Average search time bound in cue based search strategy
AAverage search time bounds in cue based search strategy
Vaibhav Wasnik ∗ Indian Institute of Technology, Goa
In this work we consider the problem of searches that utilizes past information gathered duringsearching, to evaluate the probability distribution of finding the source at each step. We start with asample strategy where the movement at each step is in the immediate neighborhood direction, witha probability proportional to the normalized difference in probability of finding the source with thepresent position source finding probability. We evaluate a lower bound for the average search timefor this strategy. We next consider the problem of the lower bound on any strategy that utilitiesinformation of the probability distribution evaluated by the searcher at any instant. We derive anexpression for the same. Finally we present an analytic expression for this lower bound in the case ofhomogeneous diffusion of particles by a source. For a general probability distribution with entropy- E , we find that the lowerbound goes as e E/ . PACS numbers:
Introduction:-
Searching for a source that emits parti-cles is a problem that is quite ubiquitous. We see thisall the way from a bacteria searching for the source ofchemoattractants [1], to a robot figuring out the source ofa gas leak in a room [2]. Search time is defined as the timerequired to find the source by a searcher. This is simi-lar to the first passage time: the first time the searcherreaches the position occupied by the source. There is a lotof theoretical work done in this area [3]. One could clas-sify search strategies into two broad categories. Searcheswith cues and searches without cues. Searches with-out cues are reviewed in [4]. As has been stated there,searches with cues can also be split up into two kinds.One of them involves chemotatic strategies that assumea sufficient concentration of cues and another categoryof strategies that involve searching through informationcoming from sparse cues. Infotaxis [5] falls in the latercategory.A searcher moving through an environment of particlesemitted by a source has a history of hits at times t , .., t n at positions (cid:126)r ( t ) , ..., (cid:126)r ( t n ). These make up the cues thatprovide all the information from the environment. Thisinformation could be utilized in deciding a future direc-tion in many ways. One important quantity that couldbe measured is the probability of finding the source atany location in space. One could use Bayes theorem toevaluate this as P ( (cid:126)r | (cid:126)r ( t ) , ..., (cid:126)r ( t n )) = P ( (cid:126)r ( t ) , ..., (cid:126)r ( t n )) | (cid:126)r ) (cid:80) x P ( (cid:126)r ( t ) , ..., (cid:126)r ( t n )) | (cid:126)x ) (1)here P ( (cid:126)r | (cid:126)r ( t ) , ..., (cid:126)r ( t n ) is the probability of finding thesource at position (cid:126)r given hits at positions (cid:126)r ( t ) , ..., (cid:126)r ( t n )and P ( (cid:126)r ( t ) , ..., (cid:126)r ( t n )) | (cid:126)x ) is probability of hits happeningat positions (cid:126)r ( t ) , ..., (cid:126)r ( t n )) given the source is at position (cid:126)x . Infotaxis [5] utilizes this probability to evaluate theentropy of the source. The motion of the searcher at eachstep is in a direction in which the expected informationgain is a maximum. In [5] it was conveyed that evaluat- ing the search time analytically for a searcher undergoingInfotaxis, was difficult, given the complexity of the searchalgorithm. Given this issue, the question arises whetherit would be possible to evaluate the search times for aclass of cue based searches and any statement be madeabout certain universal features, such as lower bound onthese search times given certain constraints. In this workwe begin with a strategy that utilizes past cues to eval-uate the probability distribution of finding the source ateach