1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244
//! Bellman-Ford algorithms.
use crate::prelude::*;
use crate::visit::{IntoEdges, IntoNodeIdentifiers, NodeCount, NodeIndexable, VisitMap, Visitable};
use super::{FloatMeasure, NegativeCycle};
#[derive(Debug, Clone)]
pub struct Paths<NodeId, EdgeWeight> {
pub distances: Vec<EdgeWeight>,
pub predecessors: Vec<Option<NodeId>>,
}
/// \[Generic\] Compute shortest paths from node `source` to all other.
///
/// Using the [Bellman–Ford algorithm][bf]; negative edge costs are
/// permitted, but the graph must not have a cycle of negative weights
/// (in that case it will return an error).
///
/// On success, return one vec with path costs, and another one which points
/// out the predecessor of a node along a shortest path. The vectors
/// are indexed by the graph's node indices.
///
/// [bf]: https://en.wikipedia.org/wiki/Bellman%E2%80%93Ford_algorithm
///
/// # Example
/// ```rust
/// use petgraph::Graph;
/// use petgraph::algo::bellman_ford;
/// use petgraph::prelude::*;
///
/// let mut g = Graph::new();
/// let a = g.add_node(()); // node with no weight
/// let b = g.add_node(());
/// let c = g.add_node(());
/// let d = g.add_node(());
/// let e = g.add_node(());
/// let f = g.add_node(());
/// g.extend_with_edges(&[
/// (0, 1, 2.0),
/// (0, 3, 4.0),
/// (1, 2, 1.0),
/// (1, 5, 7.0),
/// (2, 4, 5.0),
/// (4, 5, 1.0),
/// (3, 4, 1.0),
/// ]);
///
/// // Graph represented with the weight of each edge
/// //
/// // 2 1
/// // a ----- b ----- c
/// // | 4 | 7 |
/// // d f | 5
/// // | 1 | 1 |
/// // \------ e ------/
///
/// let path = bellman_ford(&g, a);
/// assert!(path.is_ok());
/// let path = path.unwrap();
/// assert_eq!(path.distances, vec![ 0.0, 2.0, 3.0, 4.0, 5.0, 6.0]);
/// assert_eq!(path.predecessors, vec![None, Some(a),Some(b),Some(a), Some(d), Some(e)]);
///
/// // Node f (indice 5) can be reach from a with a path costing 6.
/// // Predecessor of f is Some(e) which predecessor is Some(d) which predecessor is Some(a).
/// // Thus the path from a to f is a <-> d <-> e <-> f
///
/// let graph_with_neg_cycle = Graph::<(), f32, Undirected>::from_edges(&[
/// (0, 1, -2.0),
/// (0, 3, -4.0),
/// (1, 2, -1.0),
/// (1, 5, -25.0),
/// (2, 4, -5.0),
/// (4, 5, -25.0),
/// (3, 4, -1.0),
/// ]);
///
/// assert!(bellman_ford(&graph_with_neg_cycle, NodeIndex::new(0)).is_err());
/// ```
pub fn bellman_ford<G>(
g: G,
source: G::NodeId,
) -> Result<Paths<G::NodeId, G::EdgeWeight>, NegativeCycle>
where
G: NodeCount + IntoNodeIdentifiers + IntoEdges + NodeIndexable,
G::EdgeWeight: FloatMeasure,
{
let ix = |i| g.to_index(i);
// Step 1 and Step 2: initialize and relax
let (distances, predecessors) = bellman_ford_initialize_relax(g, source);
// Step 3: check for negative weight cycle
for i in g.node_identifiers() {
for edge in g.edges(i) {
let j = edge.target();
let w = *edge.weight();
if distances[ix(i)] + w < distances[ix(j)] {
return Err(NegativeCycle(()));
}
}
}
Ok(Paths {
distances,
predecessors,
})
}
/// \[Generic\] Find the path of a negative cycle reachable from node `source`.
///
/// Using the [find_negative_cycle][nc]; will search the Graph for negative cycles using
/// [Bellman–Ford algorithm][bf]. If no negative cycle is found the function will return `None`.
