[\\]{}:]", strings, perl = TRUE, value = TRUE) + + # compute replacements + replacements <- sprintf('"\032%i\032"', seq_along(strings)) + + # replace the strings + mapply(function(string, replacement) { + replaced <<- sub(string, replacement, replaced, fixed = TRUE) + }, strings, replacements) + + } + + # transform the JSON into something the R parser understands + transformed <- replaced + transformed <- gsub("{}", "`names<-`(list(), character())", transformed, fixed = TRUE) + transformed <- gsub("[[{]", "list(", transformed, perl = TRUE) + transformed <- gsub("[]}]", ")", transformed, perl = TRUE) + transformed <- gsub(":", "=", transformed, fixed = TRUE) + text <- paste(transformed, collapse = "\n") + + # parse it + json <- parse(text = text, keep.source = FALSE, srcfile = NULL)[[1L]] + + # construct map between source strings, replaced strings + map <- as.character(parse(text = strings)) + names(map) <- as.character(parse(text = replacements)) + + # convert to list + map <- as.list(map) + + # remap strings in object + remapped <- renv_json_remap(json, map) + + # evaluate + eval(remapped, envir = baseenv()) + + } + + renv_json_remap <- function(json, map) { + + # fix names + if (!is.null(names(json))) { + lhs <- match(names(json), names(map), nomatch = 0L) + rhs <- match(names(map), names(json), nomatch = 0L) + names(json)[rhs] <- map[lhs] + } + + # fix values + if (is.character(json)) + return(map[[json]] %||% json) + + # handle true, false, null + if (is.name(json)) { + text <- as.character(json) + if (text == "true") + return(TRUE) + else if (text == "false") + return(FALSE) + else if (text == "null") + return(NULL) + } + + # recurse + if (is.recursive(json)) { + for (i in seq_along(json)) { + json[i] <- list(renv_json_remap(json[[i]], map)) + } + } + + json + + } + + # load the renv profile, if any + renv_bootstrap_profile_load(project) + + # construct path to library root + root <- renv_bootstrap_library_root(project) + + # construct library prefix for platform + prefix <- renv_bootstrap_platform_prefix() + + # construct full libpath + libpath <- file.path(root, prefix) + + # attempt to load + if (renv_bootstrap_load(project, libpath, version)) + return(TRUE) + + # load failed; inform user we're about to bootstrap + prefix <- paste("# Bootstrapping renv", version) + postfix <- paste(rep.int("-", 77L - nchar(prefix)), collapse = "") + header <- paste(prefix, postfix) + message(header) + + # perform bootstrap + bootstrap(version, libpath) + + # exit early if we're just testing bootstrap + if (!is.na(Sys.getenv("RENV_BOOTSTRAP_INSTALL_ONLY", unset = NA))) + return(TRUE) + + # try again to load + if (requireNamespace("renv", lib.loc = libpath, quietly = TRUE)) { + message("* Successfully installed and loaded renv ", version, ".") + return(renv::load()) + } + + # failed to download or load renv; warn the user + msg <- c( + "Failed to find an renv installation: the project will not be loaded.", + "Use `renv::activate()` to re-initialize the project." + ) + + warning(paste(msg, collapse = "\n"), call. = FALSE) + +}) diff --git a/analysis/block-discovery-sim/renv/settings.json b/analysis/block-discovery-sim/renv/settings.json new file mode 100644 index 0000000..1a06760 --- /dev/null +++ b/analysis/block-discovery-sim/renv/settings.json @@ -0,0 +1,17 @@ +{ + "bioconductor.version": null, + "external.libraries": [], + "ignored.packages": [], + "package.dependency.fields": [ + "Imports", + "Depends", + "LinkingTo" + ], + "r.version": null, + "snapshot.type": "implicit", + "use.cache": true, + "vcs.ignore.cellar": true, + "vcs.ignore.library": true, + "vcs.ignore.local": true, + "vcs.manage.ignores": true +} diff --git a/analysis/block-discovery-sim/tests/testthat.R