Overview

The single-source shortest path (SSSP) problem is that of computing, for each node that is reachable from the source node, the shortest path with the minimum total edge weights among all possible paths; or in the case of unweighted graphs, the shortest path with the minimum number of edges. The cost (or distance) of the shortest path is the total edge weights or the number of edges.

The Delta-Stepping algorithm can be viewed as a variant of Dijkstra's algorithm with its potential for parallelism.

Related material of the algorithm:

• U. Meyer, P. Sanders, Δ-Stepping: A Parallel Single Source Shortest Path Algorithm (1998)

Concepts

Delta-Stepping Single-Source Shortest Path

The Delta-Stepping Single-Source Shortest Path (SSSP) algorithm introduces the concept of "buckets" and performs relaxation operations in a more coarse-grained manner. The algorithm utilizes a positive real number parameter delta (Δ) to achieve the following:

• Maintain an array B of buckets such that B[i] contains nodes whose distance falls within the range [iΔ, (i+1)Δ). Thus Δ is also called the "step width" or "bucket width".
• Distinguish between light edges with weight ≤ Δ and heavy edges with weight > Δ in the graph. Light-edge nodes are prioritized during relaxation as they have lower weights and are more likely to yield shorter paths.

The term relaxation refers to the process of updating the distance of a node v that is connected to node u to a potential shorter distance by considering the path through node u. Specifically, the distance of node v is updated to dist(v) = dist(u) + w(u,v), where w(u,v) is the weight of the edge (u,v). This update is performed only if the current dist(v) is greater than dist(u) + w(u,v).

In the Delta-Stepping SSSP algorithm, the relaxation also includes assigning the relaxed node to the corresponding bucket based on its updated distance value.

Below is the description of the basic Delta-Stepping SSSP algorithm, along with an example to compute the weighted shortest paths in an undirected graph starting from the source node b, and Δ is set to 3:

1. At the begining of the algorithm, all nodes have the distance as infinity except for the source node as 0. The source node is assigned to bucket B[0].

2. In each iteration, remove all nodes from the first nonempty bucket B[i]:

• Relax all light-edge neighbors of the removed nodes, the relaxed nodes might be assigned to B[i] or B[i+1]; defer the relaxation of the heavy-edge neighbors.
• If B[i] is refilled, repeat the above operation until B[i] is empty.
• Relax all deferred heavy-edge neighbors.

3. Repeat step 2 until all buckets are empty.

By dividing the nodes into buckets and processing them in parallel, the Delta-Stepping algorithm can exploit the available computational resources more efficiently, making it suitable for large-scale graphs and parallel computing environments.

Considerations

• The Delta-Stepping SSSP algorithm is only applicable to graphs with non-negative edge weights. If negative weights are present, the Delta-Stepping SSSP algorithm might produce false results. In this case, a different algorithm like the SPFA should be used.
• If there are multiple shortest paths exist between two nodes, all of them will be found.
• In disconnected graphs, the algorithm only outputs the shortest paths from the source node to all nodes belonging to the same connected component as the source node.

Example Graph

To create this graph:

// Runs each row separately in order in an empty graphset
create().edge_property(@default, "value", int32)
insert().into(@default).nodes([{_id:"A"}, {_id:"B"}, {_id:"C"}, {_id:"D"}, {_id:"E"}, {_id:"F"}, {_id:"G"}])
insert().into(@default).edges([{_from:"A", _to:"B", value:2}, {_from:"A", _to:"F", value:4}, {_from:"B", _to:"F", value:6}, {_from:"B", _to:"C", value:3}, {_from:"B", _to:"D", value:3}, {_from:"D", _to:"F", value:2}, {_from:"F", _to:"E", value:1}, {_from:"D", _to:"E", value:2}, {_from:"E", _to:"G", value:3}])

Creating HDC Graph

To load the entire graph to the HDC server hdc-server-1 as hdc_sssp:

CALL hdc.graph.create("hdc-server-1", "hdc_sssp", {
  nodes: {"*": ["*"]},
  edges: {"*": ["*"]},
  direction: "undirected",
  load_id: true,
  update: "static",
  query: "query",
  default: false
})

hdc.graph.create("hdc_sssp", {
  nodes: {"*": ["*"]},
  edges: {"*": ["*"]},
  direction: "undirected",
  load_id: true,
  update: "static",
  query: "query",
  default: false
}).to("hdc-server-1")

Parameters

Algorithm name: sssp

File Writeback

CALL algo.sssp.write("hdc_sssp", {
  params: {
    ids: "A",
    edge_schema_property: "@default.value",
    impl_type: "delta_stepping",
    return_id_uuid: "id"
  },
  return_params: {
    file: {
      filename: "costs"
    }
  }
})

algo(sssp).params({
  project: "hdc_sssp",
  ids: "A",
  edge_schema_property: "@default.value",
  impl_type: "delta_stepping",
  return_id_uuid: "id"
}).write({
  file: {
      filename: "costs"
  }
})

Result:

_id,totalCost
G,8
D,5
F,4
B,2
E,5
C,5

CALL algo.sssp.write("hdc_sssp", {
  params: {
    ids: "A",
    edge_schema_property: '@default.value',
    impl_type: 'delta_stepping',
    delta: 2,
    record_path: 1,
    return_id_uuid: "id"
  },
  return_params: {
    file: {
      filename: "paths"
    }
  }
})

algo(sssp).params({
  project: "hdc_sssp",
  ids: "A",
  edge_schema_property: '@default.value',
  impl_type: 'delta_stepping',
  delta: 2,
  record_path: 1,
  return_id_uuid: "id"
}).write({
  file: {
      filename: "paths"
  }
})

Result:

totalCost,_ids
8,A--[102]--F--[107]--E--[109]--G
5,A--[101]--B--[105]--D
5,A--[102]--F--[107]--E
5,A--[103]--B--[104]--C
4,A--[102]--F
2,A--[101]--B

Full Return

CALL algo.sssp("hdc_sssp", {
  params: {
    ids: 'A',
    edge_schema_property: 'value',
    impl_type: 'delta_stepping',
    delta: 3,
    record_path: 0,
    return_id_uuid: 'id',
    order: 'desc'
  },
  return_params: {}
}) YIELD r
RETURN r

exec{
  algo(sssp).params({
    ids: 'A',
    edge_schema_property: 'value',
    impl_type: 'delta_stepping',
    delta: 3,
    record_path: 0,
    return_id_uuid: 'id',
    order: 'desc'
  }) as r
  return r
} on hdc_sssp

Result:

Stream Return

CALL algo.sssp("hdc_sssp", {
  params: {
    ids: 'A',
    edge_schema_property: '@default.value',
    impl_type: 'delta_stepping',
    record_path: 1,
    return_id_uuid: 'id'
  },
  return_params: {
    stream: {}
  }
}) YIELD r
RETURN r

exec{
  algo(sssp).params({
    ids: 'A',
    edge_schema_property: '@default.value',
    impl_type: 'delta_stepping',
    record_path: 1,
    return_id_uuid: 'id'
  }).stream() as r
  return r
} on hdc_sssp

Result: