On Channel-Discontinuity-Constraint Routing in Multi-Channel Wireless Infrastructure Networks
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On Channel-Discontinuity-Constraint Routing in Multi-Channel Wireless Infrastructure Networks
Abishek Gopalan, Swaminathan Sankararaman
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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
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Omnidirectional and Directional transmission
• Omnidirectional transmission
• Directional transmission
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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
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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?
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No two consecutive links on the path are assigned the same channel
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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
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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.
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Edmonds-Szeider expansion• Node expansion
• Link expansion
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Minimum cost perfect matching (MCPM)
• Example network and expanded graph• Expand all nodes except s and d
• Complexity: O(ne)
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CDC expansion• Inspired by the channel discontinuity constraint• Node expansion
• Link expansion
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Looping with CDC expansion• Employ Dijkstra’s algorithm with CDC expansion
• May result in looping
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Modified expansions• If a link has three channels, no need to expand
that link
• Modified ES expansion
• Modified CDC expansion11
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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
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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.
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Key Observation
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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
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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
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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
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Edmonds’ Matching Algorithm
• Theorem: M is not a Maximum Matching if and only if there exists an augmenting path with respect to M
• Algorithm –
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Finding an Augmenting Path
• Modify Breadth-First-Search to follow only Alternating Paths
• Problem –
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Starting from 1 yields no path to 6 but one exists
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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
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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
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Thank You!
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