Random walk with long jumps for wireless ad hoc networks

The aim of these long jumps is reducing the spatial corre-lation among short term subsequent node selections, thus improving the search performance, namely the hitting time.. The paper s

Random walk with long jumps for wireless ad hoc networks

Roberto Beraldi*

Dipartimento di Informatica e Sistemistica, Università di Roma ‘‘La Sapienza”, via Ariosto 25, 00184 Roma, Italy

Article history:

Received 9 July 2007

Received in revised form 2 November 2007

Accepted 15 March 2008

Available online 26 March 2008

Keywords:

Random walk

Wireless networks

Probabilistic search algorithms

Random walk

a b s t r a c t

This paper considers a random walk-based search algorithm in which the random walk occasionally makes longer jumps The algorithm is tailored to work over wireless networks with uniform node distribution In a classical random walk each jump has the same mean length On the contrary, in the proposed algorithm a node may decide to double the expected jump length by increasing the nominal transmission power and picking a neigh-bor beyond the nominal range The aim of these long jumps is reducing the spatial corre-lation among short term subsequent node selections, thus improving the search performance, namely the hitting time

Two versions of the algorithm are studied, with and without lookahead A protocol for implementing each version is also proposed When there is no lookahead the proposed pro-tocol allows for a finer transmission power transmission regulation The paper studies, for three network topologies, the impact of the long jump probability on the hitting time and

on the average total power required before the target is found

Ó2008 Elsevier B.V All rights reserved

1 Introduction

1.1 Context of this study

Searching is a common problem arising in distributed

computer systems In general, a searching problem is to

be solved when an element in a given set (the search

space) needs to know the element(s) of the same set

enjoy-ing a known property Among the elements of the set a

neighbor relationship exits, so that the search space is best

modelled as a graph Searching is performed from the

‘‘in-side” of the space, i.e., the search is described as a

distrib-uted algorithm Instantiations of the above problem can be

recognized in several situations which include reactive

protocols for mobile ad hoc networks[14], service

discov-ery in service oriented architectures (SOA), e.g.,[6],

query-ing in sensor wireless networks, e.g., [17], and file

discovery in peer-to-peer architectures, e.g.,[7]

Depend-ing on the specific case, the property of the searched

ele-ment can be as simple as an IP address, like for routing

protocols, or it is given as a more sophisticated data

struc-ture; e.g., in the SOA paradigm, a tree encoded as an XML text file describing the service (its public interface, mes-sage types, bindings, etc.)

The approaches for addressing a search problem can be categorized as structured or unstructured.1In the first case, the search space has some form of deterministic organiza-tion For example, a subset of nodes may form a virtual back-bone and provide virtual access points for a search [13] More elaborated structures are implemented via overlay networks that are organized as a Distributed Hash Table (DHT), e.g., [9] In the unstructured approach the search space is not organized The lack of organization can be sim-ply inherited from the network topology, or even superim-posed via a random overlay network, [8,5] In this paper

we consider the problem of searching in a wireless network with no structure or overlay support Due to the lack of a structure, a search needs to explore the whole network Among the two opposite solutions, namely flooding and ran-dom walk, we guess that latter is more suitable As pointed out in[5], compared to flooding a random walk search has in fact the advantages of having a more fine-grained control of

1570-8705/$ - see front matter Ó 2008 Elsevier B.V All rights reserved.

* Tel.: +39 06 77274018.

E-mail address: beraldi@dis.uniroma1.it

1

We do not consider centralized solutions, e.g., central service directory, since not suitable for a dynamic distributed environment.

Contents lists available atScienceDirect

Ad Hoc Networks

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / a d h o c

the search space, a higher adaptiveness to termination

con-ditions and can naturally cope with failures or voluntary

dis-connections of nodes

1.2 Motivation and basic idea

The hitting time is the main performance metric of a

random walk-based search algorithm It is defined as the

average number of elements that has be visited, starting

from a given source, before a target is reached Having a

low hitting time is important, because this translates into

lower response time to searches and, potentially, lower

search costs, in terms of bandwidth and/or energy

requirements

To obtain a low hitting time, we propose a random walk

that occasionally makes long jumps; this is achieved

regu-lating the nodes’ transmission range A normal (short)

