A Survey on Algorithmic Problems in Wireless Systems

Simon Thelen, Klaus Volbert and Danilo Weber Nunes

Faculty of Computer Science and Mathematics, Ostbayerische Technische Hochschule Regensburg, Germany

Keywords:

Algorithms, WSNs, Survey, Network Construction, Routing, Interference, Localization, Charging, Latency.

Abstract:

Considering the ongoing growth of Wireless Sensor Networks (WSNs) and the challenges they pose due

to their hardware limitations as well as the intrinsic complexity of their interactions, specialized algorithms

have the potential to help solving these challenges. We present a survey on recent developments regarding

algorithmic problems which have applications in wireless systems and WSNs in particular. Focusing on the

intersection between WSNs and algorithms, we give an overview of recent results inside this intersection,

concerning topics such as routing, interference minimization, latency reduction, localization among others.

Progress on solving these problems could be potentially beneﬁcial for the industry as a whole by increasing

network throughput, reducing latency or making systems more energy-efﬁcient. We summarize and structure

these recent developments and list interesting open problems to be investigated in future works.

1 INTRODUCTION

Wireless Sensor Networks (WSNs) have been steadily

growing in signiﬁcance over the past two decades and,

in light of the ongoing shifts in the energy and indus-

try sector, appear to continue to do so in the future. By

2025, the number of online Internet-of-Things (IoT)

devices is expected to reach nearly 75 billion (Ikpe-

hai et al., 2019). Nodes in a WSN commonly have

many restrictions regarding their processing power,

range of communication and their limited energy sup-

ply, often being battery-powered (Singh and Kumar,

2012). These limitations impose additional compli-

cations when designing solutions to known problems

such as package routing, electromagnetic interference

(EMI), latency, among others.

Meanwhile, there have been many interesting ad-

vances in the ﬁeld of algorithms in recent years con-

cerning problems such as maximum ﬂow (Bl

asius

et al., 2021; Abboud et al., 2022), dominating set

(Abu-Khzam et al., 2022), or graph coloring (Ghaf-

fari and Kuhn, 2022; Ansari et al., 2022). It becomes

evident that these algorithmic problems have applica-

tions in problems faced in the WSN ﬁeld. Advances

in the ﬁeld of algorithms could lead to higher over-

all network efﬁciency due to better routing strategies.

They could increase the network’s throughput with

improved network architectures or reduce its energy

consumption by using more efﬁcient algorithms, to

name a few possible improvements. This overlap of

WSNs and algorithms holds promising new advance-

ments and is an area to be explored. To the best of our

knowledge, there has not been a survey work combin-

ing both areas of research.

In this survey, we present algorithmic problems

in wireless systems that researchers from both ﬁelds

are currently working on. We examine where signif-

icant progress has been made and explore open ques-

tions and possible further developments. We have se-

lected twenty papers published in the last ﬁve years

which address interesting problems that in our opin-

ion best represent the intersection between algorithms

and WSNs (refer to Table 1).

This work is structured as follows: In Sections 1

to 9 we detail the ﬁndings of our survey grouped by

topic. In Section 10 we summarize the work’s conclu-

sions and outline possible further research directions.

2 NETWORK CONSTRUCTION

AND INITIALIZATION

For problems like routing or interference reduction,

the network is often viewed as a pre-existing entity.

In practice, though, the construction/deployment of

the network can be equally important. Multiple recent

results deal with the process of building up a network

graph or extending an existing one.

Mertzios et al. analyzed a speciﬁc version of a

growth process with possible applications for WSNs

Thelen, S., Volbert, K. and Nunes, D.

A Survey on Algorithmic Problems in Wireless Systems.

DOI: 10.5220/0011791200003399

In Proceedings of the 12th International Conference on Sensor Networks (SENSORNETS 2023), pages 101-111

ISBN: 978-989-758-635-4; ISSN: 2184-4380

 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)

101

Table 1: A list of the papers which are part of this survey, grouped by topic and categorized by which algorithmic aspects they

are predominately concerned with: creating theoretical models describing real-life problems in WSNs; presenting algorithms

that solve these problems fully, approximately or heuristically; performing mostly theoretical analyses, for example regarding

the hardness of speciﬁc problems.

