exoplanet research is now a rapidly maturing branch of astronomy .in particular , a number of recently discovered planets have masses of order the earth s and offer the possibility of studying other worlds much like our own .the initial analysis of the kepler mission data indicates that the small transiting planets are even more abundant than large ones which is consistent with the statistical trend for planets discovered by the radial - velocity method .moreover , candidates of nearly earth - sized exoplanets within the habitable zones of their host stars have been already claimed ( e.g. * ? ? ?* ; * ? ? ?nevertheless a mere detection of such a planet would be far from convincing evidence for habitability .  far richer information concerningthe habitability would come from the atmospheric and surface properties of such planets , which will be probed by photometric and spectroscopic observation of planets in optical and near - ir bands .the light in the bands is dominated by starlight scattered by the planets , and the wavelength - dependent reflectivity and absorption features could provide direct indications of the planetary surface and atmospheric properties .this is why a number of on - going and future efforts including seeds , tmt ( with a future instrument for direct imaging of earth - like planets , see * ? ? ?* ) , o3 , see - coast , spica ( e.g. * ? ? ?* ) plan to develop instruments for direct observations of exoplanets .until we have the first scattered light observation for an earth - like exoplanet , we may use the earth itself as a useful test - bed for future investigations as proposed by .one approach is to extract spatially integrated scattered light from earthshine .in addition to several clear absorption lines of biologically important molecules like h , o , o and ch(e.g .* ) , a marginal signature of the red - edge due to vegetation ( sharp increase of reflectivity at wavelength longer than 0.75 m , e.g. * ? ? ?* ) has been reported ( e.g. * ? ? ?* ; * ? ? ?* ; * ? ? ?* ; * ? ? ?* ) in an amount consistent with simulations ( e.g. * ? ? ?* ; * ? ? ?* ; * ? ? ?* ) .  another approach is based on remote - sensing observation of the earth , notably multi - band photometric data obtained with the epoxi spacecraft . performed principle component analysis ( pca ) in order to reconstruct surface features from the epoxi data .their pca inversion identified two major eigen modes , the `` red '' and the `` blue '' components , which they roughly interpreted as land and ocean , respectively .they attempted to map the red component on the planetary surface using the diurnal variation , and found that it roughly reproduces the land distribution of the earth along the longitudinal direction .we note here that their analysis neglects the effect of clouds on the diurnal light - curve since they assumed that the cloud cover fraction is constant at all time . also attempted a longitudinal mapping of land and ocean components from a single - band , instead of multi - band , simulated photometry using the difference in reflectivity between land and ocean . ( hereinafter paper i ) developed another reconstruction method of the planetary surface .in contrast to model - independent approaches such as pca , paper i approximates the planetary surface by a combination of four components : `` ocean , '' `` soil , '' `` vegetation , '' and `` snow '' .the fractional areas of the four components were fit to reproduce the simulated light - curves of multi - band photometry of the _ cloudless _ earth , and the method recovers the presence of ocean , soil and even vegetation reasonably well .the present paper extends the methodology and analysis of paper i in various respects : 1 ) we now include a `` cloud '' component both in computing simulated diurnal multi - band light - curves ( forward procedure ) and in reconstructing the fractional areas of the five components ( inverse procedure ) , 2 ) the simulated light - curves are based on the radiative transfer code _ rstar6b _ instead of employing paper i s single scattering approximation , and 3 ) the reconstruction procedure is applied to the epoxi data instead of simulated light curves .the rest of the paper is organized as follows .section [ s : epoxi ] describes the technical details of the epoxi observations on which the subsequent sections are based .section [ s : sim ] discusses our forward procedure for computing the simulated light curves using _rstar6b _ with input of land albedo data obtained with the modis ( moderate resolution imaging spectroradiometer ) onboard the earth observing satellites _terra _ and _ aqua _ plus cloud data from modis onboard the _terra_. our inverse procedure , _ i.e. _ the reconstruction of the fractional areas of different surface components , is detailed in section [ s : inv ] .we also consider six different combinations of assumed soil , clouds and atmosphere properties , and discuss the resulting systematic uncertainty of the reconstruction .section [ s : dis ] uses this improved methodology to investigate the effect of clouds and the limits they impose on the interpretations of planetary light curves .section [ s : conclusion ] summarizes our main conclusions .appendix [ ap : input ] supplements the description of pre - process of the input data , and appendix [ ap : aitoff ] derives the longitudinal map reconstructed from epoxi data via our inverse procedure .m ) is short pass , where the detector and optical system components limit short wavelength sensitivity to .the ` ir ' filter ( nominally 0.95 m ) is long pass , which is cut off at by ccd