Constructive Heuristics for Min-Power Bounded-Hops Symmetric Connectivity Problem
aa r X i v : . [ c s . D S ] F e b Constructive Heuristics for Min-PowerBounded-Hops Symmetric ConnectivityProblem ⋆ Roman Plotnikov − − − and Adil Erzin , − − − X ] Sobolev Institute of Mathematics, Novosibirsk, Russia Novosibirsk State University, Novosibirsk, Russia
Abstract.
We consider a Min-Power Bounded-Hops Symmetric Con-nectivity problem that consists in the construction of communicationspanning tree on a given graph, where the total energy consumptionspent for the data transmission is minimized and the maximum num-ber of hops between two nodes is bounded by some predefined constant.We focus on the planar Euclidian case of this problem where the nodesare placed at the random uniformly spread points on a square and thepower cost necessary for the communication between two network ele-ments is proportional to the squared distance between them. Since thisis an NP-hard problem, we propose different polynomial heuristic algo-rithms for the approximation solution to this problem. We perform aposteriori comparative analysis of the proposed algorithms and presentthe obtained results in this paper.
Keywords:
Energy efficiency · Approximation algorithms · Symmetricconnectivity · Bounded hops.
Due to the prevalence of wireless sensor networks (WSNs) in human life, thedifferent optimization problems aimed to increase their efficiency remain actual.Since usually WSN consists of elements with the non-renewable power supplywith restricted capacity, one of the most important issues related to the de-sign of WSN is prolongation its lifetime by minimizing energy consumption ofits elements per time unit. A significant part of sensor energy is spent on thecommunication with other network elements. Therefore, the modern sensors of-ten have an ability to adjust their transmission ranges changing the transmitterpower. Herewith, usually, the energy consumption of a network’s element is as-sumed to be proportional to d s , where s ≥ d is the transmission range[1]. The problem of search of the optimal power assignment in WSN is well-studied. The most general Range Assignment Problem, where the goal is to ⋆ The research is supported by the Russian Science Foundation (project 18-71-00084). R. Plotnikov et al. find a strongly connected subgraph in a given oriented graph, has been consid-ered in [2,3]. Its subproblem, Minimum Power Symmetric Connectivity Problem(MPSCP), was first studied in [4]. The authors proved that Minimum Span-ning Tree (MST) is 2-approximation solution to this problem. Also, they pro-posed a polynomial-time approximation scheme with a performance ratio of1 + ln 2 + ε ≈ .
69 and 15/8-approximation polynomial algorithm. In [5] a greedyheuristic, later called Incremental Power: Prim (IPP), was proposed. IPP is sim-ilar to the Prim’s algorithm of the finding of MST. A Kruscal-like heuristic,later called Incremental Power: Kruscal, was studied in [6]. Both of these so-called incremental power heuristics have been proposed for the Minimum PowerAsymmetric Broadcast Problem, but they are suitable for MPSCP too. It isproved in [7] that they both have an approximation ratio 2, and it was shown inthe same paper that in practice they yield significantly more accurate solutionthan MST. Also, in a series of papers different heuristic algorithms have beenproposed for MPSCP and the experimental studies have been done: local searchprocedures [7,8,9], methods based on iterative local search [10], hybrid geneticalgorithm that uses a variable neighborhood descent as mutation [11], variableneighborhood search [12], and variable neighborhood decomposition search [13].Another important property of WSN’s efficiency is message transmission de-lay, i.e., the minimum time necessary for transmitting a message from one sensorto another via the intermediate transit nodes. As a rule, the data transmissiondelay is proportional to the maximum number of hops between two nodes ofa network. The general case, when the network is represented as directed arc-weighted graph, and the goal is to find a strongly connected subgraph withminimum total power consumptions and bounded path length, is called Min-Power Bounded-Hops Strong Connectivity Problem. In [3] a special Euclidiancase of this problem, when equidistant on the line, was considered. In [14] theapproximation algorithms with guaranteed estimates have been proposed forthe Euclidean case of this problem. The bi-criteria approximation algorithm forthe general case (not necessarily Euclidian) with guaranteed upper bounds hasbeen proposed in [15]. The authors of [16] propose improved constant factorapproximation for the planar Euclidian case of the problem.In this paper, we consider the symmetric case of Min-Power Bounded-HopsStrong Connectivity Problem, when the network is represented as undirectededge-weighted graph. Such a problem is known as Min-Power Bounded-HopsSymmetric Connectivity Problem (MPBHSCP) [15]. We also assume that sensorsare positioned on Euclidian two-dimensional space. Energy consumption for thedata transmission is assumed to be proportional to the area of a circle with centerin sensor position and radius equal to its transmission range d , and, therefore, s is considered to be equal 2. This problem is still NP-hard in two-dimensionalEuclidian case [17], and, therefore, the approximation heuristic algorithms thatallow obtaining the near-optimal solution in a short time, are required for it.Although MPBHSCP is known to be NP-hard, to the best of our knowledge,none research has been done to find the most efficient in practice approximationalgorithms. This paper is aimed to fill this gap. We propose six different con- onstructive Heuristics for MPBHSCP 3 structive heuristics for the approximation solution of MPBHSCP. We employ theideas of the most natural and widely spread heuristics for the Bounded-DiameterMinimum Spanning Tree (BDMST). We conducted an extensive numerical ex-periment where these algorithms have been compared. We present the results ofthe experiment in this paper.The rest of the paper is organized as follows. In Section 2 the problem is for-mulated, in Section 3 descriptions of the proposed algorithms are given, Section4 contains results and analysis of an experimental study, and Section 5 concludesthe paper. Mathematically, MPBHSCP can be formulated as follows. Given a connectededge-weighted undirected graph G = ( V, E ) and an integer value D ≥
1, findsuch spanning tree T ∗ of G , which is the solution to the following problem: W ( T ) = X i ∈ V max j ∈ V i ( T ) c ij → min T ,dist T ( u, v ) ≤ D ∀ u, v ∈ V, where V i ( T ) is the set of vertices adjacent to the vertex i in the tree T , c ij ≥ i, j ) ∈ E and dist T ( u, v ) is the number of edges in apath between the vertices u ∈ V and v ∈ V in T .Obviously, in general case, MPBHSCP may even not have any feasible solu-tion. In this paper, we consider a planar Euclidian case, where an edge weightequals the squared distance between the corresponding points and G is a com-plete graph. Also, we assume that the sensors are randomly uniformly distributedon a square with fixed side. Therefore, for example, the density of a networkgrows with increase of the number of its elements. We propose a set of heuristic algorithms that construct an approximate solutionto the MPBHSCP. Many of them use ideas that previously have been appliedto the solution of Bounded-Hops Minimum Spanning Tree (BDMST). As wellas it is done in many efficient heuristic algorithms for BDMST, we will use a center-based approach, where, at first, the center (one vertex if D is even or twovertices if it is odd) is chosen, and after that, the tree is constructed taking careof the depth of each vertex in relation to the center. The main difference betweenthe algorithms applied for BDMST and our methods is the calculation of theobjective function increment after the small modifications of a partial solution.An objective function of MST is additive, that is, adding (or removing) an edgewill increase (or decrease) the objective value exactly by the weight of an edge,which is not held for the objective function of MPSCP: if one wants to calculatethe change of an objective function value for MPSCP after adding or removing R. Plotnikov et al. an edge, then he has to take into account the weights of all adjacent edges of atree.Let us define the notations that will be used further. For the conveniencepurposes, we will construct a directed tree, rooted in a center. If the centercontains two vertices then one of them will be referred to as a root, and thesecond one — as its child. Let’s call the minimum number of edges between avertex and center in a tree as the depth of a vertex. Let V T ⊂ V stand for a set ofvertices in a tree T , E T stand for a set of edges of T . Let P arent T ( v ) ∈ V T be aparent of a vertex v ∈ V T . If v / ∈ V T then let P arent T ( v ) = ∅ . Let depth T ( v ) bethe depth of a vertex v in a tree T in relation to the center, that is the minimumnumber of hops (edges) between v and center. If v / ∈ V T then let depth T ( v ) beequal to −
