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3 Chain Construction in PEGASIS: Greedy Appending A greedy method: append the nearest non-chain node to the end of the chain BS Starting with the furthest node
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Minimizing Energy Expense for Minimizing Energy Expense for Chain-Based Data Gathering in Chain-Based Data Gathering in
Wireless Sensor Networks Wireless Sensor Networks
Li-Hsing YenLi-Hsing YenChung Hua UniversityChung Hua University
TaiwanTaiwan
EWSN 05
22
IntroductionIntroduction
Data gatheringData gathering The process of collecting sensed data from The process of collecting sensed data from
every sensor to a distance BSevery sensor to a distance BS
Power-conserving techniquesPower-conserving techniques Data fusionData fusion Multi-hop transmissionMulti-hop transmission
PEGASIS: chain-based data gatheringPEGASIS: chain-based data gathering Data fusion & multi-hop transmissionData fusion & multi-hop transmission
33
Chain Construction in PEGASIS: Chain Construction in PEGASIS: Greedy AppendingGreedy Appending
A greedy method: append the nearest A greedy method: append the nearest non-chain node to the end of the chainnon-chain node to the end of the chain
BS
Starting withthe furthest node
44
PEGASIS: Data GatheringPEGASIS: Data Gathering
The leader collects all sensed data and The leader collects all sensed data and transmit them to the BStransmit them to the BSNodes play the role of the leader by turnsNodes play the role of the leader by turns
BS
Current leader
55
Improved Chain Construction: Improved Chain Construction: Greedy Insertion [DWZ03]Greedy Insertion [DWZ03]
Non-chain nodes can be considered Non-chain nodes can be considered inserting into any position within the chaininserting into any position within the chain
Greedy appending Greedy insertion
Time complexity: O(n2) Time complexity: O(n3)
66
Energy Dissipation ModelEnergy Dissipation Model
When When xx transmits transmits kk-bit message to -bit message to yy, , xx consumes consumes
When When yy receives receives kk-bit message, -bit message, yy consumes consumes
),( yxdkkE ampelec
eleckE
nJ/bit 50elecE pJ/bit/m 100amp
d(x, y): distance between x and y: path loss exponent
77
MMdd: Costs of Node Pairs With Direct : Costs of Node Pairs With Direct CommunicationsCommunications
Cost of every node pair (Cost of every node pair (xx, , yy)) energy consumed in delivering a energy consumed in delivering a kk-bit -bit
message between message between xx and and yy
a b
c d
2
83 912
16
If direct transmissions are used
01689160123812029320
Md
88
Optimal Chain ProblemOptimal Chain Problem
Cost of a chain = total cost of all edges in Cost of a chain = total cost of all edges in that chainthat chainGiven all node-pair costs,Given all node-pair costs,finding an energy-optimal chain finding an energy-optimal chain finding a TSP tour on a complete graph finding a TSP tour on a complete graph NP-hardNP-hard
Traveling Salesperson Problem
99
Our Idea: Virtual ChainOur Idea: Virtual Chain
every edge of the chain may correspond to every edge of the chain may correspond to aa multi-hop data propagation pathmulti-hop data propagation path rather than rather than a a direct radio transmissiondirect radio transmission
Cost of the chain= 18
12
82
416
ab
c
d
Cost of the chain= 22
edge (c, d) correspondsto path c, b, d
16
2
4
8 ab
c
dTriangle inequalitydoes not hold
1010
MMpp: Costs of Node Pairs Using: Costs of Node Pairs Using Shortest Paths Shortest Paths
a b
c d
2
83 912
16
a b
c d
2
83 95
12
01689160123812029320
012891205385029320
If shortest paths are used
MdMpMp[b, c] corresponds
to path b, a, cMp[c, d] corresponds
to path c, a, b, d
1111
Constructing Virtual ChainsConstructing Virtual Chains
