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Two reductions to decremental BFS

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

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Two reductions to decremental BFS

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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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