response and optical system components . dashed : ccd response function ., width=377 ]   ( purple line ) , ( blue line ) , ( light blue ) , ( green ) , ( red ) , ( brown ) , and ( black ) in the march data ( solid ) and june data ( dashed ) .the observational uncertainty is less than 0.1 % , which is negligible in this figure.,width=264 ]   ( purple line ) , ( blue line ) , ( light blue ) , ( green ) , ( red ) , ( brown ) , and ( black ) in the march data ( solid ) and june data ( dashed ) .the observational uncertainty is less than 0.1 % , which is negligible in this figure.,width=359 ]  one significant improvement over paper i of the present work is the use of direct observations of the earth s diurnal light curves , both as a validation of our forward procedure and as input for the inverse procedure .in particular we employ observations of the earth that were obtained during nasa s epoxi mission .the epoxi mission reused the deep impact spacecraft to carry out multi - band photometry of the earth in optical / near - ir .epoxi observations of the earth covered one whole day in each of march , may and june of 2008 .the may data include a lunar transit of the earth and thus introduce complexities that are beyond the scope of the present effort ; we therefore do not use the may data .figure [ fig : snapshot ] depicts the geometry of the earth during the march and june 2008 observing runs .the spacecraft observed the earth 76.7% illuminated at a phase angle of 57.7 in march , near the northern spring equinox , and 61.5% illuminated at a phase angle of 76.6 in june , near the northern summer solstice .the data effectively mimic observations of earth - twins near quadrature in exoplanetary systems because the distance between the spacecraft and the earth in march ( ) and in june ( ) is a few thousand times the earth s radius and thus represents a good approximation to the geometry of an observation from infinite range .we use photometry obtained with the deep impact hrivis instrument at one - hour intervals in each of seven bands that span the visible range : , , , , , , and .images obtained with the hrivis camera are processed automatically for bias - subtraction , flat - fielding , an absolute calibration .data acquired during the epoxi mission ( e.g. * ? ? ?* ) indicate that the pixel - by - pixel flat - fielding correction has become inaccurate by about 1% , with random distribution across the detector array .the epoxi earth observations combine signal from thousands of pixels per frame , reducing the significance of random deviations to negligibility .photometric signal is extracted from epoxi earth images by aperture photometry , summing over a circular extraction region centered on earth s disc .background signal is estimated from an annular region of similar surface area .the calibrated image units multiplied by the known solid angle per pixel yields collected intensity in units of w / m/ m .the intensity is corrected for changes in the range from earth to spacecraft , which varies slightly over the course of each 24-hour observing sequence and varies significantly between observations .variations in earth s heliocentric distance , while minor , are corrected similarly .measurement uncertainties in the photometry are very small , of order 0.1% , evaluated from the distribution of pixel values in the annular background used in the photometry .photometry of the moon , which did not alter its aspect during a set of epoxi measurements in may 2008 , yields an independent estimate of photometric uncertainty of about 0.05% after scaling for the relative diameter of earth and moon. actual variability of the earth s signal is a much greater source of variability in the measurements , creating a range of measurements that is approximately - 8% of the mean signal measured in each filter .this range of variability is defined by the limits of the central 68th - percentile of the photometric sample population , yielding `` one sigma '' confidence limits on the expected value to be determined from a single independent measurement .the complete data set and data - reduction procedures are described by .the hrivis filter functions are shown in figure [ fig : epoxifilter ] along with the ccd response function .optical components within the telescope and spectrograph further limit the extreme blue and red sensitivity of the shortest and longest wavelengths bands , respectively .  .points represent the colors of some representative earth - surface components ., width=453 ]  in figure [ fig : lc_epoxi ] , we display the epoxi data interpreted in terms of apparent albedo .we follow the definition of in , as the ratio of total scattered intensity to the intensity of the incident solar flux scattered by a lossless lambert sphere of the planet s radius , at the observed phase angle , where is the scattered intensity as a function of the wavelength , is the incident flux , is the planetary radius , is the phase angle , and is the phase function for scattering by a lambert sphere . note that it is 3/2 times the geometric albedo .the left panel of figure [ fig : lc_epoxi ] displays the photometry of 7 epoxi bands averaged over both of the observed days , with the standard deviation of hourly measurements indicated as error bars .the variability in the measurements does not result from finite observational precision , but instead is the actual variability of the earth s signal .as mentioned above , the observational uncertainty of the epoxi measurements results in negligible noise level ( 0.1% with no exposure time longer than 100 ms ) because the source ( the