Introduction Wireless Ad-Hoc Network Set of transceivers communicating by radio

Introduction Wireless Ad-Hoc Network Each transceiver has a transmission power which results in a transmission range

Introduction Wireless Ad-Hoc Network Transceiver receives transmission from only if

Introduction Wireless Ad-Hoc Network As a result a directed communication graph is induced

Model & Problems Definition A set of transceivers

Model & Problems Definition A set of transceivers is the power assignment

Model & Problems Definition A set of transceivers is the power assignment

Model & Problems Definitions A set of transceivers is the power assignment is the communication graph

Model & Problems Definitions A set of transceivers is the power assignment is the communication graph is the cost of the assignment

Outline Connectivity problems Bounded hop broadcast Spanners Interference-free broadcast

Connectivity Definitions A graph is k-vertex-connected if for any two nodes there exist k-vertex-disjoint paths connecting to 2-vertex-connected

Connectivity Definitions For graph , a subset is a connected backbone if restricted to is strongly connected and for each there exists so that Connected backbone

Connectivity Problem 1 (k-vertex-connectivity) Input: A set of transceivers, and a parameter Output: A power assignment with minimal possible cost , where is k-vertex connected

Connectivity Problem 1 (k-vertex-connectivity) Input: A set of transceivers, and a parameter Output: A power assignment with minimal possible cost , where is k-vertex connected -approximation algorithm

Connectivity Problem 2 (connected backbone) Input: A set of transceivers Output: A subset of and a power assignment with minimal possible cost , where (restricted to ) is strongly connected, and for each , there exists , such that

Connectivity Problem 2 (connected backbone) Input: A set of transceivers Output: A subset of and a power assignment with minimal possible cost , where (restricted to ) is strongly connected, and for each , there exists , such that Constant-factor approximation algorithm in time

Fault-Tolerant Power Assignment Definitions For each , let be a set of closest nodes to

Fault-Tolerant Power Assignment Definitions For each , let be a set of closest nodes to

Fault-Tolerant Power Assignment Definitions For each , let be a set of closest nodes to Let

Fault-Tolerant Power Assignment The algorithm Assign each the range (denote ) Compute an of

Fault-Tolerant Power Assignment The algorithm Assign each the range (denote ) Compute an of

Fault-Tolerant Power Assignment The algorithm For each edge of increase the range of the nodes in such that each node can reach all nodes in , and vice versa (denote )

Fault-Tolerant Power Assignment The algorithm For each edge of increase the range of the nodes in such that each node can reach all nodes in , and vice versa (denote )

Fault-Tolerant Power Assignment Proof sketch Let In each is assigned at most Case 1:

Fault-Tolerant Power Assignment Proof sketch Let In each is assigned at most Case 1:

Fault-Tolerant Power Assignment Proof sketch Let In each is assigned at most Case 2:

Fault-Tolerant Power Assignment Proof sketch Let In each is assigned at most Case 2:

Fault-Tolerant Power Assignment Proof sketch Let In each is assigned at most Easy to see

Fault-Tolerant Power Assignment Proof sketch Let In each is assigned at most Easy to see Kirousis et al. proved

Fault-Tolerant Power Assignment Proof sketch Let In each is assigned at most Easy to see Kirousis et al. proved As a result and since degree of MST is constant

Connected Backbone Power Assignment Definitions Given the of , for any node , let be the size of the longest edge adjacent to

Connected Backbone Power Assignment Definitions Given the of , for any node , let be the size of the longest edge adjacent to

Connected Backbone Power Assignment The algorithm Compute an of

Connected Backbone Power Assignment The algorithm Compute an of

Connected Backbone Power Assignment The algorithm Compute an of Let be the set of all internal nodes of

Connected Backbone Power Assignment The algorithm Compute an of Let be the set of all internal nodes of Assign each with (denote )

Connected Backbone Power Assignment The algorithm Compute an of Let be the set of all internal nodes of Assign each with (denote )

Connected Backbone Power Assignment The algorithm Compute an of Let be the set of all internal nodes of Assign each with (denote )

Connected Backbone Power Assignment Proof sketch Construct a power assignment for which it holds and , as a result obtaining is derived from