Modeling Algorithms Analysis

Network construction

and initialization

(Mertzios et al., 2021) X X X

(Cor

o et al., 2019) X X X

(Miller, 2019) X X

Routing

(Jung et al., 2019) X X

(Kuo, 2019) X X X

otte et al., 2021) X X

(Galesi et al., 2019) X

Interference

(Abu-Affash et al., 2020) X X X

(Tsai et al., 2019) X X

(Esperet et al., 2021) X X

(Schindelhauer et al., 2019) X X

Localization

(Bose et al., 2020) X X

(Lv et al., 2019) X X

Charging and

energy harvesting

(Madhja et al., 2018) X X

(Hanschke and Renner, 2019) X

Reducing latency

(Karakostas and Kolliopoulos, 2022) X X X

(de Berg et al., 2019) X X

Population protocols (Raskin, 2021) X X

Drones

(Betti Sorbelli et al., 2022) X X

(Danilchenko et al., 2020) X X

where a graph grows from a starting vertex in a crys-

talline fashion and edges are allowed to be removed

later (Mertzios et al., 2021) (excess edges). In each

time slot, vertices and edges can be added accord-

ing to speciﬁc rules and the process terminates if all

vertices of the target graph and a super-set of the

edges have been added. The authors recognize a

trade-off between the number of time slots needed

and the number of excess edges. They show that

using poly-logarithmically many time slots, any tree

can be grown needing O(n) excess edges and any

planar graph can be grown using O(n logn) excess

edges. For the case of not allowing any excess edges,

the authors prove that, assuming P 6= NP, there is

no polynomial-time algorithm that ﬁnds the shortest

growth schedule and no polynomial-time approxima-

tion algorithm for ﬁnding the shortest growth sched-

ule with an approximation ratio of n

1−ε

(for every

ε > 1). There remain many open questions concern-

ing the problem presented in the paper: It is unclear

if the problem of growing a (general) graph using the

fewest number of (excess) edges is NP-hard. The pa-

per mainly focuses on the two extremes in terms of

the number of excess edges, namely there being zero

excess edges and there being lots (log(n)), leaving the

space between the two extremes largely unexplored.

A different kind of growth is known as graph aug-

mentation. Here, given a graph, one should add the

minimum number of edges to achieve a certain condi-

tion, like making the graph connected. This problem

is not new and has already been studied in the 70s

(Eswaran and Tarjan, 1976). Cor‘o et al. analyzed

a different variant of this problem where you are al-

lowed to add a ﬁxed number of edges to maximize the

total gained connectivity, i.e. the number of reachable

nodes, summed over all nodes (Cor

o et al., 2019). The

authors present algorithms that solve a generalization

of this problem optimally for trees and approximately

for DAGs with one source (node with in-degree zero)

and one sink (node with out-degree zero). They show

that for general graphs with only one source and one

sink the problem is NP-complete. They also show that

in general this version of the problem is NP-hard to

approximate within a factor of 1 −

. The authors are

also interested in solving the problem for DAGs for

multiple sources and sinks. It would also be inter-

esting to solve a version of the problem where each

added edge costs a speciﬁed amount and there is a

limited budget.

For many problems in WSNs, such as routing,

it is essential for nodes to know the nodes in their

neighborhood beforehand. This problem is known

as neighbor discovery. In a recent paper, Miller in-

vestigated multiple dynamic variants of neighbor dis-

covery, being able to show that all of these can be

reduced to a single problem, further named δ-local

gossip (Miller, 2019). For δ-local gossip, every node

in the network has a piece of information it wants to

send to nearby nodes. After a certain time interval,

each node v should have transmitted its information to

the nodes which were inside a δ-multiple of v’s trans-

mission range at some point during the execution of

the algorithm. After all nodes have managed to do so,

every node should terminate. Miller presents an algo-

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rithm that solves δ-local gossip for moving nodes on

a line with bounded velocity. The author is interested

in how a solution to this problem might be improved

if the movement of nodes is partially known, which is

common for practical applications like public trans-

portation or self-driving vehicles. As the work shows

the general applicability of δ-local gossip for neigh-

bor discovery problems, investigating to which other

kinds of problems δ-local gossip can be applied to

might be an interesting topic for future research.

3 ROUTING

One of the most important problems in WSNs is how

to route messages from one node to another.

3.1 Radio Holes and Ad Hoc Networks

Routing is especially challenging for dynamically

changing ad hoc networks where the overhead of

many classical routing approaches, which were de-

signed for relatively static networks, is simply too

high. Consequently, there has been a lot of algo-

rithmic research concerning ”on-demand” or ”online”

routing protocols. One issue of a greedy strategy like

sending each message to the closest neighbor is that

moving nodes can create radio holes, which might re-

quire some messages to take long detours to reach

their destination (Jung et al., 2019). There have been

multiple approaches to address this problem. Kuhn et

al. presented a routing algorithm which is both fast in

practice and proven to be asymptotically optimal in

the worst-case in the sense that it achieves an asymp-

totically optimal competitive ratio compared to the

best-possible ofﬂine algorithm under certain assump-

tions like reliance on local information only (Kuhn

et al., 2003). Another result (R

uhrup and Schindel-

hauer, 2006) proves asymptotic optimality in a grid-

like setting with possibly failed nodes.