earth ) is extremely bright .the right panel of figure [ fig : lc_epoxi ] shows the diurnal variation of each color band in march ( solid ) and in june ( dashed ) ; this figure illustrates both the common patterns of and difference between the march and june data .these data are qualitatively consistent with the expected photometric variability of the earth , as first discussed by .the shape variations between the lightcurves in different filters suggest looking more closely at the systematic diurnal color variability .we define color quantitatively as where is the reflectivity in band , following the conventional astronomical practice .we select the and bands to measure the variation in the blue region of the spectrum at short wavelengths and the and bands to quantify the variation in the red region of the spectrum at long wavelengths .figure [ fig : colorcolor_24 - 46_epoxi ] displays the resulting color - color diagram of the diurnal light curves of the earth for the two days . the colors of some representative components as seen through clear atmospheric layers are plotted as points in figure [ fig : colorcolor_24 - 46_epoxi ] for reference .  unsurprisingly , the average colors of the earth are similar to the colors of clouds , but somewhat displaced due to the surface components .our goal is to develop a methodology to reconstruct the surface properties from the diurnal scattered light curves of exoplanets in habitable zones . however , before proceeding to this inversion , we choose to gain better insights into the problem and to validate our assumptions and models by carrying out forward calculations and comparing the results to the epoxi data . in particular , we aim to determine the accuracy and limitation of the models of surfaces , clouds , and atmospheres as well as to validate our simulation code .therefore , in this section we combine the models and empirical data of the surface and the cloud distribution so as to reproduce the diurnal light curves in multi - bands observed by epoxi .in addition to the forward calculation presented in paper i , numerous other groups have simulated the scattered light of the earth .basically the computation of the light - curves of the earth requires the integration of the scattered light over the illuminated and visible surface .we have to divide the surface into two dimensional pixels and sum up the contribution from each pixel by approximating it locally as a flat plane .we use latitude / longitude pixels as a compromise between computation time and accuracy .  unlike paperi , which neglects multiple scattering and molecular absorptions , we incorporate both effects by using the line - by - line radiative transfer code _rstar6b_. this code has been developed intensively as a tool for interpreting the remote - sensing data of the earth .it incorporates the detailed modeling of optical properties of the atmosphere and clouds .scattering processes in the earth s atmosphere are complicated .since the albedos of clouds are generally much higher than those of the planetary surface components , the broad distribution of the cloud optical depths crucially affects the light curve of the earth .atmospheric pressure is the key parameter that determines the optical depth of rayleigh scattering as well as the atmospheric features due to molecules , and the spectral features depend sensitively on the composition of the atmosphere .both affect the broad - band photometry considered here .  for simplicitywe adopt a representative set of parameters called the `` us standard '' model atmosphere , and assume that the model is valid everywhere .basically the model provides the molecular composition and the temperature - pressure relation as a function of the altitude .however , the total scattered light is not sensitive to the details of those profiles .in addition to the homogeneous `` us standard '' model atmosphere , our model includes an empirical inhomogeneous cloud distribution .the spatial distribution of clouds is characterized by the cloud cover fraction and the cloud optical thickness in each pixel .we use the cloud data set from terra / modis atmosphere level 3 product .these data are daily global maps based on modis onboard the earth observing satellite _ terra _ and are given in latitude / longitude pixels .thus we merge them into pixels to match the resolution that we use in the numerical code .the cloud data are not available for some locations , and we interpolate the data according to the procedure described in the appendix [ ap : input ] to fill these gaps .  although we take account of the spatial distribution of clouds , their optical properties are assumed to be everywhere the same .these optical properties depend on various parameters including the vertical profile , composition , phase ( liquid or solid ) , radii and shapes of particles . for simplicity , we assume that all of the clouds are located at the altitude of 3.5 - 6.5 km and are composed of pure water ; we neglect aerosols .the size distribution function of cloud particles is assumed to be  & ( 1.0\times 10^{-6}{\rm cm} \le r \le 3.0 \times 10^{-3}{\rm cm } ) \\  0  & ( { \rm otherwise } )  \end{array }  \right.