On Channel-Discontinuity-Constraint Routing in Multi-Channel Wireless Infrastructure Networks Abishek Gopalan, Swaminathan Sankararaman 1

Wireless infrastructure networks Wireless infrastructure networks becoming more popular – Backbone may operate in a, while user interface may be on b/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 2

Omnidirectional and Directional transmission Omnidirectional transmission Directional transmission 3

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 4

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? 5 No two consecutive links on the path are assigned the same channel

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 6 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.

Edmonds-Szeider expansion Node expansion Link expansion 7

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

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

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

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

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 12 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.

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

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 "Paths Trees and Flowers", Canadian Journal of Math. 1965

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 15

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

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

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 18

Finding a CDC-Path Find an Augmenting Path between source s and destination d Algorithm is Distributed Communication Complexity – O(n 2 ) Possible Improvements – Improve communication complexity by using a Divide-and-Conquer approach – Transform to Weighted Case 19

Thank You! 20