///
/// If a negative cycle is found from source, return one vec with a path of `NodeId`s.
///
/// The time complexity of this algorithm should be the same as the Bellman-Ford (O(|V|·|E|)).
///
/// [nc]: https://blogs.asarkar.com/assets/docs/algorithms-curated/Negative-Weight%20Cycle%20Algorithms%20-%20Huang.pdf
/// [bf]: https://en.wikipedia.org/wiki/Bellman%E2%80%93Ford_algorithm
///
/// # Example
/// ```rust
/// use petgraph::Graph;
/// use petgraph::algo::find_negative_cycle;
/// use petgraph::prelude::*;
///
/// let graph_with_neg_cycle = Graph::<(), f32, Directed>::from_edges(&[
/// (0, 1, 1.),
/// (0, 2, 1.),
/// (0, 3, 1.),
/// (1, 3, 1.),
/// (2, 1, 1.),
/// (3, 2, -3.),
/// ]);
///
/// let path = find_negative_cycle(&graph_with_neg_cycle, NodeIndex::new(0));
/// assert_eq!(
/// path,
/// Some([NodeIndex::new(1), NodeIndex::new(3), NodeIndex::new(2)].to_vec())
/// );
/// ```
pub fn find_negative_cycle<G>(g: G, source: G::NodeId) -> Option<Vec<G::NodeId>>
where
G: NodeCount + IntoNodeIdentifiers + IntoEdges + NodeIndexable + Visitable,
G::EdgeWeight: FloatMeasure,
{
let ix = |i| g.to_index(i);
let mut path = Vec::<G::NodeId>::new();
// Step 1: initialize and relax
let (distance, predecessor) = bellman_ford_initialize_relax(g, source);
// Step 2: Check for negative weight cycle
'outer: for i in g.node_identifiers() {
for edge in g.edges(i) {
let j = edge.target();
let w = *edge.weight();
if distance[ix(i)] + w < distance[ix(j)] {
// Step 3: negative cycle found
let start = j;
let mut node = start;
let mut visited = g.visit_map();
// Go backward in the predecessor chain
loop {
let ancestor = match predecessor[ix(node)] {
Some(predecessor_node) => predecessor_node,
None => node, // no predecessor, self cycle
};
// We have only 2 ways to find the cycle and break the loop:
// 1. start is reached
if ancestor == start {
path.push(ancestor);
break;
}
// 2. some node was reached twice
else if visited.is_visited(&ancestor) {
// Drop any node in path that is before the first ancestor
let pos = path
.iter()
.position(|&p| p == ancestor)
.expect("we should always have a position");
path = path[pos..path.len()].to_vec();
break;
}
// None of the above, some middle path node
path.push(ancestor);
visited.visit(ancestor);
node = ancestor;
}
// We are done here
break 'outer;
}
}
}
if !path.is_empty() {
// Users will probably need to follow the path of the negative cycle
// so it should be in the reverse order than it was found by the algorithm.
path.reverse();
Some(path)
} else {
None
}
}
// Perform Step 1 and Step 2 of the Bellman-Ford algorithm.
#[inline(always)]
fn bellman_ford_initialize_relax<G>(
g: G,
source: G::NodeId,
) -> (Vec<G::EdgeWeight>, Vec<Option<G::NodeId>>)
where
G: NodeCount + IntoNodeIdentifiers + IntoEdges + NodeIndexable,
G::EdgeWeight: FloatMeasure,
{
// Step 1: initialize graph
let mut predecessor = vec![None; g.node_bound()];
let mut distance = vec![<_>::infinite(); g.node_bound()];
let ix = |i| g.to_index(i);
distance[ix(source)] = <_>::zero();
// Step 2: relax edges repeatedly
for _ in 1..g.node_count() {
let mut did_update = false;
for i in g.node_identifiers() {
for edge in g.edges(i) {
let j = edge.target();
let w = *edge.weight();
if distance[ix(i)] + w < distance[ix(j)] {
distance[ix(j)] = distance[ix(i)] + w;
predecessor[ix(j)] = Some(i);
did_update = true;
}
}
}
if !did_update {
break;
}
}
(distance, predecessor)
}