\ ] ] where is the normalization factor .it also uses the size parameter cm and .clouds with these properties are hereinafter called `` middle clouds '' .the scattering model for the planetary surface is basically the same as adopted in paper i. however , there are small differences in our treatment of land bidirectional reflection distribution functions ( brdf ) .these are described below .the scattering properties of land are described by the rossi - li model , a parametrized brdf .while the model has three different terms ( isotropic , geometric , and volume terms ; see paper i for details ) , we now approximate the planetary surface as lambertian ( isotropic scattering surface ) , and neglect the geometric and volume terms . at the geometry of the epoxi observation , this approximation leads to 5% differences in reflectivity compared to the light curves simulated with all three brdf terms .we also follow paper i in using the brdf coefficients from `` snow - free gap - filled modis brdf model parameters '' , which is a spatially and temporally averaged product derived from the 0.05 resolution brdf / albedo data .therefore we again merge the data into pixels by simply averaging the brdf coefficients within each pixel ( from data points ) .the brdf for ocean pixels is described in appendix b of paper i ( see also * ? ? ?the key parameter in this brdf is the wind velocity 10 m above the surface , which we set to 4 m / s everywhere .note that this model neglects the sub - surface scattering . in reality , the sub - surface scattering results in a slightly bluer ocean color than our model ( e.g. * ? ? ?finally we take account of the snow cover on the planetary surface , neglected in paper i. the fraction of the snow - covered area is very small , but it may affect the total scattered light curve because of its high albedo . for this purpose , we use a data set of the monthly mean ice / snow cover fraction from international satellite cloud climatology project ( isccp ) .the data are provided in grids , and we interpolate them into pixels according to the nearest - grid - point assignment .we regard those pixels with snow cover fraction greater than 0.5 as snow pixels , and then assign the albedo of `` fine snow '' from aster spectral library ( shown in the left panel of figure [ fig : albedo_surface ] below ) ..grid system for radiance look - up table .[ cols="^,^,^",options="header " , ]   .histograms are the estimates of the ground truth identical to those in figure [ fig : bvls_june_cldml-10_soil - inceptisol_ave ] .vertical dashed lines connect the maximum and minimum values returned by the various models .the statistical variations produced by signal - to - noise ratio 10 are shown as vertical solid lines ., width=453 ]      as described earlier , our inversion may depend heavily on the adopted albedos for the selected surface components. it will not be easy to evaluate such systematic uncertainties associated with the model assumptions for future studies of exoplanets . however, we may gain some insights into the problem by comparing inversions of the epoxi data using the different models listed in table [ tab : invmodel ] .figure [ fig : wiperange_ave ] is the same as the left panel of figure [ fig : bvls_june_cldml-10_soil - inceptisol_ave ] but for models i to vi . now the symbols indicate the estimates returned by the different models , and the vertical dashed lines indicate the range of model - to - model variations .most of the model dependence can be qualitatively understood .model ii ( cross ) , in which soil b is assumed instead of soil a , overestimates the area for soil because the effective albedo of soil b shows a lower - amplitude red portion of its spectrum ( fig [ fig : albedo_surface ] ) .models iii ( asterisk ) and iv ( open square ) assume different altitudes for cloud layer .the difference in cloud altitude changes the absorption depths ( fig [ fig : effalbd_cld ] ) , and therefore the estimated areas of all the components are affected accordingly . if we adopt model v ( plus ) with clouds , which have a reflectivity smaller than those with , we end up with a larger estimate of the cloud area . again , the estimation of all of the surface components is also affected by this change .model vi ( open circle ) assumes no ozone in the atmosphere , and the lack of ozone s broad absorption at decreases the slope of all albedo models at short wavelengths . as a result, the area of ocean is estimated to be larger to compensate it .  for reference we also plot the model i result with s / n=10 in figure [ fig : wiperange_ave ] .the corresponding error bars for the fractional area are similar to the variation among different models considered here .this implies that the model - dependent systematic errors are not dominant for an observational s / n of order 10 .this conclusion is also consistent with the result for the diurnal light curves shown in figure [ fig : wiperange ] .the shaded region in each panel of this figure indicates the range of the estimates among different models .again the basic features are similar to those shown in figure [ fig : bvls_june_cldml-10_soil - inceptisol ] ; while the amplitudes of the estimated area vary fairly significantly from model to model , the overall variation patterns are relatively stable .two of the minor surface components , `` soil '' and `` snow '' , are difficult to identify robustly , but the major components , `` cloud '' and `` ocean '' , are detectable in all the models .finally the presence of `` vegetation '' may be detectable at some phases without being confused with the other components , probably due to its conspicuous `` red edge '' feature .   