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//! The [`EdmondsKarp`] trait: the Edmonds-Karp algorithm for the maximum s-t
//! flow problem on a directed, capacitated graph.
//!
//! Reference: J. Edmonds, R. M. Karp, "Theoretical improvements in algorithmic
//! efficiency for network flow problems", Journal of the ACM 19 (1972),
//! 248-264. The same shortest-augmenting-path rule was independently given by
//! E. A. Dinic (1970).
//!
//! Edmonds-Karp is the Ford-Fulkerson method with a specific augmenting-path
//! rule: each iteration finds a *shortest* (fewest-arc) augmenting path in the
//! residual graph by breadth-first search and saturates it. Because the
//! source-sink residual distance never decreases and strictly increases every
//! `O(E)` augmentations, the algorithm runs in `O(V * E^2)` time regardless of
//! the capacity magnitudes.
//!
//! This is the same maximum-flow problem solved by [`Dinic`], and it shares the
//! residual graph, result, and error types from the `max_flow` module.
//! Dinic is asymptotically faster (`O(V^2 * E)`); Edmonds-Karp is kept as a
//! second, structurally different implementation, which makes the two strong
//! cross-checks for each other.
//!
//! [`Dinic`]: crate::traits::algorithms::Dinic
use alloc::{collections::VecDeque, vec::Vec};
use super::max_flow::{MaxFlowError, MaxFlowResult, build_residual};
use crate::traits::{Finite, Number, SparseValuedMatrix2D, SquareMatrix};
/// Repeatedly augments along shortest residual paths until the sink is
/// unreachable, returning the total flow pushed. Mutates `arc_cap` in place.
fn shortest_augmenting_paths<V: Number>(
source: usize,
sink: usize,
adjacency: &[Vec<usize>],
arc_to: &[usize],
arc_cap: &mut [V],
) -> V {
let zero = V::zero();
let order = adjacency.len();
let mut total = zero;
let mut parent_arc = alloc::vec![usize::MAX; order];
let mut visited = alloc::vec![false; order];
let mut queue: VecDeque<usize> = VecDeque::new();
loop {
parent_arc.fill(usize::MAX);
visited.fill(false);
queue.clear();
visited[source] = true;
queue.push_back(source);
// Breadth-first search records, per vertex, the arc used to reach it,
// which yields a shortest residual path to the sink.
while let Some(node) = queue.pop_front() {
if node == sink {
break;
}
for &arc in &adjacency[node] {
let next = arc_to[arc];
if !visited[next] && arc_cap[arc] > zero {
visited[next] = true;
parent_arc[next] = arc;
queue.push_back(next);
}
}
}
if !visited[sink] {
break;
}
// Bottleneck of the located path, walking back via reverse arcs. `<`
// and `<=` pick the same minimum (an equivalent mutant, skipped
// in .cargo/mutants.toml).
let mut bottleneck = V::max_value();
let mut node = sink;
while node != source {
let arc = parent_arc[node];
if arc_cap[arc] < bottleneck {
bottleneck = arc_cap[arc];
}
node = arc_to[arc ^ 1];
}
// Augment the path by the bottleneck.
let mut node = sink;
while node != source {
let arc = parent_arc[node];
arc_cap[arc] -= bottleneck;
arc_cap[arc ^ 1] += bottleneck;
node = arc_to[arc ^ 1];
}
total += bottleneck;
}
total
}
/// The Edmonds-Karp algorithm for the maximum s-t flow problem.
///
/// Entry `(i, j)` of the matrix is the capacity of the directed arc `i -> j`.
/// Capacities must be finite and non-negative, and self-loops are ignored. The
/// reverse residual arcs are created internally with zero capacity, so
/// antiparallel input arcs remain independent.
pub trait EdmondsKarp: SparseValuedMatrix2D + SquareMatrix {
/// Computes a maximum flow from `source` to `sink`.
///
/// # Arguments
///
/// * `source` - the index of the source node, in `0..order`.
/// * `sink` - the index of the sink node, in `0..order`.
///
/// # Complexity
///
/// `O(V * E^2)` time and `O(V + E)` space, independent of the capacity
/// magnitudes.
///
/// # Errors
///
/// Returns an error when `source` or `sink` is out of range, when they
/// coincide, or when a non-self-loop arc carries a non-finite or negative
/// capacity.
///
/// # Examples
///
/// ```
/// use geometric_traits::{impls::ValuedCSR2D, prelude::*};
///
/// // Diamond network: two internal paths each bottlenecked to 2 units.
/// let edges: ValuedCSR2D<usize, usize, usize, u64> =
/// GenericEdgesBuilder::<_, ValuedCSR2D<usize, usize, usize, u64>>::default()
/// .expected_number_of_edges(4)
/// .expected_shape((4, 4))
/// .edges(vec![(0, 1, 3u64), (0, 2, 2), (1, 3, 2), (2, 3, 3)].into_iter())
/// .build()
/// .unwrap();
///
/// let result = edges.edmonds_karp(0, 3).unwrap();
/// assert_eq!(result.max_flow(), 4);
/// ```
fn edmonds_karp(
&self,
source: usize,
sink: usize,
) -> Result<MaxFlowResult<Self::Value>, MaxFlowError>
where
Self::Value: Number + Finite;
}
impl<M: SparseValuedMatrix2D + SquareMatrix + ?Sized> EdmondsKarp for M {
fn edmonds_karp(
&self,
source: usize,
sink: usize,
) -> Result<MaxFlowResult<Self::Value>, MaxFlowError>
where
Self::Value: Number + Finite,
{
let mut residual = build_residual(self, source, sink)?;
let max_flow = shortest_augmenting_paths(
source,
sink,
&residual.adjacency,
&residual.arc_to,
&mut residual.arc_cap,
);
Ok(residual.finish(max_flow))
}
}