When tackling the mystical worlds of data structures and algorithms, our adventure brings us to a fascinating chapter today: shortest path algorithms in Ruby. These aren’t just any routes; we’re diving into the inner workings of Dijkstra’s and Bellman-Ford algorithms. Buckle up, as we unravel these algorithms, connecting dots in weighted graphs, with vivid coding illustrations that feel almost like stories whispering secrets to your computer.

Picture this: a sprawling network of city roads, each intersection bursting with possibility, each route offering an adventure or a shortcut. You stand there, map in hand, wondering which way will get you to Grandma’s house with your cookies still warm. Enter Dijkstra — your trusted GPS. Dijkstra’s algorithm is all about finding the minimum cost, or shortest path, from a starting node, which is basically you at this bustling intersection, to every other node, which represents every possible destination in the network.

The beauty of Dijkstra lies in its simplicity and efficiency for graphs with non-negative weights. It kicks off at your start node, setting its weight to zero (because that’s where you begin), while pegging all other nodes with a starting weight of infinity. From here, it saunters through the network, selecting the least contested paths, which are nodes with the smallest current weight, visiting neighbors, and updating paths if it finds a cheaper way. Here’s how you might code it in Ruby:

require 'set'

def dijkstra(graph, start_vertex)
  distances = {}
  previous_vertices = {}
  nodes = Set.new(graph.keys)

  graph.each do |node, _|
    distances[node] = Float::INFINITY
  end

  distances[start_vertex] = 0

  until nodes.empty?
    closest_node = nodes.min_by { |node| distances[node] }

    break if distances[closest_node] == Float::INFINITY

    nodes.delete(closest_node)
    graph[closest_node].each do |neighbor, cost|
      alternate_route = distances[closest_node] + cost
      if alternate_route < distances[neighbor]
        distances[neighbor] = alternate_route
        previous_vertices[neighbor] = closest_node
      end
    end
  end

  [distances, previous_vertices]
end

# Example usage
graph = {
  'A' => { 'B' => 5, 'C' => 1 },
  'B' => { 'A' => 5, 'C' => 2, 'D' => 1 },
  'C' => { 'A' => 1, 'B' => 2, 'D' => 4, 'E' => 8 },
  'D' => { 'B' => 1, 'C' => 4, 'E' => 3, 'F' => 6 },
  'E' => { 'C' => 8, 'D' => 3 },
  'F' => { 'D' => 6 }
}

distances, previous_vertices = dijkstra(graph, 'A')
puts "Shortest paths from A: #{distances}"

In the Ruby example, notice how each node casually joins the party, relentlessly pursuing the cheapest path. The Tentative Distance Hotel caters to passing nodes, updating their distance costs based on those luxurious new shortcuts found by Dijkstra.

Let me weave in a tale of curiosity — consider Bellman-Ford as the guardian angel for graphs where the world isn’t always rosy and bright, where negative weights lurk in the dark alleys. Unlike Dijkstra, Bellman-Ford doesn’t flinch at the presence of danger in the form of these negative numbers. Instead, it embraces them with open arms and checks paths for potential miracles that Dijkstra might ignore.

The Bellman-Ford algorithm wanders through each edge, not just from any old node, but visiting each edge multiple times. Why multiple times you ask? To ensure no sneaky negative cycles promise more than they should. Here’s how you could embrace the wisdom of Bellman-Ford in Ruby:

def bellman_ford(graph, start_vertex)
  distances = {}
  previous_vertices = {}

  graph.each_key do |node|
    distances[node] = Float::INFINITY
    previous_vertices[node] = nil
  end

  distances[start_vertex] = 0

  (graph.keys.size - 1).times do
    graph.each do |node, edges|
      edges.each do |neighbor, weight|
        if distances[node] + weight < distances[neighbor]
          distances[neighbor] = distances[node] + weight
          previous_vertices[neighbor] = node
        end
      end
    end
  end

  # Check for negative cycles
  graph.each do |node, edges|
    edges.each do |neighbor, weight|
      if distances[node] + weight < distances[neighbor]
        raise "Graph contains a negative-weight cycle"
      end
    end
  end

  [distances, previous_vertices]
end

# Example usage
graph = {
  'A' => { 'B' => 5, 'C' => 1 },
  'B' => { 'A' => 5, 'C' => 2, 'D' => 1 },
  'C' => { 'A' => 1, 'B' => 2, 'D' => 4, 'E' => 8 },
  'D' => { 'B' => 1, 'C' => 4, 'E' => 3, 'F' => 6 },
  'E' => { 'C' => 8, 'D' => 3 },
  'F' => { 'D' => 6 }
}

begin
  distances, previous_vertices = bellman_ford(graph, 'A')
  puts "Shortest paths from A: #{distances}"
rescue StandardError => e
  puts e.message
end

With this, Bellman-Ford strides through our colorful graph world, finding optimal paths even when challenged by negative numbers. It’s like a superhero in our algorithmic toolkit, making sure we don’t get lost amidst false promises.

Now let’s talk efficiency. Dijkstra’s algorithm, with its eager focus on the smaller stuff, runs briskly at a time complexity of O(E + V log V) when optimized using a priority queue, which lends it wings. Bellman-Ford takes a more cautious stroll, clocking in at O(VE), as it meticulously revisits each edge, ensuring every promise made is genuine, especially in those negative routes.

How do these algorithms write stories into our real lives? Consider global positioning systems calculating optimal shipping routes, data networks ensuring quick data packet transfers, or even game maps that efficiently lead an adventurer to their treasure, avoiding pesky pits and dragons.

Whether it’s Dijkstra politely navigating non-negative detours or Bellman-Ford fearlessly tackling negative landscapes, these algorithms find their rightful paths where the efficiency of route calculation writes its poetry across various domains. As we wrap up this journey, it’s time for you to forge onward, experimenting with these algorithms, tuning and twisting them in your own maps of nodes. Go ahead, plot your course, and let the Ruby magic guide you through the shortest paths yet untraveled.

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