L13 - 6.889 — Lecture 13 Approximate Distance Oracles...

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Unformatted text preview: 6.889 — Lecture 13: Approximate Distance Oracles Christian Sommer [email protected] October 26, 2011 Approximate Distance Oracle : given a graph G = ( V,E ) , preprocess it into a data structure such that we can compute approximate shortest-path distances efficiently (and output path if desired). (same scenario as in Lecture 12 except that paths are allowed to be approximately shortest) Assumption (all of Lecture 13) undirected planar G , non-negative edge lengths ‘ : E → R + Stretch for any given > , preprocessing algorithm constructs a data structure using which we can, queried for any pair of nodes ( v,w ) , output an estimate ˜ d ( v,w ) satisfying d G ( v,w ) 6 ˜ d ( v,w ) 6 (1 + ) d G ( v,w ) Exact Distance Oracles if query time poly (log n ) is desired, best methods use space ˜ Ω( n 2 ) inspect each and every portal on the separator between two query nodes. since separators have size O ( √ n ) , number of portals is “small” Question can we safely reduce the number of portals? if yes, how? negative: “neighbors” on cycle separator could be neighbors due to triangulation, edge length ∞ Cycle separators (Lecture 2): fundamental cycle is defined by a spanning tree T and any non-tree edge Lemma. For any planar graph G = ( V,E ) and any spanning tree T of radius d , we can partition V into A,B,S ⊆ V s.t. • [balanced] | A | , | B | ≤ 3 n/ 4 • [separation] no edge between any a ∈ A and b ∈ B ( A × B ∩ E = ∅ ) • [separator size] | S | ≤ 2 d + 1 • [efficient] A,B,S can be found in linear time. Idea apply the lemma using a shortest-path tree T rooted at an arbitrary node r . separator paths may contain many nodes (no bound on radius, increased number of potential portals) but they are shortest paths , which have good properties Approximate Distance Oracle Preprocessing (i) recursively separate G using shortest-path separators ( O (log n ) levels, 2 paths per level), (ii) each node stores distances to portals on O (log n ) paths Query given ( v,w ) , find best path through all the separator paths “shared” by v and w 1 Portals: –cover Approximate representation of shortest paths crossing a separator path using few portals. prove that, per node v and separator path Q , O (1 / ) portals suffice to guarantee (1 + ) –approximation for any shortest path crossing Q given a node v and a separator path...
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L13 - 6.889 — Lecture 13 Approximate Distance Oracles...

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