The Forcing Weak Edge Detour Number of a Graph

The forcing weak edge detour numbers of certain classes of graphs are determined

Connected Weak Edge Detour Number of a Graph

Certain general properties of the detour distance, weak edge detour set, connected weak edge detour set, connected weak edge detour number and connected weak edge detour basis of graphs are studied in this paper. Their relationship with the detour diameter is discussed. It is proved that for each pair of integers k and n with 2 <= k <= n, there exists a connected graph G of order n with cdnw(G)=k. It is also proved that for any three positive integers R,D,k such that k >= D and R < D <= 2R, there exists a connected graph G with radD (G) = R, diamD G = D and cdnw(G)=k

Bridging the Gap between Programming Languages and Hardware Weak Memory Models

We develop a new intermediate weak memory model, IMM, as a way of modularizing the proofs of correctness of compilation from concurrent programming languages with weak memory consistency semantics to mainstream multi-core architectures, such as POWER and ARM. We use IMM to prove the correctness of compilation from the promising semantics of Kang et al. to POWER (thereby correcting and improving their result) and ARMv7, as well as to the recently revised ARMv8 model. Our results are mechanized in Coq, and to the best of our knowledge, these are the first machine-verified compilation correctness results for models that are weaker than x86-TSO

Shortest Reconfiguration of Sliding Tokens on a Caterpillar

Suppose that we are given two independent sets I_b and I_r of a graph such that |I_b|=|I_r|, and imagine that a token is placed on each vertex in |I_b|. Then, the sliding token problem is to determine whether there exists a sequence of independent sets which transforms I_b into I_r so that each independent set in the sequence results from the previous one by sliding exactly one token along an edge in the graph. The sliding token problem is one of the reconfiguration problems that attract the attention from the viewpoint of theoretical computer science. The reconfiguration problems tend to be PSPACE-complete in general, and some polynomial time algorithms are shown in restricted cases. Recently, the problems that aim at finding a shortest reconfiguration sequence are investigated. For the 3SAT problem, a trichotomy for the complexity of finding the shortest sequence has been shown, that is, it is in P, NP-complete, or PSPACE-complete in certain conditions. In general, even if it is polynomial time solvable to decide whether two instances are reconfigured with each other, it can be NP-complete to find a shortest sequence between them. Namely, finding a shortest sequence between two independent sets can be more difficult than the decision problem of reconfigurability between them. In this paper, we show that the problem for finding a shortest sequence between two independent sets is polynomial time solvable for some graph classes which are subclasses of the class of interval graphs. More precisely, we can find a shortest sequence between two independent sets on a graph G in polynomial time if either G is a proper interval graph, a trivially perfect graph, or a caterpillar. As far as the authors know, this is the first polynomial time algorithm for the shortest sliding token problem for a graph class that requires detours

Arrangements Of Minors In The Positive Grassmannian And a Triangulation of The Hypersimplex

The structure of zero and nonzero minors in the Grassmannian leads to rich combinatorics of matroids. In this paper, we investigate an even richer structure of possible equalities and inequalities between the minors in the positive Grassmannian. It was previously shown that arrangements of equal minors of largest value are in bijection with the simplices in a certain triangulation of the hypersimplex that was studied by Stanley, Sturmfels, Lam and Postnikov. Here we investigate the entire set of arrangements and its relations with this triangulation. First, we show that second largest minors correspond to the facets of the simplices. We then introduce the notion of cubical distance on the dual graph of the triangulation, and study its relations with the arrangement of t-th largest minors. Finally, we show that arrangements of largest minors induce a structure of partially ordered sets on the entire collection of minors. We use the Lam and Postnikov circuit triangulation of the hypersimplex to describe a 2-dimensional grid structure of this poset

SPIDER: Fault Resilient SDN Pipeline with Recovery Delay Guarantees

When dealing with node or link failures in Software Defined Networking (SDN), the network capability to establish an alternative path depends on controller reachability and on the round trip times (RTTs) between controller and involved switches. Moreover, current SDN data plane abstractions for failure detection (e.g. OpenFlow "Fast-failover") do not allow programmers to tweak switches' detection mechanism, thus leaving SDN operators still relying on proprietary management interfaces (when available) to achieve guaranteed detection and recovery delays. We propose SPIDER, an OpenFlow-like pipeline design that provides i) a detection mechanism based on switches' periodic link probing and ii) fast reroute of traffic flows even in case of distant failures, regardless of controller availability. SPIDER can be implemented using stateful data plane abstractions such as OpenState or Open vSwitch, and it offers guaranteed short (i.e. ms) failure detection and recovery delays, with a configurable trade off between overhead and failover responsiveness. We present here the SPIDER pipeline design, behavioral model, and analysis on flow tables' memory impact. We also implemented and experimentally validated SPIDER using OpenState (an OpenFlow 1.3 extension for stateful packet processing), showing numerical results on its performance in terms of recovery latency and packet losses.Comment: 8 page

Extended Dijkstra algorithm and Moore-Bellman-Ford algorithm

Study the general single-source shortest path problem. Firstly, define a path function on a set of some path with same source on a graph, and develop a kind of general single-source shortest path problem (GSSSP) on the defined path function. Secondly, following respectively the approaches of the well known Dijkstra's algorithm and Moore-Bellman-Ford algorithm, design an extended Dijkstra's algorithm (EDA) and an extended Moore-Bellman-Ford algorithm (EMBFA) to solve the problem GSSSP under certain given conditions. Thirdly, introduce a few concepts, such as order-preserving in last road (OPLR) of path function, and so on. And under the assumption that the value of related path function for any path can be obtained in $M(n)$ time, prove respectively the algorithm EDA solving the problem GSSSP in $O(n^2)M(n)$ time and the algorithm EMBFA solving the problem GSSSP in $O(mn)M(n)$ time. Finally, some applications of the designed algorithms are shown with a few examples. What we done can improve both the researchers and the applications of the shortest path theory.Comment: 25 page