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- W2966707746 abstract "An $f(d)$-spanner of an unweighted $n$-vertex graph $G=(V,E)$ is a subgraph $H$ satisfying that $dist_H(u, v)$ is at most $f(dist_G(u, v))$ for every $u,v in V$. We present new spanner constructions that achieve a nearly optimal stretch of $O(lceil k /d rceil)$ for any distance value $d in [1,k^{1-o(1)}]$, and $d geq k^{1+o(1)}$. We show the following: 1. There exists an $f(d)$-spanner $H subseteq G$ with $f(d)leq 7k$ for any $d in [1,sqrt{k}/2]$ with expected size $O_{k}(n^{1+1/k})$. This in particular gives $(alpha,beta)$ spanners with $alpha=O(sqrt{k})$ and $beta=O(k)$. 2. For any $epsilon in (0,1/2]$, there exists an $(alpha,beta)$-spanner with $alpha=O(k^{epsilon})$, $beta=O_{epsilon}(k)$ and of expected size $O_{k}(n^{1+1/k})$. This implies a stretch of $O(lceil k/d rceil)$ for any $d in [sqrt{k}/2, k^{1-epsilon}]$, and for every $dgeq k^{1+epsilon}$. In particular, it provides a constant stretch already for vertex pairs at distance $k^{1+o(1)}$ (improving upon $d=(log k)^{log k}$ that was known before). Up to the $o(1)$ factor in the exponent, and the constant factor in the stretch, this is the best possible by the girth argument. 3. For any $epsilon in (0,1)$ and integer $kgeq 1$, there is a $(3+epsilon, beta)$-spanner with $beta=O_{epsilon}(k^{log(3+8/epsilon)})$ and $O_{k,epsilon}(n^{1+1/k})$ edges. We also consider the related graph concept of hopsets introduced by [Cohen, J. ACM '00]. We present a new family of $(alpha,beta)$ hopsets with $widetilde{O}(k cdot n^{1+1/k})$ edges and $alpha cdot beta=O(k)$. Most notably, we show a construction of $(3+epsilon,beta)$ hopset with $widetilde{O}_{k,epsilon}(n^{1+1/k})$ edges and hop-bound of $beta=O_{epsilon}(k^{log(3+9/epsilon)})$, improving upon the state-of-the-art hop-bound of $beta=O(log k /epsilon)^{log k}$ by [Elkin-Neiman, '17] and [Huang-Pettie, '17]." @default.
- W2966707746 created "2019-08-13" @default.
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- W2966707746 date "2019-07-26" @default.
- W2966707746 modified "2023-09-27" @default.
- W2966707746 title "New $(alpha,beta)$ Spanners and Hopsets" @default.
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