1. Let
P ower ( v, u ) = P ower ( u, v ) be the power cost necessary for thedirect communication between the vertices u and v . As it was mentioned before,in this paper, we assume that P ower ( u, v ) = ( pos u − pos v ) , where pos u and pos v are positions in Euclidian two-dimensional space of, correspondingly, the vertices u and v . Of course, these values may be calculated once since the positions arefixed. P ower T ( u ) will stand for the power consumptions of a vertex v in a tree T . N T ( v ) ⊂ V T { v } will stand for a set of neighbors of v ∈ V T in T . That is, P ower T ( v ) = P u ∈ N T ( v ) P ower ( u, v ), and the total power consumption of a tree T , which is the objective function value, is W ( T ) = P v ∈ V T P ower T ( v ). Many of known greedy approaches for BDMST use the Prim’s strategy [18] fortree building. Starting from a tree with the only vertex, these algorithms re-peatedly add a new edge that connects a non-tree vertex with a vertex in atree and does not violate the restriction on the diameter. Herewith, criteria ofchoosing the new non-tree vertex may vary while the in-tree vertex is alwayschosen greedily. A way of choosing the center vertices, which is rather essen-tial, may vary too. The general scheme of the Prim-Like Heuristic (PLH) ispresented in Algorithm 1. Below we will consider three different heuristics thatare based on the Prim’s strategy: Min-Power Center-Based Tree Construction,Min-Power Randomized Tree Construction, and Min-Power Center-Based LeastSum-of-Costs. The difference between these algorithms lies in the different im-plementations of the methods
ChooseF irstCenters , ChooseSecondCenter , and
ChooseEachV ertex . Min-Power Center-Based Tree Construction.
The first algorithm basedon PLH is Min-Power Center-Based Tree Construction (MPCBTC) which is sim-ilar to the Center-Based Tree Construction [19] for BDMST. In this algorithm,
ChooseF irstCenters chooses each vertex, that is, the algorithm starts n timeswith each vertex of V selected as a center. The method ChooseSecondCenter ( v )returns the vertex v = argmin v ∈ V \{ v } P ower T ( v, v ). And, finally, the method ChooseEachV ertex ( U, V , wBestN eighbor ) finds such vertex u ∈ U that wBestN eighbor ( u )is minimum. CBTC is known to perform worse with decrease of maximum hops onstructive Heuristics for MPBHSCP 5 Algorithm 1
Prim-Like Heuristic C [ . ] ← ChooseF irstCenters (); W ∗ ← ∞ ; for all v ∈ C [ . ] do V ← { v } ; U ← V \ { v } ; depth T [ . ] ← an array of size n that stores a depth for each vertex in a tree, filledwith -1; bestNeighbor [ . ] ← an array that stores the best neighbor in V for each vertex in U ; wBestNeighbor [ . ] ← an array that stores the total power increase if the vertexwill be connected with its best neighbor; depth ( v ) ← T ← ( v , ∅ ); if D is odd then v ← ChooseSecondCenter ( v ); depth T ( v ) ← v and an edge ( v , v ) to T ; end if V ← V T ; for all u ∈ U do bestNeighbor ( u ) ← argmin v ∈ V T { P ower ( u, v ) } ; wBestNeighbor ( u ) ← P ower ( u, bestNeighbor ( u )); end forwhile U is not empty do u ← ChooseEachV ertex ( U, V , wBestNeighbor );Add a vertex u and an edge ( u, bestNeighbor ( u )) to T ; depth T ( u ) ← depth T ( bestNeighbor ( u ) + 1); P ower T ( u ) ← P ower ( u, bestNeighbor ( u )); P ower T ( bestNeighbor ( u )) ← max { P ower T ( bestNeighbor ( u )) , P ower ( u, bestNeighbor ( u )) } ; U ← U \ { u } ; for all v ∈ U do w ← P ower ( bestNeighbor ( u ) , v ) + max { , P ower ( bestNeighbor ( u ) , v ) − P ower T ( v ) } ; if w < wBestNeighbor ( v ) then wBestNeighbor ( v ) ← w ; bestNeighbor ( v ) ← bestNeighbor ( u ); end ifend