MMpp can be obtained by running an can be obtained by running an all-pair shall-pair shortest path algorithmortest path algorithm on input on input MMdd
e.g., Floyd-Warshall: O(e.g., Floyd-Warshall: O(nn33) time complexity) time complexity
Virtual chains can be constructed by running Virtual chains can be constructed by running any chain construction algorithmany chain construction algorithm on input on input MMpp
Virtualchain
Greedy appendingGreedy appending
or insertionor insertionMd Mp
Floyd-WarshallFloyd-Warshall
or any otheror any other
O(n3)
1212
MMtt: Node-Pair Costs Based on MST: Node-Pair Costs Based on MST
Getting Getting MMpp is somewhat time expensive is somewhat time expensiveAn alternativeAn alternative
Md MSTMinimum-costMinimum-costspanning treespanning tree
algorithmalgorithmMt
Traverse alongTraverse alongthe MSTthe MST
Virtualchain
ChainChainconstructionconstructionalgorithmalgorithm
O(n2)
Mt[i, j] is the cost of theunique path from i to j
in the MST
O(n2)
1313
Triangle Inequality Property (TIP) of Triangle Inequality Property (TIP) of Node Cost PairsNode Cost Pairs
TIP does not hold in TIP does not hold in MMdd
MMdd[[ii, , jj]] may be larger than may be larger than MMdd[[ii, , kk] + ] + MMdd[[kk, , jj]] due to non-linear signal attenuationdue to non-linear signal attenuation
TIP does hold in TIP does hold in MMpp
due to the property of shortest pathsdue to the property of shortest paths
TIP does hold in TIP does hold in MMtt
proof is in our paperproof is in our paper
1414
MST-Based Chain ConstructionMST-Based Chain Construction
Used as an approximation to TSP problemUsed as an approximation to TSP problem
With TIP, the cost of the chain is no more than With TIP, the cost of the chain is no more than twicetwice of the MST of the MSTWithout TIP Without TIP No algorithm with constant No algorithm with constant performance ratioperformance ratio
Md, Mp, or Mt
MSTMinimum-costMinimum-costspanning treespanning tree
algorithmalgorithm
Virtualchain
Traverse the treeTraverse the treein prefix orderin prefix order
O(n2) O(n2)
1515
All Possible Cost Matrix/Chain All Possible Cost Matrix/Chain Construction CombinationsConstruction Combinations
Chain construc-Chain construc-tiontion
Cost matrixCost matrix
Greedy Greedy appendingappending
Greedy Greedy insertioninsertion
MST MST traversetraverse
MMdd PEGASISPEGASIS Direct-Direct-insertioninsertion**
Direct-Direct-MSTMST
MMpp Shortest-Shortest-appendingappending
Shortest-Shortest-insertioninsertion
Shortest-Shortest-MSTMST
MMtt MST-MST-appendingappending
MST-MST-insertioninsertion
MST-MST-MSTMST++
* [DWZ03] + can be further simplified (MST-reduced)
1616
Time Complexity of All MethodsTime Complexity of All Methods
MethodMethod Cost matrix Cost matrix computationcomputation
Chain Chain constructionconstruction
OverallOverall
PEGASISPEGASIS O(O(nn22)) O(O(nn22)) O(O(nn22))
Direct-insertion [DWZ03]Direct-insertion [DWZ03] O(O(nn22)) O(O(nn33)) O(O(nn33))
Direct-MSTDirect-MST O(O(nn22)) O(O(nn22)) O(O(nn22))
Shortest-appendingShortest-appending O(O(nn33)) O(O(nn22)) O(O(nn33))
Shortest-insertionShortest-insertion O(O(nn33)) O(O(nn33)) O(O(nn33))
Shortest-MSTShortest-MST O(O(nn33)) O(O(nn22)) O(O(nn33))
MST-appendingMST-appending O(O(nn22)) O(O(nn22)) O(O(nn22))
MST-insertionMST-insertion O(O(nn22)) O(O(nn33)) O(O(nn33))
MST-reducedMST-reduced O(O(nn22)) O(O(nn22)) O(O(nn22))
1717
Leader SchedulingLeader Scheduling
determine which node plays the role of determine which node plays the role of leader in each round of data collectionsleader in each round of data collectionsThe goal is to maximize the number of The goal is to maximize the number of data collection roundsdata collection roundsleader scheduling in PEGASISleader scheduling in PEGASIS round-robin (RR)round-robin (RR) An improvement: nodes are not allowed to be An improvement: nodes are not allowed to be
