Johnson’s All-Pairs Shortest Paths Algorithm in R

graph_algorithms/johnson_shortest_paths.r

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+# Johnson's All-Pairs Shortest Paths Algorithm
+#
+# Johnson's algorithm computes shortest paths between all pairs of vertices
+# in a sparse, weighted directed graph that may have negative edge weights
+# (but no negative cycles). It reweights edges using Bellman-Ford to remove
+# negative weights and then runs Dijkstra from each vertex.
+#
+# Time Complexity: O(V * E + V * (E log V)) with a binary heap priority queue
+# Space Complexity: O(V + E)
+#
+# Graph representation matches other files in this folder:
+# - Adjacency list: a named list where each name is a vertex index as a string
+# - Each entry is a list of edges: list(vertex = <int>, weight = <numeric>)
+
+# ----------------------------
+# Priority Queue (simple) API
+# ----------------------------
+create_priority_queue <- function() {
+  list(
+    elements = data.frame(vertex = integer(0), distance = numeric(0)),
+    size = 0
+  )
+}
+
+pq_insert <- function(pq, vertex, distance) {
+  pq$elements <- rbind(pq$elements, data.frame(vertex = vertex, distance = distance))
+  pq$size <- pq$size + 1
+  return(pq)
+}
+
+pq_extract_min <- function(pq) {
+  if (pq$size == 0) {
+    return(list(pq = pq, min_element = NULL))
+  }
+  min_idx <- which.min(pq$elements$distance)
+  min_element <- pq$elements[min_idx, ]
+  pq$elements <- pq$elements[-min_idx, ]
+  pq$size <- pq$size - 1
+  return(list(pq = pq, min_element = min_element))
+}
+
+pq_is_empty <- function(pq) {
+  return(pq$size == 0)
+}
+
+# -----------------------------------------------
+# Bellman-Ford potentials (no explicit supersource)
+# -----------------------------------------------
+# Equivalent to adding a new super-source s with 0-weight edges to all vertices,
+# by initializing h[i] = 0 for all i and relaxing edges (V-1) times.
+bellman_ford_potentials <- function(graph) {
+  # Collect all vertices appearing as sources or targets
+  all_vertices <- unique(c(names(graph), unlist(lapply(graph, function(x) sapply(x, function(e) e$vertex)))))
+  all_vertices <- as.numeric(all_vertices)
+  V <- max(all_vertices)
+
+  # Initialize h with zeros (super-source trick)
+  h <- rep(0, V)
+
+  # Relax edges V-1 times
+  for (i in 1:(V - 1)) {
+    updated <- FALSE
+    for (u_char in as.character(1:V)) {
+      if (!(u_char %in% names(graph))) next
+      u <- as.numeric(u_char)
+      for (edge in graph[[u_char]]) {
+        v <- edge$vertex
+        w <- edge$weight
+        if (h[u] + w < h[v]) {
+          h[v] <- h[u] + w
+          updated <- TRUE
+        }
+      }
+    }
+    if (!updated) break
+  }
+
+  # Check for negative-weight cycles
+  negative_cycle <- FALSE
+  for (u_char in as.character(1:V)) {
+    if (!(u_char %in% names(graph))) next
+    u <- as.numeric(u_char)
+    for (edge in graph[[u_char]]) {
+      v <- edge$vertex
+      w <- edge$weight
+      if (h[u] + w < h[v]) {
+        negative_cycle <- TRUE
+        break
+      }
+    }
+    if (negative_cycle) break
+  }
+
+  list(h = h, V = V, negative_cycle = negative_cycle)
+}
+
+# ---------------------------
+# Dijkstra on reweighted graph
+# ---------------------------
+dijkstra_on_adj <- function(graph, source, V) {
+  distances <- rep(Inf, V)
+  previous <- rep(-1, V)
+  visited <- rep(FALSE, V)
+
+  distances[source] <- 0
