Nodes are represented by •, connection points are represented by , and support points are represented by ◦. 110 6.4 (a) The set of points within distance 12 from the lines `h and `v (shown in the . loop), (b) the subset (Q0) of mutually independent points (shown in red), (c) the subset of points that are not independent with points in Q0 (shown in blue), and (d) using a recursive procedure after deleting independent points.
Scope of the Thesis
Organization of the Thesis
Note that the MDS problem is a special case of the MWDS problem and therefore all results available for the MWDS problem in the literature are also applicable to the MDS problem. The high complexity of PTAS motivated researchers to obtain efficient constant factor approximation algorithms for the DUDC problem.
Minimum Connected Dominating Set Problem
In the table, n and m represent the number of nodes and edges of the UDG, respectively. Another variant of the CDS problem known as the k-related m-dominated set problem is available in the literature.
Minimum Liar’s Dominating Set Problem
They also proposed a linear time algorithm to compute the minimum cardinality of the dominant liar set in a tree. A total liar dominating set (TLDS) is a dominating set L with the following two properties: (i) for each v ∈V,|NG(v)∩L| ≥2,.
Maximum Independent Set Problem
Unlike general graphs, there are constant factor approximation algorithms for the MIS problem in UDG. For the MIS problem on UDGs, van Leeuwen [65] proposed a tractable fixed-parameter algorithm that runs in O(t222tn) time, where the parameter t is called the thickness4 of the UDG.
A Simple 5-Factor Approximation Algorithm
We use a height-balanced binary tree to store the indices of the grid cells that contain a non-zero number of members in P0. The expected time complexity of the above procedure can be reduced to O(n) by using a hash table of size O(n) for storing the grid cells containing non-zero members of P0 instead of a height- balanced binary tree.
A 4-Factor Approximation Algorithm
Computing an optimum solution for a single septa-hexagon GMDS
Given an array containing the points in P, we can arrange these points in O(n) time such that the points in S1 are placed at the beginning, followed by the points in S2\S1. The time complexity can be reduced as follows: let the points S2 be stored at the beginning of the array containing the points in P, so that the points in each cell of H are stored sequentially.
A 14 3 -factor approximation algorithm
The time complexity result of the theorem follows from Lemma 3.3.6 and the fact that each point inP can participate in the calculation for at most a constant number of septa-hexagons. Let SOLA, SOLB, SOLC, and SOLD be the union of the solutions of all the septa-hexagons colored A, B, C, and D, respectively.
A 3-Factor Approximation Algorithm
Computing an optimum solution for a single super-cell GMDS
We decompose a supercell D into 3 regions namely D1, D2 and D3 as shown in Figure 3.3 (b), these regions are respectively demonstrated using unshaded, light shaded and dark shaded regions. A point on a boundary can be assigned to any one of the sets associated with that boundary.
A 45 13 -factor approximation algorithm
Shifting Strategy and its Application to the GMDS Problem
The shifting strategy
For a given set of points P in R2, if there exists an algorithm A that can produce an α-factor approximation result for the GMDS problem defined for the points inside a monotone strip of width `d, where each pair of consecutive monotone chains is at distance d ≥ 2 and ` is an integer (displacement parameter), then there exists an α(1 + 1`)-factor approximation algorithm for the GMDS problem for the point set P. Since the monotone stripes have width `d and the distance between a pair of monotone strips are d≥ 2. The resolution of one strip does not affect the resolution of any other strip.
A 5 2 -factor approximation algorithm
We divide the supercell E into three regions, as in the case of the supercell in section 3.4. The algorithm for calculating an optimal solution for a single duper cell is similar to the algorithm for a single supercell (see section 3.4).
A PTAS for the GMDS problem
In the first level of division, we consider 3 consecutive rows of hexagons as a single horizontal strip, and therefore the distance between the monotonic chains, bottom and top, bounding the strip is 2. Calculate an optimal solution for the rest of the points in the quadrants separately using the same procedure (note that any unit disk centered in the optimal solution of any quadrant does not intersect L1 and L2, and so we can solve them independently).
Conclusion
Next, we check whether the obtained solutions in the quadrants together with the selected combination cover the points in P ∩ F. We assume that all nodes (i.e. the units) are inserted in a rectangular region in the plane and they have a uniform transmission range of a unit.
Definitions and Preliminaries
In the rest of the chapter we refer to the DS obtained in (i) as D, and the connecting nodes chosen in (ii) as C. A special path between two nodes in SD(di) is a path with the connecting nodes in the same cells as that of the dominators (see figure 4.3).
Local Variables
Thus, the connections on the special paths between di and the dominators in the outermost hexagons to Hi in the triplets also connect the dominators in the six adjacent cells in Hi. Each entry in array DP is an ordered triplet (di, c1, c2) where di is a dominator (different from v), c1 and c2 are the connector nodes of di such that c1 lies in the same cell as v and c2 lies in same cell as di and adjacent to c1.
Scheduling Scheme
The sub-temporal space assigned to a septa-hexagon is further divided into 7 parts, one for each cell. The assignment of ∆ time slots in a cell is based on the ascending order of node IDs, i.e., a node with a smaller ID is assigned an earlier time slot.
Distributed Algorithm
- Distributed implementation
- Proof of correctness
- Approximation factor
- Time and message complexities
Let S be the set of angles of the regular hexagon(s) in the region of intersection of δ(p) and δ(q). If o lies in the center of the cell (call it m), then it covers some points in cells numbered 1 through 7.
Conclusion
Geometric Minimal Lie Dominating Set Problem (GMLDS): Given a set P of n points in R2, find a subset D of P of minimal cardinality that satisfies the following two conditions: (i) for each point pi ∈ P there exist at least two points in D which are at most one distance from pi, and (ii) for each distinct pair of points pi. ForA⊆V,LD(A) denotes the dominating set of a liar and LDopt(A) denotes the dominating set of an optimal liar of A in G.