jump is performed by setting the transmission range to R

and picking one node at random among those at distance

r 6 R, whereas a long jump requires to set the transmission

range to R0> R and picking a node at distance R < r 6 R0,

i.e., from a ring The percentage of long jumps is

deter-mined by the (long) jumping probability q, which is a

pro-tocol’s parameter The basic idea is that shuffling long and

short jumps should help to explore the searching space

more effectively, thus reducing the hitting time of a search

(recall that any selection is blind) In particular, by

occa-sionally making long jumps the spatial correlations among

subsequent choices, i.e., the chances of visiting a same

node, is reduced To provide the reader with a rough

expla-nation of the idea, let’us consider a set of uniformly

distrib-uted nodes and assume that the random walk always

makes short jumps, see Fig 1a This figure shows two

nodes, A and B, visited in sequence by the walk If node B

sends the walk to any node other than the ones located

in the dashed area, then such a new node, say C (not shown

in the figure), can again send the walk to A because it’s near

enough to it Node C belongs to the dashed area with

prob-ability given by the ratio of the surface of the dashed area

with pR2, and then the event of reselecting A is determined

by such a ratio However, we can reduce such a probability

by allowing B to perform a long jump, seeFig 1b In this

case, in fact, C is in the A’s transmission range with

proba-bility given by the ratio of the dashed area with pðR02

 R2Þ, and this probability can be made lower than the previous

one by properly setting R0 Another advantage of using long

jumps is more intuitive Long jumps allow to move faster

in the network and this is useful when the target node is far from the source one

1.3 Contribution of this work

In this study the long jump length is twice the normal one, in expectation Doubling the jump length appears to

be a good compromise between the need of speeding up the search and avoiding a high increase in the transmission power, which – among other things – will increase inter-ference (collisions) among concurrent transmissions and reduce the network capacity Moreover, this assumption simplifies the analysis of the protocol The contribution

of the paper can be summarized as follows

The main contribution is a study about the relationship between frequency of double jumps and hitting time We have considered two types of random walks, characterized

by the lack or presence of lookahead The hitting time, as a function of the long jumping probability, is derived for three different topologies, namely a line, 2D grid, and random geo-metric graph (RGG) For the first topology we have derived

an analytical upper bound, while for the grid one we present numerical results based on Markov chains RGG are studied via simulations In wireless networks it is often important to reduce the total energy spent to run a protocol Although long jumps reduce the hitting time, the energy spent to per-form each long jump is higher than the one spent for a nor-mal jump The net effect of long jumps on the total energy spent to reach the target is then also studied in the paper The last contribution is the description of two distrib-uted and efficient implementations of the random walk algorithms These protocols can run directly atop the data link layers and are characterized by the presence or lack of look-ahead In particular, the implementation of the ran-dom walk with lookahead is suitable for searching objects whose description is short enough to fit into a single

pack-et, as opposite to the implementation of the random walk search with no lookahead, which however allows for a fi-ner power transmission regulation

1.4 Related works Random walks (RWs) have been studied as a query/ searching or gathering mechanism over ad hoc or sensor wireless networks and many variants of the basic algo-rithm have been proposed,[19] The aim of these variants

is to improve the overall effectiveness of the algorithm A first way to achieve this result is to provide the RW with

a memory of the visited nodes For example, in[2]Avin and Britto used a biased random walk that gives priority

to unvisited neighbors instead of choosing uniformly at random The strength of bias is a protocol’s parameter that can be regulated according to the required needs Since the random walk is forced to visit new parts of the network, the hitting time is reduced For a bias level equal to one,

a same node is never selected again, unless this is the only option; thus, the random walk aims at behaving as a self-avoiding walk (SAW), see[4]

Memory is also used in random walks with choice, RWC, proposed in[1] At each step of RWCðdÞ, instead of selecting

Fig 1 Basic idea for speeding up random walk using long jumps

Cons-tant jump length (left), variable length (right).

