Algorithms for Wireless Network Design

1
Algorithms for Wireless Network Design
MohammadTaghi HajiAghayi Labs
Research
2
Purposes of this Talk

• Real-world applications with deep algorithmic
  underpinnings and consequences
• Present several problems motivated by (wireless
  sensor) networks
• Show how we can tie together theory and practice
• Demonstrate nice intersections of wireless
  multi-hop networks, algorithmic graph theory,
  probability theory, computational geometry,
  computational economics and finally computational
  complexity

3
Outline

• Focus on two real-world applications
• Power optimization in fault-tolerant topology
  control and related problems
• The low coverage problem and related problems
• Conclusion

4
 Power Optimization in Fault-Tolerant Topology
Control

• Wireless multihop networks
• Simple low-power devices
• Power is the main limitation
• Power assignment
• A power setting for each device
• Defines possible communication links
• Power versus distance It takes power rc to
  transmit a message to distance r for some power
  attenuation exponent c between 2 and 4.

5
Power Optimization in Fault-Tolerant Topology
Control

• Goal Minimize power usage while maintaining key
  network properties
• Connectivity There is a communication path
  between any pair of nodes
• k-Fault tolerance Connectivity is maintained in
  light of at most k-1 failures
• Device failures (our focus)
• Communication link failures
• By k-Fault tolerance, we also have k-disjoint
  paths and thus higher network capacity

6
Model

• A wireless network is modeled as a graph G(V,E)
  with cost functions d and p on E
• V is the set of mobile devices
• E is the set of pairs of devices which can
  communicate bi-directionally
• duv is the distance between device u and v
• puv is the power needed to transmit between
  device u and v (usually it is distance to the
  power attenuation exponent)

7
Model

• Conversely, a subgraph H(V,E) of the network
  graph G defines an assignment of power settings
  device u transmits at
• p(u) max (u,v) in E puv
• The power used by a wireless network with power
  settings defined by H is
• P(H) S u in V p(u)

8
Problem Formulation

• Given
• A wireless network
• Find
• An assignment of power settings that guarantees
  k-fault tolerance while minimizing power usage
• Recall k-fault tolerance means the network
  remains connected even when up to k-1 devices (or
  communication links) fail

9
Related Results for Power Minimization

• Connectivity
• Cone-based local heuristics
• Rodoplu, Meng 99 Wattenhofer, Li, Bahl,
  Halpern, Wang 02
• A 2-approximation based on minimum weight
  spanning tree
• Kerousis, Kranakis, Krizanc, Pelc 00
• A 1.69-approximation based on minimum weight
  Steiner tree and a more practical
  1.875-approximation Calinescu, Mandoiu,
  Zelikovsky 02

10
Related Results for Power Minimization

• 2-Fault tolerance
• Heuristic to minimize maximum transmit power
• Ramanathan, Rosales-Hain 00 (the only
  previous result)
• Fault tolerance for general k
• Pioneered in Bahramgiri, H., Mirrokni, WINET
  and H., Immorlica, Mirrokni,IEEE/ACM TON
• More than 100 references

11
Cone-Based Heuristic

• Algorithm
• Input A set of nodes on the plane, with max.
  power P
• Each node increases its power until the angle
  between any two consecutive neighbors is less
  than some threshold or it reaches its maximum
  power P.
• Output two nodes are connected if both can hear
  each other with the new power assignment
• Theorem BHM02 If the network of max. powers
  is k-connected and the angle between any
  pair of adjacent neighbors is at most 2p/3k, then
  the new network is k-connected (2p/3k is almost
  tight)
• Main disadvantage The algorithm is local and
  thus does not give any bound on the global goal
  of minimizing sum of the powers (or the average
  power)

12
Approximating Connectivity

• Recall the power P(H) of subgraph H is
• P(H) S u in V p(u)
• where p(u) max (u,v) in H(E) puv
• Define the weight W(H) of subgraph H as
• W(H) S (u,v) in H(E) puv

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Approximating Connectivity

• Theorem KKKP 00 The minimum weight spanning
  tree MST of G uses at most twice as much power as
  the minimum power connected subgraph OPT of G.
• Lemma 1 For any graph G, P(G) 2W(G).
• Lemma 2 For any tree T, W(T) P(T).
• Lemma 3 OPT is a tree
• Proof (of Thm) From the above lemmas,
• P(MST) 2W(MST) 2W(OPT) 2P(OPT).