A recent paper (Jung et al., 2019) approaches the

problem of radio holes by, in addition to the regu-

lar short-range communication, giving nodes limited

access to long-distance communication (e.g. satellite,

cellular). They present an O(log

n) communication-

round algorithm to compute an abstract version of the

network (called an overlay network) which includes

both the short- and long-distance connections and re-

sults in c-competitive routing paths for some constant

c. Their algorithm heavily relies on the convex hulls

of the radio holes in the network and assumes that the

convex hulls do not overlap. It would be interesting to

know how the problem can be solved if this restriction

is lifted. The authors also mainly focus on the case of

a static network, although they present a dynamic so-

lution based on recomputing the overlay network pe-

riodically. It seems likely that this relatively simple

method can be improved.

There is a strong algorithmic connection between

routing and many classical graph-theoretic problems.

Much attention in recent research has been on the

DOMINATINGSET problem, which in its simplest

form can be formulated as follows Given an undi-

rected graph G = (V,E), ﬁnd the smallest vertex sub-

set S ⊆V , such that for every vertex v ∈V \S, there

exists an adjacent vertex v

∈ S. This NP-complete

problem has useful applications for routing: As every

node is adjacent to at least one node in the dominating

set, routing can be simpliﬁed by only having nodes in

the dominating set function as routers (Kuhn and Wat-

tenhofer, 2003). A common variant of this problem is

CONNECTEDDOMINATINGSET where the subgraph

induced by the dominating set must be connected.

A connected dominating set in a network is espe-

cially useful for routing as it can be used as a virtual

backbone: For any two nodes s,t, there exists a path

from s to t such that each node on that path (except for

possibly s and t) is part of the virtual backbone. Moti-

vated by this idea, a natural problem to study is: What

is the smallest virtual backbone (connected dominat-

ing set) such that the number of internal nodes neces-

sary to get from any node s to any node t increases at

most by a factor of α?. This is known as the Con-

nected Dominating Set Problem with Routing Cost

Constraint (CDR-α). As discovered by (Ding et al.,

2011), you do not actually need to consider all pairs

(s,t), but only those with a shortest distance of 2,

resulting in the 1-DR-α problem: Any approxima-

tion algorithm for 1-DR-α applies CDR-α as well,

retaining its approximation ratio. The difﬁculty of

CDR-α largely depends on the choice of α: For

α = 1, there exists an O(log n)-approximation algo-

rithm, by reducing the problem to SETCOVER (Ding

et al., 2011). For sufﬁciently large α, the problem

can be reduced to regular CONNECTEDDOMINAT-

INGSET, again yielding an O(log n)-approximation

algorithm (Guha and Khuller, 1998). So the question

arises: Does there exist a polynomial-time O(log n)-

approximation algorithm for every α ≥ 2 (Du and

Wan, 2013)? In a recent paper, Kuo was able to

show that the answer to this questions is no by prov-

ing that no polynomial-time 2

log

1−ε

-approximation

algorithm exists for any constant ε > 0, assuming

that NP * DTIME(n

poly logn

) (Kuo, 2019). How-

ever, the authors do present an O



1−

(logn)



approximation algorithm for any constant α > 1 as

well as an O(

√

nlog n)-approximation algorithm for

α ≥ 5. They also show that 1-DR-α (and therefore

A Survey on Algorithmic Problems in Wireless Systems

103

CDR-α) can be viewed as a special case of the SUB-

MODULARCOSTSETCOVER problem and the MIN-

IMUMRAINBOWSUBGRAPH problem. If better ap-

proximation algorithms for those problems are found

in the future, this might lead to better approximation

algorithms for CDR-α.

An NP-complete problem which is closely related

to DOMINATINGSET is SETCOVER. Here, a universe

of elements U and a set of subsets S of U is given

(∀s ∈ S : s ⊆U) and the objective is to ﬁnd the small-

est number of sets in S whose union equals U, i.e.

these subsets cover U completely. In terms of approx-

imability, DOMINATINGSET and SETCOVER are in

some sense almost the same problem as they can be

converted into each other using L-reductions (Kann,

1992). Approximating these problems in a distributed

setting has been studied extensively: e.g. DOMINAT-

INGSET in (Kuhn and Wattenhofer, 2003; Jia et al.,

2002) and SETCOVER in (Kuhn et al., 2006).

A recent paper (G

otte et al., 2021) tackles SET-

COVER in a distributed model where each node rep-

resents either an element of U or of S and edges rep-

resent ∈-relations. Previous results focused on the

LOCAL the CONGEST model where in each time step

nodes can send distinct messages to all their neigh-

bors. This new result considers two different models:

In the KT

-CONGEST model, each of the n nodes can

send one O(logn) size message per time step to some

or all of its neighbors. In the BEEPING model, at ev-

ery time step a node can either beep or receive. Re-

ceiving nodes only learn whether or not at least one

neighboring node has beeped in the current time step.

The KT

-CONGEST algorithm presented achieves an

expected approximation ratio of O(log ∆), requiring

O(log

∆) time and

√

∆ · n) messages with high

probability where ∆ is the maximum node degree in

the communication graph (

O hides poly-logarithmic

factors). The BEEPING algorithm takes O(k

) for

any chosen k > 3, achieving an approximation ratio

of O



log

(∆) ·

√

∆



. The parameter k allows ﬁne-

tuning the trade-off between approximation ratio and

runtime. The known lower bound for reaching an

approximation ratio of O(

√

∆) is O(k) (Kuhn et al.,

2006). If the problem can be solved in o(k

) with the

same approximation ratio is still an open question.

