Compact Routing on Internet-Like Graphs

The Thorup-Zwick (TZ) routing scheme is the first generic stretch-3 routing scheme delivering a nearly optimal local memory upper bound. Using both direct analysis and simulation, we calculate the stretch distribution of this routing scheme on random graphs with power-law node degree distributions, $P_k \sim k^{-\gamma}$. We find that the average stretch is very low and virtually independent of $\gamma$. In particular, for the Internet interdomain graph, $\gamma \sim 2.1$, the average stretch is around 1.1, with up to 70% of paths being shortest. As the network grows, the average stretch slowly decreases. The routing table is very small, too. It is well below its upper bounds, and its size is around 50 records for $10^4$-node networks. Furthermore, we find that both the average shortest path length (i.e. distance) $\bar{d}$ and width of the distance distribution $\sigma$ observed in the real Internet inter-AS graph have values that are very close to the minimums of the average stretch in the $\bar{d}$- and $\sigma$-directions. This leads us to the discovery of a unique critical quasi-stationary point of the average TZ stretch as a function of $\bar{d}$ and $\sigma$. The Internet distance distribution is located in a close neighborhood of this point. This observation suggests the analytical structure of the average stretch function may be an indirect indicator of some hidden optimization criteria influencing the Internet's interdomain topology evolution.Comment: 29 pages, 16 figure

On Efficient Distributed Construction of Near Optimal Routing Schemes

Given a distributed network represented by a weighted undirected graph $G=(V,E)$ on $n$ vertices, and a parameter $k$, we devise a distributed algorithm that computes a routing scheme in $(n^{1/2+1/k}+D)\cdot n^{o(1)}$ rounds, where $D$ is the hop-diameter of the network. The running time matches the lower bound of $\tilde{\Omega}(n^{1/2}+D)$ rounds (which holds for any scheme with polynomial stretch), up to lower order terms. The routing tables are of size $\tilde{O}(n^{1/k})$, the labels are of size $O(k\log^2n)$, and every packet is routed on a path suffering stretch at most $4k-5+o(1)$. Our construction nearly matches the state-of-the-art for routing schemes built in a centralized sequential manner. The previous best algorithms for building routing tables in a distributed small messages model were by \cite[STOC 2013]{LP13} and \cite[PODC 2015]{LP15}. The former has similar properties but suffers from substantially larger routing tables of size $O(n^{1/2+1/k})$, while the latter has sub-optimal running time of $\tilde{O}(\min\{(nD)^{1/2}\cdot n^{1/k},n^{2/3+2/(3k)}+D\})$

Simpler, faster and shorter labels for distances in graphs

We consider how to assign labels to any undirected graph with n nodes such that, given the labels of two nodes and no other information regarding the graph, it is possible to determine the distance between the two nodes. The challenge in such a distance labeling scheme is primarily to minimize the maximum label lenght and secondarily to minimize the time needed to answer distance queries (decoding). Previous schemes have offered different trade-offs between label lengths and query time. This paper presents a simple algorithm with shorter labels and shorter query time than any previous solution, thereby improving the state-of-the-art with respect to both label length and query time in one single algorithm. Our solution addresses several open problems concerning label length and decoding time and is the first improvement of label length for more than three decades. More specifically, we present a distance labeling scheme with label size (log 3)/2 + o(n) (logarithms are in base 2) and O(1) decoding time. This outperforms all existing results with respect to both size and decoding time, including Winkler's (Combinatorica 1983) decade-old result, which uses labels of size (log 3)n and O(n/log n) decoding time, and Gavoille et al. (SODA'01), which uses labels of size 11n + o(n) and O(loglog n) decoding time. In addition, our algorithm is simpler than the previous ones. In the case of integral edge weights of size at most W, we present almost matching upper and lower bounds for label sizes. For r-additive approximation schemes, where distances can be off by an additive constant r, we give both upper and lower bounds. In particular, we present an upper bound for 1-additive approximation schemes which, in the unweighted case, has the same size (ignoring second order terms) as an adjacency scheme: n/2. We also give results for bipartite graphs and for exact and 1-additive distance oracles

Distributed computing of efficient routing schemes in generalized chordal graphs

International audienceEfficient algorithms for computing routing tables should take advantage of the particular properties arising in large scale networks. Two of them are of particular interest: low (logarithmic) diameter and high clustering coefficient. High clustering coefficient implies the existence of few large induced cycles. Considering this fact, we propose here a routing scheme that computes short routes in the class of $k$-chordal graphs, i.e., graphs with no induced cycles of length more than $k$. In the class of $k$-chordal graphs, our routing scheme achieves an additive stretch of at most $k-1$, i.e., for all pairs of nodes, the length of the route never exceeds their distance plus $k-1$. In order to compute the routing tables of any $n$-node graph with diameter $D$ we propose a distributed algorithm which uses messages of size $O(\log n)$ and takes $O(D)$ time. The corresponding routing scheme achieves the stretch of $k-1$ on $k$-chordal graphs. We then propose a routing scheme that achieves a better additive stretch of $1$ in chordal graphs (notice that chordal graphs are 3-chordal graphs). In this case, the distributed computation of the routing tables takes $O(\min\{\Delta D , n\})$ time, where $\Delta$ is the maximum degree of the graph. Our routing schemes use addresses of size $\log n$ bits and local memory of size $2(d-1) \log n$ bits per node of degree $d$

OREGAMI: Software Tools for Mapping Parallel Computations to Parallel Architectures

22 pagesThe mapping problem in message-passing parallel processors involves the assignment of tasks in a parallel computation to processors and the routing of inter-task messages along the links of the interconnection network. We have developed a unified set of software tools called OREGAMI for automatic and guided mapping of parallel computations to parallel architectures in order to achieve portability and maximal performance from parallel systems. Our tools include a description language which enables the programmer of parallel algorithms to specify information about the static and dynamic communication behavior of the computation to be mapped. This information is used by the mapping algorithms to assign tasks to processors and to route communication in the network topology. Two key features of our system are (a) the ability to take advantage of the regularity present in both the computation structure and the interconnection network and (b) the desire to balance the user's knowledge and intuition with the computational power of efficient combinatorial algorithms