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Approximating k-Connectivity

• Theorem The minimum weight
• k-connected subgraph of G is a
• 2k-approximation to minimum power
• k-connected subgraph
• Proof sketch use Maders theorem to decompose a
  graph into k forests and then use the previous
  results for forests (trees)
• Minimum weight k-connected subgraph an LP-based
  algorithm gives a solution of weight at most
  O(log k) times optimal weight (n is at least
  6k2) Cheriyan, Vempala, Vetta, STOC02,
  Kortsarz, Nutov, STOC04

15
Approximating k-Connectivity

• Thus there is an O(k log k) approximation for
  minimum power k-connected subgraph
• A more complicated combinatorial algorithm gives
  an O(k) approximation
• First we find a 2-approximation to the minimum
  weight k-outconnected sub-graph using matroid
  matching
• We can add k-2 disjoint paths to this graph in
  order to make it k-connected via a min-cost
  k-flow algorithm
• We pay a factor O(k) in each of the above two
  steps to go from weight to power

16
Approximating k-Connectivity

• Furthermore, a more subtle approach gives
  O(log4n) approximation, an improvement when k is
  large H., Kortsarz, Mirrokni, Nutov, IPCO05,
  also Math. Prog.07
• Theorem If there are
• c-approximation for min. weight k-connected sub.
• d-approximation for min. power k-edge cover
• Then we have 2cd-approximation for min power
• k-edge cover sub-graph with min. degree k
• c is in O(log n) by previous results
• d is in O(log4n) by a new involved algorithm
• There are some hardness results for the
  k-edge-connectivity case (the best factor is in
  O(vn))

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Distributed (Local) Approximation

• Algorithm H., Immorlica, Mirrokni, MOBICOM03
• Construct minimum weight spanning tree with O(n
  log n m) messages Gallager, Humbler, Spira,
  83
• Use local augmentation to create a
• k-connected sub-graph with O(n) messages
• Theorem If puv (duv)c for all pairs of nodes
  and we have metricity of distances, then the
  algorithm is an O(1)-approximation when k is
  constant.
• Comparing via simulations on random or real
  inputs
• Some other variants Bredin, Demaine, H., Rus,
  MobiHoc05, Demaine, H., Mahini, Oveis, Sayedi,
  Zadi,SODA07

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A Related Problem Repairing Fault-Tolerance

• Problem Formulation given an initial placement
  of nodes (the unit disk graph model)
• Objective
• Add minimum number of nodes to obtain
  connectivity
• Or minimize maximum/average movement of the
  current nodes without adding any new node
• More generally obtaining k-fault tolerance or
• k-connectivity of the whole network

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Approximation Algorithms for Repairing the Network

• Minimum number of added nodes to obtain
  connectivity 5/2-approximation Du, Wang, Xu,
  2001
• More generally obtaining k-fault tolerance or
  k-connectivity O(k4)-approximation (Simple
  Algorithm, Complicated Analysis)
• Bredin, Demaine, H., Rus, MobiHoc05
• More generally minimizing movement to obtain a
  new configuration with a property P (e.g. being
  connected being independent, having a perfect
  matching, etc.)
• More formally the goal of movement problem is to
  move the agents into a configuration containing
  at most h vertices that contain all k agents and
  induce a good target patterns, i.e., an induced
  graph, in the given set G. Agents can have even
  different colors.

20
Minimizing MovementFixed-Parameter Tractability

• Fixed-parameter algorithms
• Parameterize problem by parameter P
• (typically, the cost of the optimal solution)
• and aim for f(P) nO(1) (or even f(P) nO(1))
• There is an FPT algorithm for Vertex Cover
• There is no FPT algorithm for Dominating Set

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Minimizing Movement FPT (contd)

• CONNECTIVITY. Move the pebbles (agents) so that
  they are connected and on distinct vertices
• GRID. Move the k2 pebbles so that they form a
• k k grid.
• s-t CONNECTIVITY. Move the pebbles to form a path
  of pebbled vertices between fixed vertices
• s and t.
• STEINER CONNECTIVITY. Connect the red pebbles by
  moving the blue pebbles to form a Steiner tree.