of different models as a function of time .the curve labeled `` reference '' assumes ocean , soil a and middle clouds of optical thickness 10 but no vegetation component .the curve labeled `` ref '' means that the model omits clouds but otherwise is identical to the reference model , and the other labels are to be understood in the same manner ., width=377 ]  so far we have shown that our inversion procedure can identify ocean and clouds reasonably well but that the other components are not securely identified .nevertheless , we may ask if an additional , unknown surface component is required to achieve a satisfactory fit . this is a retreat in a sense butmay be a less model - dependent approach .  for this purpose ,we compute and compare the goodness - of - fit to the observed data with and without specific surface types .figure [ fig : barchi ] shows per square - root of the degree - of - freedom for various models ( without vegetation for simplicity ) .the reference model plotted in black solid line assumes that the planetary surface comprises of clouds , ocean and soil a. the other curves correspond to the four variants of the reference model : models with clouds , ocean and soil b instead of soil a ( dot - dashed ) , with cloud and soil a without ocean ( dotted ) , with ocean and soil a without clouds ( long - dotted ) , and with clouds and ocean without soil ( short - dotted ) .the resulting residuals of may be roughly interpreted as the inverse of the signal - to - noise ratio required to detect the discrepancy between the model and the observation . since figure [ fig : barchi ] indicates that models with both ocean and clouds have values as low as 0.01 , the presence of an additional component would be revealed by the data only with s / n 100 . on the other hand ,models without either clouds or ocean show the difference in values of 0.020.03 at certain phases of the light curves .it provides a rough idea of the signal - to - noise ratio ( or the amount of the exposure time ) required to identify such surface components in future exoplanet observations .incidentally it is interesting to note that the time variation pattern of essentially follows that of the _ missing _ component ; for instance , the model without ocean shows the largest discrepancy at when almost all of the illuminated and visible area is covered by ocean ( fig.[fig : snapshot ] ) .our primary goal in developing an improved treatment of forward and inverse procedures is to allow us to examine the effects of clouds .the presence of clouds affects the recoverable fractional areas of the different surface components in at least three ways : blocking the direct reflection spectra from the underlying surface areas , the nonlinear modification of the underlying surface reflection spectra by semi - transparent clouds , and the additional and strong time variability they introduce .we examine these three effects in turn below .approximately half of the surface of the earth is covered by clouds with . to the extent that the corresponding regions are completely unseen, the total surface area that one can directly observe is reduced by the same amount .moreover the relative fractions of surface components contributing the light curves are modified because ocean and vegetation surfaces are more likely to be cloud - covered than soil ( see also * ? ? ?.    this effect of clouds may be evaluated more quantitatively with the forward calculation . to be specific ,we consider hypothetical earth - twins with different cloud cover fractions ; cloudless , reduced cloudiness , and realistic cloudiness cases .the realistic cloud case is based on the modis cloud data as described in section [ s : sim ] , while the reduced cloudiness case has roughly 50 % of the cloud cover fraction .the latter is created by multiplying both the observed cloud cover fraction and cloud optical thickness at each pixel by a uniformly distributed random number between 0 and 1 .the cloudless case neglects clouds entirely .the resultant color - color plot for and versus and is shown in figure [ fig : colorcolor_24 - 46 ] .the color trajectories over a diurnal period of the earth are shown by the three curves . for reference ,the colors of hypothetical planets with uniform surfaces of ocean ( plus ) , soil a ( open square ) , soil b ( painted square ) , vegetation ( asterisk ) , and snow ( cross ) with the us standard atmosphere are plotted as points .this figure indicates clearly why clouds dramatically limit our ability to recover the surface components from light curves . as the cloud cover fraction increases , the overall color approaches that of the clouds , and the amplitude of color variations is significantly reduced .the qualitative behavior is similar for color - color plots in other bands .it is , however , encouraging that the clouds do not much change the overall shape of the trajectory in color - color space .this implies that the recovery of surface components is still possible given sufficient photometric precision .our inverse procedure assumes that the observed total albedo spectrum of the earth can be approximated by a linear combination of five distinct spectra corresponding to clouds and the four surface components . in reality , the reflection spectrum of a particular surface component covered by clouds is not necessarily composed of a sum of the clouds and the surface spectra .  