Connected Backbone Power Assignment Proof sketch Let be the connected backbone in For each node let be the transmission range of in

Connected Backbone Power Assignment Proof sketch For each node let be all the nodes within distance from

Connected Backbone Power Assignment Proof sketch For each node let be all the nodes within distance from

Connected Backbone Power Assignment Proof sketch For each node let be all the nodes within distance from For each node compute of

Connected Backbone Power Assignment Proof sketch For each node let be all the nodes within distance from For each node compute of

Connected Backbone Power Assignment Proof sketch In : Each node is assigned

Connected Backbone Power Assignment Proof sketch In : Each node is assigned

Connected Backbone Power Assignment Proof sketch In : Each node is assigned Each node is assigned

Connected Backbone Power Assignment Proof sketch Carmi et al. showed that

Connected Backbone Power Assignment Proof sketch Carmi et al. showed that

Connected Backbone Power Assignment Proof sketch Carmi et al. showed that

Connected Backbone Power Assignment Proof sketch Carmi et al. showed that + + +

Connected Backbone Power Assignment Proof sketch Carmi et al. showed that Using this and is at least longest edge in we obtain

Connected Backbone Power Assignment is at least longest edge in and Thus (summing over all v),

Connected Backbone Power Assignment Proof sketch Kirousis et al. proved that given an assigning each node with yields a 2-factor approximation for strong-connectivity (denote )

Connected Backbone Power Assignment Proof sketch Kirousis et al. proved that given an assigning each node with yields a 2-factor approximation for strong-connectivity (denote ) Using this fact and that B gives us strong connectivity, we obtain

Connected Backbone Power Assignment Proof sketch Therefore,

Broadcast A graph is a broadcast graph rooted at if there is a path from to any

Broadcast A graph is a broadcast graph rooted at if there is a path from to any

Broadcast A graph is a h-bounded-hop broadcast graph rooted at if there is a path from to any and the number of hops is limited by 4-bounded-hop broadcast

Broadcast A graph is a k-h-broadcast graph if it remains h-bounded-hop broadcast graph even with the removal of up to nodes 2-4-bounded-hop broadcast

Broadcast A graph is a k-h-broadcast graph if it remains h-bounded-hop broadcast graph even with the removal of up to nodes 2-vertex disjoint paths under 4 hops

Broadcast A graph is a k-h-broadcast graph if it remains h-bounded-hop broadcast graph even with the removal of up to nodes 2-vertex disjoint paths under 4 hops

Problem 3 (k-h-bounded broadcast) Input: A set of transceivers in , root node and parameters Output: A power assignment so that is k-h-broadcast and is minimized

Planar Case The Algorithm Take a power assignment so that is 1-h-bounded hop graph

Planar Case The Algorithm Take a power assignment so that is 1-h-bounded hop graph Let be a directed spanning tree of Max distance – h hops

Planar Case The Algorithm Take a power assignment so that is 1-h-bounded hop graph Let be a directed spanning tree of Max distance – h hops

Planar Case The Algorithm Add edges from to its grandchildren

Planar Case The Algorithm Add edges from to its grandchildren Remove edges from the children of

Planar Case The Algorithm Add edges from to its grandchildren Remove edges from the children of Denote the resulting tree Max distance – h-1 hops

Planar Case The Algorithm No power is assigned yet! We have a skeleton with a bounded cost

Planar Case The Algorithm Assign

Planar Case The Algorithm Assign to reach k closest neighbors.

Planar Case The Algorithm For each directed edge in increase the range of all nodes in to reach all nodes in

Planar Case The Algorithm For each directed edge in increase the range of all nodes in to reach all nodes in

Planar Case The Algorithm For each directed edge in increase the range of all nodes in to reach all nodes in

Planar Case The Algorithm For each directed edge in increase the range of all nodes in to reach all nodes in

Planar Case The Algorithm For each directed edge in increase the range of all nodes in to reach all nodes in

Planar Case The Algorithm For each directed edge in increase the range of all nodes in to reach all nodes in

Planar Case The Algorithm For each directed edge in increase the range of all nodes in to reach all nodes in

Planar Case The Algorithm For each directed edge in increase the range of all nodes in to reach all nodes in