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Minimizing Movement FPT (contd)

• 2-CONNECTIVITY. Move the pebbles so that they
  induce a 2-connected graph and the pebbles are on
  distinct vertices.
• s-t d-CONNECTIVITY. Move the pebbles so that
  there are d vertex-disjoint paths using pebbled
  vertices between two fixed vertices s and t.
• s-t d-EDGE-CONNECTIVITY. Move the pebbles so that
  there are d edge-disjoint paths of pebbled
  vertices between s and t.
• FACILITY LOCATION. Move client and facility
  pebbles so that each client pebble is within
  distance at most d from at least one facility
  pebble.
• MATCHING. Move the pebbles so that the pebbles
  are on distinct vertices and there is a perfect
  matching in the graph induced by the pebbles.

23
Minimizing Movement

• Essentially all these problems are polynomialy
  hard to approximate or at least there is no
  algorithm better than vn-approximatoin
• (for max connectivity and max path there are
  such algorithnms) Demaine, H., Mahini, Oveis,
  Sayedi, Zadi, SODA07
• But all of them except GIRD, 2-CONNECTIVITY, and
  FACILITY LOCATION with unbounded number of
  clients have FPT algorithms which are quite
  surprising and interesting for practical purposes
  Demaine, H.,Marx, Submitted07

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Outline

• Focus on two real-world applications
• Power optimization in fault-tolerant topology
  control and related problems
• The low coverage problem and related problems
• Conclusion

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Interference Problem
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The Low Coverage Problem, MotivationDemaine,
H., Feige, Salavatipur, SODA06, SICOMP

• The problem is a bit simplified formulation of a
  real-world application in Bell-Labs
• Input
• Decomposition of the plane into regions with
  various client densities
• n base station locations, each with a set of
  options for power/directional cones/etc and a
  cost for each one
• Budget limiting the total cost of options
• Satisfaction sk for covering a region by kgt 0
  base stations where s1 s2 s3 . . . 0
  (simplified version s11 and s2s3. . .0 is
  called budgeted unique cover.)
• Goal Find a set of base stations and options
  within the total budget which maximizes the total
  satisfaction weighted by client densities

27
The Unique Coverage Problem Demaine, H., Feige,
Salavatipur, SODA06

• The Unique Coverage Problem
• Given a universe U of n elements, and
• Given a collection S of subsets of U.
• Find a sub-collection S , a subset of S, which
  maximizes the number of elements that are
  uniquely covered, i.e., appear in exactly one set
  of S
• The Budgeted Unique Coverage Problem
• Given profits for elements and costs for subsets,
  and
• Given also a budget B
• Find a sub-collection S, a subset of S, whose
  total cost is at most B and maximizes the total
  profit of elements that are uniquely covered

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The Unique Coverage Problem

• The budgeted case is almost equivalent to the
  wireless network problem
• It has the same flavor of the maximum coverage
  problem (has e/(e-1)-approximation)
• Unique coverage is a generalization of MAX-CUT
• It has very close connections to the radio
  broadcast problem (considered extensively)
• For simplicity, we focus on the (un-budgeted)
  unique coverage problem in the rest of the talk

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The Unique Coverage Problem

• Simple O(log n) approximation algorithm
• Partition the elements into log n classes
  according to their degrees, i.e., the number of
  sets that cover an element
• Let i be the class of maximum cardinality
• Choose a set in S to be in S with prob. 1/2 i
• Proof Sketch we uniquely cover 1/e2 fraction of
  elements of class i in expectation
• Can be de-randomized by the standard method of
  conditional expectation
• Can be generalized for budgets, real-world
• O(log B) where B is the max size of a subset
  (nontrivial)

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Hardness Result

• The algorithm seems naïve
• Several other problems have the same solution
• Can we do better?
• It seems combination sometimes can be hard
• Theorem This problem is hard to approximate
  within a factor better than O(logc n), 0ltclt1,
  unless NP has a sub-exponential algorithm.
• O(log1/3 n) hard or even O(log n) hard under
  stronger but plausible complexity assumptions

31
Proof Ideas
B Elements

• A bad instance that we cannot uniquely cover gt
  1/log n fraction

B1
A Sets
2 i random subsets
O(log n)
n
Bi Class i
In this graph, at most O(n) elements of B can
be uniquely covered by sets of A
n
Bp
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Proof Ideas