just one neighbor, the walk moves to the next node after

examining a small number d of neighbors sampled at

ran-dom Again, the random walk process is enriched with

memory In our work memory is not considered

Another strategy for reducing the hitting time is to

ex-ploit lookahead In this case the RW uses information

about the neighbors at distance L hops from the current

position For example, in the ACQUIRE protocol[18]L is a

tunable lookahead parameter to combine random walks

with controlled floods In our work, lookahead L Ử 1 is

con-sidered However, the technique we propose to reduce the

hitting time is orthogonal to lookahead

Still other strategies for improving the search

perfor-mance are based on concurrent RWs For example,

Shak-kottai in[16] has analyzed different variants of random

walk-based query mechanisms that include source and

sink-driven sticky searches Our solution can be extended

to multiple RWs as well

The core methodology employed throughout this paper

is borrowed from the a recent paper from Zuniga et al., see

[22], where it has been used to study a pushỜpull

mecha-nism for enhancing the performance of random walk-based

querying in heterogeneous sensor networks Our work is

also related to the aforementioned paper because both

works leverage heterogeneity to improve the performance

More precisely, it considers a sensor network with two

kinds of nodes, i.e., normal nodes with low communication

capability (low degree) and cluster head nodes with higher

communication capability (high degree) When an event is

generated at a node in the network, the event is forwarded

to a cluster head A query for that event is implemented as a

simple random walk The presence of cluster heads

intro-duces heterogeneity in the topology; and, this reintro-duces the

hitting time considerably Although not detailed in the

pa-per, for a reliable communication between a low and high

degree node to occur, the transmission power of both nodes

should be increased; thus, transmission power regulation

at each node is implicity assumed in the work There are

three main differences with our work First, our work

as-sumes heterogeneity in the behavior of each node, whereas

the paper assumes that a fixed subset of nodes, the cluster

heads, has different communication capabilities Second,

we search for an object that Ờ unlike the event Ờ cannot

be moved from a node to another; finally, in our study the

cost is not limited to the hitting time, but it also includes

the average power per hit

Other important applications of the RW are as a

sam-pling technique [3] and routing [5,11,17] However, the

goal of these random walks is different from ours

The rest of the paper is organized as follows The next

section describes the proposed protocol; Section3gives a

background on the used methods; Section4discusses the

performance results for the deterministic topologies and

Section 5 for random geometric graph Conclusions are

summarized in Section6

2 Proposed protocol

In the following, the terms device and node are used

interchangeably We assume that each device can vary

the transmission power in order to modify its transmission range In particular, the nominal transmission range, R, is covered using power Pmax 1 while the extended range,

R0> R, using power Pmax2 The neighbors of a node u vary according to the range The nodes located at distance r 6 R are the uỖs close neigh-bors, whereas the ones at distance R < r 6 R0are called the far neighbor of u The wireless link between two nodes is always bidirectional A node of the network stores un-iquely identified objects, so that a node is associated with two IDs: the network address, which is a low level ID used

to send a message, and a high level more abstract one, rep-resenting the stored objects For example, in the context of

a SOA, o can be a software service whose ID is a verbose text file describing the service in XML

The searching problem arises when node s is interested

in discovering the node currently hosting a given object o

To this end, s issues an asynchronous searchđObject : oỡ primitive that triggers a search for o Once the node t stor-ing o is reached, t is in charge of notifystor-ing s that it is the owner of o For example, t can use some routing protocol

to notify s or even trigger another search for s carrying the reply How the notification is actually performed is however out of the scope of our analysis

2.1 Protocol description

We consider two algorithms for the searchđỡ primitive, which are based on a random walk The first one Ờ Algo1 Ờ assumes that a node only knows the low level ID of the neighbors The second one Ờ Algo2 - assumes that a node

is also aware of their high level ID The pseudo-code is re-ported in Fig 2 A more detailed description of how to implement the algorithms is given later in this section Random walk is performed by a searching message m carrying o The random walk terminates when the node storing the object receives m, i.e., we assume that the searched object exists In Algo1, if the node receiving the message does not store the object, then with probability qơ1  q it sends the message m to a far [close] neighbor

at random

Algo2 exploits lookahead If an intermediate node, say i, decides to forward m via a long jump it first checks if a close or far node stores o If such is the case, i sends m to the node storing the object, otherwise it acts as in the pre-vious algorithm

We will now drill down to the implementation of the two search algorithms The key aspect is the use an effi-cient distributed selection protocol for the next node to visit We describe two protocol implementations The first one can be adopted for both algorithms and it is character-ized by two transmission power levels, namely Pmax 1 and

Pmax2, corresponding to the normal transmission range R and extended one, R0; the second protocol allows for a finer transmission power regulation, but it is only suitable for Algo1

2.2 Implementation description The key point when implementing the search primitive

is how the neighbor node is selected Perhaps the most

straightforward solution is for each node i to announce

it-self via a periodic beacon packet, sent at the maximum

power Pmax 2, i.e., range R0 A node j classifies i either as a

close or far node, according to the strength of the beacon

signal received.2Since j is aware of the ID of all the current

neighbors and their distance attribute, it can easily make the

right selection

Beside the clear drawback of sending beacons at an

ade-quate rate, this solution is not efficient to implement

look-ahead Beacons sent by the target node are in fact required

to carry the high level description of objects it stores, e.g.,

XML files.3Since beacons are sent at the maximum power

Pmax2 the energy drain out for their transmissions can

be-come quite high, especially when the node handles many

objects We now describe two distributed selection

proto-cols that leverage the broadcast nature of wireless

transmis-sions and do not require beacons

2.2.1 Solution with two fixed Power Levels (2PL)

This protocol is suitable to implement both algorithms,

Algo1 and Algo2

It is a variant of the classical RTS/CTS/DATA/ACK

mes-sage exchange protocol, in which DATA is sent first to

al-low, if necessary, lookahead (recall the high level ID is stored inside the packet which is going to be forwarded) The protocol works as follows

