Dynamic Light Spanners in Doubling Metrics

Abstract

A $t$-spanner of a point set $X$ in a metric space $(\mathcal{X}, δ)$ is a graph $G$ with vertex set $P$ such that, for any pair of points $u,v \in X$, the distance between $u$ and $v$ in $G$ is at most $t$ times $δ(u,v)$. We study the problem of maintaining a spanner for a dynamic point set $X$ -- that is, when $X$ undergoes a sequence of insertions and deletions -- in a metric space of constant doubling dimension. For any constant $\varepsilon>0$, we maintain a $(1+\varepsilon)$-spanner of $P$ whose total weight remains within a constant factor of the weight of the minimum spanning tree of $X$. Each update (insertion or deletion) can be performed in $\operatorname{poly}(\log Φ)$ time, where $Φ$ denotes the aspect ratio of $X$. Prior to our work, no efficient dynamic algorithm for maintaining a light-weight spanner was known even for point sets in low-dimensional Euclidean space.

Figures (5)

• Figure 1: On bottom: The path graph $P_n$. All edges are unit weight. On top: a depiction of the net tree $T$ of $P_n$. There is a laminar hierarchy of nets; the $2^i$-net is $N_i=\{v_j\in P_n\mid j\mod 2^i=0\}$ (all vertices at hight $i$ in the tree and above). The net-tree spanner contains all the tree edges, and many additional edges. The weight of the edge $\{v_i,v_j\}$ is $|i-j|$. The tree $T$ has depth $\log n$, and the total weight of the edges going from depth $i$ vertices to depth $i+1$ is $\Theta(n)$. Thus $w(T)=\Theta(n\cdot\log n)$. As the weight of the MST of $P_n$ is $n-1$, the net-tree spanner has lightness $\Omega(\log n)$.
• Figure 2: Pseudocode for gao2004deformable algorithm to update net $\mathcal{N}$ after inserting point $x$ into $X$
• Figure 3: Pseudocode for gao2004deformable algorithm to update net $\mathcal{N}$ after deleting point $x$ from $X$
• Figure 5: Deletion procedure with bounded recourse
• Figure 6: A depiction of the upper bound proof of Lemma \ref{['lem:estimate-correct']}. The graph $L[i]$, and the sketch graph $H$. The net points $N_{i'}$ are outlined in red, in both $L[i]$ and $H$. The "large-scale" edges of $L[i]$, with scale $(i-1)$ and $(i-2)$, are drawn in blue. A pair $(u,v)$ and the shortest path $P$ between them in $L[i]$ is marked in green, and the corresponding path $P'$ in $H$ is marked in green.

Theorems & Definitions (31)

• Theorem 1.2
• Lemma 2.2: gao2004deformable
• Lemma 2.3: Rephrasing of gao2004deformable
• Lemma 2.4
• proof
• Claim 2.5: gao2004deformable
• Corollary 2.6
• Lemma 2.8
• proof
• Lemma 2.8