TY - GEN

T1 - Local algorithms for sparse spanning graphs

AU - Levi, Reut

AU - Ron, Dana

AU - Rubinfeld, Ronitt

N1 - Publisher Copyright:
©.

PY - 2014/9/1

Y1 - 2014/9/1

N2 - We initiate the study of the problem of designing sublinear-time (local) algorithms that, given an edge (u, v) in a connected graph G = (V,E), decide whether (u, v) belongs to a sparse spanning graph G0 = (V,E0) of G. Namely, G0 should be connected and |E0| should be upper bounded by (1 + ∈)|V | for a given parameter ∈ > 0. To this end the algorithms may query the incidence relation of the graph G, and we seek algorithms whose query complexity and running time (per given edge (u, v)) is as small as possible. Such an algorithm may be randomized but (for a fixed choice of its random coins) its decision on different edges in the graph should be consistent with the same spanning graph G0 and independent of the order of queries. We first show that for general (bounded-degree) graphs, the query complexity of any such algorithm must be Ω (√ |V |). This lower bound holds for graphs that have high expansion. We then turn to design and analyze algorithms both for graphs with high expansion (obtaining a result that roughly matches the lower bound) and for graphs that are (strongly) non-expanding (obtaining results in which the complexity does not depend on |V |). The complexity of the problem for graphs that do not fall into these two categories is left as an open question.

AB - We initiate the study of the problem of designing sublinear-time (local) algorithms that, given an edge (u, v) in a connected graph G = (V,E), decide whether (u, v) belongs to a sparse spanning graph G0 = (V,E0) of G. Namely, G0 should be connected and |E0| should be upper bounded by (1 + ∈)|V | for a given parameter ∈ > 0. To this end the algorithms may query the incidence relation of the graph G, and we seek algorithms whose query complexity and running time (per given edge (u, v)) is as small as possible. Such an algorithm may be randomized but (for a fixed choice of its random coins) its decision on different edges in the graph should be consistent with the same spanning graph G0 and independent of the order of queries. We first show that for general (bounded-degree) graphs, the query complexity of any such algorithm must be Ω (√ |V |). This lower bound holds for graphs that have high expansion. We then turn to design and analyze algorithms both for graphs with high expansion (obtaining a result that roughly matches the lower bound) and for graphs that are (strongly) non-expanding (obtaining results in which the complexity does not depend on |V |). The complexity of the problem for graphs that do not fall into these two categories is left as an open question.

KW - Local

KW - Spanning graph

KW - Sparse

UR - http://www.scopus.com/inward/record.url?scp=84920122402&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.APPROX-RANDOM.2014.826

DO - 10.4230/LIPIcs.APPROX-RANDOM.2014.826

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AN - SCOPUS:84920122402

T3 - Leibniz International Proceedings in Informatics, LIPIcs

SP - 826

EP - 842

BT - Leibniz International Proceedings in Informatics, LIPIcs

A2 - Jansen, Klaus

A2 - Moore, Cristopher

A2 - Devanur, Nikhil R.

A2 - Rolim, Jose D. P.

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 17th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2014 and the 18th International Workshop on Randomization and Computation, RANDOM 2014

Y2 - 4 September 2014 through 6 September 2014

ER -