#directed-graph #path #graph-algorithms #shortest #algorithm #calculations #edge

fast_paths

Fast shortest path calculations on directed graphs made possible by pre-processing the graph using Contraction Hierarchies

4 releases (1 stable)

1.0.0 May 4, 2024
0.2.0 Apr 25, 2021
0.1.1 Jun 17, 2019
0.1.0 Jun 17, 2019

#80 in Algorithms

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MIT/Apache

150KB
3K SLoC

Fast Paths

The most famous algorithms used to calculate shortest paths are probably Dijkstra's algorithm and A*. However, shortest path calculation can be done much faster by preprocessing the graph.

Fast Paths uses Contraction Hierarchies, one of the best known speed-up techniques for shortest path calculation. It is especially suited to calculate shortest paths in road networks, but can be used for any directed graph with positive, non-zero edge weights.

Installation

In Cargo.toml

[dependencies]
fast_paths = "0.2.0"

Basic usage

// begin with an empty graph
let mut input_graph = InputGraph::new();

// add an edge between nodes with ID 0 and 6, the weight of the edge is 12.
// Note that the node IDs should be consecutive, if your graph has N nodes use 0...N-1 as node IDs,
// otherwise performance will degrade.
input_graph.add_edge(0, 6, 12);
// ... add many more edges here

// freeze the graph before using it (you cannot add more edges afterwards, unless you call thaw() first)
input_graph.freeze();

// prepare the graph for fast shortest path calculations. note that you have to do this again if you want to change the
// graph topology or any of the edge weights
let fast_graph = fast_paths::prepare(&input_graph);

// calculate the shortest path between nodes with ID 8 and 6 
let shortest_path = fast_paths::calc_path(&fast_graph, 8, 6);

match shortest_path {
    Some(p) => {
        // the weight of the shortest path
        let weight = p.get_weight();
        
        // all nodes of the shortest path (including source and target)
        let nodes = p.get_nodes();
    },
    None => {
        // no path has been found (nodes are not connected in this graph)
    }
}


Batch-wise shortest path calculation

For batch-wise calculation of shortest paths the method described above is inefficient. You should keep the PathCalculator object to execute multiple queries instead:

// ... see above
// create a path calculator (note: not thread-safe, use a separate object per thread)
let mut path_calculator = fast_paths::create_calculator(&fast_graph);
let shortest_path = path_calculator.calc_path(&fast_graph, 8, 6);

Calculating paths between multiple sources and targets

We can also efficiently calculate the shortest path when we want to consider multiple sources or targets:

// ... see above
// we want to either start at node 2 or 3 both of which carry a different initial weight
let sources = vec![(3, 5), (2, 7)];
// ... and go to either node 6 or 8 which also both carry a cost upon arrival
let targets = vec![(6, 2), (8, 10)];
// calculate the path with minimum cost that connects any of the sources with any of the targets while taking into 
// account the initial weights of each source and node
let shortest_path = path_calculator.calc_path_multiple_sources_and_targets(&fast_graph, sources, targets);

Serializing the prepared graph

FastGraph implements standard Serde serialization.

To be able to use the graph in a 32bit WebAssembly environment, it needs to be transformed to a 32bit representation when preparing it on a 64bit system. This can be achieved with the following two methods, but it will only work for graphs that do not exceed the 32bit limit, i.e. the number of nodes and edges and all weights must be below 2^32.

use fast_paths::{deserialize_32, serialize_32, FastGraph};

#[derive(Serialize, Deserialize)]
struct YourData {
    #[serde(serialize_with = "serialize_32", deserialize_with = "deserialize_32")]
    graph: FastGraph,
    // the rest of your struct
}

Preparing the graph after changes

The graph preparation can be done much faster using a fixed node ordering, which is just a permutation of node ids. This can be done like this:

let fast_graph = fast_paths::prepare(&input_graph);
let node_ordering = fast_graph.get_node_ordering();

let another_fast_graph = fast_paths::prepare_with_order(&another_input_graph, &node_ordering);

For this to work another_input_graph must have the same number of nodes as input_graph, otherwise prepare_with_order will return an error. Also performance will only be acceptable if input_graph and another_input_graph are similar to each other, say you only changed a few edge weights.

Benchmarks

FastPaths was run on a single core on a consumer-grade laptop using the road networks provided for the DIMACS implementation challenge graphs. The following graphs were used for the benchmark:

area number of nodes number of edges
New York 264.347 730.100
California&Nevada 1.890.816 4.630.444
USA 23.947.348 57.708.624
graph metric preparation time average query time out edges in edges
NY city distance 9 s 55 μs 747.555 747.559
CAL&NV distance 36 s 103 μs 4.147.109 4.147.183
USA distance 10.6 min 630 μs 52.617.216 52.617.642
NY city time 6 s 26 μs 706.053 706.084
CAL&NV time 24 s 60 μs 3.975.276 3.975.627
USA time 5.5 min 305 μs 49.277.058 49.283.162

The shortest path calculation time was averaged over 100k random routing queries. The benchmarks were run on a Macbook Pro M1 Max using Rust 1.74.1. The code for running these benchmarks can be found on the benchmarks branch.

There are also some benchmarks using smaller maps included in the test suite. You can run them like this:

export RUST_TEST_THREADS=1; cargo test --release -- --ignored --nocapture

Graph limitations

  • loop-edges (from node A to node A) will be ignored, because since we are only considering positive non-zero edge-weights they cannot be part of a shortest path
  • in case the graph has duplicate edges (multiple edges from node A to node B) only the edge with the lowest weight will be considered

Special Thanks

Thanks to Dustin Carlino from A/B Street!

License

This project is licensed under either of

at your option.

Contribution

Unless you explicitly state otherwise, any contribution intentionally submitted for inclusion in fast_paths by you, as defined in the Apache-2.0 license, shall be dual licensed as above, without any additional terms or conditions.

Dependencies

~1.3–2MB
~40K SLoC