Implicitly defined graphs (possibly infinite)
Author scheinerman
3 Stars
Updated Last
2 Years Ago
Started In
February 2021


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An ImplicitGraph is a graph in which the vertices and edges are implicitly defined by two functions: one that tests for vertex membership and one that returns a list of the (out) neighbors of a vertex.

The vertex set of an ImplicitGraph may be finite or (implicitly) infinite. The (out) degrees, however, must be finite.

Creating Graphs

An ImplicitGraph is defined as follows:

ImplicitGraph{T}(has_vertex, out_neighbors)


  • T is the data type of the vertices.
  • has_vertex(v::T)::Bool is a function that takes objects of type T as input and returns true if v is a vertex of the graph.
  • out_neighbors(v::T)::Vector{T} is a function that takes objects of type T as input and returns a list of the (out) neighbors of v.

For example, the following creates an (essentially) infinite path whose vertices are integers (see the iPath function):

yes(v::Int)::Bool = true 
N(v::Int)::Vector{Int} = [v-1, v+1]
G = ImplicitGraph{Int}(yes, N)

The yes function always returns true for any Int. The N function returns the two neighbors of a vertex v. (For a truly infinite path, use BigInt in place of Int.)

Note that if v is an element of its own neighbor set, that represents a loop at vertex v.

Undirected and directed graphs

The user-supplied out_neighbors function can be used to create both undirected and directed graphs. If an undirected graph is intended, be sure that if {v,w} is an edge of the graph, then w will be in the list returned by out_neighbors(v) and v will be in the list returned by out_neighbors(w).

To create an infinite directed path, the earlier example can be modified like this:

yes(v::Int)::Bool = true 
N(v::Int)::Vector{Int} = [v+1]
G = ImplicitGraph{Int}(yes, N)

Predefined Graphs

We provide a few basic graphs that can be created using the following methods:

  • iCycle(n::Int) creates an undirected cycle with vertex set {1,2,...,n}; iCycle(n,false) creates a directed n-cycle.

  • iPath() creates an (essentially) infinite undirected path whose vertex set contains all integers (objects of type Int); iPath(false) creates a one-way infinite path ⋯ → -2 → -1 → 0 → 1 → 2 → ⋯.

  • iGrid() creates an (essentially) infinite grid whose vertices are ordered pairs of integers (objects of type Int).

  • iCube(d::Int) creates a d-dimensional cube graph. The vertices are all d-long strings of 0s and 1s. Two vertices are adjacent iff they differ in exactly one bit.

  • iKnight() creates the Knight's move graph on an (essentially) infinite chessboard. The vertices are pairs of integers (objects of type Int).

  • iShift(alphabet, n::Int) creates the shift digraph whose vertices are n-tuples of elements of alphabet.


  • To test if v is a vertex of an ImplicitGraph G, use has(G). Note that the data type of v must match the element type of G. (The function eltype returns the data type of the vertices of the ImplicitGraph.)

  • To test if {v,w} is an edge of G use G[v,w] or has(G,v,w). Note that v and w must both be vertices of G or an error is thrown.

  • To get a list of the (out) neighbors of a vertex v, use G[v].

  • To get the degree of a vertex in a graph, use deg(G,v).

julia> G = iGrid()

julia> has_vertex(G,(1,2))

julia> G[(1,2)]
4-element Array{Tuple{Int64,Int64},1}:
 (1, 1)
 (1, 3)
 (0, 2)
 (2, 2)

julia> G[(1,2),(1,3)]

julia> deg(G,(5,0))

Path Finding

Shortest path

The function find_path finds a shortest path between vertices of a graph. This function may run without returning if the graph is infinite and disconnected.

julia> G = iGrid()

julia> find_path(G,(0,0), (3,5))
9-element Array{Tuple{Int64,Int64},1}:
 (0, 0)
 (0, 1)
 (0, 2)
 (0, 3)
 (0, 4)
 (0, 5)
 (1, 5)
 (2, 5)
 (3, 5)

The function dist returns the length of a shortest path between vertices in the graph.

julia> dist(G,(0,0),(3,5))

Guided path finding

The function guided_path_finder employs a score function to try to find a path between vertices. It may be faster than find_path, but might not give a shortest path.

This function is called as follows: guided_path_finder(G,s,t,score=sc, depth=d, verbose=0) where

  • G is an ImplicitGraph,
  • s is the starting vertex of the desired path,
  • t is the ending vertex of the desired path,
  • sc is a score function that mapping vertices to integers and should get smaller as vertices get closer to t (and should minimize at t),
  • d controls amount of look ahead (default is 1), and
  • verbose sets how often to print progess information (or 0 for no diagnostics).
julia> G = iKnight();

julia> s = (9,9); t = (0,0);

julia> sc(v) = sum(abs.(v));  # score of (a,b) is |a| + |b|

julia> guided_path_finder(G,s,t,score=sc,depth=1)
9-element Vector{Tuple{Int64, Int64}}:
 (9, 9)
 (8, 7)
 (7, 5)
 (6, 3)
 (5, 1)
 (3, 0)
 (1, -1)
 (-1, -2)
 (0, 0)

# With better look-ahead we find a shorter path

julia> guided_path_finder(G,s,t,score=sc,depth=3)
7-element Vector{Tuple{Int64, Int64}}:
 (9, 9)
 (8, 7)
 (7, 5)
 (6, 3)
 (4, 2)
 (2, 1)
 (0, 0)

Greater depth can find a shorter path, but that comes at a cost:

julia> using BenchmarkTools

julia> @btime guided_path_finder(G,s,t,score=sc,depth=1);
  52.361 μs (1308 allocations: 81.05 KiB)

julia> @btime guided_path_finder(G,s,t,score=sc,depth=3);
  407.546 μs (8691 allocations: 696.47 KiB)


The extras directory contains additional code and examples that may be useful in conjunction with the ImplicitGraph type. See the README in that directory.

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