Planar Case The Algorithm For each directed edge in increase the range of all nodes in to reach all nodes in

Planar Case The Algorithm For each directed edge in increase the range of all nodes in to reach all nodes in

Planar Case The Algorithm Denote the resulting power assignment

Planar Case The Algorithm Denote the resulting power assignment Along each path in there are vertex-disjoint paths in of at most hops

Analysis For a single edge in the power increase of is bounded by:

Analysis For a single edge in the power increase of is bounded by:

Analysis For a single edge in the power increase of is bounded by: Power assignment in

Planar Case Analysis For a single edge in the power increase of is bounded by: Node can be in many -s

Planar Case Analysis For a single edge in the power increase of is bounded by: Node can be in many -s, with many edges

Planar Case Analysis For a single edge in the power increase of is bounded by: Node can be in many -s, with many edges But eventually only one ‘dominates’ the bound

Planar Case Analysis A node can be dominated only by the outgoing edges of in

Planar Case Analysis A node can be dominated only by the outgoing edges of in A single edge can dominate at most nodes (those in )

Analysis A node can be dominated only by the outgoing edges of in A single edge can dominate at most nodes (those in ) Recall,

Analysis A node can be dominated only by the outgoing edges of in A single edge can dominate at most nodes (those in ) As a result,

Analysis

Analysis Due to

Analysis PTAS due to Funke and Laue [24]

Analysis Let be the optimal power assignment for the k-h-broadcast problem From ,

Analysis Let be the optimal power assignment for the k-h-broadcast problem From , We need to bound

Analysis Let be a power assignment so that each node has at least neighbors Clearly,

Analysis - Hamiltonian cycle based power assignment for the k-(n-1)-broadcast problem, so that

Analysis - Hamiltonian cycle based power assignment for the k-(n-1)-broadcast problem, so that In each node has at least neighbors

Analysis – Hamiltonian cycle based power assignment for the k-(n-1)-broadcast problem, so that In each node has at least neighbors From ,

k-(n-1)-broadcast The Algorithm

k-(n-1)-broadcast The Algorithm Compute an MST of

k-(n-1)-broadcast The Algorithm Compute an MST of Construct a Hamiltonian cycle with cost

k-(n-1)-broadcast The Algorithm Compute an MST of Construct a Hamiltonian cycle with cost Assign each node to reach nodes in both directions of the cycle  Example: k=4

k-(n-1)-broadcast The Algorithm Compute an MST of Construct a Hamiltonian cycle with cost Assign each node to reach nodes in both directions of the cycle  As a result,

k-(n-1)-broadcast Hamiltonian Cycle Stage Compute an MST of

k-(n-1)-broadcast Hamiltonian Cycle Stage Compute an MST of Apply MST-Augmentation (Calinescu and Wan)

k-(n-1)-broadcast Hamiltonian Cycle Stage Compute an MST of Apply MST-Augmentation (Calinescu and Wan)

k-(n-1)-broadcast Hamiltonian Cycle Stage Compute an MST of Apply MST-Augmentation (Calinescu and Wan)

k-(n-1)-broadcast Hamiltonian Cycle Stage Compute an MST of Apply MST-Augmentation (Calinescu and Wan)

k-(n-1)-broadcast Hamiltonian Cycle Stage Compute an MST of Apply MST-Augmentation (Calinescu and Wan)

k-(n-1)-broadcast Hamiltonian Cycle Stage Compute an MST of Apply MST-Augmentation (Calinescu and Wan)

k-(n-1)-broadcast Hamiltonian Cycle Stage Compute an MST of Apply MST-Augmentation (Calinescu and Wan) 2-strongly connected undirected graph

k-(n-1)-broadcast Hamiltonian Cycle Stage Compute an MST of Apply MST-Augmentation (Calinescu and Wan) Apply TSP-Approx (Bender and Checkuri) Square of every biconnected graph is Hamiltonian (Fleischner)

k-(n-1)-broadcast Hamiltonian Cycle Stage Compute an MST of Apply MST-Augmentation (Calinescu and Wan) Apply TSP-Approx (Bender and Checkuri) As a result, The cost of the Hamiltonian cycle

Back to k-h-broadcast Analysis - A simple approximation due to: For any it holds:

Back to k-h-broadcast Analysis - Take as before

Back to k-h-broadcast Analysis - Take as before The most distant node at most hops away

Back to k-h-broadcast Analysis - Take as before The most distant node at most hops away Assign the root to reach all!

