We were discussing the parallel Breadth-First-Search using the 2D partitioning approach.
This is the latter case of two highly tuned parallel approaches for BFS on large parallel systems:  
1) a level synchronous strategy that relies on a simple vertex based graph partitioning and
2) a two-dimensional sparse matrix partitioning-based approach that mitigates parallel communication overhead.
The first approach was about partitioning 
The second approach was about mitigating communication overhead
In the second approach, the idea is that with each BFS iteration is equivalent to a sparse matrix -sparse vector multiplication (SpMSV). The matrix is the adjacency matrix which is decomposed equally into a number of processors. The vector is the frontier with integer variables. The algorithm does not store the previous frontiers explicitly as multiple sparse vectors. Instead it computes the breadth first spanning tree by returning a dense parents array.  The parent is always chosen to be a vertex with the highest label. The parents array is used in the element-wise  with the result of SpMSV. The inner loop performs a single level traversal 
All vectors and input matrix are distributed 2D wise. For example, f which is an initially sparse vector. Each n/p sized piece of the vector takes a progressive section of the adjacency matrix. The vector operations are also parallelized using loop-level parallelism.
In the 2D processor assignment, the subvectors are accumulated at all the processors on the jth column and each processor after performing its SpMSV scatters the corresponding piece of its intermediate vector to its owner along the ith processor row.
#codingexercise
Write a function to reduce the keys with 0 values in a javascript object:
var items = { "1" : 0, "2": 0, "3": 0, "4":0, "5": 55, "6": 0, "7":0, "8" : 0, "9": 0  }
Object.entries(items).forEach(([key, value]) => {
    if (value === "" || value === null || value === 0){
        delete items[key];
    }

});
document.write(JSON.stringify(items));
{"5":55}