3.2 Tomography

In order to route messages from one node to another,

it is integral to know if all nodes along the way are

up and running correctly. An interesting approach

for doing this is network tomography, speciﬁcally

Boolean network tomography where one only con-

siders the paths that past messages were sent along.

If one message does not reach its destination, one

can deduce that at least one of the nodes along the

path does not work correctly. Using multiple of these

kinds of deductions, failed nodes can be identiﬁed.

There has been lots of research concerning tomogra-

phy (Kakkavas et al., 2020; Gray et al., 2020). Galesi

et al. studied the inﬂuence of vertex connectivity on

Boolean network tomography (Galesi et al., 2019).

Vertex connectivity of a graph is deﬁned as the min-

imum number of vertices which need to be removed

until the graph is not connected anymore. The au-

thors use the measure of maximal identiﬁability to de-

scribe how well Boolean tomography identiﬁes non-

working nodes in a network. They prove tight bounds

on the relationship between vertex connectivity and

maximal identiﬁability for so-called Line of Sight net-

works and weaker bounds for general networks. They

also extend these results to random graphs.

4 INTERFERENCE

Minimizing interference is clearly one of the most

fundamental and most relevant problems in WSNs.

4.1 Optimizing Transmission Ranges

In theoretic works regarding interference minimiza-

tion, researchers distinguish the asymmetric model

(Agrawal and Das, 2013), where a node u can send

messages to some other node v if and only if v is in-

side u’s transmission range, and the symmetric model

(Halld

orsson and Tokuyama, 2008), where u addi-

tionally has to be inside v’s transmission range.

In a widely studied interference-related problem,

nodes are located at ﬁxed positions in the 2D plane,

and the task is to select the transmission range of each

node to achieve minimum interference while retain-

ing the connectedness of the network. In the sender-

centric model (Moaveninejad and Li, 2005), the in-

terference of each node is deﬁned as the number of

nodes in its transmission range and the network’s in-

terference is deﬁned as the maximum node interfer-

ence. The receiver-centric model (Von Rickenbach

et al., 2009) deﬁnes the network’s interference anal-

ogously in terms of receiving nodes. This receiver-

centric notion seems to be the most widely accepted

in the literature. While sender-centric interference

can be minimized in polynomial time (Moaveninejad

and Li, 2005) in both the symmetric and asymmet-

ric model, minimizing receiver-centric interference is

NP-hard (Buchin, 2008; Brise et al., 2014). Interfer-

ence can also be modelled in terms of graph edges

instead of vertices. In (Meyer auf der Heide et al.,

SENSORNETS 2023 - 12th International Conference on Sensor Networks

104

2004), the interference of an edge (u,v) is deﬁned as

the number of other edges whose endpoints lie within

the transmission range of u or of v.

Instead of minimizing the maximum node inter-

ference, some researchers focus on minimizing the

sum of of all node interferences: total interference.

Since the sum of all in-degrees in a graph is equal

to the sum of all out-degrees, sender-centric and

receiver-centric total interference is identical. Sig-

niﬁcant progress has been made in analyzing the to-

tal interference in the symmetric network model for

both the 2D and the 1D case, where all nodes lie on

a straight line. While the 1D case can be solved opti-

mally in O(n

) time (Tan et al., 2011), the 2D case is

NP-hard (Lam et al., 2010). However, for the 2D case,

an O(log n)-approximation algorithm exists (Mosci-

broda and Wattenhofer, 2005) as well as an algorithm

that guarantees O(log n) total interference for all in-

puts (Fussen et al., 2005).

Abu-Affash et al. consider the problem of mini-

mizing total interference in the asymmetric network

model (Abu-Affash et al., 2020). Analogously to the

previous results on the symmetric variant, this new

paper shows that solving the 1D case optimally is

feasible in polynomial time by presenting an O(n

)

algorithm. The authors prove that the 2D case is

NP-hard, but they do present a 2-approximation algo-

rithm. They conjecture that techniques of their 1D al-

gorithm could be used to improve the running time of

the corresponding algorithm for the symmetric model.

In general, many questions regarding interference

minimization still seem to not have satisfying an-

swers. The problem of minimizing maximum node

interference in the receiver-centric model seems to be

particularly challenging. Apart from its NP-hardness,

not much seems to be known about this problem.