in order to show the difference more quantitatively , we use _rstar6b _ to compute the reflectivity of vegetation entirely covered by clouds of .as plotted in the left panel of figure [ fig : cldml-10+grass ] , the resulting spectrum is best - fit by a sum of clouds , vegetation and soil with fractional areas of 0.888 , 0.279 and 0.135 , respectively .the cloud - diluted red - edge feature of the vegetation is partially misidentified as soil in this case .the estimated total area exceeds unity so as to compensate for the lower reflectivities of vegetation and soil compared to clouds .we repeat the exercise for vegetation covered by clouds of ; the result is plotted in the right panel of figure [ fig : cldml-10+grass ] . in this case , the spectrum is decomposed as a sum of clouds , vegetation and ocean .these misidentifications are the result of our simplified treatment of the template surface albedos assumed in the inverse procedure . in reality , the wide range of cloud properties results in a variety of cloud spectra ( fig . [fig : effalbd_cld ] ) that are partially degenerate with the spectra of the other surface components . in principle , this issue could be addressed with a more elaborate and realistic model , but doing so is not justified without quite high signal - to - noise ratio observations .   case .it is best - fit by a combination of 0 % ocean , 0 % snow , 27.9% vegetation ( cross ) , 13.5% soil ( open circle ) , 88.8% cloud ( plus ) .right : the case .it is best - fit by a combination of 10.4% ocean ( open circle ) , 0 % snow , 80.5% vegetation ( cross ) 12.0% , 0 % soil , and cloud ( plus ) ., width=302 ]   case .it is best - fit by a combination of 0 % ocean , 0 % snow , 27.9% vegetation ( cross ) , 13.5% soil ( open circle ) , 88.8% cloud ( plus ) .right : the case .it is best - fit by a combination of 10.4% ocean ( open circle ) , 0 % snow , 80.5% vegetation ( cross ) 12.0% , 0 % soil , and cloud ( plus ) ., width=302 ]   if an exoplanet is an oblique rotator like the earth , the overall color of its surface may exhibit seasonal variations including those due to autumn leaves and snow cover . otherwise its surface components are taken to be independent of time .clouds can introduce light - curve fluctuations due to their spatial and temporal variations , but the good agreement of the diurnal epoxi light curves and our forward simulations based on a static cloud map ( see section [ s : sim ] ) indicates that the variability due to clouds is not important on a 24 hour time scale , as discussed in e.g. .since we may have to stack the data of exoplanets over a period of a few weeks or more to improve the photon statistics , the cloud variations on those longer time scales may well be more important .nevertheless , we may hope that the decomposition of the stacked light curves will simply provide us an average measure of each component because the light curves are essentially linear combinations of the observing days .in turn , the time variability among the light curves folded according to the diurnal period may act as a useful probe of cloud pattern and thus weather on an exoplanet . found that the apparent rotation period is shifted by the net movement of cloud distribution .also it might be possible to use the time variability to separate the contributions from clouds and those from surface components by e.g. looking for the day with the lowest reflectivity because it is likely to correspond to the least cloudy day ; generally surface components have lower reflectivity than clouds ( fig .[ fig : albedo_surface ] ) .we have presented a methodology to interpret future photometric data of earth - like exoplanets .the methodology is an improvement over the one presented in paper i in various ways .in particular we have incorporated the presence of clouds , utilized the radiative transfer code _rstar6b _ for the atmosphere and based the reconstruction on a more diverse set of template albedo spectra .in addition , we have validated the forward procedure using the epoxi observations of the earth s light curves as well as using them as an input for the inverse procedure .our main conclusions are summarized below .  1 .our forward procedure reproduces the epoxi light curves well using the spatial distribution of the cloud cover fraction and optical depth on each observing day .this scheme will be useful to predict the light curves of earth - like exoplanets , potentially detectable with future ground - based and space facilities .2 .  while our inverse procedure is approximate and model - dependent , the presence of ocean and clouds is routinely detectable from the diurnal light curves of an earth - twin with s / n 10 .however , s / n 20 is required for a high confidence detection .3 .  for the necessarily limited suite of models considered in section [ ss : modeldepend ] ,the systematic uncertainties associated with the assumptions and input template spectra are similar in amplitude to the statistical errors at s / n .thus , systematic uncertainties are not necessarily dominant for the signal - to - noise levels realistically expected for early exoplanet light - curve data .however , it is obviously possible for a terrestrial planet to have important surface components without counterparts on the earth ; if so , very serious systematic errors could ensue .an earth - twin exoplanet , s / n may enable us to detect the presence of components other than ocean and clouds in a fairly model - independent fashion .  as discussed in section [ s : dis ] , cloud cover adds complexity and uncertainty to the interpretation of planetary light curves .fortunately , clouds have their unique features such as the grey reflection spectra , time variability ( e.g. * ? ? ?* ; * ? ? ?* ) , the effect on thermal emission ( e.g. * ? ? ?* ; * ? ? ?* ; * ? ? ?* ) , the anisotropic nature of scattering ( e.g. * ? ? ?* ; * ? ? ?* ) , and polarization ( e.g. * ? ? ?