Improved and Parameterized Algorithms for Online Multi-level Aggregation: A Memory-based Approach

TL;DR

This work tackles online MLAP-D, where requests arrive over time on a rooted, vertex-weighted tree with deadlines and shared transmission costs. It introduces a memory-based framework that informs dynamic aggregation via expansion and investment stages, yielding two parameterized online algorithms: an $e(D+1)$-competitive method and an $e(4H+2)$-competitive method based on caterpillar dimension with heavy-path decomposition. The analysis combines a careful accounting scheme with memory to bound the algorithm cost against OPT, achieving improved competitive ratios in regimes where the caterpillar dimension H is small. The framework provides direct applicability to general tree structures, surpassing previous depth-based results in certain regimes and offering a natural measure for evaluating performance on structured trees.

Abstract

We study the online multi-level aggregation problem with deadlines (MLAP-D) introduced by Bienkowski et al. (ESA 2016, OR 2020). In this problem, requests arrive over time at the vertices of a given vertex-weighted tree, and each request has a deadline that it must be served by. The cost of serving a request equals the cost of a path from the root to the vertex where the request resides. Instead of serving each request individually, requests can be aggregated and served by transmitting a subtree from the root that spans the vertices on which the requests reside, to potentially be more cost-effective. The aggregated cost is the weight of the transmission subtree. The goal of MLAP-D is to find an aggregation solution that minimizes the total cost while serving all requests. We present improved and parameterized algorithms for MLAP-D. Our result is twofold. First, we present an $e(D+1)$-competitive algorithm where $D$ is the depth of the tree. Second, we present an $e(4H+2)$-competitive algorithm where $H$ is the caterpillar dimension of the tree. Here, $H \le D$ and $H \le \log_2 |V|$ where $|V|$ is the number of vertices in the given tree. The caterpillar dimension remains constant for rich but simple classes of trees, such as line graphs ($H=1$), caterpillar graphs ($H=2$), and lobster graphs ($H=3$). To the best of our knowledge, this is the first online algorithm parameterized on a measure better than depth. The state-of-the-art online algorithms are $6(D+1)$-competitive by Buchbinder, Feldman, Naor, and Talmon (SODA 2017) and $O(\log |V|)$-competitive by Azar and Touitou (FOCS 2020). Our framework outperforms the state-of-the-art ratios when $H = o(\min\{D,\log_2 |V|\})$. Our simple framework directly applies to trees with any structure and differs from the previous frameworks that reduce the problem to trees with specific structures.

Figures (1)

• Figure 1: $\mathcal{T}$ consists of the solid and dashed edges. Components connected by the solid edges define the elements of $\mathcal{P}$. The heavy path decomposition $\mathcal{P}$ of $\mathcal{T}$ has six elements: $\{v_d,v_h\}$, $\{r,v_a,v_e,v_i\}$, $\{v_j\}$, $\{v_f\}$, $\{v_b,v_g\}$, and $\{v_c\}$. When the leaf vertex $u=v_h, v_j, v_f, v_g,$ or $v_c$, the $r$-$u$ path intersects two elements in $\mathcal{P}$ which maximizes $d(\mathcal{P},u)$, so the dimension of $\mathcal{P}$ is two. A decomposition with a lower dimension does not exist, so the caterpillar dimension of $\mathcal{T}$ is two.

Theorems & Definitions (41)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1: Expansion vertices, Investment vertices
• Definition 2.2: $I_{v, i}, I_{v, i}', next_{v, i}, next_{v, i}'$
• Definition 2.3: Anticipated vertices, unanticipated vertices
• Lemma 2.3
• Lemma 2.4
• Lemma 2.4
• proof
• Lemma 2.4