4.2 Graph Coloring

Interference in WSNs can be reduced by using mul-

tiple frequencies/channels. This motivates the prob-

lem of selecting the right channel for any given mes-

sage. In practice, nodes in a WSN often have multiple

network interfaces and can consequently send/receive

on multiple channels at the same time. To reduce in-

terference, neighboring nodes can send/receive mes-

sages using different frequencies. The connection be-

tween problems regarding channel selection in WSNs

and the graph theoretic problems EDGECOLORING

and STRONGEDGECOLORING has long been known

(Barrett et al., 2006). EDGECOLORING is the prob-

lem of coloring the edges of a graph, using as few

colors as possible, such that no two adjacent edges

have the same color. STRONGEDGECOLORING ad-

ditionally requires that the endpoints for no pair of

same-color edges are connected via a single edge.

In the context of WSNs, these two graph problems

are strongly related to problems of avoiding primary

and secondary interference respectively. Although

the NP-hardness of both EDGECOLORING (Holyer,

1981) and STRONGEDGECOLORING (Stockmeyer

and Vazirani, 1982) has been proven, there is still ac-

tive research concerning the two problems (Saberi and

Wajc, 2021; Balliu et al., 2022). There are also many

open questions regarding the strong variant. It has

been conjectured that the optimal strong edge color-

ing of any graph uses at most 5∆

/4 colors where ∆

is the maximum node degree (Hal

asz and S

os, 1989).

Despite some signiﬁcant progress, this conjecture re-

mains unproven (Deng et al., 2019).

Tsai et al. studied the multi-channel assignment

problem. Here, each node in the network has r radio

interfaces which for each time slot can send/receive

on one of k possible channels (Tsai et al., 2019). The

goal is to ﬁnd a channel schedule for all nodes that op-

timizes a certain objective function. In contrast to pre-

vious results, which focused on maximizing the total

number of connections per time slot or on minimizing

the time needed for sending a given list of messages

(Chaporkar et al., 2008; Kumar et al., 2004), Tsai et

al. focus on minimizing latency. More speciﬁcally,

they try to ﬁnd a periodic schedule which minimizes

max

i∈E

, where E is the set of edges, T

is the max-

imum message delay of any message sent along edge

i for that schedule, and w

is edge i’s weight. The

authors show that for unit weights and r = 1, their

channel assignment problem reduces to EDGECOL-

ORING for k ≥ bn/2c and to STRONGEDGECOLOR-

ING for k = 1. In addition, they present an approxi-

mation algorithm for general weights which is based

on the idea of duplicating edges depending on their

weight. Simulations conﬁrm that the proposed algo-

rithms work well even if they are generalized to r > 1.

In the future, this result might be extended to a more

realistic interference model, such as SINR (Signal-to-

Interference-plus-Noise Ratio), or even to other kinds

of scheduling problems.

As seen above, channel selection has clear algo-

rithmic connections to edge coloring problems. If

we consider the one-channel-case, avoiding simulta-

neously sending, neighboring nodes has a surprising

link to coloring a graph’s vertices: If all nodes with

color A send ﬁrst, then all nodes with color B and so

on, no two nodes within transmission distance of each

other will send simultaneously. Esperet et al. tack-

led the problem of coloring a unit disk graph with as

few colors as possible in a distributed setting with lo-

cal communication, speciﬁcally in the LOCAL model

A Survey on Algorithmic Problems in Wireless Systems

105

where only the communication rounds count towards

the running time (Esperet et al., 2021). In the case that

nodes know their location, the authors present an al-

gorithm which produces a (3 + ε)ω(G) + 6-coloring

in O(1) rounds of communication for any constant

ε > 0 where ω(G) is the size of the largest clique.

For certain kinds of graphs (McDiarmid and Reed,

1999; McDiarmid, 2003), this is an improvement over

a previously-known 3-approximation algorithm. In

the case that nodes do not know their location, Es-

peret et al. demonstrate how to achieve a 5.68ω(G)-

coloring in O(log

logn) rounds with high probability

and, assuming that ω(G) = O(1), how to achieve such

a coloring in O(log

∗

n) rounds deterministically. Adi-

tionally, the authors show that the average degree of

any unit disk graph G is at most 5.68ω(G). They con-

jecture that this can be improved to 4ω(G). The au-

thors are interested in whether algorithms might exist

that achieve cω(G)-colorings for a lower c under dif-

ferent round constraints. Additionally, it might be in-

teresting to know if similar techniques can be applied

to general disk graphs where nodes can have different

transmission ranges.

In general, there are many open questions in the

ﬁeld of graph coloring. For unit graphs in particular,

we know that the minimum number of colors needed

for coloring a graph is at most 3ω(G) −2, but little

progress has been made in improving the constant 3

in this result (Esperet et al., 2021).

