Energy-Efficient Algorithms for Path Coverage in Sensor Networks
Abstract
1. Introduction
2. Related Work
3. Least Movement of Sensors on Path Coverage
3.1. Problem Description
3.2. The Least Movement Algorithm
3.2.1. Path Disjunction
Algorithm 1: Curve disjunction |
1. Let step-size d be (big-coordinate − min-coordinate)/n; 2. Let the collection Q be NULL; 3. For i from 1 to n do 4. xi = (i − 1/2)d; 5. Insert the fetched point (h(xi), xi) into collection Q; 6. Return collection Q. |
3.2.2. Initial Movement
Algorithm 2: Initializing movement |
/* Input: points in collection Q with size n in path P, deployed node collection S with size m. Output: move S shortest such that for si in S, there exists pj in Q, where |si, pj| ≤ R2; */ 1. For i from 1 to m do 2. If |si, every point in Q| > R2, then 3. move si to its closest point pj such that |si, pj| = R2; 4. Return. |
3.2.3. Last Movement
Algorithm 3: Find-redundant-nodes |
/* Input: points collection Q with size n in P, nodes collection S with size m, point i. Output: the closest redundant node in S(i) to N(pos(i)).*/ 1. For (1≤ i ≤ n) & (Pi ≥ d), then 2. If Pi < d after moving s, then 3. Return Ø; 4. If orient = left, then 5. Set N(pos(i)) as Mj(pi) 6. Else if orient = right, then 7. Set N(pos(i)) as Nj(pi) 8. Else if orient = self 9. Set N(pos(i)) as i; 10. Take i out of Q; 11. If exist a closest redundant node s to N(pos(i)), then 12. Insert i into Q; 13. Return s. |
Algorithm 4: PCP solution |
/* Input: points collection Q with size n in P, nodes collection S with size m. Output: move S shortest distance to cover P, or report failure.*/ 1. Rank all points in line with x-axis; 2. Compute covering probability of all elements in collection Q; 3. Let mv_len = 0; 4. Initializing node movement; /*Algorithm 2*/ 5. For (0 ≤ i ≤ n) & For (1 ≤ j ≤ n) 6. If exists redundant node s in S(Mi(pj)) or S(Ni(pj)), then 7. When (i = 0), move s to s0 according to Regulation 1; 8. mv_len = mv_len + |s0, s|; else 9. If s in S(Mi(pj)), then let orient be left 10. Else let orient be right. 11. If x0 in Mi(pj) or Ni(pj) possesses a redundant node; 12. Note path (pf, pf−1, …, p1); 13. For (r = f, r--, r ≥ 1) 14. If exists redundant node s in S(pr), then 15. When (r ≥ 2) then move node according to Regulation 2. 16. Else move s according to Regulation 1; 17. Move node to s0; add |s0, s| to mv_len. |
3.2.4. Simulations
4. Largest Path Coverage Lifetime
4.1. Problem Analysis
4.1.1. Marks
- E(s): The remaining battery level of node s, which is also called the lifetime of s;
- S(v): The collection of nodes that covered point v;
- C(s): The collection of points within the covering region of node s.
