# Scale-Free Compact Routing Schemes in Networks of Low Doubling Dimension

## Abstract

In this work, we consider compact routing schemes in networks of low doubling dimension, where the doubling dimension is the least value α such that any ball in the network can be covered by at most 2 ^{α} balls of half radius. There are two variants of routing-scheme design: (i) labeled (name-dependent) routing, in which the designer is allowed to rename the nodes so that the names (labels) can contain additional routing information, for example, topological information; and (ii) name-independent routing, which works on top of the arbitrary original node names in the network, that is, the node names are independent of the routing scheme. In this article, given any constant ε ϵ (0, 1) and an n-node edge-weighted network of doubling dimension α ϵ O(loglog n), we present —a (1 + ε)-stretch labeled compact routing scheme with Γlog n-bit routing labels, O(log ^{2} n/loglog n)-bit packet headers, and ((1/ε) ^{O(α)} log ^{3} n)-bit routing information at each node; —a (9 + ε)-stretch name-independent compact routing scheme with O(log ^{2} n/loglog n)-bit packet headers, and ((1/ε) ^{O(α)} log ^{3} n)-bit routing information at each node. In addition, we prove a lower bound: any name-independent routing scheme with o(n ^{(ε/60)2}) bitsmore »

- Authors:

- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
- Arizona State Univ., Tempe, AZ (United States)
- Google, Kirkland, WA (United States)

- Publication Date:

- Research Org.:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1390021

- Report Number(s):
- LLNL-JRNL-738008

Journal ID: ISSN 1549-6325

- Grant/Contract Number:
- AC52-07NA27344

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- ACM Transactions on Algorithms

- Additional Journal Information:
- Journal Volume: 12; Journal Issue: 3; Journal ID: ISSN 1549-6325

- Publisher:
- Association for Computing Machinery

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; Scale free; labeled routing; name-independent routing; compact routing; doubling dimension

### Citation Formats

```
Konjevod, Goran, Richa, Andréa W., and Xia, Donglin.
```*Scale-Free Compact Routing Schemes in Networks of Low Doubling Dimension*. United States: N. p., 2016.
Web. doi:10.1145/2876055.

```
Konjevod, Goran, Richa, Andréa W., & Xia, Donglin.
```*Scale-Free Compact Routing Schemes in Networks of Low Doubling Dimension*. United States. doi:10.1145/2876055.

```
Konjevod, Goran, Richa, Andréa W., and Xia, Donglin. Wed .
"Scale-Free Compact Routing Schemes in Networks of Low Doubling Dimension". United States.
doi:10.1145/2876055. https://www.osti.gov/servlets/purl/1390021.
```

```
@article{osti_1390021,
```

title = {Scale-Free Compact Routing Schemes in Networks of Low Doubling Dimension},

author = {Konjevod, Goran and Richa, Andréa W. and Xia, Donglin},

abstractNote = {In this work, we consider compact routing schemes in networks of low doubling dimension, where the doubling dimension is the least value α such that any ball in the network can be covered by at most 2α balls of half radius. There are two variants of routing-scheme design: (i) labeled (name-dependent) routing, in which the designer is allowed to rename the nodes so that the names (labels) can contain additional routing information, for example, topological information; and (ii) name-independent routing, which works on top of the arbitrary original node names in the network, that is, the node names are independent of the routing scheme. In this article, given any constant ε ϵ (0, 1) and an n-node edge-weighted network of doubling dimension α ϵ O(loglog n), we present —a (1 + ε)-stretch labeled compact routing scheme with Γlog n-bit routing labels, O(log2 n/loglog n)-bit packet headers, and ((1/ε)O(α) log3 n)-bit routing information at each node; —a (9 + ε)-stretch name-independent compact routing scheme with O(log2 n/loglog n)-bit packet headers, and ((1/ε)O(α) log3 n)-bit routing information at each node. In addition, we prove a lower bound: any name-independent routing scheme with o(n(ε/60)2) bits of storage at each node has stretch no less than 9 - ε for any ε ϵ (0, 8). Therefore, our name-independent routing scheme achieves asymptotically optimal stretch with polylogarithmic storage at each node and packet headers. Note that both schemes are scale-free in the sense that their space requirements do not depend on the normalized diameter Δ of the network. Finally, we also present a simpler nonscale-free (9 + ε)-stretch name-independent compact routing scheme with improved space requirements if Δ is polynomial in n.},

doi = {10.1145/2876055},

journal = {ACM Transactions on Algorithms},

number = 3,

volume = 12,

place = {United States},

year = {Wed Jun 15 00:00:00 EDT 2016},

month = {Wed Jun 15 00:00:00 EDT 2016}

}