The selecting node, say i, sends the packet m by a broad-cast primitive, either at the power Pmax 1or Pmax 2(range R or

R0) The packet carries a flag D, indicating wherever the far (D Ử 1) of close (D Ử 0) nodes are allowed to process m On receiving m, a node j first determines if it is a far or close neighbor (this is achieved as explained before) If j is a iỖs close (far) neighbor and D Ử 0 (D Ử 1) j can process m This consists in storing m and scheduling the transmission of a short control packet, Request To Forward (RTF), after a ran-dom delay dt With the transmission of this control packet,

j candidates itself as the next node that has to forward m

As soon as i hears the first control packet associated to

m, say from node k, the node sends another control packet, clear to forward CTS, carrying k The aim of this new packet

is to inform that the selected node is k All nodes but k de-lete m and de-schedule the transmission of their RTS The strength of such a solution is that it allows for the look-ahead implementation If a node stores the searched ob-ject, the sends the RTS packet immediately, i.e., dt Ử 0 Note that when the transmission range is R0, both close and far nodes are allowed to reply

Time-outs are included to protect the above protocol against collisions For, example if no CTF/RTF are received after a given time interval, the node repeats the same ac-tions, resp sending an RTS or m Finally, we remark that the transmission of m by k acts as an implicit ACK for i 2.2.2 Solution with full Transmission Power Control (TPC) This protocol is only suitable to implement Algo1 and it allows for transmission power control, TPC It can be used when the ID of the searched object fits a single packet TPC means that the selecting node is not constrained to set its power either to Pmax1 or Pmax2; rather, it can also tune the power level to the minimum one required for a correct packet detection Such a finer transmission power regula-tion allows to reduce the average power required by the whole random walk The next section analyzes such a reduction in more details

The protocol is similar to the previous one, but it encompasses a simple variant of the power control MAC protocol (PCM), see[20] It works as follows

The selecting node i sends a short request to send con-trol packet (RTS), using either Pmax1 or Pmax2 (for range R and R0, respectively) The RTS packet carries the power le-vel Ptx used for its transmission and a flag D indicating wherever far (D Ử 1) or close (D Ử 0) nodes are enabled

to reply On receiving an RTS packet, a node j acts as fol-lows If j is a close neighbor and D Ử 1, then j ignores the control packet Otherwise, it schedules the transmission

of a clear to send packet (CTS) at the maximum transmis-sion power, after a random delay dt.4The CTS packet con-tains the power level, Pmin, with which i is required to send the data packet m, where PminỬP tx

P rxS0 Prxis the received signal strength and S0is the minimum power required for a correct signal detection

Fig 2 The random walk algorithms for the Searchđỡ primitive.

2

The received signal S j decays as S j Ử kP i =daij , where d ij is the distance

among i and j, 2 6 a 6 4 a the path loss parameter and k a constant (it can

be estimated when a signal S  S 0 is received, S 0 being the minimum value

required for a correct signal detection) Hence, j classifies i as a far node if

S j > R a

and close otherwise Usually, a Ử 2.