step and where the searcher at each step moves inthe immediate neighborhood, with a probability propor-tional to the normalized difference in probability of find-ing the source with the present position source findingprobability. We then attempt to evaluate a lower boundon the search times in case of homogeous diffusion of par-ticles by a source. We then consider the problem of thelower bound on any strategy that utilities informationof the probability distribution evaluated by the searcherat any instant. We evaluate an analytical expression forlower bound in case of homogeneous diffusion of particlesby a source. For a general probability distribution withentropy - E , we find that the lowerbound goes as e E/ . Narrowing the source:-
Let us assume that the source emitting particles is lo-cated at the origin. A searcher moving through an en-vironment of particles emitted by a source has a historyof hits at times t , .., t n at positions (cid:126)r ( t ) , ..., (cid:126)r ( t n ). Wehave, P ( (cid:126)r | (cid:126)r ( t ) , ..., (cid:126)r ( t n )) =exp[ − (cid:82) t P ( (cid:126)r ( t (cid:48) ) | (cid:126)r )] dt (cid:48) P ( (cid:126)r ( t ) , ..., (cid:126)r ( t n )) | (cid:126)r ) (cid:80) x exp[ − (cid:82) t P ( (cid:126)r ( t (cid:48) ) | (cid:126)x )] dt (cid:48) P ( (cid:126)r ( t ) , ..., (cid:126)r ( t n )) | (cid:126)x ) (2)Here, P ( (cid:126)r ( t ) , ..., (cid:126)r ( t n )) | (cid:126)r ) is the probability of havinghits at positions (cid:126)r ( t ) , ..., (cid:126)r ( t n )) given the source is at po-sition r . The exponentials correspond to no hits happen-ing at the other locations along the trajectory. Becausethe hits are independent of each other and can happen a r X i v : . [ c ond - m a t . o t h e r] F e b at any time. We could write the above as P ( (cid:126)r | (cid:126)r , ..., (cid:126)r n )= exp[ − (cid:82) t S ( (cid:126)r ( t (cid:48) ) | (cid:126)r ) dt (cid:48) ] S ( (cid:126)r | (cid:126)r ) ...S ( (cid:126)r n | (cid:126)r ) (cid:80) x exp[ − (cid:82) t S ( (cid:126)r ( t (cid:48) ) | (cid:126)x ) dt (cid:48) ] S ( (cid:126)r | (cid:126)x ) ..S ( (cid:126)r n | (cid:126)x ) (3)Where above we (cid:126)r .. (cid:126)r n are just the positions in space,implying that the probability evaluations are only de-pendent on the positions in space where hits happen ir-respective of the time they happen. The S ( (cid:126)r | (cid:126)x ) is usedabove, to imply that the probability of having hits issimply the probability of having particles at location (cid:126)r assuming source is at (cid:126)x . This assumes that the searcherhas an analytical expression for how the particle distri-bution would be, given the source location.The probability that the hits happened at these posi-tions is simply e − (cid:82) t [ S ( (cid:126)r ( t (cid:48) ) | dt (cid:48) S ( (cid:126)r | (cid:126) ...S ( (cid:126)r n | (cid:126)
0) (4)Hence, the average probability of finding the source at (cid:126)r would be P ( (cid:126)r ) = (cid:88) trajectories (cid:88) n n ! (cid:88) r ,..r n e − (cid:82) t [ S ( (cid:126)r ( t (cid:48) ) | (cid:126)r )+ S ( (cid:126)r ( t (cid:48) ) | dt (cid:48) S ( (cid:126)r | (cid:126) ...S ( (cid:126)r n | (cid:126) × S ( (cid:126)r | (cid:126)r ) ...S ( (cid:126)r n | (cid:126)r ) (cid:80) x e − (cid:82) t S ( (cid:126)r ( t (cid:48) ) | (cid:126)x ) dt (cid:48) S ( (cid:126)r | (cid:126)x ) ...S ( (cid:126)r n | (cid:126)x ) (5)It is obvious that if our trajectory took an infinite timewe would have the best narrowing of the source location.Hence, the best possible average probability distributionpossible telling us the probability of locating the sourceat position (cid:126)r is P ∞ ( (cid:126)r ) = (cid:88) trajectories (cid:88) n n ! (cid:88) r ,..r n e − (cid:82) ∞ [ S ( (cid:126)r ( t (cid:48) ) | (cid:126)r )+ S ( (cid:126)r ( t (cid:48) ) | dt (cid:48) S ( (cid:126)r | (cid:126) ...S ( (cid:126)r n | (cid:126) × S ( (cid:126)r | (cid:126)r ) ...S ( (cid:126)r n | (cid:126)r ) (cid:80) x e − (cid:82) ∞ S ( (cid:126)r ( t (cid:48) ) | (cid:126)x ) dt (cid:48) S ( (cid:126)r | (cid:126)x ) ...S ( (cid:126)r n | (cid:126)x ) (6)Let us assume for illustrative purposes that S ( (cid:126)r | (cid:126)x ) = S ( | (cid:126)r − (cid:126)x | ). Also, let us assume that S is appreciableonly up to a distance L away from the source. One im-mediately see’s from the above expression that because ofpresence of terms like S ( (cid:126)r | (cid:126) S ( (cid:126)r | (cid:126)r ), implies that the av-erage probability distribution of finding the source eval-uated above is appreciable over a distance 2 L , as long as we are considering trajectories of lengths or order largerthan L . This implies that the probability distributionmeasured by the searcher will not narrow the source bet-ter compared to S . If we consider the limit in which t → t . This implies that the measuredprobability distribution by the searcher cannot narrowthe source better than S ( x ). Example Strategy:-
Let us consider a search strategyin which the probability to jump to a neighboring loca-tion is proportional to the difference in the probabilityof finding the source from its own location. The prob-ability for the searcher to jump to the nearest neighbor( x +2 dx, y ) on an average would go as β θ ( P ( x +2 dx, y ) − P ( x, y )) ( P ( x +2 dx,y ) − P ( x,y )) P ( x,y ) . P ( x, y ) is the average prob-ability of finding the source at position ( x, y ) that hasbeen evaluated by the searcher using the Baye’s theoremas talked in Eq.5. This could depend on the starting po-sition of the searcher. β is a rate at which this jumpshappen and Θ( x ) is defined as,Θ( x ) = (cid:40) , if x > . , otherwise . (7)Let us consider the average time to reach the source fromposition ( x, y ) as T ( x, y ). As derived in appendix0 = − P ( x, y ) − β ∇ T ( x, y ) · ∇ P ( x, y ) − βT ( x, y ) ∇ P ( x, y )+ αβ ∇ T ( x, y ) (8)For simplicity let probability distribution have radialsymmetry with the source located at r = 0. Then theabove equation simply becomes0 = − P ( r ) − β ∂T (¯ r ) ∂r ∂P (¯ r ) ∂r − βT (¯ r )[ ∂ P (¯ r ) ∂r + 1 r ∂P (¯ r ) ∂r ]+ αβP ( r )[ ∂ T (¯ r ) ∂r + 1 r ∂T (¯ r ) ∂r ] (9)For α = 0 the solution with T ( r = 0) = 0 is T ( r ) = − βrP (cid:48) ( r ) (cid:90) r xP ( x ) dx (10)As talked above, the probability distribution P ( x )would never be as localized near the source as S ( x ).In case we are considering homogeneous diffusion by asource at the origin, in two dimensions, the equilibriumparticle concentration at r goes as K ( r/l ). Hence thelower bound on search time simply is r s LB ( r s ) FIG. 1: LB ( r ) plotted against r for l = 1. We see that thelower bound goes exponentially as r at larger values of r . T ( r ) > LB ( r ) = − βrK (cid:48) ( r/l ) (cid:90) r xK ( x/l ) dx (11)This is plotted in fig.1. One can see that for largetimes the LB ( r ) increases exponentially with r . Thiswould be the lower bound even if α (cid:54) = 0, because α onlyadds randomness to the search and hence would increasethe search times. Generic lower bound:-
We can simply use the fact thatthe probability distribution evaluated by Baye’s theoremis not as concentrated near the source as S ( x, y ) to simplyevaluate a lower bound on search time as follows. First letus assume that the searcher knows that the source is lo-cated at the origin with a probability 1. Then the small-est time taken by the searcher to reach the source goesas r , the distance between the source and the searcher.In case the searcher instead knowns that the source islocated at two points (cid:126)x and (cid:126)x with probability p and p . Then, the smallest possible search time would simplygo as p | (cid:126)x − (cid:126)x s | + p | (cid:126)x − (cid:126)x s | where (cid:126)x s is the searchersposition. This is obvious because