Section: Application Domains

Compact Routing

Project-team positioning

In this axis, CEPAGE mainly works on the design on distributed and light data structures. One of the techniques consists in summarizing the topology and metric of the networks allowing to route or to approximate the original distances within the network. Such structures, often called spanners, does not require the storage of all the original network links. Then we get economic distributed data structures that can be updated without a high communication cost. Our main collaborations are done with the best specialists world-wide, in particular: Israel (Weizmann), USA (MIT, Microsoft, Chicago), Belgium (Alcatel Lucent-Bell), France (Paris, Nice).

Algorithms and Routing are also intensively studied in research labs in the USA (CAIDA). Our contributions appear regularly at all of the major conferences in Distributed Computing (PODC, DISC, SPAA), as well as at top venues with a more general algorithmic audience (STOC, SODA, ICALP, ESA). Members of CEPAGE actively participate in these events (ICALP 2010 and DISC 2009 were organized by members of CEPAGE).

Within Inria, studies of mobile agents are also performed in the GANG project and to some extent also within MASCOTTE within the european project EULER.

Scientific achievements

There are several techniques to manage sub-linear size routing tables (in the number of nodes of the platform) while guaranteeing almost shortest paths. Some techniques provide routes of length at most $1+ϵ$ times the length of the shortest one while maintaining a poly-logarithmic number of entries per routing table. However, these techniques are not universal in the sense that they apply only on some class of underlying topologies. Universal schemes exist. Typically they achieve $O\left(\sqrt{n}\right)$-entry local routing tables for a stretch factor of 3 in the worst case. Some experiments have shown that such methods, although universal, work very well in practice, in average, on realistic scale-free or existing topologies.

The space lower bound of $O\left(\sqrt{n}\right)$-entry for routing with multiplicative stretch 3 is due to the existence of dense graphs with large girth. Dense graphs can be sparsified to subgraphs (spanners), with various stretch guarantees. There are spanners with additive stretch guarantees (some even have constant additive stretch) but only very few additive routing schemes are known.

In (SPAA 2012 [101] ), we give reasons why routing in unweighted graphs with additive stretch is difficult in the form of space lower bounds for general graphs and for planar graphs. On the positive side, we give an almost tight upper bound: we present the first non-trivial compact routing scheme with $o\left({lg}^{2}n\right)$-bit addresses, additive stretch $O\left(n^{1/3}\right)$, and table size $O\left(n^{1/3}\right)$ bits for planar graphs.

We have recently considered the forbidden-set extension of distance oracles and routing schemes. Given an arbitrary set of edge/node failure $F$, a source $s$ and a target $t$ such that $s,t\notin F$, the goal is to route (or evaluate the distance) between $s$ and $t$ in the graph $G\setminus F$, so avoiding $F$. The classical problem is for $F=\varnothing$. This extension is considered as a first step toward fully dynamic data-structures, a challenging goal. For graphs of low doubling dimension we have shown in (PODC 2012 [58] ) that it is possible to route from $s$ to $t$ in $G\setminus F$ with stretch $1+ϵ$, for all $s,t,F$, given poly-logarithmic size labels of all the nodes invoked in the query $(s,t,F)$. This has been generalized to all planar graphs achieving similar stretch and label size performences. As a byproduct we have designed a fully dynamic algorithm for maintaining $1+ϵ$ approximate distances in planar graphs supporting edge/node addition/removal within update and query time $\sqrt{n}$ in the worst-case (STOC 2012 [57] ).

${\Theta }_k$-graphs are geometric graphs that appear in the context of graph navigation. The shortest-path metric of these graphs is known to approximate the Euclidean complete graph up to a factor depending on the cone number $k$ and the dimension of the space. We have introduced in (WG 2010 [79] ) a specific subgraph of the ${\Theta }_6$-graph defined in the 2D Euclidean space, namely the $\text{half-}{\Theta }_6$-graph, composed of the even-cone edges of the ${\Theta }_6$-graph. Our main contribution is to show that these graphs are exactly the TD-Delaunay graphs, and are strongly connected to the geodesic embeddings of orthogonal surfaces of coplanar points in the 3D Euclidean space. We also studied the asymptotic behavior of these spanners  (Adv. in Appl. Proba.  [116] ) and in collaboration with Ljubomir Perković, we worked on the question of bounded degree planar spanner. We proposed an algorithm that computes a plane 6-spanner of degree at most 6 in (ICALP 2010 [80] ). The previous best bound on the maximum degree for constant stretch plane spanners was only 14.

In order to cope with network dynamism and failures, and motivated by multipath routing, we introduce a multi-connected variant of spanners. For that purpose we introduce in (OPODIS 2011 [102] ) the $p$-multipath cost between two nodes $u$ and $v$ as the minimum weight of a collection of $p$ internally vertex-disjoint paths between $u$ and $v$. Given a weighted graph $G$, a subgraph $H$ is a $p$-multipath $s$-spanner if for all $u,v$, the $p$-multipath cost between $u$ and $v$ in $H$ is at most $s$ times the $p$-multipath cost in $G$. The $s$ factor is called the stretch. Building upon recent results on fault-tolerant spanners, we show how to build $p$-multipath spanners of constant stretch and of $O\left(n^{1+1/k}\right)$ edges, for fixed parameters $p$ and $k$, $n$ being the number of nodes of the graph. Such spanners can be constructed by a distributed algorithm running in $O\left(k\right)$ rounds. Additionally, we give an improved construction for the case $p=k=2$. Our spanner $H$ has $O\left(n^{3/2}\right)$ edges and the $p$-multipath cost in $H$ between any two node is at most twice the corresponding one in $G$ plus $O\left(W\right)$, $W$ being the maximum edge weight.

We also worked on compact coding in data warehouses: in order to get quick answer in large data, we have to estimate, select and materialize (store) partial data structures. We got several solutions with a prescribed guarantee in different models for the following problems: view size estimation with small samples, view selection, parallel computation of frequent itemsets. In (Theor. Comp. Sci.  [105] ) a new algorithm that allow the administrator or user of a DBMS to choose which part of the data cube to optimize (known as the the views selection problem), that takes as input a fact table and computes a set of views to store in order to speed up queries.

Perspectives: The compact coding activity in data-warehouse is promising since the amount of data collected keeps on increasing and being able to answer in real-time complex requests (data mining) is still challenging.

Some robust data structures already exist which, given a small number of $k$ changes of topology or $k$ faults, tolerate these faults, i.e., alternative routes with bounded stretch can be provided without any updates. This is a first step toward dynamic networks but the updates of these data structures are currently still quite complicated with a high communication cost.