Spanners What is a spanner? A spanning subgraph that approximates some measure of the original graph

Spanners What is a spanner? A spanning subgraph that approximates some measure of the original graph E.g., Euclidean distance

Spanners What is a spanner? A spanning subgraph that approximates some measure of the original graph E.g., Euclidean distance

Spanners What is a spanner? A spanning subgraph that approximates some measure of the original graph E.g., Euclidean distance Shortest path is at most times longer than in

Spanners What is a spanner? A spanning subgraph that approximates some measure of the original graph E.g., Euclidean distance Shortest path is at most times longer than in stretch factor

Spanners We propose two spanner optimization measures Distance – reducing transmission latency Energy – increasing network lifetime

Spanner optimization measures The original graph Let be the wireless nodes in the plane Let be a weighted complete graph Weight function: The Euclidean distance

Spanner optimization measures The original graph Let be the wireless nodes in the plane Let be a weighted complete graph Weight function: The Euclidean distance Proportional to the energy required to transmit from to

Spanner optimization measures The original graph Let be the wireless nodes in the plane Let be a weighted complete graph Weight function:

Spanner optimization measures The spanner Let p be a power assignment

Spanner optimization measures The spanner Let p be a power assignment is an induced directed graph, where

Spanner optimization measures The spanner Let p be a power assignment is an induced directed graph, where The cost:

Spanner optimization measures Energy measure (stretch factor) The energy of some path is its weight

Spanner optimization measures Energy measure (stretch factor) The energy of some path is its weight The minimum energy from to in

Spanner optimization measures Energy measure (stretch factor) The energy of some path is its weight The minimum energy from to in The minimum energy from to in

Spanner optimization measures Energy measure (stretch factor) The energy stretch factor of

Spanner optimization measures Energy measure (stretch factor) The energy stretch factor of We aim to minimize both and

Spanner optimization measures Energy measure (stretch factor) The energy stretch factor of We aim to minimize both and Clear benefits Prolonged network lifetime Low cost Low interference…

Spanner optimization measures Distance measure (stretch factor) The distance of some path

Spanner optimization measures Distance measure (stretch factor) The distance of some path The minimum distance from to in

Spanner optimization measures Distance measure (stretch factor) The distance stretch factor of

Spanner optimization measures Distance measure (stretch factor) The distance stretch factor of We aim to minimize both and

Spanner optimization measures Distance measure (stretch factor) The distance stretch factor of We aim to minimize both and Clear benefits Low delay in message delivery Low cost

Main results Preliminaries We consider a random, independent, and uniform node distribution in a unit square The probability of our results converges to 1 as the number of nodes, n, increases

Main results Preliminaries We consider a random, independent, and uniform node distribution in a unit square Spanners make sense only if the induced graph is strongly connected

Main results Preliminaries We consider a random, independent, and uniform node distribution in a unit square Spanners make sense only if the induced graph is strongly connected Otherwise, the stretch factor is infinity Path does not exist

Main results Preliminaries We consider a random, independent, and uniform node distribution in a unit square Spanners make sense only if the induced graph is strongly connected The cost of any spanner is at least the minimum cost of strong connectivity

Main results Preliminaries We consider a random, independent, and uniform node distribution in a unit square Spanners make sense only if the induced graph is strongly connected The cost of any spanner is at least the minimum cost of strong connectivity (denote this cost )

Main results Energy spanner Develop power assignment so that where , ,

Main results Distance spanner Develop a power assignment so that

Lower bound on the cost of any spanner Technical details Some bounds… Using [Zhang and Hou ‘05] Lower bound on the cost of any spanner

Minimum spanning tree of G Technical details Some bounds… Using [Zhang and Hou ‘05] From [Kirousis et al. ‘00] Minimum spanning tree of G The weight of the tree

Technical details Some bounds… Using [Zhang and Hou ‘05] From [Kirousis et al. ‘00] Using [Berend et al. ‘08] & [Penrose ‘97] Maximum length edge of MST