3

XML uses a text encoding; even the description of a simple service can

4

With the transmission of the CTS control packet, j

can-didates itself as the next node that has to forward m (this

is similar to the candidature done in the previous protocol

by the RTS packet) Node i sends m to the node, say k, form

which the the first CTS packet is received, using the

mini-mum power indicated in the packet.5 All nodes that hear

such a transmission de-schedule their CTS; any reception

of other CTS packets, if any, is ignored Node i uses the

trans-mission of a CTS packet received from k as an implicit ACK of

its previous transmission (if k is the target of m, k

acknowl-edges i explicitly) As for the previous protocol, the critical

operations are made guarded via timers

2.3 Energy consideration

Let Pjbe the average power required to perform a step

(jump) of the walk The mean energy spent by the whole

random walk before the target is reached can be written

as E ¼ hDTPj, where h is the average hitting time and DT

the physical time taken to perform a step, assumed equal

for each step (the hitting time is independent from the

powers used at each step) To compare Algo1 against Algo2

under the energy efficiency point of view we compute

Phit¼ E

DT¼ hPj, namely the average power per hit The

hit-ting time h will be derived in the next sections for the

deterministic topologies The average power per step, Pj,

depends on the actual protocol implementation used and

can be roughly computed as follows The energy spent in

both protocols, 2PL and TCP, is dominated by the energy

spent to send the data packet, m (the size of the control

packets is, in fact, usually very small compared to the data

packet size) We set the nominal transmission range to

R ¼ 1 and assume that no retransmissions due to collisions

occur; furthermore, the transmitting power P and the

max-imum range r are assumed to be related through the well

known decay function P  ra, where 2 6 a 6 4 is the path

attenuation factor Fading is not modelled in our analysis

Let P1[P2] be the average power required for making a

short [long] jump Then, Pj¼ ð1  qÞP1þ qP2 For the sake

of simplicity, we neglect border effects, i.e., the selecting

node is not at the edge of the topology

As far as 2PL is concerned, for all the topologies

consid-ered in the paper, P1¼ 1 and P2¼ R0a; thus, Pj¼ 1 þ q

ðR0a 1Þ For the TPC protocol, Pj is computed as follows

First, the line topology does not allow for power regulation

and thus Pj¼ 1 þ qðR0a 1Þ For the grid topology, P1¼ 1,

while

P2¼122a

þ12 ðpffiffiffi2

Þa

In fact, half of the far neighbors are at distance ffiffiffi

2 p Thus, the average power per step in the grid topology is

Pj¼ 1 þ qð2a1þ 2a22  1Þ

Finally, for the random geometric graph model, the

se-lected node is at distance ðr; r þ drÞ with probability

2prdr; hence

P1¼1p

Z 1 0

2prra

dr ¼a2

þ 2 Similarly, we have

pðR02 1Þ

Z R 0

1

2prra

ða þ 2Þ

ðR0aþ2 1Þ

ðR02 1Þ from which

Pj¼ð2  2qÞ ð2 þ aÞ þ

2q ð2 þ aÞ

ðR0aþ2 1Þ

ðR02 1Þ

2.3.1 Discussion

It is worth measuring the benefit of a transmission power control per se More precisely, let suppose we want

to implement a natural random walk (q ¼ 0) with no look-ahead and assume that RGG is a reasonable model for our network What is the energy we can save by regulating the transmission power? Since the hitting time is independent from the transmission power nodes use, the net reduction

in the energy spent by the whole random walk is given by the ratio of the power per step under the TPC with the power per step under 2PL, calculated for q ¼ 0 in the RGG case (seeTable 1) This ratio is a

2þa For a ¼ 2 the reduc-tion is then1and it raises to1for a ¼ 4

In the rest of this paper, we assume that Algo1 is imple-mented via TPC and Algo2 by 2PL

3 Background on the used methods The hitting time huv of a random walk executed on a graph G ¼ ðV; EÞ is defined as the expected number of steps before node v is visited starting from node u Before to present the results, we summarize the method used to de-rive the hitting time

3.1 Resistance method This method is borrowed from graph theory,[10], and it will be used to derive an upper bound on the hitting time for the line topology The commute time is by definition the sum Cuv¼ huvþ hvu, which represents the expected num-ber of steps in a random walk starting from u before node

v is visited and then node u is reached again Although in general huv–hvu, we restrict ourself to consider the special class of symmetric graphs, for which huv¼ hvu and thus

Cuv¼ 2huv(a graph is symmetric if it is both vertex transi-tive and edge transitransi-tive)

The reason of using the commute time Cuvis that it is related to the effective resistance ruvof the electrical net-work of G in which each edge represents a 1-X resistor

In particular, we have:

Table 1 Average power per step, R ¼ 1, a is path attenuation factor

 1Þ

 1Þ 1 þ qð2 a 1 þ 2a22  1Þ

0aþ2

1Þ

5

As explained in [20] , the sending node may periodically raises the

Cuv¼ 2jEjruv

Thus, for a symmetric graph:

huv¼ jEjruv

from which the problem of computing the hitting time is

reduced to the computation of ruv This method is valid

for a natural random walk To accommodate long jumps

as a natural behavior of the walk we will connect nodes

with multiple arcs If fact, the token performing the walk

follows one edge at random In this way, if muvis the

num-ber of edges connecting u to v the transition probability

from u to v becomes

puv¼ Pmuv

k2dðuÞ

muk

where d(u) is the set of neighbors of node u The transition

probability can be modified as long as the constrain

muk¼ mkuis satisfied, while the above resistance result is

still valid

3.2 Markov chains

This method is more general than the previous one It

exploits the definition of a random walk as a Markov chain,

[12] Let N ¼ jVj be the number of nodes of the network

and P a N  N transition matrix of a Markov chain, whose

entry pijis the probability that the walk moves from node

i to j Let now t be our target node and construct the matrix

Q from P by removing row t and column t The matrix

ðI  Q Þ1, where I is identity matrix, exists and it is called

the fundamental matrix The hitting time of state t starting

from i is the ith element of the column vector w

w ¼ ðI  Q Þ11

where 1 is a column vector all of whose entries are 1 The

numerically

4 Results for deterministic topology

In this section we compute the hitting time for the

pro-posed algorithm when it is executed on three different

topologies, namely a line, square grid and random

geomet-ric graph

4.1 One dimension case

In this topology n þ 1 wireless devices, numbered

0; 1; ; n, are deployed along a straight line, at distance

R from each other while the extended transmission range

is R0

¼ 2R, seeFig 3 Our aim is to calculate the maximum

hitting time h0nas a function of q, denoted hnðqÞ

Consider the graph with n þ 1 nodes such that i is con-nected to its one hop neighbors with one edge and to the two-hops neighbors with m edges, where m is such that

q ¼ m mþ1, seeFig 4a Unless for nodes 1 and n  1, the natu-ral random walk on such a muti-edge graph makes a long jump with probability 2m