forif depth T ( u ) < ⌊ D/ ⌋ then V ← V ∪ { u } ; for all v ∈ U do w ← P ower ( u, v ) + max { , P ower ( u, v ) − P ower T ( v ) } ; if w < wBestNeighbor ( v ) then wBestNeighbor ( v ) ← w ; bestNeighbor ( v ) ← u ; end ifend forend ifend whileif W ( T ) < W ∗ then W ∗ ← W ( T ); T ∗ ← T ; end ifend forreturn T ∗ ; R. Plotnikov et al. number and increase of the points density, since the nodes that lie far from acenter (let’s call them far nodes ) often have the maximum allowable depth and,therefore, once added, they cannot be connected with any other node. And, forthis reason, far nodes cannot be connected with any node in their proximitywithout violating the hops restriction, and they are forced to be connected witha tree by long arcs. Obviously, in MPCBTC, as well as in CBTC, the closestto the center nodes are added sooner, and in a case of large density and small D MPCBTC will have the same disadvantage as CBTC: far nodes will be con-nected with a tree via long edges. Due to this fact solution obtained by MPCBTCshould appear extremely inefficient for the cases when n is large and D is small.The computational complexity of MPCBTC is O ( n ) since it is repeated n timesfor each vertex chosen as the center, and each iteration requires O ( n ) time. Min-Power Randomized Tree Construction.
One simple approach aimedto overcome the mentioned disadvantage of CBTC is Randomized Tree Con-struction (RTC) proposed in [19]. As well as CBTC, RTC chooses a centervertex (or two center vertices if D is odd), then it iteratively chooses a vertexoutside a tree and connects it with some vertex in a tree. But in contrast toMPCBTC, each time the vertex is chosen at random. The process is repeated n times, and the best tree is returned. We adapted this algorithm to MPBH-SCP. Let’s call the obtained heuristic as Min-Power Randomized Tree Con-struction (MPRTC). Since this algorithm is also based on PLH, the only partsthat should be mentioned are the special implementations of the subroutines ChooseF irstCenters , ChooseSecondCenter , and
ChooseEachV ertex , whichare extremely simple in this case: the method
ChooseF irstCenters n timeschooses a vertex v ∈ V at random, as well as it is done in RTC [19]. The bothmethods ChooseSecondCenter and
ChooseEachV ertex choose a vertex v ∈ U at random (where U is a set of non-tree vertices, see Algorithm 1). This cir-cumstance theoretically should cause better results of MPRTC comparing withMPCBTC on high-dense graphs constructed on uniformly spread set of points,because on each step of MPRTC the constructed partial solution consists ofrandom subset of V . If D is not too small, then the positions of the backbonevertices are also uniformly spread on each step, and therefore, on average, theweight of added edge should be rather small after some appropriate numberof steps. Because of the fact that MPRTC is repeated n times with differentrandomly chosen center, its total computational complexity is O ( n ). Min-Power Center-based Least Sum-of-Costs.
Another greedy algorithmfor BDMST was proposed in [23], it is called Center-based Least Sum-of-Costs.In similar manner to CBTC and RTC, it constructs a tree iteratively adding avertex and an edge to the current tree. The difference of this algorithm fromthe mentioned above heuristics is that it chooses a vertex outside a tree withthe minimum sum of costs of edges with other non-tree vertices. We employeda similar strategy and called the obtained algorithm Min-Power Center-basedLeast Sum-of-Costs (MPCBLSoC). But instead of minimizing the sum of the onstructive Heuristics for MPBHSCP 7 edge weights, we minimized the sum of the power costs in a star-like subgraphwith a center in a given vertex what is more suitable for MPBHSCP. As well asthe methods described above, MPCBLSoC is based on PLH. In this case, themethods
ChooseF irstCenters , ChooseSecondCenter , and
ChooseEachV ertex have the same implementation: given an already constructed partial tree T , thereis selected a such vertex v ∈ V \ V T , that a star graph on remaining verticesrooted in v has minimum total power. The algorithm that chooses the best stargraph center is called F indBestStarCenter , and its pseudo-code is given inAlgorithm 2. Thus, from the one hand, since