b/analysis/block-discovery-sim/tests/testthat.R new file mode 100644 index 0000000..41a568e --- /dev/null +++ b/analysis/block-discovery-sim/tests/testthat.R @@ -0,0 +1,11 @@ +# This file is part of the standard setup for testthat. +# It is recommended that you do not modify it. +# +# Where should you do additional test configuration? +# Learn more about the roles of various files in: +# * https://r-pkgs.org/tests.html +# * https://testthat.r-lib.org/reference/test_package.html#special-files + +library(testthat) + +test_check("blockdiscoverysim") diff --git a/analysis/block-discovery-sim/tests/testthat/test-partition.R b/analysis/block-discovery-sim/tests/testthat/test-partition.R new file mode 100644 index 0000000..b1115ca --- /dev/null +++ b/analysis/block-discovery-sim/tests/testthat/test-partition.R @@ -0,0 +1,18 @@ +test_that( + "should partition into linearly scaled buckets", { + samples <- c(1, 100, 500, 800, 850) + + partitions <- partition( + block_array = 1:5, + network_size = 4, + distribution = function(n) samples[1:n] + ) + + expect_equal(partitions, list( + c(1, 2), + c(3), + c(4), + c(5)) + ) + } +) diff --git a/analysis/block-discovery-sim/tests/testthat/test-stats.R b/analysis/block-discovery-sim/tests/testthat/test-stats.R new file mode 100644 index 0000000..e264de5 --- /dev/null +++ b/analysis/block-discovery-sim/tests/testthat/test-stats.R @@ -0,0 +1,17 @@ +test_that( + "should collect stats as they are input", { + stats <- Stats$new() + + stats$add_stat(a = 1, b = 2, name = 'hello') + stats$add_stat(a = 1, b = 3, name = 'world') + + expect_equal( + stats$as_tibble(), + tribble( + ~a, ~b, ~name, + 1, 2, 'hello', + 1, 3, 'world', + ) + ) + } +) diff --git a/analysis/block-discovery.Rmd b/analysis/block-discovery.Rmd new file mode 100644 index 0000000..03fd057 --- /dev/null +++ b/analysis/block-discovery.Rmd @@ -0,0 +1,104 @@ +--- +title: "Block Discovery Problem" +output: + bookdown::gitbook: + number_sections: false +--- + +$$ +\newcommand{\rv}[1]{\textbf{#1}} +\newcommand{\imin}{\rv{I}_{\text{min}}} +$$ + +## Problem Statement + +Let $F = \left\{b_1, \cdots, b_m\right\}$ be an erasure-coded file, and let $O = \left\{o_1, \cdots, o_n\right\}$ be a set of nodes storing that file. We define a _storage function_ $s \longrightarrow O \times 2^F$ as a function mapping subsets of $F$ into nodes in $O$. + +In the simplified block discovery problem, we have a _downloader node_ which is attempting to construct a subset $D \subseteq F$ of blocks by repeatedly sampling nodes from $O$. "Discovery", in this context, can be seen as the downloader node running a round-based protocol where, at round $i$, it samples a random contact $o_i$ and learns about $s(o_i)$. + +To make this slightly more formal, we denote $D_i \subseteq F$ to be the set of blocks that the downloader has learned after $i^{th}$ contact. By the way we state the protocol to work, we have that: + +$$ +\begin{equation} + D_i = D_{i - 1} \cup s(o_i) + (\#eq:discovery) +\end{equation} +$$ + +Since the file is erasure coded, the goal of the downloader is to learn some $D_i$ such that: + +$$ +\begin{equation} + \left|D_i\right| \geq c \times \left|F\right| + (\#eq:complete) +\end{equation} +$$ + +When $D_i$ satisfies Eq. \@ref(eq:complete), we say that $D_i$ is _complete_. We can then state the problem as follows. + +**Statement.