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Theoretical Computer Science: 8th IFIP TC 1/WG 2.2 by Josep Diaz, Ivan Lanese, Davide Sangiorgi

By Josep Diaz, Ivan Lanese, Davide Sangiorgi

This booklet constitutes the refereed complaints of the eighth FIP WG 2.2 foreign convention, TCS 2014, held in Rome, Italy, in September 2014. The 26 revised complete papers offered, including invited talks, have been rigorously reviewed and chosen from seventy three submissions. [Suggestion--please cost and upload extra if wanted] TCS-2014 consisted of 2 tracks, with separate application committees, which dealt respectively with: - music A: Algorithms, Complexity and versions of Computation, and - song B: common sense, Semantics, Specification and Verification

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Extra info for Theoretical Computer Science: 8th IFIP TC 1/WG 2.2 International Conference, TCS 2014, Rome, Italy, September 1-3, 2014. Proceedings

Example text

We will prove Theorem 1 by giving a Karp reduction from the CLIQUE problem. Recall that in the CLIQUE problem, we are given an undirected graph G = V , E , and an integer k, and the goal is to find whether there exists a complete subgraph of G with k vertices. Assume we are given an arbitrary undirected graph G = V , E , where n and m denote |V | and |E | respectively, and an integer k. We construct a corresponding bipartite graph G = V1 ∪ V2 , E as explained below. For every vertex vi ∈ V , G has a corresponding vertex vi ∈ V1 .

Applying the lemma with S = VˆR and t = k − |VˆL | and t = k yield the bounds from (1) and (2), respectively. As in the proof of Theorem 2, the budget constraint (2) from the LP (ignoring Vˆ1 ) yields k ≥ α · |VˆL | + (1 − α) · |VˆR |. Substituting this bound for k into the above coverage bounds yield the following lower bounds on the respective coverage obtained by our two solutions: 1. W1 = w(A) + w(BL ) + (1 − α) · β · w(BR ), and 2. W2 = (1 − α · β) · (w(A) + w(BR )), ˆ where β = 1 − ||VVˆL || .

2. For any node u, the number of nodes reachable from u grows exponentially as the distance increases. 3. For any two nodes u and v, if both of them have many reachable nodes within a distance of O(log n), then it is very likely that there exists a common node w that is reachable from both u and v by paths with length O(log n). Next we consider the finishing time of multiple-piece streaming in Preferential Attachment graphs (PA-graphs). The notion of preferential attachment graphs was first introduced by Barabási and Albert [2], and they have been used to model social and peer-to-peer networks.

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