* ) . in future work, it would be desirable to develop techniques for removal of clouds by making use of these characteristics in order to extract uncontaminated information on the surface .our identification of surface components depends on the differences in their reflection spectra .for example , the presence of ocean is identified as a `` dark '' components in our inversion procedure .although there might be other `` dark '' components than ocean , this approach will provide a useful check for the presence of oceans in combination with other indicators , such as `` ocean glint '' or polarimetry .  in this paper as well as paper i ,we have focused on the case of an earth - twin , but in order to further investigate the potential of photometric light curves for the characterization of terrestrial exoplanets , it will be necessary to consider models of planets unlike the earth in various aspects .we plan to investigate such models in future work .we thank atsushi taruya for useful discussion and comments , giovanna tinetti for providing the reflection spectra shown in figure 7 as a reference , and nicolas b. cowan for his advice concerning the epoxi data processing .we are grateful to openclastr project for providing the _ rstar6b _ package .thanks the staff of the department o f astrophysical sciences of princeton university for their warm hospitality during her visit , in particular dmitry savransky , tyler groff , and david spergel for their helpful discussions . y.f . and y.s .gratefully acknowledge support from the global collaborative research fund ( gcrf ) `` a world - wide investigation of other worlds '' grant , and the global scholars program of princeton university , respectively .is supported by jsps ( japan society for the promotion of science ) fellowship for research ( pd:22 - 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optical instrumentation engineers ( spie ) conference , vol .7731 , society of photo - optical instrumentation engineers ( spie ) conference series  , j. , et al .2009 , experimental astronomy , 23 , 357  , s. , turner , e. l. , schafer , j. , & ford , e. b. 2005 , astrobiology , 5 , 372  , m. 2009 , in american institute of physics conference series , vol . 1158 , american institute of physics conference series , ed . t. usuda , m. tamura , & m. ishii , 11  , g. , meadows , v. s. , crisp , d. , fong , w. , fishbein , e. , turnbull , m. , & bibring , j. 2006a , astrobiology , 6 , 34  , g. , meadows , v. s. , crisp , d. , kiang , n. y. , kahn , b. h. , fishbein , e. , velusamy , t. , & turnbull , m. 2006b , astrobiology , 6 , 881  , s. s. , butler , r. p. , rivera , e. j. , haghighipour , n. , henry , g. w. , & williamson , m. h. 2010 , , 723 , 954  , d. m. , & gaidos , e. 2008 , icarus , 195 , 927  , n. j. , smith , p. s. , traub , w. a. , & jucks , k. w. 2002 , , 574 , 430  , m. e. , kasting , j. f. , williams , d. m. , kane , t. j. , & philbrick , c. r. 2010 , , 723 , 1168the modis global cloud data available online require pre - processing to deal with two of their limitations : pixels with invalid cloud cover fraction data and those with invalid cloud optical thickness data .for the former case , we assign a _ cloud cover fraction _ equal to the average value of adjacent pixels with valid values , if any .we iterate this procedure to assign values to pixels which initially have no adjacent pixels with valid values .re - binning to resolution is accomplished by averaging .  in the latter case, pixels may be invalid for one of three reasons : ( 1 ) the cloud optical thickness is too small to measure ; these pixels are assigned a value of 0 . ( 2 ) the sun illuminates regions near the poles at too low angle of incidence to allow measurements of cloud optical thickness ; these pixels are assigned values via the same iterative averaging of adjacent pixels used for cloud cover fraction ( see above ) .( 3 ) a small fraction of the earth was not observed by modis on the days of the epoxi observations ; the resulting gaps were filled by the same iterative procedure .because the radiance is related to the cloud optical thickness in a non - linear way , it is inappropriate to average in the coarse - bining process .instead , we assign the value of cloud optical thickness in one of the sub - pixels to the whole pixel .diurnal variations may be translated into a longitudinal map or even a 2-dimensional map if combined with yearly variations . herewe map the diurnal variation of each component reconstructed from june data ( the left panel of figure [ fig : bvls_june_cldml-10_soil - inceptisol ] ) into a 7-slice model and display the result in figure [ fig : aitoff_june_cldml-10_soil - inceptisol ] .it is apparent that some of the major geographical features of the earth e.g. two oceans , the sahara desert , and the two largest land masses are properly located .

Abstract: as a test - bed for future investigations of directly imaged terrestrial exoplanets , we present the recovery of the surface components of the earth from multi - band diurnal light curves obtained with the epoxi spacecraft . we find that the presence and longitudinal distribution of ocean , soil and vegetation are reasonably well reproduced by fitting the observed color variations with a simplified model composed of _ a priori _ known albedo spectra of ocean , soil , vegetation , snow and clouds . the effect of atmosphere , including clouds , on light scattered from surface components is modeled using a radiative transfer code . the required noise levels for future observations of exoplanets are also determined . our model - dependent approach allows us to infer the presence of major elements of the planet ( in the case of the earth , clouds and ocean ) with observations having s / n in most cases and with high confidence if s / n . in addition , s / n 100 enables us to detect the presence of components other than ocean and clouds in a fairly model - independent way . degradation of our inversion procedure produced by cloud cover is also quantified . while cloud cover significantly dilutes the magnitude of color variations compared to the cloudless case , the pattern of color changes remains . therefore , the possibility of investigating surface features through light curve fitting remains even for exoplanets with cloud cover similar to the earth s .