ChooseF irstCenters returns asingle vertex, the algorithm MPCBLSoC contains the only iteration. But, fromthe other hand,
F indBestStarCenter runs in time O ( n ), and, therefore, thetotal computational complexity of MPCBLSoC is O ( n ). Algorithm 2
FindBestStarCenter
Input: U ⊂ V ;Output: center ∈ U ; center ← ∅ ; minCost ← ∞ ; for all u ∈ U do leavesCostSum ← centerCost ← for all v ∈ U \ u do leavesCostSum ← leavesCostSum + P ower ( u, v ); centerCost ← max( P ower ( u, v ) , centerCost ); end forif centerCost + leavesCostSum < minCost then center ← u ; minCost ← centerCost + leavesCostSum ; end ifend forreturn center ; Authors of [22] suggest another greedy heuristic called Center-based RecursiveClustering (CBRC) for BDMST. This algorithm starts with a spanning startree rooted in the center, chosen in such a way that the sum of edge weights isminimum. Then the leaves, whose depth is less than ⌊ D/ ⌋ , are iteratively reor-ganized into a cluster with a center in some node. On each iteration, the leavesare reattached to a center if this improves solution and the restriction on thenumber of hops is held. We called our implementation for MPBHSCP Min-PowerCenter-based Recursive Clustering (MPCBRC). As a center choosing subroutinethe previously described algorithm F indBestStarCenter is used. The pseudo-code of MPCBRC is presented in Algorithm 3. Each iteration of the algorithm
R. Plotnikov et al. takes O ( n ) operations because of the complexity of F indBestStarCenter , and,since there are O ( n ) iterations, the algorithm runs in time O ( n ). Algorithm 3
Min-Power Center-based Recursive Clustering v ← F indBestStarCenter ( V ); V ← { v } ; T ← a star graph rooted in v ; U ← V \ { v } ; depth T [ . ] ← an array of size n that stores a depth for each vertex in a tree, filledwith -1; depth ( v ) ← if D is odd then v ← F indBestStarCenter ( V ); depth T ( v ) ← v and an edge ( v , v ) to T ; end ifwhile U is not empty do U ← { v ∈ U : depth T ( v ) < ⌊ D/ ⌋} center ← F indBestStarCenter ( U ); if center == ∅ thenbreak ; end if U ← U \ { center } ; for all u ∈ U do Set powerIncrease ← { power increase after reassigning a parent of u from P arent T ( u ) to center } ; if powerIncrease < then T ← ( T \ { ( u, P arent T ( u )) } ) ∪ { ( u, center ) } ; depth T ( u ) = depth T ( center ) + 1; end ifend forend while One of the most efficient heuristics applied to BDMST in planar Euclidian casewith uniformly distributed vertices consists in recursive splitting the given regioninto equal parts ( quadrants ) and search of their centers [23]. We implementeda variant of the similar approach for MPBHMSCP and called it Min-PowerQuadrant Center-based Heuristic (MPQCH). The pseudo-code of this algorithmis given in Algorithm 4. As well as in some of the previous heuristics, it startswith choosing a center by the algorithm
F indBestStarCenter . But this timein order to reach central symmetry we choose the only start center despite theparity of D . Then inside the main loop the region is iteratively split into thesquared cells of equal size. For each cell, its center is chosen by the algorithm onstructive Heuristics for MPBHSCP 9 F indBestStarCenter and then it is added to the tree with an edge that connectsit with a center of a previous iteration’s cell that contains it, or with v at thefirst iteration. At each iteration the number of cells four times greater than thenumber of cells in the previous iteration, that is, each cell consists of four cells ofthe next iteration. At each iteration the height of a constructed tree is increasedby 1, and, since stepsCount is bounded by ⌊ D/ ⌋ , the diameter constraint is notviolated.In our implementation, for the speed purposes, a regular rectangular grid ofsize qsize × qsize is initially set on the given region, and a corresponding grid cellis assigned to each vertex. Then, due to this grid, during the main loop the subsetof vertices that belong to each cell c ∈ C are found in constant time. Actually, qsize is a parameter of the algorithm, and the greater value of qsize allowsto obtain better solution but increases the running time. The computationalcomplexity of the algorithm is O ( qsize + min {⌊ D/ ⌋ , log( qsize ) } n ). Algorithm 4
