Non-minimal Paths Research Articles

HAFTA: Highly adaptive fault‐tolerant routing algorithm for two‐dimensional network‐on‐chips

AbstractLearning‐based routing algorithms are good candidates for two‐dimensional Network‐on‐Chip (NoC) architectures since they can give good path selection decisions by combining current and past state of the network traffic. On the other hand, they generally send packets through nonminimal paths in order to detour congested areas in an attempt to minimize the communication cost. Since adapting to the traffic changes takes time to gather enough feedback information by the applied learning model, non‐minimal paths may take more time than the congested minimal paths. In this work, we incorporate a probabilistic method to a Q‐learning‐based NoC routing algorithm for selecting a minimal or nonminimal path to minimize the negative effect of learning duration. We also consider the errors in the architecture and propose a fault‐tolerance mechanism that detects both transient and permanent link errors. We compared our method against a standard Q‐learning‐based routing algorithm on several traffic models in terms of throughput and latency. The results show that our method outperforms its counterpart up to 30% in latency and 7% in throughput.

The Case for Dynamic Bias in Global Adaptive Routing

Global adaptive routing is a critical component of high-radix networks in large-scale systems and is necessary to fully exploit the path diversity in high-radix topologies. The routing decision in global adaptive routing is made between minimal and non-minimal paths, often based on local information (e.g., queue occupancy) and rely on “approximate” congestion information through backpressure. Different heuristic-based adaptive routing algorithms have been proposed for high-radix topologies but they often rely on local-only information that can lead to inefficient routing decisions. In this letter, we propose DGB – <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">De</b> coupled, <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</b> radient descent-based <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</b> iasing routing algorithm to address the limitation of global adaptive routing. With DGB, both the local and global congestion information are <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">decoupled</i> in the routing decision. In particular, we propose to leverage a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dynamic</i> bias in the global adaptive routing where gradient descent approach is leveraged to adjust the adaptive routing bias appropriately. Our results show that DGB can effectively adjust the bias dynamically to outperform previously proposed routing algorithms on high-radix topologies.

High-Performance Routing With Multipathing and Path Diversity in Ethernet and HPC Networks

The recent line of research into topology design focuses on lowering network diameter. Many low-diameter topologies such as Slim Fly or Jellyfish that substantially reduce cost, power consumption, and latency have been proposed. A key challenge in realizing the benefits of these topologies is routing. On one hand, these networks provide shorter path lengths than established topologies such as Clos or torus, leading to performance improvements. On the other hand, the number of shortest paths between each pair of endpoints is much smaller than in Clos, but there is a large number of non-minimal paths between router pairs. This hampers or even makes it impossible to use established multipath routing schemes such as ECMP. In this article, to facilitate high-performance routing in modern networks, we analyze existing routing protocols and architectures, focusing on how well they exploit the diversity of minimal and non-minimal paths. We first develop a taxonomy of different forms of support for multipathing and overall path diversity. Then, we analyze how existing routing schemes support this diversity. Among others, we consider multipathing with both shortest and non-shortest paths, support for disjoint paths, or enabling adaptivity. To address the ongoing convergence of HPC and “Big Data” domains, we consider routing protocols developed for both HPC systems and for data centers as well as general clusters. Thus, we cover architectures and protocols based on Ethernet, InfiniBand, and other HPC networks such as Myrinet. Our review will foster developing future high-performance multipathing routing protocols in supercomputers and data centers.

Application of Logical Sub-networking in Congestion-aware Deadlock-free SDmesh Routing

An adaptive routing helps in evading early network saturation by steering data packets through the less congested area at the oppressive loaded situation. However, performances of adaptive routing are not always promising under all circumstances. Say for, given more freedom in choosing an alternate route on non-minimal paths for a substantially loaded network even may result in worsening network performances due to following longer route under adaptive routing. Here, underlying topology facilitates routing by offering more alternate short-cut routes on minimal or quasi-minimal paths. This work presents a congestion-aware (CA) adaptive routing for one-hop diagonally connected subnet-based mesh (SDmesh) network aiming to facilitate both performances and routing flexibility simultaneously. Our proposed technique on the selected system facilitates packet routing, offering more options in choosing an output link from minimal or quasi-minimal paths and hence helps in lowering packet delay by shortening the length of traversed traffic under the oppressive loaded situation. Furthermore, we have also employed a congestion-aware virtual input crossbar router aiming to split the entire network into two distinct logically separated sub-networks. It facilitates preserving important routing properties like deadlock, live-lock fairness, and other essential routing constraints. Experiments, conducted over two 8×8- and 12×12-sized networks, show an average improvement of 25--87.5% saturated latency and 60--83% throughput improvement under uniform traffic patterns for the proposed CA routing compared to centralized adaptive XY routing. Experimental results on application-specific PARSEC and SPLASH2 benchmark suites show an average of 22--50% latency and 23--30% throughput improvements by the proposed technique compared to centralized XY routing on the baseline mesh network. Moreover, experiments were also carried out to check the performance of the proposed routing method with different newly proposed deadlock-free adaptive routing approaches over the same subnet-based diagonal mesh (SDmesh) network and reported.

