Modify non_backtracking_matrix function's Return Type (#328)

* modified to use a sparse matrix in non_backtracking_matrix

* modified to reserve a capacity of size 'nz' & used 'append!' instead of 'push!'

* added tests for non_backtracking_matrix

* modified to achieve zero allocation

* reverted

* Apply suggestions from code review

---------

Co-authored-by: Guillaume Dalle <22795598+gdalle@users.noreply.github.com>

src/linalg/nonbacktracking.jl

@@ -26,17 +26,25 @@ function non_backtracking_matrix(g::AbstractGraph)
         end
     end
 
-    B = zeros(Float64, m, m)
-
+    nz = if is_directed(g)
+        sum(indegree(g, v) * outdegree(g, v) for v in vertices(g))
+    else
+        sum(degree(g, v) * (degree(g, v) - 1) for v in vertices(g))
+    end
+    rowidx = sizehint!(Vector{Int}(), nz)
+    colidx = sizehint!(Vector{Int}(), nz)
     for (e, u) in edgeidmap
         i, j = src(e), dst(e)
         for k in inneighbors(g, i)
             k == j && continue
             v = edgeidmap[Edge(k, i)]
-            B[v, u] = 1
+
+            push!(rowidx, v)
+            push!(colidx, u)
         end
     end
 
+    B = sparse(rowidx, colidx, ones(Int, length(rowidx)), m, m)
     return B, edgeidmap
 end
 

test/linalg/nonbacktracking.jl

@@ -0,0 +1,59 @@
+using SparseArrays
+
+@testset "nonbacktracking" begin
+    # Case: simple undirected
+    ug = path_graph(5)
+    B, edgemap = non_backtracking_matrix(ug)
+    #      | 1->2 | 2->3 | 3->4 | 4->5 | 2->1 | 3->2 | 4->3 | 5->4
+    # -------------------------------------------------------------
+    # 1->2 |    0 |    1 |    0 |    0 |    0 |    0 |    0 |    0
+    # 2->3 |    0 |    0 |    1 |    0 |    0 |    0 |    0 |    0
+    # 3->4 |    0 |    0 |    0 |    1 |    0 |    0 |    0 |    0
+    # 4->5 |    0 |    0 |    0 |    0 |    0 |    0 |    0 |    0
+    # 2->1 |    0 |    0 |    0 |    0 |    0 |    0 |    0 |    0
+    # 3->2 |    0 |    0 |    0 |    0 |    1 |    0 |    0 |    0
+    # 4->3 |    0 |    0 |    0 |    0 |    0 |    1 |    0 |    0
+    # 5->4 |    0 |    0 |    0 |    0 |    0 |    0 |    1 |    0
+    B_ = [
+        0 1 0 0 0 0 0 0
+        0 0 1 0 0 0 0 0
+        0 0 0 1 0 0 0 0
+        0 0 0 0 0 0 0 0
+        0 0 0 0 0 0 0 0
+        0 0 0 0 1 0 0 0
+        0 0 0 0 0 1 0 0
+        0 0 0 0 0 0 1 0
+    ]
+    egs = Edge.([(1, 2), (2, 3), (3, 4), (4, 5), (2, 1), (3, 2), (4, 3), (5, 4)])
+    @test typeof(B) <: SparseMatrixCSC
+    @test B == B_
+
+    # Case: simple directed
+    dg = SimpleDiGraph(5)
+    add_edge!(dg, 1, 2)
+    add_edge!(dg, 2, 3)
+    add_edge!(dg, 1, 3)
+    add_edge!(dg, 3, 4)
+    add_edge!(dg, 3, 5)
+    add_edge!(dg, 4, 3)
+    B, edgemap = non_backtracking_matrix(dg)
+    #      | 1->2 | 1->3 | 2->3 | 3->4 | 3->5 | 4->3
+    # -----------------------------------------------
+    # 1->2 |    0 |    0 |    1 |    0 |    0 |    0
+    # 1->3 |    0 |    0 |    0 |    1 |    1 |    0
+    # 2->3 |    0 |    0 |    0 |    1 |    1 |    0
+    # 3->4 |    0 |    0 |    0 |    0 |    0 |    0
+    # 3->5 |    0 |    0 |    0 |    0 |    0 |    0
+    # 4->3 |    0 |    0 |    0 |    0 |    1 |    0
+    B_ = [
+        0 0 1 0 0 0
+        0 0 0 1 1 0
+        0 0 0 1 1 0
+        0 0 0 0 0 0
+        0 0 0 0 0 0
+        0 0 0 0 1 0
+    ]
+    egs = Edge.([(1, 2), (1, 3), (2, 3), (3, 4), (3, 5), (4, 3)])
+    @test typeof(B) <: SparseMatrixCSC
+    @test B == B_
+end

test/linalg/runtests.jl

@@ -4,7 +4,7 @@ using SparseArrays
 using LinearAlgebra
 const linalgtestdir = dirname(@__FILE__)
 
-tests = ["graphmatrices", "spectral"]
+tests = ["graphmatrices", "spectral", "nonbacktracking"]
 
 @testset "Graphs.LinAlg" begin
     for t in tests