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v4.5
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    English
    v4.5

      Delta-Stepping Single-Source Shortest Path

      ✓ File Writeback ✕ Property Writeback ✓ Direct Return ✓ Stream Return ✕ Stats

      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:

      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.
      Remove node b from B[0]:
      Relax light-edge neighbors a with dist(a) = min(0+2,∞) = 2, and d with dist(b) = min(0+3,∞) = 3.

      Remove node a from B[0]:
      Light-edge neighbor b cannot be relaxed.
      Relax heavy-edge neighbor c with dist(c) = min(0+5,∞) = 5, d cannot be relaxed.

      3. Repeat step 2 until all buckets are empty.

      Remove nodes d and c from B[1]:
      Relax light-edge neighbor g with dist(g) = min(5+2,∞) = 7, b, c and d cannot be relaxed.
      Relax heavy-edge neighbor e with dist(e) = min(5+4,∞) = 9, a and b cannot be relaxed.

      Remove node g from B[2]:
      Light-edge neighbor c cannot be relaxed.
      Relax heavy-edge neighbor f with dist(f) = min(7+5,∞) = 12.

      Remove node e from B[3]:
      Relax light-edge neighbor f with dist(f) = min(9+1,12) = 10.

      Remove node f from B[3]:
      Light-edge neighbor e cannot be relaxed.
      Heavy-edge neighbor g cannot be relaxed.
      The algorithm ends here since 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.

      Syntax

      • Command: algo(sssp)
      • Parameters:
      Name
      Type
      Spec
      Default
      Optional
      Description
      ids / uuids _id / _uuid / / No ID/UUID of the single source node
      direction string in, out / Yes Direction of the shortest path, ignore the edge direction if not set
      edge_schema_property []@schema?.property Numeric type, must LTE / Yes One or multiple edge properties to be used as edge weights, where the values of multiple properties are summed up; treat the graph as unweighted if not set
      record_path int 0, 1 0 Yes 1 to return the shortest paths, 0 to return the shortest distances
      sssp_type string delta_stepping dijkstra No To run the Delta-Stepping SSSP algorithm, keep it as delta_stepping
      delta float >0 2 Yes The value of delta
      limit int ≥-1 -1 Yes Number of results to return, -1 to return all results
      order string asc, desc / Yes Sort nodes by the shortest distance from the source node

      Examples

      The example graph is as follows:

      File Writeback

      Spec record_path Content Description
      filename 0 _id,totalCost The shortest distance/cost from the source node to each other node
      1 _id--_uuid--_id The shortest path from the source node to each other node, the path is represented by the alternating ID of nodes and UUID of edges that form the path
      algo(sssp).params({
        uuids: 1,
        edge_schema_property: '@default.value',
        sssp_type: 'delta_stepping',
        delta: 2
      }).write({
        file: {
          filename: 'costs'
        }
      })
      

      Results: File costs

      G,8
      F,4
      E,5
      D,5
      C,5
      B,2
      A,0
      
      algo(sssp).params({
        uuids: 1,
        edge_schema_property: '@default.value',
        sssp_type: 'delta_stepping',
        delta: 2,
        record_path: 1
      }).write({
        file: {
          filename: 'paths'
        }
      })
      

      Results: File paths

      A--[102]--F--[107]--E--[109]--G
      A--[102]--F--[107]--E
      A--[101]--B--[105]--D
      A--[101]--B--[104]--C
      A--[102]--F
      A--[101]--B
      A
      

      Direct Return

      Alias Ordinal record_path Type Description Columns
      0 0 []perNode The shortest cost/distance from the source node to each other node _uuid, totalCost
      1 []perPath The shortest path from the source node to each other node, the path is represented by the alternating UUID of nodes and UUID of edges that form the path /
      algo(sssp).params({
        uuids: 1,
        edge_schema_property: '@default.value',
        sssp_type: 'delta_stepping',
        delta: 2,
        order: 'desc'
      }) as costs
      return costs
      

      Results: costs

      _uuid totalCost
      7 8
      5 5
      4 5
      3 5
      6 4
      2 2
      1 0
      algo(sssp).params({
        uuids: 1,
        edge_schema_property: '@default.value',
        direction: 'out',
        record_path: 1,
        sssp_type: 'delta_stepping',
        delta: 2,
        order: 'asc'
      }) as paths
      return paths
      

      Results: paths

      1
      1--[101]--2
      1--[102]--6
      1--[102]--6--[107]--5
      1--[101]--2--[105]--4
      1--[101]--2--[104]--3
      1--[102]--6--[107]--5--[109]--7

      Stream Return

      Alias Ordinal record_path Type Description Columns
      0 0 []perNode The shortest cost/distance from the source node to each other node _uuid, totalCost
      1 []perPath The shortest path from the source node to each other node, the path is represented by the alternating UUID of nodes and UUID of edges that form the path /
      algo(sssp).params({
        uuids: 1,
        edge_schema_property: '@default.value',
        sssp_type: 'delta_stepping'
      }).stream() as costs
      where costs.totalCost <> [0,5]
      return costs
      

      Results: costs

      _uuid totalCost
      6 4
      2 2
      algo(sssp).params({
        uuids: 1,
        edge_schema_property: '@default.value',
        sssp_type: 'delta_stepping',
        record_path: 1
      }).stream() as p
      where length(p) <> [0,3]
      return p
      

      Results: p

      1--[102]--6--[107]--5
      1--[101]--2--[105]--4
      1--[101]--2--[104]--3
      1--[102]--6
      1--[101]--2
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