1. 1 Algorithmic aspects of radio access network design in 4G cellular networks David Amzallag Computer Science Department, Technion Joint work with Seffi Naor (Microsoft Research) and Danny Raz (Technion)

2. 2 B A Radio Network Controllers (RNCs) Base Stations Mobile Stations Star Topology Access Network Tree Topology Access Network

3. 3 The routing cost of a tree expected traffic loadrouting cost Given a spanning tree T of a set of base stations, each with its expected traffic load, rooted at RNC r, the routing cost,, between the RNC r and a base station i is the weighted sum of the costs along the (unique) path between then in T. routing cost of a tree The routing cost of a tree T is defined as Routing cost of 5+3+2 = 10 Routing cost of 1+2 = 3 1 2 3 3 4 5 6 Root — Routing cost of the whole tree is 46 — The cost of a minimum spanning tree is 24 Illustrated with

4. 4 The bounded-degree minimum routing cost spanning tree problem – A set of base stations, and an RNC r – A symmetric connection cost is defined for all – Another given cost is of connecting base station to the RNC r triangle inequality – Costs satisfying the triangle inequality ( ) – Degree-constraints for every – The (single-RNC) BDRT problem – The (single-RNC) BDRT problem is to find a minimum routing cost spanning tree on rooted at r that meets for every

5. 5 Previous work – Multiple RNC tree-topology for Access Network Design (AND) – [Harmatos, Jüttner, Szentesi 1999, Harmatos, Szentesi, Gódor 2000, Jüttner, Orbán, Fiala 2005] – Single RNC tree-topology AND – [Szlouencsák et al. 2002] – All published solutions are Simulated Annealing-based – No performance guarantee – Without traffic load considerations

6. 6 Our results are the same – The problems of designing multiple-RNCs and a single-RNC RAN are the same – NP-hard hardness – Study of family of greedy heuristics approximation algorithms – First time using approximation algorithms – A linear-time -approximation algorithm for BDRT with for every NP-hardPNP – In general, BDRT is NP-hard to approximate, unless P=NP

7. 7 The natural greedy approach best – Start at the root vertex r, pick the best vertices as its children, from the left most child to the rightmost one, and then move to the next level in the constructed tree. – For every vertex v ( ), the algorithm pick the best unpicked vertices and assign them as its children, until spanning all the vertices of G. RNC with degree bound b(r) The first best vertex The b(r)th best vertex

8. 8 Examples Picks the vertices v in an increasing order of their ratio where is the cost of the shortest-path connecting v to r in the constructed tree T. RNC with degree bound 3 The best 4 th vertex (with ratio 8/3) — Greedy1 cost is 31 Picks the vertices v in an increasing order of their ratio. — Greedy2 cost is 30 RNC with degree bound 3 The best 4 th vertex (with ratio 3/2)

9. 9 It is not always good to be a greedy 3 vertices with degree bounds n Greedy1 solution (with cost O(n 2 )) n vertices with degree bounds 3 3 vertices of distance n/2 n vertices of distance O(nlogn) Optimal solution (with cost O(n))

10. 10 NP-hard – When speaking of NP-hard problems this can be a (very) difficult to answer can be difficult – Relating directly to can be difficult – Instead we can relate the two in two steps: and where is a lower bound on the optimal solution – How to lower bounding BDRT? shortest-path tree rooted at the RNC node – using the shortest-path tree rooted at the RNC node Lower bounding the optimum The 3rd closest vertex to the root v2v2 v3v3 v4v4 v1v1 i.e. cost ( ) cost ( )

11. 11 An approximation algorithm shortest-path tree – Construct a shortest-path tree, with root r, on the input complete (metric) graph; renumber the vertices so that is the i th closest vertex to the root. – Set as the root of the tree T ; – While T is not a spanning tree do – Pick the vertices of least indices and assign them, from the left most son to the rightmost, as the children of vertex where for and otherwise. – Set

12. 12 Back to the example RNC with degree bound 3 The 4 th closest vertex to the root (with distance 3) — Greedy1 cost is 31 — Greedy2 cost is 30 — Our algorithm’s cost is 29 The cost of the shortest path tree, rooted at a, is 25

13. 13 Bounding the contribution of a vertex – the basic idea The last non-leaf vertex

14. 14 – In general, if is the last non-leaf vertex, then – Since the cost of the SPT of G (with as its root), as computed by the algorithm, is a lower bound of the optimal solution, – In addition, remember that T has leaves and hence, – So,

15. 15 Behavior in practice – Mid-scale networks simulation (up to 200 base stations) – Wide variation of degree-constraints Curve 3: Greedy 2 Curve 4: approximation algorithm (min{w(u,v}/b(v)}) related costs Curve 1: Greedy 1 (min{d(r,v}/b(v)})

16. 16 Future work – Is it possible to improve the approximation factor of BDRT? depth – The depth of the tree is a measure of the time needed for sending information from a base station to the RNC. What if we also have an upper bound on the depth of the tree? How then the problem can be approximated?