• Bipartite Independent Set (BIS) problem
• Given a bipartite graph G(A UB,E) where
  ABn
• Find a bipartite sub-graph G(A UB, E) where
  Aa, Bb and E is an empty set.
• Theorem Unless NP has sub-exponential
  algorithm, it is hard to decide between (n
  c,n/logdn)-BIS and (n c ,n/logd n)-BIS where
  0ltcltc 1 and 0 dltd 1
• Now between A and Bi we put a random matching
  and an instance of BIS where the edges remain
  with probability 1/2i-1
• For Yes instance, we have a unique cover of size
  O(nlog1-d n)
• For No instance, we have a unique cover of size
  at most O(nlog1-dn)
• The inapproximability threshold can be improved
  under other stronger but still plausible
  complexity assumptions

33
Other Aspects of the Results

• Unique coverage is the first natural maximization
  problem with almost logarithmic-hardness
• We believe that it can be a central problem for
  maximization problems like set cover for
  minimization problems
• Toward this end, we obtain the same almost
  logarithmic hardness for envy-free pricing, an
  important problem in computational economics
  considered by Guruswami, Hartline, Karlin,
  Kempe, Kenyon, McSherry, SODA05
• Other maximization candidates deadline TSP
  BBCM, STOC04 and budgeted connected sub-graph
  Moss, Rabani, STOC01 where both have O(log n)
  approximation but not better so far
• Finally, better models of interference for the
  real-world application also has been considered
  via some primal-dual schemas and also simulations
  on real-world inputs Bahl, H., Mirrokni, Qui,
  Saberi,IEEE TMC

34
Application of Market Equilibrium in Distributed
Load BalancingBahl, H., Jain, Mirrokni, Qui,
Saberi,IEEE TMC

• Wireless devices
• Cell-phones, laptops with WiFi cards
• Referred as clients or users interchangeably
• Demand connections to access points
• Uniform for cell-phones (voice connection)
• Non-uniform for laptops (application dependent)

35
Application of Market Equilibrium in Distributed
Load BalancingBahl, H., Jain, Mirrokni, Qui,
Saberi,IEEE TMC

• Access points
• Cell-towers, Base stations, Wireless routers
• Capacities
• Total traffic they can serve
• Integer for cell-towers
• Variable transmission power
• Capable of operating at various power levels
• Assume levels are continuous real numbers

36
Clients to APs assignment

• Assign clients to APs in an efficient way
• No over-loading of APs
• Assigning the maximum number of clients and thus
  satisfying maximum demand

37
One Heuristic Solution

• A client connects to the AP with reasonable
  signal and then the lightest load
• Requires support both from AP and Clients
• APs have to communicate their current load
• Clients have WiFi cards from various vendors
  running legacy software
• Overall it has limited benefit in practice

38
Ideal Case

• We would like a client connects to the AP with
  the best received signal strength
• If an AP j transmitting at power level Pj then a
  client i at distance dij receives signal with
  strength
• Pij a.Pj.dij-c
• where a and c are constants capturing various
  models of power attenuation

39
Cell Breathing Heuristic

• An overloaded AP decreases its communication
  radius by decreasing power
• A lightly loaded AP increases its communication
  radius by increasing power
• Hopefully an equilibrium would be reached
• Will show that an equilibrium exist
• Can be computed in polynomial time
• Can be reached by a tatonnement process
• Lets start with economics and game theory

40
Market Equilibrium A distributed load balancing
mechanism.

• Fisher setting with linear Utilities
• m buyers (each with budget Bi) and n goods for
  sale
• (each with quantity qj)
• Each buyer has linear utility ui, i.e. utility
  of i is
• sumj uij xij where uijgt 0 is the utility of
  buyer i for good j and xij is the amount of good
  j bought by i.
• A market equilibrium or market clearance is a
  price vector p that
• maximizes utility sumj uij xij of buyer i subject
  to his budget sumj pj xij lt Bi
• The demand and supply for each good j are equal
• sumj xij qj (and thus the budgets are totally
  spent).