Uniform Minimal First: Latency Reduction in Throughput-Optimal Oblivious Routing for Mesh-Based Networks-on-Chip

Mesh-based networks-on-chips (NoCs) are increasingly used for on-chip communications in embedded multicore processors. O1TURN is a well-known oblivious routing algorithm for mesh-based NoCs that has been shown to be worst-case throughput optimal for even network radices, but not for odd radices. More recently, another oblivious routing algorithm called U2TURN has been shown to be worst-case throughput optimal for both odd and even radices. This is accomplished by load-balancing among 2-turn paths in XYX or YXY routing, including nonminimal paths. Besides being worst-case throughput optimal, U2TURN achieves higher throughput than O1TURN under adversarial traffic. However, for random traffic, where the traffic is inherently load-balanced, O1TURN achieves lower network latency since it only considers minimal paths. In this letter, we propose a hybrid oblivious routing algorithm called uniform-minimal-first (UMF) routing. UMF works by exploiting any inherent load-balancing characteristics of the traffic pattern to reduce packet latency, but it retains the throughput optimality of U2TURN for both odd and even radices, and it achieves the performance of U2TURN under adversarial traffic. UMF is very inexpensive to implement as it just requires incrementing two small counters, but only for the head flit when a node injects a new packet (not for any pass-through traffic). These simple updates can be performed one cycle ahead of packet injection to avoid slowing down any router pipeline stage. Despite its simplicity, UMF outperforms U2TURN by 23.7% under random traffic and O1TURN by 12.6% under adversarial traffic.

Distributed Shortcut Networks: Low-Latency Low-Degree Non-Random Topologies Targeting the Diameter and Cable Length Trade-Off

Low communication latency becomes a main concern in highly parallel computers and supercomputers that reach millions of processing cores. Random network topologies are better suited to achieve low average shortest path length and low diameter in terms of the hop counts between nodes. However, random topologies lead to two problems: (1) increased aggregate cable length on a machine room floor that would become dominant for communication latency in next-generation custom supercomputers, and (2) high routing complexity that typically requires a routing table at each node (e.g., topology-agnostic deadlock-free routing). In this context, we first propose low-degree non-random topologies that exploit the small-world effect, which has been well modeled by some random network models. Our main idea is to carefully design a set of various-length shortcuts that keep the diameter small while maintaining a short cable length for economical passive electric cables. We also propose custom routing that uses the regularity of the various-length shortcuts. Our experimental graph analyses show that our proposed topology has low diameter and low average shortest path length, which are considerably better than those of the counterpart 3-D torus and are near to those of a random topology with the same average degree. The proposed topology has average cable length drastically shorter than that of the counterpart random topology, which leads to low cost of interconnection networks. Our custom routing takes non-minimal paths to provide lower zero-load latency than the minimal custom routings on different counterpart topologies. Our discrete-event simulation results using SimGrid show that our proposed topology is suitable for applications that have irregular communication patterns or non-nearest neighbor collective communication patterns.

Towards High-Performance Bufferless NoCs with SCEPTER

In the many-core era, the network on-chip (NoC) is playing a larger role in meeting performance, area and power goals, as router buffers contribute greatly to NoC area and power usage. Proposals have advocated bufferless NoCs, however a performance wall has been reached such that high throughput performance has not been extracted. We present SCEPTER, a high-performance bufferless mesh NoC that sets up single-cycle virtual express paths dynamically across the chip, allowing deflected packets to go through non-minimal paths with no latency penalty. For a 64 node network, we demonstrate an average $62$ percent reduction in latency and an average 1.3 $\times$ higher throughput over a baseline bufferless NoC for synthetic traffic patterns; with comparable performance to a single-cycle multihop buffered mesh network with six flit buffers, per input port, in each router.

