2. Outline Introduction Network Model and Problem Formulation
NP-Completeness of DPMP
Upper and lower bounds on optimal solution
Conslusion
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3. Introduction-1
Wireless sensor networks are ad hoc networks composed of a large number of small nodes. Such networks are typically used for monitoring and surveillance functions
One of the most fundamental functionalities of the network is to maintain its full connectivity
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4. Introduction-2
The sensor nodes in wireless sensor networks have a limited energy that usually cannot be renewed
Thus the life time of a network is constrained by the amount of energy that is spent by sensor nodes in performing their operation of sensing, processing, and transmitting data to the control center
Furthermore, the power consumption in sensor network is dominated by the radio transmission/reception circuitry
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5. Introduction-3
The power propagation models described in articles imply that when the distance increases the power required to cover the distance increases dramatically
If there are two transmit power levels available for nodes to make the network connected, the higher power level used to connect the nodes far away is much larger than the low power level that is used to connect the nodes in the neighborhood
Hence, the dual power management must be done to minimize the total number of nodes assigned high-power-transmission
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6. Figure 1. Clusters with high- and low- power nodes
High Power Level
Low Power Level
Figure 1. Clusters with high- and low- power nodes
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7. Introduction-4
In 2003, By Sholander, Frank, Yankopolus, nodes are grouped into two categories: high-power(HP) nodes and low-power(LP) nodes, and the assignment is done to minimize the overall power while maintaining the full connectivity of the network
This is the same problem that we consider in this paper
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8. Network Model and Problem Formulation-1
We assume that N nodes are placed in a sensor field and the location of each mode is given as x-y coordinate
Two power levels are given: PH and PL
The transmission range using the high- and low-power radios are denoted by rH and rL, respectively, where rL < rH
It is assumed that the network is fully connected if each node is assigned the high-power; hence, a feasible solution always exists
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9. Network Model and Problem Formulation-2
Suppose each node v is assigned high- or low-power. Then a directed graph G=(V,E) is constructed as follows. The vertex set V represents the N sensor nodes and the edge set E is defined such that a directed edge (u,v)E iff the distance from u to v is no larger than rL(or rH) if u is assigned the lower-power(or high-power)
A directed graph G is called strongly connected iff there exists a directed path from an arbitrary node to an arbitrary node in G
In this paper, fully connected graph is the same as strongly connected graph
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10. Network Model and Problem Formulation-3
In order to minimize the overall power consumption in the network as much as possible, the overall transmit power used by all the N nodes should be as little as possible
Therefore, our problem us to assign each node a power level PH or PL such that the summation of the assigned power to each node is minimum while maintaining the full connectivity of the network
We call this is a Dual Power Management Problem(DPMP) which is formally defined as follows
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11. Network Model and Problem Formulation-4
DPMP
Instance: A set {(xi,yi)|1iN} of the x-y coordinates of the N nodes; two transmit power levels PH and PL; rH+ represents the transmission range using PH and rL+ represents the transmission range using PL
Objective: To assign a power level to each A:V{PH or PL} such that the corresponding directed graph G=(V,E) is strongly connected, and \sum_{v \in V}{A(v)} is minimized
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12. NP-Completeness of DPMP-1
In this section, we show a polynomial time transformation from a known NP-complete problem, namely the vertex cover problem, to a decision vision of the DPMP called D-DPMP
Vertex Cover Problem: G=(V,E), SV
Instance: Given a graph G=(V,E) and an positive integer k
Question: Does G have a vertex cover of sizek ?
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13. NP-Completeness of DPMP-2
Given an arbitrary graph G=(V,E) and an integer k, we will transform G to an instance to the DPMP: a set T of x-y coordinates of N nodes, rH and rL, and an integer c, such that G has a vertex cover of size k iff there is a feasible solution to the DPMP with the number of high-power nodes being no more than c(the actual value of c will be discussed later)
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14. t e1 v1 v2 v3 v4 v1 v2 e5 e2 e3 e1 e2 e3 e4 e5 v4 e4 v3 r Figure 2
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15. NP-Completeness of DPMP-2
Any directed edge discussed in figure 2 corresponds to the transmission using the high-power transmission
: c=Nhigh=number of components in T + the size of S
: k=c-the number of components in T
The selected vertex components that are responsible for connecting the edge components have two assigned high-power nodes; all other components have only one high-power node
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16. Upper and lower bounds on optimal solution
Initially, we assign low-power to all of the N nodes, and let K denote the number of components
Theorem: The optimal number of high-power nodes O is such that KO2(K-1)
The lower bound is trivial
We construct a virtual undirected graph G’ such that G’ has K vertices C1, C2,…,CK and the edge set E’ is defined as follows. There exists a pair of nodes uCi and vCj such that u (v) can reach v (u) using high-power transmission at u (v). By assumption, G’ must be single component
Let T’ be an arbitrary spanning tree of G’, and we construct a set of high power nodes as follows. For each edge (Ci, Cj)T’, pick an arbitrary node uCi and an arbitrary vCj, and assign high-power to both u and v. Repeat the process for each edge in T’
Then the resulting set of high-power nodes provide a feasible solution to the DPMP, and the number of high-power nodes is at most 2(T-1)
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17. A near-optimal heuristic algorithm
1.Divide the network into several components by restricting every node to transmit at PL
2.Construct a virtual graph G’ on the components
3.Choose a spanning tree T’ of G’ in terms of components
4.M(M denote the set of high-power nodes)
5.Choose a leaf component Cu of T’ and its corresponding internal components Cv
6.Consider all the pair of assigned high-power node i and j in Cu and Cv respectively. If i and j can reach each other with PH, go to step 8
7.If a pair of nodes i and j in Cu and Cv respectively can reach each other with PH, assign this pair of nodes PH. Mi, Mj
8.Delete the component Cu from T’
9.While T’ has more than one components left, go back to step 5
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18. Conclusion
We introduce the DPMP and prove that it is a NP-Completeness problem
We also give a near-optimal heuristic algorithm
For future work
Develop distributed version of heuristic algorithm
Reallocate high- and low-power nodes after a certain periods of time to maximize the network lifetime
More than two levels are introduced
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