Hardness of the GMLDS Problem
Note that the edge e(vi, vj) is represented as a sequence of line segments in the embedding. If any edge e(vi, vj) in G has none of its endpoints in D, we do the following: consider the sequence of segments representing the edge e(vi, vj) in the embedding.
Approximation Algorithms
A 63 2 -factor approximation algorithm
The pseudocode to find a liar's dominant set for the points in P is given in Algorithm 5.1. The points pk and pj cover more cells when they covered less number of common cells.
Improving the approximation factor
If we choose at most three points for each cell, we get a liar's dominant set. In Algorithm 5.2, for a 37-hexagon H, we find a liar's dominating set SH for S0 from S00 (refer the definitions of S0 and S00 defined previously in this section).
Further improvement of the approximation factor
The running time result follows from Lemma 5.3.7 and the fact that a point in P participates in the algorithm a finite number of times. Using the technique in Subsection 5.3.2, we can obtain a 282k -factor approximation algorithm for points lying in a 66-hexagon, where 3 ≤ k ≤ 282, and have the following theorem.
A PTAS for the GMLDS problem
Algorithm
Liar's dominating set of the r-th neighborhood of a vertex v, LD(Nr[v]), can be calculated with respect to G as follows. For a given v ∈ V, liar's dominating set LDopt(Ni) of Ni can be computed in polynomial time.
Conclusion
Geometric Maximum Independent Set (GMIS) problem: Given a set P of n points in R2, find a maximal cardinality set P0 ⊆ P such that the points in P0 are pairwise independent, i.e. the Euclidean distance between any two points in P0 is greater than 1. Geometric Maximum Weighted Independent Set (GMWIS) problem : Given a set P of n points in R2 and a weight function w : P → R+, find a set P0 ⊆ P such that that the points in P0 are pairwise independent and the sum of the weights of the points in P0 is maximized.
Geometric Maximum Independent Set Problem
Preliminaries
After processing all the points in the H strip we get a GMIS ofQ corresponding to a longer chain. We first set ni = 0 for each point pi on the bar, indicating that the maximum length of a chain ending atpi is zero.
A 2-factor approximation algorithm
The correctness of Algorithm 6.2 follows from: (i) for each point pi in the strip, Algorithm 6.2 calculates a longest chain ending in pi by calling Algorithm 6.1 and (ii) Algorithm 6.2 returns the longest chain after processing all points in the strip. The space complexity of Algorithm 6.2 follows from Lemma 6.1.6, since we can reuse the matrix M for each call to Algorithm 6.1.
Geometric Maximum Weighted Independent Set Problem
Data structures
Given a set of points, there exists a data structure that answers counter-disk-emptiness queries in O(logn) time with O(n) storage and O(nlogn) preprocessing time. Given a set of n weighted points, there exists a semi-dynamic data structure that answers anti-disk maximum queries in O(log3n) time with O(nlogn) storage and O(log3n) insertion time.
The GMWIS Problem on Small Strips
The O(m) inserts (line 10) and queries (line 7) take O(mlog3m) time together, and the space is O(mlogm) due to the data structure's storage. Therefore, using the data structure from Lemma 6.2.8, the O(m) inserts and queries take O(mlog2m) time and the space is O(m).
The GMWIS Problem on Large Strips
The data structure Di stores a point pj with weight Wi,j for each quantity Si,j ∈ Si. For each point pi, the data structure Di stores the point pj for each set Si,j ∈ Si.
A PTAS for the GMIS Problem
Given a set P of n points in the plane and an integer k > 1, the proposed algorithm computes an independent set of size at least 1.
Conclusion
Liar’s dominating set ( P )
By definition, the points in S0\S lie in a maximum of 24 cells around H (i.e. one layer around H). The detailed pseudocode to find a liar's dominant set for the points lying in a given 37-hexagon H is given in Algorithm 5.2.
Liar’s dominating set (H, k)
If any of these combinations satisfies the liar's dominance conditions, Algorithm 5.2 reports the combination of points. Nk} in G such that Si ⊆Ni ⊆ V and, using Algorithm 5.4 as a subroutine, computes the liar's dominating set for G.
Liar’s dominating set (V )
Liar’s dominating set(N r [v])
Note that in inequality (5.1) the right-hand part is a function with exponential in r and the left-hand part is a polynomial in r. These sets are needed to process the remaining points (ie the points equal to pi+1) of the strip.
Processing the point p i+1
The worst case time complexity of lines 4 - 21 is O(logi) due to flat point location in line number 9.
Maximum independent set strip ( Q )
The correctness of Algorithm 6.3 follows from the fact that M ISodd and M ISeven are independent sets since all strips are 1 unit wide and the points in any two even (respectively odd) strips are independent.
Maximum independent set ( P )
Given a set of n weighted points, there exists a data structure that answers the largest disk queries in O(log2n) time with O(nlogn) and O(nlog2n) storage and preprocessing time, respectively. To efficiently determine the index j, we use the data structure from Lemma 6.2.7 (weighted) or 6.2.8 (unweighted).
Small strip( Q )
Note that the Dj data structure stores only pk points, so that d(pj, pk)>1. To reduce the running time toO(m2logm) and space toO(m2) for the unweighted version, we use the data structure from Lemma 6.2.4 and the same argument as in the proof of Theorem 6.3.5 shows that wmax−wret ≤3.
Large strip( Q )
A new bound on maximum independent set and minimum connected dominating set in unit disk graphs. A new distributed approximation algorithm for building minimum connected dominant set in wireless ad hoc networks.