out of N possible mea-surements, the source is seen at N p times at (cid:126)x and N p times at (cid:126)x . One could extend this to say that fora source probability distribution P ( (cid:126)x ) as understood bythe searcher, the shortest time to reach the source on anaverage should go as (cid:82) d(cid:126)x | (cid:126)x s − (cid:126)x | P ( (cid:126)x ) .Since the fact that the probability distribution eval-uated by Baye’s theorem is not as concentrated nearthe source as S ( (cid:126)x ), the search time evaluated usingany strategy that utilizes the probability distributionas measured by a searcher could never be smaller than v s (cid:82) d(cid:126)x | (cid:126)x s − (cid:126)x | S ( (cid:126)x ) ( v s is the speed of the searcher, whichwe take to be equal to 1 below), which for S ∼ K ( r/l )is LB ( r s ) ∼ (cid:90) rdθdrK ( r/l ) (cid:113) ( r s − r cos θ ) + r sin θ = (cid:90) rdθdrK ( r/l ) (cid:112) r s + r − rr s cos θ (12)Now since1 (cid:112) r s + r − rr s cos θ = (cid:88) l =0 , ∞ r l r l +1 s P l (cos θ ) , r s > r = (cid:88) l =0 , ∞ r ls r l +1 P l (cos θ ) , r s < r (13)implies12 r s − r cos θ ddr s (cid:112) r s + r − rr s cos θ = (cid:88) l =0 , ∞ r l r l +1 s P l (cos θ ) , r s > r = (cid:88) l =0 , ∞ r ls r l +1 P l (cos θ ) , r s < r (14)Hence dLB ( r s ) dr s = (cid:90) r s (cid:90) π rdθdrK ( r/l ) (cid:88) l =0 , ∞ r s − r cos θ ) r l r l +1 s P l (cos θ ) + (cid:90) ∞ r s (cid:90) π rdθdrK ( r/l ) (cid:88) l =0 , ∞ r s − r cos θ ) r ls r l +1 P l (cos θ ) (15)For large values of r s , the second integral would con-tribute minisculely. Also majority contribution in firstterm would only show from the l = 0. Hence dLB ( r s ) dr s ≈ × π (cid:90) r s rdrK ( r/l ) × r s r s ≈ πl (16)As r s is made smaller, other contributions start appear-ing. However, we note that as r s becomes larger andlarger, the lower bound on search time goes simply as r s . This simply states the fact that as r s becomes large,the range over which the region of size l surrounding thesource looks like a point object to the searcher. This be-havior is seen by solving Eq.12 for l = 1 as plotted in fig.2. r s ( r s ) FIG. 2: LB ( r s ) plotted as a function of r s solving Eq.12 with l = 1. As can be seen at large values of r s > l we have anexpected lower bound going as r s . Also note that for r s = 0the lower bound on the search time is not zero. For small values of r s one could simply expand LB ( r s ) ∼ (cid:90) rdθdrK ( r/l ) (cid:112) r s + r − rr s cos θ = (cid:90) r dθdrK ( r/l ) (cid:114) r s r − r s r cos θ = (cid:90) ∞ (cid:90) π dθdrK ( r/l )( r s r − r s θ )= (cid:90) ∞ (cid:90) π dθdrK ( r/l )( r s sin θ r )= 2 π π l + 12 π πl r s = π l + . π lr s (17)which is the behavior for r s << l . Note that LB ( r s ) ∼ (cid:90) rdθdrK ( r/l ) (cid:113) ( r s − r cos θ ) + r sin θ = l (cid:90) rl dθd rl K ( r/l ) (cid:114) ( r s l ) + ( rl ) − rl r s l cos θ = l (cid:90) xdθdxK ( x ) (cid:114) ( r s l ) + x − x r s l cos θ (18)Hence all that matters is how r s compares to l . From fig.2we can see that when r s > l , the behavior of LB ( r s ) islinear. From Eq.16 we can see that slope of this line is4 πl . One can hence say that LB ( r s ) > π l πl r s (19) If we instead consider the lower bound on search timegiven a particular entropy of the source probability dis-tribution we have to minimize LB = (cid:90) πrdrdθrS ( r, θ ) − λ (cid:90) πrdrdθ ( S ( r, θ ) ln S ( r, θ ) + E ) − β ( (cid:90) πrdrdθS ( r, θ ) −
1) (20)We have assumed the searcher is located at r = 0. Here λ is the Lagrange multiplier that sets E to the entropyof the probability distribution S ( r, θ ). Minimizing w.r.t S ( r, θ ) gives r − λ (ln S ( r, θ ) + 1) − β = 0 (21)which solves to S ( r, θ ) = e r/λ − β/λ − (22) λ < (cid:90) πrdrdθ ( S ( r, θ ) ln S ( r, θ )) = − E → (2 π ) λe βλ − ( β − λ ) = − E (cid:90) πrdrdθS ( r, θ ) = 1 → e β/λ − (2 π ) λ = 1(23)which implies ( β − λ ) = − Eλ → ( βλ ) = 3 − E and λ = − π e E/ − Hence the lower bound is LB = (cid:90) πrdrdθrS ( r, θ ) = − π ) λ e βλ − = − λ = e E/ − π (24) Conclusion:-
