feat-hamilitonian_cycle

graph_algorithms/hamilitonian_cycle.r

@@ -0,0 +1,102 @@
+# Hamiltonian Cycle Detection (Backtracking)
+#
+# The Hamiltonian Cycle problem determines whether there exists a cycle
+# in a graph that visits each vertex exactly once and returns to the start.
+# It is an NP-complete problem.
+#
+# This implementation uses backtracking to explore possible paths.
+#
+# Time Complexity: O(N!)
+# Space Complexity: O(N)
+#
+# Input: adjacency matrix (square matrix where adj[i][j] = 1 if edge exists)
+# Output: list containing:
+#   - has_cycle: TRUE/FALSE
+#   - cycle: list of vertices forming the Hamiltonian cycle (if found)
+#
+# Example usage at the end of this file.
+
+hamiltonian_cycle <- function(graph) {
+  num_vertices <- nrow(graph)
+  path <- rep(-1, num_vertices)
+
+  # Start at vertex 1 (can be any vertex)
+  path[1] <- 1
+
+  # Helper function to check if vertex v can be added at position pos
+  is_safe <- function(v, graph, path, pos) {
+    # Check if current vertex is adjacent to previous vertex
+    if (graph[path[pos - 1], v] == 0) return(FALSE)
+
+    # Check if vertex is already in the path
+    if (v %in% path[1:(pos - 1)]) return(FALSE)
+
+    return(TRUE)
+  }
+
+  # Recursive utility to build Hamiltonian cycle
+  ham_cycle_util <- function(graph, path, pos) {
+    if (pos == num_vertices + 1) {
+      # If last vertex connects to the first → cycle found
+      if (graph[path[num_vertices], path[1]] == 1) {
+        return(TRUE)
+      } else {
+        return(FALSE)
+      }
+    }
+
+    # Try adding each vertex as next candidate
+    for (v in 2:num_vertices) {
+      if (is_safe(v, graph, path, pos)) {
+        path[pos] <- v
+        if (ham_cycle_util(graph, path, pos + 1)) return(TRUE)
+        # Backtrack
+        path[pos] <- -1
+      }
+    }
+    return(FALSE)
+  }
+
+  # Start backtracking from vertex 1
+  if (ham_cycle_util(graph, path, 2)) {
+    path <- c(path, path[1])  # complete the cycle
+    return(list(has_cycle = TRUE, cycle = path))
+  } else {
+    return(list(has_cycle = FALSE, cycle = NULL))
+  }
+}
+
+# ============================
+# Example Usage and Test
+# ============================
+
+cat("=== Hamiltonian Cycle Detection ===\n")
+
+# Example graph (Adjacency Matrix)
+# Graph:
+# 1 - 2 - 3
+# |   |   |
+# 4 - 5 - 6
+# This graph contains a Hamiltonian cycle
+ham_graph <- matrix(
+  c(0,1,0,1,0,0,
+    1,0,1,1,1,0,
+    0,1,0,0,1,1,
+    1,1,0,0,1,0,
+    0,1,1,1,0,1,
+    0,0,1,0,1,0),
+  nrow = 6, byrow = TRUE
+)
+
+cat("Adjacency Matrix:\n")
+print(ham_graph)
+
+cat("\nRunning Hamiltonian Cycle detection...\n")
+result <- hamiltonian_cycle(ham_graph)
+
+if (result$has_cycle) {
+  cat("Hamiltonian Cycle found:\n")
+  cat("Cycle:", paste(result$cycle, collapse = " -> "), "\n")
+} else {
+  cat("No Hamiltonian Cycle exists in this graph.\n")
+}