4.3 Constructive Interference

Although interference in wireless systems should usu-

ally try to be avoided, there exists the concept of con-

structive interference where multiple signals overlap

in such a way that they amplify each other. Schin-

delhauer et al. study a problem where one node needs

to broadcast a signal to every other node as quickly

as possible using constructive interference (Schindel-

hauer et al., 2019). In particular, they are interested

in the multiple input/multiple output model (MIMO)

where multiple nodes can cooperate to produce a

stronger signal. Instead of trying to broadcast a real

message, the main focus lies on the collaboration as-

pect to achieve the signal strength needed so the sig-

nal can be detected in a noisy environment. The algo-

rithm presented needs O(log logn −log log ρ) rounds

to broadcast a message where n is the number of

nodes and ρ is the node density. The authors ignore

interference effects which would arise when sending

an actual message. Message encodings that would

prevent these effects might be an interesting area of

research. The authors also assume a path loss expo-

nent of α = 2 in their work. Despite some conjectures,

the situation for larger values of α is left unexplored.

5 LOCALIZATION

Another interesting topic in wireless networks is lo-

calizing nodes. Global protocols like GPS guarantee

accurate localization in many cases. However, they

have relatively high power-consumption and do not

work in indoor situations. Alternatively, the known

location of some nodes (often called anchors) is used

to determine the positions of all nodes. Multiple

methods have been proposed, including triangulation

(Savarese et al., 2001), multi-dimensional scaling (Ji

and Zha, 2004) and trilateration (Moore et al., 2004)

where, given the positions of two points of a triangle

and the lengths of all three sides, the position of the

third point can be deduced.

Bose et al. study the problem of localizing nodes

in a unit-disk graph by starting at only three anchors

and localizing other nodes using trilateration (Bose

et al., 2020). Assuming RP 6= NP where RP is the

complexity class commonly known as ”Randomized

Polynomial-time”, this localization problem is NP-

hard even if it is known that there exists a unique solu-

tion (Aspnes et al., 2004). The authors therefore focus

on localizing only some of the nodes. They deﬁne a

node to be interior if every point on the boundary of

its unit disk is covered by at least one other unit disk.

A node is strongly interior if it and all its neighbors

are interior. The authors propose a distributed algo-

rithm that localizes at least all strongly interior nodes

given three strongly interior anchor nodes, assuming

all strongly interior nodes are connected. There are

multiple ways to generalize their approach: What if

the graph of all strongly interior nodes consists of

multiple connected components? What if the trans-

mission range of different nodes is different? etc.

Approaches like (Bose et al., 2020) rely on the

assumption that distances between nodes can be de-

termined accurately. In practice, these distances can

only be approximated. But in some scenarios even

that is not possible. The authors of (Lv et al., 2019)

design a localization model, which they call BSLoc,

based on telecommunication localization. It uses base

stations of a telecommunication network (e.g. cellu-

lar), which the device is connected to. Typically, the

radio signal strength indications (RSSI) of the base

stations are used as a distance estimator to localize de-

vices (Vagheﬁ et al., 2011; Zhu et al., 2016; Margolies

et al., 2017). Lv et al. are interested in how to locate

devices if these signal strengths are not known. They

propose a two-level solution where a Hidden Markov

Model in the ﬁrst level produces a rough location esti-

SENSORNETS 2023 - 12th International Conference on Sensor Networks

106

mate which is improved by a machine learning model

in the second level. Their method uses both historical

data as well as speed information of the devices and

achieves comparable results to state-of-the-art RSSI-

based methods in their experiments.

6 CHARGING AND ENERGY

HARVESTING

Nodes in WSNs are usually battery-powered. To re-

duce maintenance, they are often capable of recharg-

ing their batteries. The authors of (Hanschke and

Renner, 2019) study the scheduling of tasks which

are performed on WSN nodes under time and energy

constraints, with the additional complication that sen-

sors are powered using energy harvesting, speciﬁcally

using solar power. Sensors perform tasks like mea-

suring humidity or ﬁne dust with (time) dependencies

between tasks, such as ”to measure ﬁne dust, a re-

cent humidity measurement must exist”. Sensors can

charge their internal capacitor if the current energy

is insufﬁcient to complete the next task. Given an

energy prediction for a future time period, the goal

is to maximize the number of (periodically executed)

tasks that are performed. Using a concept called task

graphs, the authors solve the problem with an integer

linear program solver. Using a simple performance

comparison, the authors conjecture that their algo-

rithm works even on low-end hardware under reason-

able time constraints. They do not, however, provide

an actual implementation to verify this claim.

Another concept in WSNs is Wireless Power

Transfer. Madhja et al. investigated the situation

of an ad hoc network consisting of multiple mobile

agents, moving on a random-walk-like path, and a

single static charging device with a speciﬁed ﬁnite

energy supply, capable of charging the batteries of

nearby agents (Madhja et al., 2018). The objective

is to have the network up and running for as long as

possible using the given energy supply. Unlike other

publications, they allow for changing the charging

power (and therefore charging range) dynamically.

The energy received by each device is determined by

a simpliﬁed version of the Friis transmission equation

(Friis, 1946). The authors demonstrate the theoret-

ical beneﬁt of using an adaptive charging range by

showing that any ﬁxed charging range is sub-optimal

for at least one possible agent scenario. They also

show that, under slightly simpliﬁed conditions, ﬁnd-

ing the optimal charging range schedule ofﬂine is NP-

hard. Then, they present several heuristic approaches

to solving the problem online, showing the practical-

ity of these approaches using experiments. It might

be interesting to investigate if better results can be

achieved by considering the predictable behavior of

real-life agents (e.g. using machine learning).