41
Fisher Setting with Linear Utilities
Goods
Buyers
42
Market Equilibrium A distributed load balancing
mechanism.

• Static supply
• corresponding to capacities of APs
• Prices
• corresponding to powers at APs
• Utilities
• Analogous to received signal strength function
• Either all clients are served or all APs are
  saturated
• Analogous to the market clearance(equiblirum)
  condition
• Thus our situation is analogous to Fisher setting
  with linear utilities

43
Clients assignment to APs
APs
Clients
44
Analogousness Is Only Inspirational

• Get inspiration from various algorithms for the
  Fisher setting and develop algorithms for our
  setting
• Though we do not know any reduction in fact
  there are some key differences

45
Differences from the Market Equilibrium setting

• Demand
• Price dependent in Market equilibrium setting
• Power independent in our setting
• Is demand splittable?
• Yes for the Market equilibrium setting
• No for our setting
• Market equilibrium clears both sides but our
  solution requires clearance on either client side
  or AP side
• This also means two separate linear programs for
  these two separate cases

46
Three Approaches for Market Equilibrium

• Convex Programming Based
• Eisenberg, Gale 1957
• Primal-Dual Based
• Devanur, Papadimitriou, Saberi, Vazirani 2004
• Auction Based
• Garg, Kapoor 2003

47
Three Approaches for Load Balancing

• Linear Programming
• Minimum weight complete matching
• Primal-Dual
• Uses properties of bipartite graph matching
• Auction
• Useful in dynamically changing situation

48
Another Application of Market Equilibria in
Networking

• Fleisher, Jain, Mahdian 2004 used market
  equilibrium inspiration to obtain Toll-Taxes in
  Multi-commodity Selfish Routing Problem
• This is essentially a distributed load balancing
  i.e., distributed congestion control problem

49
Linear Programming Based Solution

• Create a complete bipartite graph
• One side is the set of all clients
• The other side is the set of all APs,
  conceptually each AP is repeated as many times as
  its capacity (unit demand)
• The weight between client i and AP j is
• wij c.ln(dij) ln(a)
• Find the minimum weight complete matching

50
Theorem

• Minimum weight matching is supported by a power
  assignment to APs
• Power assignment are the dual variables
• Two cases for the primal program which is known
  at the beginning
• Solution can satisfy all clients
• Solution can saturate all APs

51
Case 1 Complete matching covers all clients
52
Case 1 Pick Dual Variables
53
Write Dual Program
xij
54
Optimize the dual program

• Choose Pj e pj
• Using the complementary slackness condition we
  will show that the minimum weight complete
  matching is supported by these power levels

55
Proof

• Dual feasibility gives
• -?i pj wij ln(Pj) c.ln(dij) ln(a)
  ln(a.Pj.dij-c)
• Complementary slackness gives
• xij1 implies -?i ln(a.Pj.dij-c)
• (Remember if an AP j transmitting at power
  level Pj then a client i at distance dij receives
  signal with strength Pij a.Pj.dij-c)
• Together they imply that i is connected to the
  AP with the strongest received signal strength

56
Case 2 Complete matching saturates all APs
57
Case 2 The rest of the proof is similar
58
Optimizing Dual Program

• Once the primal is optimized the dual can be
  optimized with the Dijkstra algorithm for the
  shortest path

59
Primal-Dual-Type Algorithm

• Previous algorithm needs the input upfront
• In practice, we need a tatonnement process
• The received signal strength formula does not
  work in case there are obstructions
• A weaker assumption is that the received signal
  strength is directly proportional to the
  transmitted power true even in the presence of
  obstructions

60
Cell-towers
Cell-phones
61
Start with arbitrary non-zero powers
62
Powers and Received Signal Strength
8
8
4
7
63
Equality Edges
8
8
64
Equality Graph
Desirable APs for each Client
65
Maximum Matching
Maximum Matching, Deficiency 1
66
Neighborhood Set
T
S
Neighborhood Set
67
Deficiency of a Set
T
S
Deficiency of S Capacities on T - S
68
Simple Observation

• Deficiency of a Set S Deficiency of the Maximum
  Matching
• Maximum Deficiency over Sets Minimum Deficiency
  over Matching

69
Generalization of Halls Theorem

• Maximum Deficiency over Sets Minimum
  Deficiency over Matching
• Maximum Deficiency over Sets Deficiency of the
  Maximum Matching