Energy spanner [power assignment] Technical details Energy spanner [power assignment]

Energy spanner [power assignment] Technical details Energy spanner [power assignment] Find the minimum spanning tree (MST)

Energy spanner [power assignment] Technical details Energy spanner [power assignment] Find the minimum spanning tree (MST) Lemma: We can find nodes so that any node is within hops from some node in U

Energy spanner [power assignment] Technical details Energy spanner [power assignment] Find the minimum spanning tree (MST) Lemma: We can find nodes so that any node is within hops from some node in U Take diameter

Energy spanner [power assignment] Technical details Energy spanner [power assignment] Find the minimum spanning tree (MST) Lemma: We can find nodes so that any node is within hops from some node in U Take diameter Add the -th node to U

Energy spanner [power assignment] Technical details Energy spanner [power assignment] Find the minimum spanning tree (MST) Lemma: We can find nodes so that any node is within hops from some node in U Take diameter Add the -th node to U Remove first nodes from the diameter

Energy spanner [power assignment] Technical details Energy spanner [power assignment] Find the minimum spanning tree (MST) Lemma: We can find nodes so that any node is within hops from some node in U Take diameter Add the -th node to U Remove first nodes from the diameter

Energy spanner [power assignment] Technical details Energy spanner [power assignment] Find the minimum spanning tree (MST) Lemma: We can find nodes so that any node is within hops from some node in U Take diameter Add the -th node to U Remove first nodes from the diameter

Energy spanner [power assignment] Technical details Energy spanner [power assignment] Find the minimum spanning tree (MST) Lemma: We can find nodes so that any node is within hops from some node in U Take diameter Add the -th node to U Remove first nodes from the diameter

Energy spanner [power assignment] Technical details Energy spanner [power assignment] Find the minimum spanning tree (MST) Lemma: We can find nodes so that any node is within hops from some node in U Take diameter Add the -th node to U Remove first nodes from the diameter

Energy spanner [power assignment] Technical details Energy spanner [power assignment] Find the minimum spanning tree (MST) Lemma: We can find nodes so that any node is within hops from some node in U

Energy spanner [power assignment] Technical details Energy spanner [power assignment] Find the minimum spanning tree (MST) Lemma: We can find nodes so that any node is within hops from some node in U Let be a LAST rooted at LAST [Khuller et al. ’93] is a spanning tree T of G, rooted at some so that and

Energy spanner [power assignment] Technical details Energy spanner [power assignment] Define the power assignment p so that

Energy spanner [power assignment] Technical details Energy spanner [power assignment] Define the power assignment p so that Let

Energy spanner [power assignment] Technical details Energy spanner [power assignment] Define the power assignment p so that Let Finally, For technical reasons

Energy spanner [cost analysis] Technical details Energy spanner [cost analysis]

Energy spanner [stretch analysis] Technical details Energy spanner [stretch analysis] If , there is a path P in G, so that and

Energy spanner [stretch analysis] Technical details Energy spanner [stretch analysis] If , there is a path P in G, so that and Therefore, since for every u, path P also exists in

Energy spanner [stretch analysis] Technical details Energy spanner [stretch analysis] Otherwise,

Energy spanner [stretch analysis] Technical details Energy spanner [stretch analysis] For any two nodes, s and t, the path in first arrives at some LAST origin by using the MST edges (denote P’)

Energy spanner [stretch analysis] Technical details Energy spanner [stretch analysis] For any two nodes, s and t, the path in first arrives at some LAST origin by using the MST edges (denote P’)

Energy spanner [stretch analysis] Technical details Energy spanner [stretch analysis] For any two nodes, s and t, the path in first arrives at some LAST origin by using the MST edges (denote P’) second travels through the edges of from to t (denote P’’)

Energy spanner [stretch analysis] Technical details Energy spanner [stretch analysis] For any two nodes, s and t, the path in first arrives at some LAST origin by using the MST edges (denote P’) second travels through the edges of from to t (denote P’’)

Energy spanner [stretch analysis] Technical details Energy spanner [stretch analysis] Otherwise, We bound the weight of P’ and P’’ Lemma Maximum edge of MST

Energy spanner [stretch analysis] Technical details Energy spanner [stretch analysis] Otherwise, We bound the weight of P’ and P’’ A possible path goes through s

