Online Multi-level Aggregation with Delays and Stochastic Arrivals

TL;DR

A deterministic online algorithm is presented which achieves a constant ratio of expectations, meaning that the ratio between the expected costs of the solution generated by the algorithm and the optimal offline solution is bounded by a constant.

Abstract

This paper presents a new research direction for online Multi-Level Aggregation (MLA) with delays. In this problem, we are given an edge-weighted rooted tree $T$, and we have to serve a sequence of requests arriving at its vertices in an online manner. Each request $r$ is characterized by two parameters: its arrival time $t(r)$ and location $l(r)$ (a vertex). Once a request $r$ arrives, we can either serve it immediately or postpone this action until any time $t > t(r)$. We can serve several pending requests at the same time, and the service cost of a service corresponds to the weight of the subtree that contains all the requests served and the root of $T$. Postponing the service of a request $r$ to time $t > t(r)$ generates an additional delay cost of $t - t(r)$. The goal is to serve all requests in an online manner such that the total cost (i.e., the total sum of service and delay costs) is minimized. The current best algorithm for this problem achieves a competitive ratio of $O(d^2)$ (Azar and Touitou, FOCS'19), where $d$ denotes the depth of the tree. Here, we consider a stochastic version of MLA where the requests follow a Poisson arrival process. We present a deterministic online algorithm which achieves a constant ratio of expectations, meaning that the ratio between the expected costs of the solution generated by our algorithm and the optimal offline solution is bounded by a constant. Our algorithm is obtained by carefully combining two strategies. In the first one, we plan periodic oblivious visits to the subset of frequent vertices, whereas in the second one, we greedily serve the pending requests in the remaining vertices. This problem is complex enough to demonstrate a very rare phenomenon that ``single-minded" or ``sample-average" strategies are not enough in stochastic optimization.

Figures (5)

• Figure 1: Here is an example to show how Algorithm \ref{['alg:plan']} works on an heavy instance. Given the tree consisting of 7 vertices (with $w_i \ge 1/ \lambda_i$ for each vertex $i \in [7]$ marked in different color), we use the length of the colored line to denote the saturated amount (i.e., $\boldsymbol{\lambda}_i / 2 \cdot t^2$) of a vertex $i$ at any time $t$. At time $p_1$, the subtree $T_1$ including vertices 1 and 3 is determined; similarly, $T_2$ includes vertices 2 and 5 at time $t_2$; $T_3$ includes vertices 4 and 6 at time $p_3$; and $T_4$ includes vertex 7 at time $p_4$.
• Figure 2: An example of a balanced partition (Definition \ref{['definition:balanced_partition']}). The weight of each edge is shown in black, and the arrival rate of each vertex is shown in red. Green subsets corresponds to parts of type-I while purple ones correspond to parts of type-II. Some value of $\pi$ are shown for the top-left type-I part and for the top-right type-II part.
• Figure 3: Construction of a balanced partition in the proof of Lemma \ref{['lemma:balanced_partition']}. The weights of the edges and the arrival rates of the vertices are the same as in Figure \ref{['fig:example_partition']}. The numbers represent an ordering of the vertices. The gray sets corresponds to $U_i$, for $i\in [40]$. We illustrate the step $i=30$ of the algorithm. We have $C_{30}=\{24,26\}$ and $U_{30}=\{u_{30}\}\cup U_{24}\cup U_{30}=\{u_{30},u_{24},u_{19}, u_{20},u_{26}\}$. Since $\pi(U_{30}\cup \{u_{25}\})=2.65>1$, we add $U_{30}$ into $\mathcal{P}^{(29)}$ to create $\mathcal{P}^{(30)}$ (red stoke). We remark that $U_{30}$ is a balanced subset of type-II and $U_{27}$ is a balanced subset of type-II, while $U_{25}$ is not a balanced subset.
• Figure 4: The construction of the augmented tree associated with the instance and the balanced partition of Figure \ref{['fig:example_partition']}. The new edges and vertices are shown in red. The illustrate the calculation of the length of these edges for a part $U_1$ of type-I and for a part $U_2$ of type-II. For each part $U$ of the partition, we indicate the values of $\boldsymbol{\lambda}(U)$ and $\pi(U)$. For simplicity, we have rounded the values to their second decimal.
• Figure :

Theorems & Definitions (49)

• Theorem 1.1
• Definition 2.1: Poisson arrival model
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• Proposition 2.5
• Proposition 2.6
• Proposition 2.7
• Definition 2.8: ratio of expectations
• Lemma 3.1