2mþ2¼ m mþ1¼ q and a short one with probability 1

mþ1¼ p ¼ 1  q The random walk will in fact select one edge at random The probability of a long (short) jump from node node 1 to node 3 (0 or 2) is m

mþ2< q ( 2

mþ2> p) The same is valid for node n 1 This means that long jumps at such a node are emulated with a lower prob-ability This error becomes negligible as n increases Let now calculate the effective resistance, seeFig 4 We assume n ¼ 4k The multiple edges connecting 4i  1 to 4i þ 1 are eliminated so that the graph in (a) is transformed

in (b), which in turns is a series of elementary subgraphs (c) The effective resistance of the graph (b) is higher than the original one

an elementary graph First, we observe that the electrical network associated to the elementary graph can be drawn

as inFig 5a, where the unlabelled edges are 1-X resistors, and r is the resistance associated to the m multiple edges This resistance corresponds to m 1-X resistors in parallel, i.e., r ¼1

m¼1qq By applying the Kennelly’s Delta–Star transformation, the network is now transformed into

which is obtained exploiting the series/parallel resistor rules

It is now easy to see that ð2 þ rÞRAE¼ 2r þ 2r==ð2r þ r2

þ 2Þ;

where a==b denotes the effective resistance of resistors a and b connected in parallel We have

2r==ð2r þ r2

þ 2Þ ¼ 2rð2r þ r

2þ 2Þ 2r þ 2r þ r2þ 2: Thus,

RAE¼ 2r

2 þ rþ

ð2rÞðr2þ 2r þ 2Þ ð2 þ rÞðr2þ 4r þ 2Þ:

We have Fig 3 An example of linear deployment of 15 nodes The transmission

a

b

c

Fig 4 (a) Multi edge infinite linear graph (m represents the number of multiple links); (b) elimination of multiple links; (c) elementary recurrent subgraph.

2 þ r¼ 2

1  q

1 þ q

and

ðr2

þ 2r þ 2Þ

ðr2þ 4r þ 2Þ¼

1 þ q2

1  q2þ 2q;

from which

RAE¼ 4ð1  qÞ

1 þ 2q  q2:

Consider now k > 1 subgraphs in series The original

graph has mðn  1Þ þ n links, while its resistance is kRAE

Then:

hnðqÞ 6 h0nðqÞ ¼n4 ½mðn  1Þ þ n

2

1 þ 2q  q21 þ 2q  qnq 2

Note that for q ¼ 0 the analysis provides the exact hitting time of n2 Thus, the percentage reduction in the hitting time compared to a natural walk is 1 h n ðqÞ

n 2 > 1h0n ðqÞ

n 2 4.1.1 Hitting time under Algo1

walk with no lookahead The first plot (top, left) shows the hitting time for n ¼ 12 as a function of q The hitting time first decreases with q, and then it increases again This

Fig 5 Computation of the equivalent resistance of the electrical network Edges with no labels are 1-X resistors.

0

20

40

60

80

100

120

140

Jumping probability

Upper bound Markov

0 0.2 0.4 0.6 0.8 1

Jumping proabability

n=16,Bound n=16,Markov n=32,Bound n=32,Markov n=64,Bound n=64,Markov

0 20 40 60 80 100 120 140

Starting point

q=0 q=0.2 q=0.7

Fig 6 Performance of Algo1 Absolute hitting time for 13 nodes as a function of q (top, left), relative reduction in the hitting time as a function of q (top,

is due to the fact that when the packet falls in an odd

no-des,i.e., it occupies an even node and then makes a short

jump, it cannot reach the target unless it makes a short

jump again The value of q regulates the frequency of such

events – a switch from odd to even nodes and vice versa

occurs with probability 1  q Once the walk falls into an

odd node, the expected number of steps before the token

returns back to an even position is 1=ð1  qÞ, a penalty

on the hitting time that increases with q On the other

hand, having q high is beneficial because the random walk

moves faster towards to target The combined effect

ex-plains the variation of the plot

The relative reduction as a function of q is given in the

next plot (top, right) The line size is a parameter The

anal-ysis now provides a lower bound For q ¼ 1 the probability

to fall into an odd node is zero For any value of q ¼ 1  

such a probability is however higher than zero, affecting

the hitting time negatively This is more evident in this

plot, in which a discontinuity at q ¼ 1 is visible

The lost plot ofFig 6(bottom) shows the hitting time as

a function of the starting point of the walk Results are

ob-tained using Markov chains One can wondering whenever

making long jumps is always beneficial regardless from

where the walk starts The plot answer to this question

As expected the highest reduction in the hitting time due

to long jumps is obtained when the source is far from the

target since the walker move ‘‘faster” in the line The reduction decreases as the walk starts nearby the target

