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Enhancements to Decremental BFS for Efficient Graph Query and Matrix Multiplication

This document explores two significant reductions to decremental BFS (Breadth-First Search) algorithms, aimed at improving efficiency in graph processing tasks such as shortest path queries and boolean matrix multiplication. We analyze the performance implications of edge deletions and query strategies on both static and dynamic undirected graphs. By leveraging insights from works by Even-Shiloach (1981) and Roditty-Zwick (2004), we establish a connection between decremental BFS operations and boolean matrix manipulations, highlighting the potential for optimization in algorithms involving all-pairs shortest path (APSP) and single-source shortest path (SSSP) problems.

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Enhancements to Decremental BFS for Efficient Graph Query and Matrix Multiplication

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  1. Two reductions to decremental BFS Camil Demetrescu University of Rome “La Sapienza”

  2. delete(S): delete all edges in S from graph G query(v): return distance from s to v in graph G (if it is at most d) Each non-tree edge can fall down at most 2d times overall… depth d O(md) total timeover any deletion sequence O(d) time per deletion (amortized over Ω(m) deletions) Decremental BFS[Even-Shiloach’81]

  3. Boolean matrix multiplication (off-line) decremental undirected BFS Weighted (static) undirected APSP (off-line) decremental undirected SSSP Can we do any better than O(mn)? Roditty and Zwick in ESA 2004 have shown two reductions:

  4. B A y x Bipartite graph with an edge (x,y) for each A[x,y]=1 Bipartite graph with an edge (x,y) for each B[x,y]=1 Matrix mult. decremental BFS A and B boolean matrices We wish to compute C=A·B C[x,y]=1 iff there is z such that A[x,z]=1 and B[z,y]=1 C[x,y]=1 iff path of length 2between x on first layer and y on last layer

  5. C B A s 1 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 First row: C[1,x]=1 iff dist(s,x)=3 Second row: C[2,x]=1 iff dist(s,x)=4 Third row: C[3,x]=1 iff dist(s,x)=5 … … Matrix mult. decremental BFS x n deletions and n2 queriesDecremental BFS in o(mn) total time would yield boolean matrix multiplication in o(mn)

  6. … n2W nW 2nW Undirected APSP decremental SSSP n deletions and n2 queries Decremental SSSP in o(mn) total time would yield undirected APSP in o(mn) 1 2 n 1 G 153 272 W = largest edge weight Exercise 8: does this reduction work in directed graphs?

  7. More details in Decremental BFS:[Even-Shiloach’81]S. Even and Y. Shiloach, An On-line Edge Deletion Problem, J. Assoc. Comput. Mach. (JACM), Vol. 28, pp. 1-4, 1981 Reductions to decremental BFS:[Roditty-Zwick’04]Liam Roditty, Uri Zwick, On dynamic shortest paths problems Proc. of 12th ESA (2004), 580-591.

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