7 REDUCING LATENCY

The paper (Tsai et al., 2019) mentioned in Section 4

deals with the problem of selecting radio channels to

reduce latency. Interestingly, multiple other recent

works have latency reduction as a common theme.

Karakostas and Kolliopoulos study a problem

with applications in 2-way synchronization for Digi-

tal Twins (Karakostas and Kolliopoulos, 2022). For

Digital Twins, data typically should be kept fresh,

meaning as up-to-date as possible. One challenge

when trying to achieve this is that critical resources

like CPUs or are often shared by multiple Twins. If

a Twin task takes c clock cycles and the allowed time

constraint equals T cycles, then the total number of

simultaneous tasks on that CPU cannot exceed

The authors formulate a generalized version of this

problem where n tasks are distributed onto m ma-

chines. For each task i and machine j, a so-called

tolerance constraint u

i j

is deﬁned as follows: If task

i is assigned to machine j, then machine j can have

at most u

i j

tasks in total, including task i. The au-

thors analyze two optimization variants of this prob-

lem: In the ﬁrst variant, each job has a weight and the

goal is to maximize the total weight of the assigned

jobs assigned under the given constraints. The au-

thors present a (1 −

)-approximation algorithm for

this variant. For the second variant, tolerance con-

straints can be loosened by a global scaling factor

ρ ≥ 1. The goal is to minimize ρ so all tasks can be

assigned. The authors prove that, assuming P 6= NP,

there is no polynomial-time (n

1/2−ε

)-approximation

for this variant, even if all machines share tolerance

constraints. They prove even stronger hardness re-

sults for speciﬁc approaches, namely the conﬁgura-

tion linear program (LP) and quadratic programming.

The gap between the general hardness result and these

speciﬁc hardness results is mentioned by the authors

as an open problem.

Another latency-focused result deals with broad-

casting messages. Broadcasting algorithms are often

evaluated using metrics like throughput. De Berg et

al. instead focus on displacement: How much do the

packets of the message arrive out of order (de Berg

et al., 2019)? For any node and any time step, the au-

thors deﬁne displacement as the difference between

the number of the latest packet received and the num-

ber of the earliest packet which has not been received

yet. Low displacement can reduce buffer sizes as well

A Survey on Algorithmic Problems in Wireless Systems

107

as latency. The authors analyze different broadcasting

algorithms that try to minimize displacement, show-

ing that these algorithms tend to maximize through-

put as well. However, which kinds of applications

are most positively affected by displacement-focused

broadcast algorithms still needs to be investigated.

8 POPULATION PROTOCOLS

Many methods and frameworks can be used to de-

ﬁne how the elements in a network are connected and

communicate. One of them is population protocols.

Population protocols were ﬁrst proposed by (An-

gluin et al., 2004) and are a powerful and ﬂexible

communication and computing model, especially for

mobile ad hoc networks (Aspnes and Ruppert, 2009).

A population protocol consists of a collection of sim-

ple and mobile agents, which can assume a ﬁnite set

of states. Like in a WSN, these agents are allowed

to communicate if they are close. This communica-

tion is somewhat limited, as a scheduler determines

which agents will interact. These interactions cause

a state update for the two agents involved and are

based on a deﬁned transition table. Agents have a

constant amount of memory to hold their state. An

agent’s movement is unpredictable but is subject to

constraints to keep the system fair, in other words, all

agents must have a chance to be selected for interac-

tion. As all agents have the same set of states and

are updated using the same set of rules, they can be

considered to be virtually anonymous.

The interactions between the nodes carry the com-

putation along. As shown by (Angluin et al., 2007b),

in a setup where there are no restrictions on which

agents can interact (i.e. fair scheduling), if the given

predicates are semilinear, meaning that they are deﬁn-

able in ﬁrst-order Presburger arithmetic (Presburger,

1929), these predicates are always computable by

population protocols. This allows population proto-

cols to perform quite powerful computations.

A population protocol is formally deﬁned by the

tuple (Q, Σ, Y , ι, ω, δ), where Q is the set of states,

Σ the set of input and Y the set of output symbols,

ι : Σ → Q the function that maps the input to an inter-

nal state, ω : Q →Y the function that maps the agent’s

internal state to an output and δ : Q ×Q → Q ×Q the

function that deﬁnes the state transitions of two in-

teracting agents. Eventually the output of all agents

will converge to the same value. This structure de-

ﬁnes a very generic and ﬂexible framework. Different

tasks, network topologies, communication protocols,

and other aspects of a WSN can be modelled and even

integrated in the network itself. Due to their simplic-

ity, population protocols can be implemented using

essentially just three lookup tables. This allows them

to run on even extremely low-powered hardware.