In this case in fact long jumps also helps the walk to move away from the target This behavior is clearly visible for

q ¼ 0:2 However, as q is further increased, starting from nearby nodes can even be worst than avoiding long jumps Odd positions are particularly penalized since the target cannot be reached before a short jump is made, an event that happens with probability 1  q For example, starting from node 11 and setting q ¼ 0:9 provides the highest hit-ting time

4.1.2 Hitting time under Algo2 Let now consider the case when lookahead is in place The presence of lookahead reduces the actual length of the line The target is in fact reached the one step after the random walk visits node n  1 Also, with probability

q, the walk moves from node n  2 to the target We can then exploit the previous bound on the maximum hitting time by simply reducing the line length of one unit The ex-act values are obtained via Markov chains

shows the hitting time as a function ofq Lookahead avoids that the walk remains stacked visiting odd nodes The hit-ting time thus always decreases with q In particular, for

q ¼ 1, the walker runs over 6 points and it hits the target

0

20

40

60

80

100

120

140

Jumping probability

Markov Upper bound

0 0.2 0.4 0.6 0.8 1

Jumping proabability

n=16 n=64

0 20 40 60 80 100 120

Starting point

q=0 q=0.2 q=0.7

Fig 7 Performance of Algo2 Absolute hitting time for 13 nodes as a function of q (top, left), relative reduction in the hitting time as a function of q (top,

one step after it has reached point 5 Thus, the hitting time

becomes 26ð¼ 52

þ 1Þ

The improvement over the natural random walk is

dis-played in the next plot (top, right) for different line sizes

Lookahead also allows the hitting time to be reduced

regardless the initial starting point, see the last plot

(bottom)

4.1.3 Power

func-tion of the jumping probability, for the two algorithms

The attenuation factor is a ¼ 2 and the number of edges

is n ¼ 12 Such a power is then equal to hðqÞð1 þ 3qÞ, where

hðqÞ is the hitting time for the jumping probability q (see

In Algo1 the power increases with q whereas in Algo2 it

decreases This means that for Algo2 the reduction in the

hitting time completely compensates the higher cost per

step of long jumps Note that for q ¼ 1 and n an even

num-ber, the hitting time of Algo1 is ðnÞ2¼n 2

4 On the other hand, the average power per step is 4 Thus, the total

power spent making always long jumps (q ¼ 1) is the same

of the total power spent making always short jumps

(q ¼ 0), i.e., n2 However, as soon as q becomes less than

1 the hitting time increases and, consequently, the average

power increases too

4.2 Two dimension case

To derive the hitting time for the grid topology, we

ex-ploit the Markov chain approach,[15] To this end we need

to define the transition probability matrix Points of the

grid are assigned coordinates w.r.t a Cartesian axis, with

origin at the bottom-leftmost point of the grid Let

s ¼ ði; jÞ be a grid point and ks1; s2k ¼ ji1 i2j þ jj1 j2j

the distance between s1and s2 They are:

ps

1 s 2¼

ð1  qÞ=d1ðs1Þ ks1; s2k ¼ 1

q=d2ðs1Þ ks1; s2k ¼ 2

8

>

>

ð1Þ

where diðuÞ is the number of u’s i-hops neighbors For

Algo2 they also include:

ps

1 s 2¼ 1q ks1; s2k ¼ 1; s2¼ t

ks1; s2k ¼ 2; s2¼ t



ð2Þ This approach is also used to obtain the exact numerical re-sults for the line topology

4.2.1 Hitting time under Algo1 and Algo2

a natural random walk (q ¼ 0) as a function of the jumping probability for the two algorithms, for different grid sizes L

A L  L grid contains ðL þ 1Þ2nodes The source is placed at ð1; L=2Þ while the target to ðL  1; L=2Þ With no lookahead the improvement increases until roughly q  0:8 and then

it decreases The reason is that since the distance between the source and the destination is L  2, i.e., an even num-ber, the walk needs to perform an even number of short jumps for the target being reached And, the probability

of such an event decreases with q On the other hand, long jumps alone reduces the hitting time because nodes at odd distance from the target are not visited, i.e., the searching space is halved The combination of these two effects ex-plains the presence of a maximum in the plot Note that for q ¼ 1, nodes at an odd distance from the target are never visited, i.e., the searching space is halved