Min-Power Quadrant Center-based Heuristic v ← F indBestStarCenter ( V ); T ← ( { v } , ∅ ); U ← V \ { v } ;Construct rectangular grid of size qsize × qsize on a given square; stepsCount ← min( ⌊ D/ ⌋ , log ( qsize )); cellCenter — an array of size n that stores a cell center for each vertex;Fill cellCenter with v (initially the whole square is a single cell and the root is acenter); for all step ∈ { , ..., stepsCount } do Split grid into 2 step × step cells C of equal size; for all c ∈ C do U c ← vertices of U located in c ; center ← F indBestStarCenter ( U c ); T ← T ∪ { ( bestCenter, cellCenter ( center )) } ; U ← U \ { center } ; for all u ∈ U c \ { center } do cellCenter ( u ) ← center ; end forend forend for Another good approach for building spanning tree with bounded diameter is,first, construction a tree without restriction on diameter and, after that, itera-tively decrease depths of vertices until the restriction on diameter is satisfied.The iterative algorithm that reduces the diameter of an input spanning tree forBDMST has been proposed in [24]. We propose the heuristic for MPBHSCP called Min-Power Iterative Refinement (MPIR), which is based on the similaridea. The pseudo-code of this algorithm is presented in Algorithm 5. At first,a center v is chosen by the F indBestStarCenter subroutine. Then, a near-optimal solution for an unbounded problem rooted in v is constructed by IPP Algorithm 5
Min-Power Iterative Refinement v ← F indBestStarCenter ( V );Construct spanning tree T rooted in v by IPP; V ← { v } ; U ← V \ { v } ; depth ( v ) ← if D is odd then v ← most remote neighbor of v in T ; depth T ( v ) ← v and an edge ( v , v ) to T ; end if Calculate the values of depth T ; U ← { v ∈ V \ { s } : depth T ( v ) > h } ; while U is not empty do bestChild ← ∅ ; bestP arent ← ∅ ; minP owerIncrease ← ∞ ;Mark all vertices in U as not considered; for all u ∈ U do C ← { u } ∪ { v ∈ V : depth T ( v ) > v is predecessor of u in T } for all c ∈ { not considered elements of C } doif c is considered thencontinue ; end if Mark c as considered; P ← { v ∈ V : depth T ( v ) < min( ⌊ D/ ⌋ − , depth T ( c ) − } ; for all p ∈ P do powerIncrease ← maximum power costs change of vertices c , P arent T ( c ),and p after assigning p as a parent of c in T ; if powerIncrease < minP owerIncrease then minP owerIncrease ← powerIncrease ; bestChild ← c ; bestP arent ← p ; end ifend forend forend for T ← T \ ( { ( bestChild, P arent T ( bestChild )) } ) ∪ { ( bestChild, bestP arent ) } ;Decrease Level T for all the vertices in the branch rooted in bestChild by Level T ( bestChild ) − depth T ( bestP arent ) − U ← U \ { v ∈ U : depth T ( v ) ≤ ⌊ D/ ⌋} ; end while onstructive Heuristics for MPBHSCP 11 [5]. If D is odd, then the most remote neighbor of v in T is selected as secondcenter. The algorithm works with a set of vertices U whose depth exceeds ⌊ D/ ⌋ .For each u ∈ U the best removing of an edge from the path from u to v andsubsequent adding another edge that decreases a depth of u and increases the W ( T ) at least is found. The best of such edge exchanges among all vertices of U is performed. After each modification of a tree depth of some vertices in U maybe decreased, therefore, the vertices whose depth is less than ⌊ D/ ⌋ are removedthen from U . The computational complexity of the algorithm is O ( n ). We have implemented all the described algorithms in C++ programming lan-guage and run them on the data sets that are given in Beasley’s OR-Libraryfor Euclidian Steiner Problem (http://people.brunel.ac.uk/ ∼ mastjjb/jeb/orlib).These test cases present the random uniformly distributed points in the unitsquare. For the same dimension 15 different instances are provided. We tested4 variants of dimension: n = 100, 250, 500, and 1000, 15 instances for each. Wealso took different values of D for each dimension. The experiment was launchedon the Intel Core i5-4460 3.2GHz processor with 8Gb RAM. n D MPCBTC MPRTC MPCBLSoC MPCBRC MPQBH MPIRav err time av err time av err time av err time av err time av err time100 5 8.17 0.47 0 Table 1: Comparison of the experiment’s results obtained by different heuristics.For the algorithm MPQCH we chose qsize = n since such value does not slowdown the algorithm match, while the solution quality is significantly greater thanin the case qsize = √ n .The results of the experiment are presented in Table 1. For each algorithmand each tested combination of n and D the average objective value (av), averagetime in seconds (t), and standard deviation (err) are shown. In average, whenthe diameter bound is low, the best solution is constructed by MPRTC. Withlarge values of D MPIR constructs the best solution. Note that MPBTC and