leaders if their distances to neighbors are leaders if their distances to neighbors are beyond some thresholdbeyond some threshold
1818
Formulating the ProblemFormulating the Problem
NotationNotation eeii: energy consumed by node : energy consumed by node ii in transmits a in transmits akk-bit message to the BS-bit message to the BS
ii,,jj: energy consumed by : energy consumed by ii in transmitting to in transmitting to jj eerr: energy consumed by : energy consumed by ii in receiving a msg in receiving a msg EEii: the amount of energy : the amount of energy ii initially has initially has xxii: the number of times : the number of times ii has been selected to has been selected to
be the leaderbe the leader
1919
Optimal Leader SchedulingOptimal Leader Scheduling
Find positive integer values of Find positive integer values of xxii’s as to’s as tomaximizemaximize xxiisubject tosubject to
nnrnnnnnnn
rrrr
rrrr
r
E
EEE
x
xxx
ee
eeeeeeeeee
ee
3
2
1
3
2
1
1,1,1,
4,332,32,3
3,23,221,2
2,12,12,11
22
A Linear Programming Problem
2020
MRPF: MRPF: Maximum Residual Power FirstMaximum Residual Power First
In each round of data collection, selects In each round of data collection, selects the node that has the maximum residual the node that has the maximum residual power to be the leaderpower to be the leaderResidual power information is Residual power information is piggybacked with data message as a part piggybacked with data message as a part of the aggregated dataof the aggregated dataEach node compares its power level with Each node compares its power level with that attached with incoming data message that attached with incoming data message and sends only the larger oneand sends only the larger one
2121
Simulation Results: Number of Simulation Results: Number of Rounds Before 1st Node DiesRounds Before 1st Node Dies
02000400060008000
100001200014000
50 100 200Number of nodes
Rou
nds
PEGASIS
Direct-MST
Others
02000400060008000
100001200014000
50 100 200
Number of nodes
Rou
nds
PEGASIS
Direct-MSTOthers
50 x 50 network 100 x 100 network
BS is located at (50, 150)
2222
Number of Rounds BeforeNumber of Rounds Before1st Node Dies (Cont.)1st Node Dies (Cont.)
0
1000
2000
3000
4000
5000
6000
7000
50 100 200
Number of nodes
Roun
ds
PEGASIS
Direct-MST
Others
0
1000
2000
3000
4000
5000
6000
7000
50 100 200
Number of nodes
Roun
ds
PEGASIS
Direct-MST
Others
50 x 50 network 100 x 100 network
BS is located at (50, 200)
2323
Number of Rounds BeforeNumber of Rounds Before1st Node Dies (Cont.)1st Node Dies (Cont.)
0200400600800
100012001400160018002000
50 100 200
Number of nodes
Rou
nds
PEGASIS
Direct-MST
Others
0200
400600800
1000
120014001600
18002000
50 100 200
Number of nodes
Rou
nds
PEGASIS
Direct-MST
Others
BS is located at (50, 300)
50 x 50 network 100 x 100 network
2424
Performance of Leader Scheduling Performance of Leader Scheduling Algorithms (50x50 Network)Algorithms (50x50 Network)
BS is located at (25, 150) BS is located at (25, 250)
2525
Variance of Residual PowerVariance of Residual PowerWhen 1st Node DiesWhen 1st Node Dies
BS is located at (25, 150) BS is located at (25, 250)
2626
ConclusionsConclusions
Optimal chain problem is NP-hardOptimal chain problem is NP-hardMST-appending and MST-reduced both MST-appending and MST-reduced both have the merits of lower time cost and have the merits of lower time cost and better resultsbetter resultsOptimal leader scheduling is a linear Optimal leader scheduling is a linear programming problemprogramming problemMRPF performs nearly the same as the MRPF performs nearly the same as the optimal schedulingoptimal scheduling