4.1.2. Preliminaries
4.1.3. Problem Description
4.2. Largest Path Coverage Lifetime Algorithm
4.2.1. Nodes Partitioning
Algorithm 5: Nodes-partition (S, Q) |
/* Input: deploy nodes collection S = {v1, v2, …, vn}, and points collection Q = {p1,…,pm} in P. Output: partitions collection of nodes. */ 1. Note S(pi) be the collection of nodes covering point pi; 2. Note T1, T2, …, Th be collection of node partitions; 3. S1 = Ø; 4. For (1 ≤ i ≤ h) & (1 ≤ j ≤ m) 5. randomly pitch a subset S0 of S(pj); 6. Ti = Ti + (S0 − S1); 7. S(pj) = S(pj) − (S′ − S1); 8. S1 = S1 + (S0 − S1); 9. Return T1, T2, …, Th. |
4.2.2. Combine Partitions
Algorithm 6: Combine-partition (T, Q) |
/* Input: collection T = {T1, T2,…, Th} of node partitions, collection Q = {p1,…,pm} of points in P Output: new partition collections W1, W2, …, Wr (r ≤ h), each covers all points in collection Q. */ 1. Arrange T by quantity of covered points by Ti, denoted by T = {T1, T2, …, Th}. 2. r = 1; T0 = T; S0 = Q; 3. If (r ≤ h) 4. Set Wr be zero, j be one, respectively; 5. If (T0 is not null) & (S0 is not null) & (j ≤ h) & (|C(Tj) − (Q − S0)| ≤ 0) 6. j++; Wr = Wr + { Tj }; T0 = T0 − Tj; S0 = S0 − C(Tj); j++; 7. If (S0 is null collection) 8. r++; else if (r ≤ 1) 9. Return zero; else 10. Merge all collections in T0 with Wr − 1; Merge all collections in Wr with Wr − 1; r--; 11. Return W1, …, Wr. |
4.2.3. Partition Schedule
Algorithm 7: Partition schedule (W) |
/* Input: partition collections W = {W1, W2, …, Wr} (r ≤ h), each covers all points in Q = {p1,…,pm}. Output: the schedule of collections in W */ 1. Calculate expected lifetime for each Wi; 2. For (1 ≤ i ≤ r) 3. Command nodes in collection Wi covering points in collection P; 4. Return. |
4.2.4. Intra-Schedule
Algorithm 8: Intra-partition scheduling (Wi, Q) |
/* Input: node collection Wi covering all points in collection Q = {p1,..,pm}. Output: schedule of nodes */ 1. While nodes in Wi cover all points in Q 2. Produce G = (V1,V2,E), where V1 denotes nodes set in Wi, and V2 denotes points set in Q; 3. Find a LRM (noted as M) in G; 4. For each vertex without matching in V2 5. Denote unmatched vertices set by V2′ in V2; 6. Induce a new sub-graph G0 from V1 and V2′; 7. Find a LRM in G0; 8. For element v in V2, note M(v) as the element in collection V1 matching some element of v in collection V2, and the one in M(v) with least battery be ver0; 9. For element v in collection V2, schedule M(v) to cover point v and point M(v), thus each element in collection Vi cost ver0 battery; 10. Delete elements in Wi out of service. 11. Return. |
4.3. Simulations
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Algorithms | Application Field | Object |
---|---|---|
MND [5] | point coverage | nodes movement |
MTPCA [7] | point/region coverage | nodes movement |
MSCP [8] | point/region coverage | nodes movement |
MSPA [9] | barrier coverage | nodes movement |
ABC [11] | point coverage | network lifetime |
MC-MIP [13] | point coverage | network lifetime |
MLCS [14] | point coverage | network lifetime |
MSPA [16] | barrier coverage | network lifetime |
HBOA [18] | point coverage | network lifetime |
Our proposal | path coverage | nodes movement/network lifetime |
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Liu, Z.; Zhou, W. Energy-Efficient Algorithms for Path Coverage in Sensor Networks. Sensors 2023, 23, 5026. https://doi.org/10.3390/s23115026
Liu Z, Zhou W. Energy-Efficient Algorithms for Path Coverage in Sensor Networks. Sensors. 2023; 23(11):5026. https://doi.org/10.3390/s23115026
Chicago/Turabian StyleLiu, Zhixiong, and Wei Zhou. 2023. "Energy-Efficient Algorithms for Path Coverage in Sensor Networks" Sensors 23, no. 11: 5026. https://doi.org/10.3390/s23115026
APA StyleLiu, Z., & Zhou, W. (2023). Energy-Efficient Algorithms for Path Coverage in Sensor Networks. Sensors, 23(11), 5026. https://doi.org/10.3390/s23115026