Consider the following IO population proto-

col: Σ=Y =Q={0,?,1}, ι and ω are the iden-

tity function, δ = {(0, 0) → (0, 0), (?,0) → (0,0),

(1,0) → (?,0),(0,?) →(0,?), (?,?) → (?,?), (1, ?) →

(1,?),(0, 1) → (?,1), (?,1) → (1, 1),(1,1) → (1,0)}.

In O(n logn) interactions, this simple population pro-

tocol is able to determine with high probability which

state (1 or 0) the majority of the agents were in at the

beginning of the execution (Angluin et al., 2007a).

Populations protocols can also be applied to more

complex tasks, such as counting. By deﬁning δ =

{(a,b) → (b

a+b

c,d

a+b

e)}, the population protocol

can not only determine the majority state, but also by

what margin it was the majority in O(n logn) interac-

tions (Mocquard et al., 2015).

Michail and Spirakis studied another important,

real-world application of population protocols. By al-

lowing each agent to store not only its current state,

but also its connection state, it enables populations

protocols to eventually generate a stable spanning star

network topology (Michail and Spirakis, 2014). The

method works under dynamic conditions, enabling a

network to adapt over time in response to an increased

network load and thus improving its total capacity.

Message loss can also be added to the agent’s

communication dynamic by simply adding the pos-

sibility of only one agent updating its local state and

the state of the other agent involved in the communi-

cation remaining unchanged, as if the communication

did not happen (Raskin, 2021).

The combination of the last two works could lead

to interesting further developments not only for the

ﬁeld of population protocols but also to the broader

WSN ﬁeld, improving dynamic topology generation

and the network’s resiliency to network instabilities.

9 DRONES

Multiple recent works deal with algorithmic problems

motivated by applications for aerial drones.

Sorbelli et al. considered a smart agriculture

scenario where sensors are deployed in an area

(Betti Sorbelli et al., 2022). Instead of using a multi-

hop approach to transmit the data to a depot, a drone

is used. This drone starts at the depot, ﬂies across

the area, collecting data from the sensors, and returns

to the depot. The drone’s storage and energy sup-

ply is limited. Some data like images might have

a higher priority or weight. The authors deﬁne the

Single-drone Data-collection Maximization Problem

SENSORNETS 2023 - 12th International Conference on Sensor Networks

108

as ﬁnding the drone ”mission” (consisting of a path

and the data to collect) which maximizes the mis-

sion’s reward, i.e. the sum of the collected data’s

weights. The authors show that the problem is NP-

hard. They also present an approximation algorithm,

based on known approximation algorithms for KNAP-

SACK, MINIMUMSETCOVER and TSP, as well as

two simple heuristic solutions. All three algorithms

perform well on synthetically generated data. In-

teresting extensions to this problem include multiple

drones or a more realistic model for the drone’s en-

ergy. It might be interesting to know in which scenar-

ios this unusual drone setup is advantageous.

Danilchenko et al. studied the (NP-hard) problem

of covering points in the 2D plane with a ﬁxed number

circular disks or squares of identical size, maximizing

the weighted sum of all covered points (Danilchenko

et al., 2020). As motivation, the authors name is a

surveillance application where a small and ﬁxed num-

ber of aerial drones with a certain view radius cover

a large number of ground users. The authors study

both the static case and the dynamic case under user

additions and deletions. Two models are considered:

In the case that all drones can communicate, the au-

thors use a known PTAS for the static version (Jin

et al., 2018; Khuller et al., 2014) to solve the dy-

namic version with approximition ratio 4 for disks and

7 for squares, needing O(n log n) initiliaztion time and

O(logn) update time. For the second model, drones

can only communicate if they are within a distance of

COM

, a constant parameter, under the `

norm. As-

suming the number of drones is increased from m to

O(m

√

m), the authors present an O(1)-approximation

algorithm dynamic case with the same running times.

10 CONCLUSIONS

The ﬁelds of WSNs and algorithms are constantly

evolving. Many advances have been made in recent

years, not only by solving old problems but also by

creating entire new ﬁelds of research. In this survey,

we presented some important challenges and current

developments in algorithmic problems with potential

applications in WSN systems. We also detailed pos-

sible further research paths that have the potential to

signiﬁcantly advance and improve current WSNs.

Particularly interesting are in our opinion the de-

velopments concerning interference reduction (Sec-

tion 4) and population protocols (Section 8). There

are many open interference-related problems which

are worth exploring, including minimizing receiver-

centric maximum interference as well as some open

questions concerning vertex and edge coloring in a

graph. Population protocols present an interesting ap-

proach to model a wireless system. We believe them

to have many practical applications left to explore.

In conclusion, there is a clear connection between

advances in algorithmic research and WSNs improve-

ments and we believe it should be further explored.

ACKNOWLEDGEMENTS

This work was supported by the Regensburg Center of

Energy and Resources (RCER). Further information

under www.rcer.de.

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