MPCBLCoS results are very poor when D is small, but with large values of D their average objective values are close to minimum. MPBTC and MPRTC ap-peared to be the most time consuming on large dimension cases, while MPQCHalways runs significantly faster than other algorithms. Besides, MPQCH per-formance almost does not depends on D . Most probably this is because themaximum diameter of the constructed solution is much less than D , — thisgives us a possibility for the further improvements of this algorithm. (a) MPCBTC. W ( T ) = 7 . (b) MPRTC. W ( T ) = 1 . (c) MPCBLSoC. W ( T ) = 4 . (d) MPCBRC. W ( T ) = 3 . (e) MPIR. W ( T ) = 2 . (f) MPQCH. W ( T ) = 2 . Fig. 1: Algorithms results on the same instance. D = 15 , n = 250As an illustration, we also present in Fig. 1 the solutions that were obtainedby different algorithms on the same instance when D = 15 , n = 250. For theconvenience, the edges that remote from a center by an equal distance (i.e., hopscount) are colored in the same color. Since the diameter bound is odd in this case,there are two centers (connected by a black edge) in solutions constructed byall algorithms except MPQCH, which always builds a tree with the only center.The difference in the behaviour of the algorithms is seen in these pictures. Thediameter bound is still not enough for MPCBTC and MPCBLSoC to constructgood solutions: in both cases the backbone is too small and there are manyleaves far from a center that are coincident with long edges (colored in red).MPCBRC constructs a tree with a lot of long edges in backbone, since the onstructive Heuristics for MPBHSCP 13 backbone vertices are always chosen as center of the current set of leaves duringthe tree construction. MPIR result contains a lot of vertices with large degreethat are coincident with rather long edges, that slightly deteriorate solution.The remained two algorithms, MPRTC and MPQCH, that performed the best,have the following common features: (1) the number of vertices increases withincreasing of their depths; (2) the average edge weight decreases with increaseof the depth. MPRTC always chooses a vertex at random, and, in average,the distance to the closest in-tree vertex becomes less while the constructedtree size grows. MPQCH constructs a tree whose backbone vertices are locatedclose to the quadrants geometric centers. Note that MPQCH built a tree withmaximum depth equal 6 while the depth upper bound is 7. This allows to improvesolution in this case: each of the longest edges that connect a center with its fourchildren could be replaced by two shorter edges with intermediate vertex that islocated close to edge’s geometric center. We assume that such modification willsignificantly improve the solution, and we plan to implement it in future. In this paper, the NP-hard Min-Power Bounded Hops Symmetric ConnectivityProblem was considered. We proposed six different constructive heuristics for itsapproximation solution. As main ideas of our approaches, we used some of theknown heuristics that were previously developed for BDMST. We implementedall the proposed algorithms and conducted the numerical experiment on differentrandomly generated test instances. The simulation shows that in cases withlarge diameter the algorithm MPIR yields the best results, while the usage ofMPRTC is more preferable when the diameter is small. If one needs to obtain asolution of rather good quality in shortest time, then MPQCH could be the bestchoice. Besides, the experiment results show that MPQCH can be significantlyimproved. In future we plan to develop different variants of local search andother metaheuristics that appeared to be efficient for BDMST, such as variableneighborhood search, genetic algorithm, and ant colony optimization, where thetrees obtained by different algorithms proposed in this paper will serve as startsolutions.