Section: New Results

Compact and distributed data structures

Query optimization in databases

Participants : Nicolas Hanusse, Sofian Maabout.

Datacubes are data structures designed to query optimization in databases. In [42] we provide some algorithmic solutions in a user centric setting: the request time is guaranteed and the amount of memory space is minimized.

Parallel computations of Borders

Participants : Nicolas Hanusse, Sofian Maabout.

Borders are fundamental building blocks in data mining. They are used to find frequent patterns, dependencies between attributes, ... In[43] we provide an algorithm that computes borders with a speedup of $p$ (under reasonable hypothesis) for $p$ cores.

Node-disjoint multipath spanners and their relationship with fault-tolerant spanners

Participants : Cyril Gavoille, Quentin Godfroy.

Motivated by multipath routing, we introduce a multi-connected variant of spanners. For that purpose we introduce in [40] 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.

On approximate distance labels and routing schemes with affine stretch

Participant : Cyril Gavoille.

For every integral parameter $k>1$, given an unweighted graph $G$, we construct in polynomial time in [25] , for each vertex $u$, a distance label $L\left(u\right)$ of size $O\left(n^{2/(2k-1)}\right)$. For any $u,v\in G$, given $L\left(u\right),L\left(v\right)$ we can return in time $O\left(k\right)$ an affine approximation $ð(u,v)$ on the distance $d(u,v)$ between $u$ and $v$ in $G$ such that $d(u,v)\le ð(u,v)\le (2k-2)d(u,v)+1$. Hence we say that our distance label scheme has affine stretch of $(2k-2)d+1$. For $k=2$ our construction is comparable to the $O\left(n^{5/3}\right)$ size, $2d+1$ affine stretch of the distance oracle of Pǎtraşcu and Roditty (FOCS '10), it incurs a $o(logn)$ storage overhead while providing the benefits of a distance label.

For any $k>1$, given a restriction of $o\left(n^{1+1/(k-1)}\right)$ on the total size of the data structure, our construction provides distance labels with affine stretch of $(2k-2)d+1$ which is better than the stretch $(2k-1)d$ scheme of Thorup and Zwick (J. ACM '05).

Our second contribution is a compact routing scheme with poly-logarithmic addresses that provides affine stretch guarantees. With $O\left(n^{3/(3k-2)}\right)$-bit routing tables we obtain affine stretch of $(4k-6)d+1$, for any $k>1$.

Given a restriction of $o\left(n^{1/(k-1)}\right)$ on the table size, our routing scheme provides affine stretch which is better than the stretch $(4k-5)d$ routing scheme of Thorup and Zwick (SPAA '01).

Sparse spanners vs. compact routing

Participant : Cyril Gavoille.

Routing with multiplicative stretch 3 (which means that the path used by the routing scheme can be up to three times longer than a shortest path) can be done with routing tables of $\Theta \left(\sqrt{n}\right)$ bits(Tilde-big-$O$ notation is similar to big-$O$ notation up to factors poly-logarithmic in $n$.) per node. The space lower bound is due to the existence of dense graphs with large girth. Dense graphs can be sparsified to subgraphs, called 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 [39] , 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. We prove that any routing scheme using routing tables of size $M$ bits per node and addresses of poly-logarithmic length has additive stretch $\Omega \left(\sqrt{n/M}\right)$ for general graphs, and $\Omega (\sqrt{n}/M)$ for planar graphs. Routing with tables of size $O\left(n^{1/3}\right)$ thus requires a polynomial additive stretch of $\Omega \left(n^{1/3}\right)$, whereas spanners with average degree $O\left(n^{1/3}\right)$ and constant additive stretch exist for all graphs. Spanners, however sparse they are, do not tell us how to route. These bounds provide the first separation of sparse spanner problems and compact routing problems.

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. Actually, our scheme applies to general graphs, and the above bound of $n^{1/3}$ on the stretch and the routing table size holds for all graphs of linear local tree-width, a class of graphs including bounded-genus graphs and apex-minor-free graphs.