+  pq <- create_priority_queue()
+  pq <- pq_insert(pq, source, 0)
+
+  while (!pq_is_empty(pq)) {
+    res <- pq_extract_min(pq)
+    pq <- res$pq
+    cur <- res$min_element
+    if (is.null(cur)) break
+
+    u <- cur$vertex
+    if (visited[u]) next
+    visited[u] <- TRUE
+
+    u_char <- as.character(u)
+    if (u_char %in% names(graph)) {
+      for (edge in graph[[u_char]]) {
+        v <- edge$vertex
+        w <- edge$weight
+        if (!visited[v] && distances[u] + w < distances[v]) {
+          distances[v] <- distances[u] + w
+          previous[v] <- u
+          pq <- pq_insert(pq, v, distances[v])
+        }
+      }
+    }
+  }
+
+  list(distances = distances, previous = previous)
+}
+
+# ---------------------
+# Johnson's main driver
+# ---------------------
+johnson_shortest_paths <- function(graph) {
+  # Ensure all vertices 1..V exist as keys (even if empty list)
+  all_vertices <- unique(c(names(graph), unlist(lapply(graph, function(x) sapply(x, function(e) e$vertex)))))
+  all_vertices <- as.numeric(all_vertices)
+  V <- max(all_vertices)
+  for (u in as.character(1:V)) {
+    if (!(u %in% names(graph))) graph[[u]] <- list()
+  }
+
+  # Step 1: Bellman-Ford to get potentials h
+  bf <- bellman_ford_potentials(graph)
+  if (bf$negative_cycle) {
+    return(list(
+      distances = NULL,
+      negative_cycle = TRUE,
+      message = "Graph contains a negative-weight cycle; shortest paths undefined"
+    ))
+  }
+  h <- bf$h
+
+  # Step 2: Reweight edges to eliminate negative weights
+  reweighted <- vector("list", V)
+  names(reweighted) <- as.character(1:V)
+  for (i in 1:V) reweighted[[i]] <- list()
+  for (u_char in as.character(1:V)) {
+    u <- as.numeric(u_char)
+    for (edge in graph[[u_char]]) {
+      v <- edge$vertex
+      w <- edge$weight
+      w_prime <- w + h[u] - h[v]
+      reweighted[[u_char]] <- append(reweighted[[u_char]], list(list(vertex = v, weight = w_prime)))
+    }
+  }
+
+  # Step 3: Run Dijkstra from each vertex on the reweighted graph
+  dist_matrix <- matrix(Inf, nrow = V, ncol = V)
+  for (s in 1:V) {
+    dj <- dijkstra_on_adj(reweighted, s, V)
+    # Convert distances back to original weights
+    for (v in 1:V) {
+      if (!is.infinite(dj$distances[v])) {
+        dist_matrix[s, v] <- dj$distances[v] - h[s] + h[v]
+      }
+    }
+  }
+
+  list(
+    distances = dist_matrix,
+    negative_cycle = FALSE
+  )
+}
+
+# -----------------
+# Example / Demo
+# -----------------
+cat("=== Johnson's All-Pairs Shortest Paths Algorithm ===\n")
+
+# Convert the Java example (0-based) to 1-based indices used here
+# Java edges:
+# 0->1(3), 0->2(8), 0->4(-4), 1->3(1), 1->4(7), 2->1(4), 3->0(2), 3->2(-5), 4->3(6)
+j_graph <- list(
+  "1" = list(list(vertex = 2, weight = 3), list(vertex = 3, weight = 8), list(vertex = 5, weight = -4)),
+  "2" = list(list(vertex = 4, weight = 1), list(vertex = 5, weight = 7)),
+  "3" = list(list(vertex = 2, weight = 4)),
+  "4" = list(list(vertex = 1, weight = 2), list(vertex = 3, weight = -5)),
+  "5" = list(list(vertex = 4, weight = 6))
+)
+
+cat("Running Johnson on 5-vertex graph (with negative edges, no negative cycles) ...\n")
+j_res <- johnson_shortest_paths(j_graph)
+if (isTRUE(j_res$negative_cycle)) {
+  cat("Negative cycle detected. Shortest paths undefined.\n")
+} else {
+  cat("All-pairs shortest path distance matrix (rows = sources, cols = targets):\n")
+  print(j_res$distances)
+}