Energy spanner [stretch analysis] Technical details Energy spanner [stretch analysis] Otherwise, We bound the weight of P’ and P’’ Eventually,

Distance spanner [power assignment] Technical details Distance spanner [power assignment] The general idea is that for uniformly distributed nodes, we can always find “good” relays between any pair of nodes

Distance spanner [power assignment] Technical details Distance spanner [power assignment] To find these relays, for any pair of nodes, s and t, we start a recursive process

Distance spanner [power assignment] Technical details Distance spanner [power assignment] To find these relays, for any pair of nodes, s and t, we start a recursive process At step i, we place adjacent disks along the edge The diameter of a disk at step i is

Distance spanner [power assignment] Technical details Distance spanner [power assignment] To find these relays, for any pair of nodes, s and t, we start a recursive process At step i, we place adjacent disks along the edge The diameter of a disk at step i is

Distance spanner [power assignment] Technical details Distance spanner [power assignment] To find these relays, for any pair of nodes, s and t, we start a recursive process At step i, we place adjacent disks along the edge The diameter of a disk at step i is

Distance spanner [power assignment] Technical details Distance spanner [power assignment] To find these relays, for any pair of nodes, s and t, we start a recursive process At step i, we place adjacent disks along the edge The diameter of a disk at step i is The process ends when one of the disks has no relay nodes

Distance spanner [power assignment] Technical details Distance spanner [power assignment] To find these relays, for any pair of nodes, s and t, we start a recursive process At step i, we place adjacent disks along the edge Finally, we use relay nodes to obtain a path We use an arbitrary node in each disk at the last non-empty step

Distance spanner [power assignment] Technical details Distance spanner [power assignment] The power assignment p is obtained by ensuring that all paths are in

Distance spanner [power assignment] Technical details Distance spanner [power assignment] The power assignment p is obtained by ensuring that all paths are in Let be the constructed path from s to t

Distance spanner [power assignment] Technical details Distance spanner [power assignment] The power assignment p is obtained by ensuring that all paths are in Let be the constructed path from s to t And be all the edges from u in all the paths

Distance spanner [power assignment] Technical details Distance spanner [power assignment] The power assignment p is obtained by ensuring that all paths are in Let be the constructed path from s to t And be all the edges from u in all the paths Finally,

Distance spanner [analysis] Technical details Distance spanner [analysis] Lemma: Let D be the maximum radius disk which can be placed inside the unit square, so there are no nodes in D Let r be the radius of D

Distance spanner [analysis] Technical details Distance spanner [analysis] Lemma: Let D be the maximum radius disk which can be placed inside the unit square, so there are no nodes in D Then, Let r be the radius of D

Distance spanner [analysis] Technical details Distance spanner [analysis] From Lemma,

Distance spanner [analysis] Technical details Distance spanner [analysis] From Lemma, Clearly,

Extended wireless network model Power assignment Nodes have no fixed power supply Each node has an initial battery charge b(v) The lifetime of node v is The network lifetime is

Wireless network model Power assignment Interference is a direct consequence of a power assignment p ?

Wireless network model Power assignment Interference is a direct consequence of a power assignment p Several interference models exist Number of nodes affected by transmission Number of edges affected by transmission

Wireless network model Power assignment Interference is a direct consequence of a power assignment p Several interference models exist Number of nodes affected by transmission Number of edges affected by transmission We combine several common models by defining the interference to be

Main results Contribution We develop power assignment: can be computed in time where n is the number of nodes and

Technical details The construction The power assignment is computed by dividing the unit square into k grid cells

Technical details The construction The power assignment is computed by dividing the unit square into k grid cells Then we compute a k shortest path trees rooted at an arbitrary node in each cell

Technical details The construction The power assignment is computed by dividing the unit square into k grid cells Then we compute a k shortest path trees rooted at an arbitrary node in each cell The power assignment of nodes is increased to assure all these k trees are included

Technical details The construction The power assignment is computed by dividing the unit square into k grid cells Then we compute a k shortest path trees rooted at an arbitrary node in each cell The power assignment of nodes is increased to assure all these k trees are included The power assignment of nodes is increased again to be at least