An effective methodology to improve the performance of the up*/down* routing algorithm

Networks of workstations (NOWs) are being considered as a cost-effective alternative to parallel computers. Most NOWs are arranged as a switch-based network and provide mechanisms for discovering the network topology. Hence, they provide support for both regular and irregular topologies, which makes routing and deadlock avoidance quite complicated. Current proposals use the up*/down* routing algorithm to remove cyclic dependencies between channels and avoid deadlock. However, routing is considerably restricted and most messages must follow nonminimal paths, increasing latency and wasting resources. We propose and evaluate a simple and effective methodology to compute up*/down* routing tables. The new methodology is based on computing a depth-first search (DPS) spanning tree on the network graph that decreases the number of routing restrictions with respect to the breadth-first search (BFS) spanning tree used by the traditional methodology. Additionally, we propose different heuristic rules for computing the spanning trees to improve the efficiency of up*/down* routing. Evaluation results for several different topologies show that computing the up*/down* routing tables by using the new methodology increases throughput by a factor of up to 2.48 in large networks with respect to the traditional methodology, and also reduces latency significantly.

High-performance routing in networks of workstations with irregular topology

Networks of workstations are rapidly emerging as a cost-effective alternative to parallel computers. Switch-based interconnects with irregular topology allow the wiring flexibility, scalability, and incremental expansion capability required in this environment. However, the irregularity also makes routing and deadlock avoidance on such systems quite complicated. In current proposals, many messages are routed following nonminimal paths, increasing latency and wasting resources. In this paper, we propose two general methodologies for the design of adaptive routing algorithms for networks with irregular topology. Routing algorithms designed according to these methodologies allow messages to follow minimal paths in most cases, reducing message latency and increasing network throughput. As an example of application, we propose two adaptive routing algorithms for ANI (previously known as Autonet). They can be implemented either by duplicating physical channels or by splitting each physical channel into two virtual channels. In the former case, the implementation does not require a new switch design. It only requires changing the routing tables and adding links in parallel with existing ones, taking advantage of spare switch ports. In the latter case, a new switch design is required, but the network topology is not changed. Evaluation results for several different tapologies and message distributions show that the new routing algorithms are able to increase throughput for random traffic by a factor of up to 4 with respect to the original up*/down* algorithm, also reducing latency significantly. For other message distributions, throughput is increased more than seven times. We also show that most of the improvement comes from the use of minimal routing.

Software-based rerouting for fault-tolerant pipelined communication

This paper presents a software-based approach to fault-tolerant routing in networks using wormhole or virtual cut-through switching. When a message encounters a faulty output link, it is removed from the network by the local router and delivered to the messaging layer of the local node's operating system. The message passing software can reroute this message, possibly along nonminimal paths. Alternatively, the message may be addressed to an intermediate node, which will forward the message to the destination. A message may encounter multiple faults and pass through multiple intermediate nodes. The proposed techniques are applicable to both obliviously and adaptively routed networks. The techniques are specifically targeted toward commercial multiprocessors where the mean time to repair (MTTR) is much smaller than the mean time between router failures (MTBF), i.e., it is sufficient to tolerate a maximum of three failures. This paper presents requirements for buffer management, deadlock freedom, and livelock freedom. Simulation results are presented to evaluate the degradation in latency and throughput as a function of the number and distribution of faults. There are several advantages of such an approach. Router designs are minimally impacted, and thus remain compact and fast. Only messages that encounter faulty components are affected, while the machine is ensured of continued operation until the faulty components can be replaced. The technique leverages existing network technology, and the concepts are portable across evolving switch and router designs. Therefore, we feel that the technique is a good candidate for incorporation into the next generation of multiprocessor networks.

Linear Steiner Trees for Infinite Spirals

A full Steiner tree T for a given set of points P is defined to be linear if all Steiner points lie on one path called the trunk of T. A (nonfull) Steiner tree is linear if it is a degeneracy of a full linear Steiner tree. Suppose P is a simple polygonal line. Roughly speaking, T is similar to P if its trunk turns to the left or right when P does. P is a left-turn (or right-turn) polygonal spiral if it always turns to the left (or right) at its vertices. P is an infinite spiral if n tends to infinity. In this paper we first prove some results on nonminimal paths and the decomposition of Steiner minimal trees, and then, based on these results, we study the case in which an infinite spiral P has a Steiner minimal tree that is linear and similar to P itself.

The use of the state space to record the behavioral effects of subproblems and symmetries in the Tower of Hanoi problem

This research was designed to focus on the effects of problem structure on the behavior of subjects solving that problem. The behavior of forty-five adult subjects solving the 4-ring Tower of Hanoi problem was exhibited as paths through the “state space representation” of the problem. Four hypotheses concerning the effects of the problem's structure were tested experimentally. (1) Subjects' paths were both non-random and goal-directed through the problem and its subproblems and the special role of subgoal states was identified. (2) “Episodes” were seen to occur during problem solving corresponding to the consistent solution throughout the problem of subproblems with identical or isomorphic structure, however, (3) the evidence for congruence of subjects' non-minimal solution paths through isomorphic subproblems was inconclusive. (4) The special effect on behavior of symmetries within the structure of the problem was delineated. Directions for further research are outlined.