70
Maximum Matching
Maximum Matching, Deficiency 1
71
Most Deficient Sets
Two Most Deficient Sets
72
Smallest Most Deficient Set
S
Pick the smallest. Use Super-modularity!
73
Neighborhood Set
T
S
Neighborhood Set
74
Complement of the Neighborhood Set
S
Tc
Complement of the Neighborhood Set
75
Initialize r.
S
Tc
Initialize r 1
76
About to start raising r.
S
Tc
Start Raising r
77
Equality edges about to be lost.
S
Tc
Equality edge which will be lost
78
Useless equality edges.
S
Tc
Did not belong to any maximum matching
79
Equality edges deleted.
S
Tc
Let it go
80
All other equality edges remain.
S
Tc
All other equality edges are preserved!
81
A new equality edge added
S
Tc
At some point a new equality appears. r 2
82
Subcase A Deficiency Decreases
S
Tc
New equality edge gives an augmenting path
83
Subcase B Deficiency does not decrease
S
Tc
New edge does not create any augmenting path
84
Smallest most deficient set increases
S
New S is a strict super set of old S!
85
Eventually Subcase A will happen
S
Eventually the size of the matching increases
86
Case 1 Deficiency Reaches Zero
S
All Clients are served!
87
All APs are saturated
S
Or the algorithm will prove that none exist!
88
Unsplittable Demand
89
Unsplittable Demand

• The integer program is APX-hard in general
  (because of knapsack)
• Assuming that the number of clients is much
  larger than the number of APs, a realistic
  assumption, we can obtain a nice approximation
  heuristic.
• First we compute a basic feasible solution

90
Analysis of Basic Feasible Solution
91
Approximate Solution

• All xijs but a small number of xijs are
  integral
• Theorem Number of xij which are integral is at
  least the number of clients the number APs
• Most clients are served unsplittably
• Clients which are served splittably do not
  serve them
• The algorithm is almost optimal

92
Discrete Power Levels

• Over the shelf APs have only fixed number of
  discrete power levels
• Equilibrium may not exist
• In fact it is NP-hard to test whether it exists
  or not
• If every client has a deterministic tie breaking
  rule then we can compute the equilibrium if
  exists under the tie breaking rule

93
Discrete Power Levels

• Start with the maximum power levels for each AP
• Take any overloaded AP and decrease its power
  level by one notch
• If an equilibrium exist then it will be computed
  in time mk, where m is the number of APs and k is
  the number of power levels
• This is a distributed tatonnement process!

94
Proof.

• Suppose Pj is an equilibrium power level for the
  jth AP.
• Inductively prove that when j reaches the power
  level Pj then it will not be overloaded again.
• Here we use the deterministic tie breaking rule.

95
Conclusion.

• Theory of market equilibrium is a good way of
  synchronizing independent entitys to do
  distributed load balancing.
• We simulated these algorithm and observed
  meaningful results.
• Thanks Kamal Jain for the main part of slides.

96
Reconstruction Problem

• Geometric graph structure
• Given Information about local geometry
• E.g., approximate distances
• Goal Reconstruct global geometry

Sensor network
97
Motivation Cricket Location SystemMIT SLAM
Project since 2000 Badoiu, Demaine, H., Indyk,
SOCG04, DCG

• Cricket devices chirp radio ultrasound
• Radio travels at speed of light
• Ultrasound travels at speed of sound
• Measuredistances
• Localized
• Accuracy 1cm

?
?
?
?
Cricket v2
98
General Embedding Problem

• Given graph with edge weights
• Embed into desired metric space(e.g., 2D or 3D)
  with edge lengths matching specified weights
• Weights may be exact orapproximate
• Simple only if we know all
• the distances exactly

99
Embedding with Approx. Distances

• Exact embedding likely impossible
• Goal Minimize maximum distortion
• Additive embed length - true length
• Multiplicative embed length / true length
• Sample results
• Badoiu, Demaine, H., Indyk, SOCG04, Embed any
  metric on n points into Euclidean 2D plane with
  multiplicative/additive distortion (by knowing
  angles) or quasi-polynomial time if we do not
  have the angles
• Alon, Badoiu, Demaine, Farach-Colton, H.,
  Sidiropoulos, SODA05, Embed any metric on n
  points into low dimensions such that we preserve
  the order approximately
• Approximating best embedding into plane in
  polynomial time is a very important open problem

100
Conclusions

• Introduction and motivations
• Focused on two real-world applications
• Power optimization in fault-tolerant topology
  control and related problems
• The low coverage problem and related problems
• Several other related areas to wireless networks
• Geometric embedding Reconstruction Problem
• Market equiblirum Load Balancing
• Game theory Low Coverage
• Questions ?

101
Thanks for your attention
????
Obrigado