On Channel-Discontinuity-Constraint Routing in Multi-Channel Wireless Infrastructure Networks

1. On Channel-Discontinuity-Constraint Routing in Multi-Channel Wireless Infrastructure Networks AbishekGopalan, SwaminathanSankararaman

2. Wireless infrastructure networks • Wireless infrastructure networks becoming more popular • Backbone may operate in 802.11a, while user interface may be on 802.11b/g • Increasing throughput in wireless infrastructure networks • Simultaneous transmission on multiple orthogonal channels • Use of directional antenna for improved spatial throughput • Inter-flow and Intra-flow interference • Inter-flow: Two links belonging to different flows cannot be scheduled at the same time • Intra-flow: Two links belonging to the same flow cannot be scheduled at the same time • Routing and channel assignment • Compute path and channel assignment that avoids inter- and intra-flow interference

3. Omnidirectional and Directional transmission • Omnidirectional transmission • Directional transmission

4. Collinearity (distance-2) constraint • Two non-adjacent links cannot be scheduled at the same time • X-Y and Z-W transmission cannot take place simultaneously • Distance-2 dependency • Logical distance-2; not physical distance-2 • Channel assignment problem is equivalent to distance-2 coloring problem (NP-Hard) • Eliminating distance-2 dependency • Use directional transmission • Use power control • Space the nodes sufficiently apart to eliminate side and back lobe interference • Use of metamaterials for shaping the electromagnetic radiation

5. Link and path bandwidth • Consider wireless infrastructure network with no distance-2 constraint • Wireless interference constraints • A node cannot receive from two different transmitters on the same channel • A node cannot transmit and receive on the same channel • Assume bandwidth of a link (for a channel) is B • When is the bandwidth of a multi-hop path B? No two consecutive links on the path are assigned the same channel

6. Routing and channel assignment • Channel discontinuity constraint (CDC) • No two consecutive links in a path are assigned the same channel • A path that obeys the constraint is called CDC path • Goal: To obtain the minimum-cost CDC path • Example Given a multi-channel wireless network with no collinear interference, the set of available channels at every node, the cost of the links, and a node pair (s, d) find the minimum cost path between s and d along with channel assignment on every link of the path such that no two consecutive links in the path are assigned the same channel.

7. Edmonds-Szeider expansion • Node expansion • Link expansion

8. Minimum cost perfect matching (MCPM) • Example network and expanded graph • Expand all nodes except s and d • Complexity: O(ne)

9. CDC expansion • Inspired by the channel discontinuity constraint • Node expansion • Link expansion

10. Looping with CDC expansion • Employ Dijkstra’s algorithm with CDC expansion • May result in looping

11. Modified expansions • If a link has three channels, no need to expand that link • Modified ES expansion • Modified CDC expansion

12. Finding CDC Paths for Unweighted Graphs • No Cost associated with each edge • Geometric Setting – • Unit-Disk-Graph Model • Each node has range 1 • Two nodes u and v are connected by an edge if the disks of radius 1 centered at u and v overlap Given a multi-channel wireless network with no collinear interference, the set of available channels at every node, and a node pair (s, d) find the minimum length path between s and d along with channel assignment on every link of the path such that no two consecutive links in the path are assigned the same channel.

13. Key Observation A Minimum Length Alternating Path between s and d gives the Minimum Length CDC path between s and d We have a matching M where every vertex except s and d are matched Expand nodes as before

14. Cardinality Matching Problem • Maximum Matching • A matching M of Maximum Cardinality • General Graphs • Needs to work for both Bipartite and Non-Bipartite Graphs • Solved by Jack Edmonds in 19651 1 "Paths Trees and Flowers", Canadian Journal of Math. 1965

15. Edmonds’ Matching Algorithm • Preliminaries • Free Vertices • A vertex u is free with respect to a matching M if it is not incident with any edge in M • Alternating Path • A path is alternating with respect to a matching M if its edges are alternately in M and not in M • Augmenting Path • Alternating Path between two free vertices

16. Edmonds’ Matching Algorithm • Theorem: M is not a Maximum Matching if and only if there exists an augmenting path with respect to M • Algorithm –

17. Finding an Augmenting Path • Modify Breadth-First-Search to follow only Alternating Paths • Problem – Starting from 1 yields no path to 6 but one exists

18. Solution • During the modified BFS, if a cycle of odd number of vertices is encountered, it is termed as a blossom • Shrink the blossom to a single macrovertex • Continue BFS

19. Finding a CDC-Path • Find an Augmenting Path between source s and destination d • Algorithm is Distributed • Communication Complexity – O(n2) • Possible Improvements • Improve communication complexity by using a Divide-and-Conquer approach • Transform to Weighted Case

20. Thank You!