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 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

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 expansion11

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 beforeWe have a matching M where every vertex

except s and d are matchedA Minimum Length Alternating Path between s and dgives 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 19651

141 "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 BFS18

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

19

Thank You!

20