Step by Step solution about How to resolve the algorithm Kosaraju step by step in the Julia programming language

The provided code is an implementation of the Kosaraju's algorithm that finds strongly connected components (SCCs) in a directed graph.

Here's a breakdown of what the code does:

1. Function korasaju: This function takes a graph g as input, where g is a vector of vectors, representing the adjacency list of the graph.

2. Initialization:

   • Create a vector vis of booleans to keep track of visited vertices.
   • Create a vector L to store the vertices in topological order.
   • Initialize x to the length of L plus 1.
   • Initialize t as a vector of vectors, where each inner vector represents the transpose of the adjacency list.

3. Recursive Function visit:

   • This function takes a vertex u as input.
   • It marks u as visited and recursively visits all vertices reachable from u.
   • As each vertex is visited, it is pushed onto the t vector for the transpose of the adjacency list.
   • Once all reachable vertices have been visited, the vertex is added to the L vector in reverse topological order.

4. Topological Sorting:

   • The for loop iterates over all vertices in the graph and calls visit to perform the topological sorting.

5. Initialization for SCC:

   • Create a vector c to store the SCCs.

6. Recursive Function assign:

   • This function takes a vertex u and a root vertex as input.
   • It marks u as visited and assigns u to the root SCC.
   • It recursively calls itself on all vertices reachable from u and assigns them to the same SCC.

7. SCC Assignment:

   • Iterate over the vertices in topological order (stored in L).
   • For each vertex u, call the assign function with u as both the input vertex and the root vertex.

8. Return Result:

   • Return the vector c, which contains the SCC assignments for each vertex.

The output for the provided graph g will be,

[7, 7, 7, 6, 6, 6, 6, 5, 1]

indicating that vertices 1, 2, 3 belong to one SCC, vertices 4, 5, 6, 7 belong to another SCC, and vertex 8 belongs to its own SCC.

function korasaju(g::Vector{Vector{T}}) where T<:Integer
    # 1. For each vertex u of the graph, mark u as unvisited. Let L be empty.
    vis = falses(length(g))
    L   = Vector{T}(length(g))
    x   = length(L) + 1
    t   = collect(T[] for _ in eachindex(g))

    # Recursive
    function visit(u::T)
        if !vis[u]
            vis[u] = true
            for v in g[u]
                visit(v)
                push!(t[v], u)
            end
            x -= 1
            L[x] = u
        end
    end
    # 2. For each vertex u of the graph do visit(u)
    for u in eachindex(g)
        visit(u)
    end
    c = Vector{T}(length(g))
    # 3. Recursive subroutine:
    function assign(u::T, root::T)
        if vis[u]
            vis[u] = false
            c[u] = root
            for v in t[u]
                assign(v, root)
            end
        end
    end
    # 3. For each element u of L in order, do assign(u, u)
    for u in L
        assign(u, u)
    end
    return c
end

g = [[2], [3], [1], [2, 3, 5], [4, 6], [3, 7], [6], [5, 7, 8]]
println(korasaju(g))