In [5] the difficulty in evaluating thesearch time for Infotaxis was highlighted. and instead acalculation for a different search strategy, which does notutilize information about past hits, was presented. Theyevaluated the lower limit for search time for this strat-egy in certain limits as ∼ e E , where − E is the entropyof the probability distribution of finding the source. Wehowever have in this work we evaluated a lower boundon the average search time in a search strategy thatseeks to evaluate the probability distribution of findingthe source, given the information of past hits, such thatrate of jumps to a neibhouring site is proportional to thenormalized difference of evaluated probability of findingthe source with the present site of the searcher. Thislowerbound goes as the exponential of distance from thesource for large distances. We then provided an expres-sion for the lower bound for the search time for any cuesbased search strategy. For a general probability distri-bution with entropy - E , we showed that the lowerboundgoes as e E/ . We see that the lowerbound again goesas e E/ which is similar to e E in ref.[5], which was eval-uated for a non cue based search strategy for the limitin which the search time as well as entropy are muchlarger than 1. The similarity almost begs a conjecturethat for probability distributions that do not narrow thesource position well, cue based searches do not performappreciably better than non cue based searches. It wouldbe interesting to further explore this statement throughfurther research. Appendix:-
To simplify things, let us consider the sys-tem in one dimension. The final result can be easilygeneralized to higher dimensions. We have T ( x ) = − dt + T ( x + dx )[ β Π( x + dx ) + α ∆( x + dx )]+ T ( x − dx )[ β Π( x − dx ) + α ∆( x − dx )]+ T ( x )[1 − β (Π( x + dx ) − + Π( x − dx ) − + 2 α ∆( x ))](25)where Π( i ) = Θ( P ( x ) − P ( i )) ( P ( x ) − P ( i )) P ( x )Π( i ) − = Θ( − P ( x ) + P ( i )) ( − P ( x ) + P ( i )) P ( x )∆( i ) = 1 P ( x ) = P ( i )= 0 P ( x ) (cid:54) = P ( i ) (26)The eq. 25 simply states that we can reach point x fromany of its neighbors x + dx and x − dx , which subtractstime dt from times T ( x + dx ) , T ( x − dx ) to reach sourcefrom these sites. Each of the times T ( x + dx ) , T ( x − dx )are multiplied by the probabilities to make the jump from x + dx and x − dx to x respectively. The term multiplying T ( x ) on the RHS is simply the probability of not makinga jump to the neighbors x + dx , x − dx . α is the probabilityof making a jump randomly in case the neibhouring sitehas the same probability of finding the source as presentsite.This eq. 25 becomes0 = − dt + β [ T ( x + dx )Π( x + dx ) + T ( x − dx )Π( x − dx )] − T ( x ) β [(Π( x + dx ) − + Π( x − dx ) − )] + βαdx ∇ T ( x )(27)or0 = − dt + β [( T ( x ) + dx∂ x T ( x ))Π( x + dx )+ ( T ( x ) − dx∂ x T ( x ))Π( x − dx )] − T ( x ) β [(Π( x + dx ) − + Π( x − dx ) − )] + βαdx ∇ T ( x )(28)or 0 = − dt − β dx ∂ x T ( x )[ ∂ x P ( x ) P ( x ) Θ( P ( x ) − P ( x + dx ))+ ∂ x P ( x ) P ( x ) Θ( P ( x ) − P ( x − dx ))]+ T ( x ) β [(Π( x + dx ) + Π( x − dx ))] − T ( x ) β [(Π( x + dx ) − + Π( x − dx ) − )] + βαdx ∇ T ( x )(29)nowΠ( i ) − Π( i ) − = [Θ( P ( x ) − P ( i )) + Θ( − P ( x ) + P ( i )))]( P ( x ) − P ( i )) P ( x ) = ( P ( x ) − P ( i )) P ( x ) (30)hence0 = − dt − β dx ∂ x T ( x ) ∂ x P ( x ) P ( x ) − T ( x ) β [ ( P ( x + dx ) + P ( x − dx ) − P ( x ) P ( x ) ] + βαdx ∇ T ( x )(31)or 0 = − P ( x ) dt − β dx ∂ x T ( x ) ∂ x P ( x ) − T ( x ) β dx ∇ P ( x ) + βαdx P ( x ) ∇ T ( x ) (32)which becomes in higher dimensions0 = − P ( x ) − β ∇ T ( x ) · ∇ P ( x ) − βT ( x ) ∇ P ( x ) + βαP ( x ) ∇ T ( x ) (33)we have redefined βdx dt → β above. ∗∗