The advantage of using lookahead is that the target is reached as soon as one of its four one hop neighbor is reached (one hop lookahead) Moreover, depending on q the random walk can sometimes even exploit two-hops lookahead (recall that when a two-hops target’s neighbor performs a long jump, the packet is delivered to the target) Thus, the lookahead capability increases with q As the grid size increases, lookahead becomes less effective The num-ber of target’s neighbors will in fact be lower compared to the total number of nodes in the network This explain why the improvement decreases as L increases As L ! 1 the lookahead advantage becomes negligible, so that the hit-ting time reduction is the same as Algo-1

start-ing point x, for L ¼ 10, the target placed at ðL  1; L=2Þ and the source at ðx; L=2Þ, q given as a parameter For a given q, the hitting time decreases with the starting point for both

0

50

100

150

200

250

300

0 0.2 0.4 0.6 0.8 1

Jumping probability

Algo1

Fig 8 Average power per hit, for the line topology; 13 nodes, a ¼ 2.

0 0.2 0.4 0.6 0.8 1

Jumping probability

6 X 6,

10 X 10

6 X 6, Lookaheaed

10 X 10, Lookaheaed

Fig 9 Reduction in the hitting time as a function of the jumping prob-ability, different L  L grids; source placed at ð1; L=2Þ target at ðL  1; L=2Þ.

protocols As expected, this reduction is however much

stronger in Algo1 (left plot) than Algo2 (right)

Moreover, in Algo1 when starting very close to the

tar-get, i.e., x ¼ 8, an increase in q is beneficial only until a

gi-ven value For example, the left side plot shows how the

hitting time for q ¼ 0:8 is higher than for q ¼ 0:2

4.2.2 Power

grid, starting point (1,5), target point (9,5) and a ¼ 2 Algo2

provides the lowest power per hit; its average power is

al-most constant The reduction in the hitting time then

var-ies with q as ð1 þ 2qÞ1(seeTable 1) Thus, we can reduce

the hitting time by keeping the total power constant

How-ever, for a > 2 the power will increase with the jumping

probability

5 Simulation results for the random geometric graph

The final set of results are given for a random geometric

graph For this kind of topology we have used simulations

A given number of points, n, are drawn at random inside a

square region of edge L Any two points at distance at most

R are connected via a short range wireless link, while two points at distance r, R < r < R0 are connected through a long range link A short [long] wireless link is used with probability 1  q½q

Without loss of generality, to calculate R0 we set R ¼ 1 The expected length of a short range link and then of a short jump is



n¼1p

Z 1 0

2pr2

dr ¼23 while the one of a long range link is

n0¼p 1

ðR02 1Þ

Z R 0

1

2pr2dr ¼23ðR

03

 1Þ

ðR02 1Þ:

by setting n0

¼ 2n we get R0 1:613 If we are observing the walk at a random instant of time, then its distance, given that a short jump occurs, varies of 2/3, whereas under a long jumps this variation is twice, i.e., 4/3

5.1 Results

We have studied Algo1 and Algo2 by lunching 1000 independent runs of the algorithms over 300 nodes, scat-tered at random inside a square of edge L ¼ ffiffiffiffiffiffi300

5

q (node density 5) The nominal transmission range of each node

is R ¼ 1; The source and the destination are placed at height L=2 and distance 1 from the left and right borders, respectively The extended range is R0 The average power per hit is estimated as the ratio of the total power used during all the walks with the number of random walks Algo1 uses the TPC protocol, while Algo2 2PL The path attenuator factor is a ¼ 2 The evaluation assumes an ideal transmission system with no collisions

5.1.1 Extending the nominal transmission range The first set of experiments are obtained for R0¼ 1:613

of q for Algo1 (left) and Algo2 (right) The innermost graph

is the average power per step estimated during the simula-tions, which matches the value of analysis (seeTable 1) The other internal plot reports the total power per hit In Algo1 the hitting time decreases only due to the beneficial

0

50

100

150

200

250

300

350

Starting point

q=0 q=0.2 q=0.8

0

50 100 150 200 250

Starting point

q=0 q=0.2 q=0.8

Fig 10 Hitting time as a function of the starting point Target at (L  1; L=2), Initial point at ðx; L=2Þ L ¼ 10 Algo1 (left), Algo2 (right).

0

100

200

300

400

500

600

Jumping probability

Algo1

Fig 11 Average power per hit Source at ð1; L=2Þ, target at ðL  1; L=2Þ,

a ¼ 2.