** Let $\imin$ be a random variable representing the first round at which $D_i$ is complete. We want to estimate $F(i) = \mathbb{P}(\imin \leq i)$; namely, the probability that the downloader has discovered all relevant blocks by round $i$. + +## Case (1) - Erasure Coding but no Replication + +If we assume there is no replication then, unless we contact the same node twice, every node we contact contributes with new information. Indeed, the absence of replication implies, necessarily, that: + +$$ +\begin{equation} + \bigcap_{o \in O} s(o) = \emptyset + (\#eq:disjoint) +\end{equation} +$$ + +So that if we are contacting a new node at round $i$, we must necessarily have that: + +$$ +\begin{equation} + \left|D_{i}\right| \stackrel{1}{=} \left|D_{i - 1} \cup s(o_i)\right| \stackrel{2}{=} \left|D_{i - 1}\right| + \left|s(o_i)\right| + (\#eq:monotonic) +\end{equation} +$$ +Where (1) follows from Eq. \@ref(eq:discovery), and (2) follows from the $s(o_i)$ being disjoint (Eq. \@ref(eq:disjoint)). This leads to the corollary: + +**Corollary 1.** After $\lceil c \times n\rceil$ rounds, the downloader will necessarily have learned enough blocks to download $F$. + +which follows trivially from Eq. \@ref(eq:monotonic) and the implication that $D_{\lceil c \times n\rceil}$ must be complete. $\blacksquare$ + +As for $F(i)$, note that we can estimate the probability of completion by estimating the probability that $|D_i|$ is bigger than the completion number (Eq. \@ref(eq:complete)). How exactly that looks like and how tractable it is, however, depends on the formulation we give it. + +### Independent Partition Sizes + +Suppose we knew the distribution for partition sizes in $O$, i.e., we knew that the number of blocks assigned to a node in $O$ follows some distribution $\mathcal{B}$ (e.g., a truncated Gaussian). + +If we have a "large enough" network, this means we would be able to approximate the number of blocks assigned to each node as $m$ independent random variables $\rv{Y}_i$, where $\rv{Y}_i \sim \mathcal{B}$. In that case, we would be able to express the total number of blocks learned by the downloader by round $i$ as a random variable $\rv{L}_i$ which represents the sum of the iid random variables $\rv{Y}_j \sim \mathcal{B}$: + +$$ +\begin{equation} + \rv{L}_i \sim \sum_{j = 1}^{i} \rv{Y}_j + (\#eq:learning-sum) +\end{equation} +$$ + +The shape of the distribution would be the $i$-fold convolution of $\mathcal{B}$ with itself, which can be tractable for some distributions. + +More interestingly, though, Eq. \@ref(eq:learning-sum) allows us to express a $\mathcal{B}$-independent estimate of the average number of rounds a downloader will undergo before completing a download. We have that: + +$$ +\mathbb{E}(\rv{L}_i) \sim \sum_{j = 1}^i \mathbb{E}(\rv{Y}_j) = i\mathbb{E}(\rv{Y}) = i\times \mu_{\rv{Y}} +$$ + +We can then solve for $i$ and the completion condition to get: + +$$ +\begin{equation} + i \times \mu_{\rv{Y}} \geq c \times |F| \iff i \geq \frac{c \times |F|}{\mu_{\rv{Y}}} + (\#eq:average-completion) +\end{equation} +$$ + +note that this is intuitive to the point of being trivial. If we let $c = 1$, we get $i \geq |F|/\mu_{\rv{Y}}$, which just means that on average the node will have to sample a number of nodes equal to the number of blocks over the average partition size. In practice we can use $\overline{\mu_\rv{Y}} = \frac{1}{m}\sum_i \left|s(o_i)\right|$ instead of $\mu_{\rv{Y}}$ to estimate what $i$ can look like. + +### Non-Independent Partition Sizes + +If we cannot approximate partition sizes and independent random variables, then the problem changes. Stripping it down, we can cast it as follows. We have a set of integers $P = \{p_1, \cdots, p_m\}$ representing the sizes of each partition. We then want to understand the distribution of the partial sums for random permutations of $P$. + +As I understand it, there is no good way of addressing this without running simulations. The difference is that if we assume disjoint partitions then the simulations are a lot simpler as we do not need to track the contents of $D_i$. \ No newline at end of file