SpiderDAN: Matching Augmentation in Demand-Aware Networks

TL;DR

This work designs and evaluates a series of heuristics that can deal with arbitrary graphs as the underlying network structure and proves that this problem is NP-hard, even if the physical network is a cycle.

Abstract

Graph augmentation is a fundamental and well-studied problem that arises in network optimization. We consider a new variant of this model motivated by reconfigurable communication networks. In this variant, we consider a given physical network and the measured communication demands between the nodes. Our goal is to augment the given physical network with a matching, so that the shortest path lengths in the augmented network, weighted with the demands, are minimal.We prove that this problem is NP-hard, even if the physical network is a cycle. We then use results from demand-aware network design to provide a constant-factor approximation algorithm for adding a matching in case that only a few nodes in the network cause almost all the communication. For general real-world communication patterns, we design and evaluate a series of heuristics that can deal with arbitrary graphs as the underlying network structure. Our algorithms are validated experimentally using real-world traces (from e.g., Facebook) of data centers.

Figures (4)

• Figure 2: Steps of SpiderDAN considering a graph with $14$ nodes (Figure \ref{['fig: main Graph']}), and super-nodes of size $\alpha = 3$. We first run a depth-first search (DFS) from the node in the upper-right, shown in blue (Figure \ref{['fig: DFS']}). We then do a pre-order traversal of the tree from one of the deepest leaves, collecting nodes in groups of $3$(Figure \ref{['fig: positioning']}, we consider the last group to have size only $2$). Figure \ref{['fig: dan on super']} shows a DAN is built on top of the super-graph, and lastly in Figure \ref{['fig: inside super']} we show how this DAN can be transformed back into matchings.
• Figure 3: The running time of all of our algorithms is displayed. The number of vertices are powers of two (up to 4096 vertices). We have considered randomly generated demand matrices with sparsity value $0.9$. We capped off the running times at $10$ seconds, and showed it in log-log format for better visibility.
• Figure 4: Left: the results on the three datasets from Facebook. Right: Results for $66$ instances from SuiteSparse matrix collection. The instances are sorted by the quality of SpiderDAN.
• Figure 5: Effect of various parameters on the approximation ratio of algorithms (the cost of the respective algorithm divided by the cost of ring). Figure \ref{['fig: infra']} considers on $4096$ node (as it is even and a power of six, so we can have both 2D and 3D Torus with this size), and we used sparse instances with $\gamma =0.9$. The two other figures are based on $4096$ and $1024$ nodes, due to the time limit that we set for each instance.

Theorems & Definitions (6)

• Theorem 3.1
• Lemma 4.1
• Lemma 4.2
• Lemma 4.3
• Lemma 4.4: DAN lowerbound avin2017demand
• Theorem 4.1