leader election is one of the fundamental problems in distributed computing , first stated in .every node of a network has a boolean variable initialized to 0 and , after the election , exactly one node , called the _ leader _ , should change this value to 1 .all other nodes should know which node is the leader .if nodes of the network have distinct labels , then leader election is always possible ( e.g. , the node with the largest label can become a leader ) .however , nodes may refrain from revealing their identities , e.g. , for security or privacy reasons .hence it is desirable to have leader election algorithms that do not rely on node identities but exploit asymmetries of the network due to its topology and to port labelings . with unlabeled nodes ,leader election is impossible in symmetric networks  .a network is modeled as an undirected connected graph .we assume that nodes are unlabeled , but ports at each node have arbitrary fixed labelings  , where is the degree of the node . throughout the paper, we will use the term `` graph '' to mean a graph with the above properties .we do not assume any coherence between port labelings at various nodes .nodes can read the port numbers . when sending a message through port , a node adds this information to the message , and when receiving a message through port , a node is aware that this message came through this port .the topology of the network is unknown to the nodes , but depending on the specification of the problem , nodes may know some numerical parameters of the network , such as the number of nodes ( size ) , and/or the diameter .we consider two versions of the leader election problem ( le ) : _ strong le _ and _ weak le_. in strong le one leader has to be elected whenever this is possible , while all nodes must terminate declaring that leader election is impossible otherwise .weak le differs from strong le in that no requirement on the behavior of nodes is imposed , if leader election is impossible . in both cases , upon election of the leader , every non - leader is required to know a path ( coded as a sequence of ports ) from it to the leader .  in this paperwe investigate the time of leader election in the extensively studied model . in this model ,communication proceeds in synchronous + rounds and all nodes start simultaneously . in each roundeach node can exchange arbitrary messages with all its neighbors and perform arbitrary local computations .the time of completing a task is the number of rounds it takes . in particular ,the time of weak le is the number of rounds required by the last node to elect a leader if leader election is possible , and the time of strong le is the number of rounds required by the last node to elect a leader if this is possible , and to terminate by declaring that this is impossible , otherwise .it should be observed that the synchronous process of the model can be simulated in an asynchronous network , by defining for each node separately its asynchronous round in which it performs local computations , then sends messages stamped to all neighbors , and waits until getting messages stamped from all neighbors . ( to make this work , every node is required to send a message with all consecutive stamps , until termination , possibly empty messages for some stamps . )all our results concerning time can be translated for asynchronous networks by replacing `` time of completing a task '' by `` the maximum number of asynchronous rounds to complete it , taken over all nodes '' .if nodes have distinct labels , then time in the model is enough to solve any problem solvable on a given network , as after this time all nodes have an exact copy of the network .by contrast , in our scenario of unlabeled nodes , time is often not enough , for example to elect a leader , even if this task is feasible .this is due to the fact that after time each node learns only all paths of length originating at it and coded as sequences of port numbers .this is far less information than having a picture of the radius neighborhood .a node may not know if two paths originating at it have the same other endpoint or not .it turns out that these ambiguities may force time much larger than to accomplish leader election .we show that the time of leader election depends on three parameters of the network : its diameter , its size , and its _ level of symmetry _ .the latter parameter is defined for any network ( see section 2 for the formal definition ) and , if leader election is feasible , this is the smallest depth at which some node has a unique view of the network .the view at depth from a node ( formally defined in section 2 ) is the tree of all paths of length originating at this node , coded as sequences of port numbers on these paths .it is the maximal information a node can gain after rounds of communication in the model .it turns out that the time of leader election also crucially depends on the _ knowledge _ of parameters and/or by the nodes .on the other hand , it does not depend on _ knowing _ , although it often depends on this parameter as well .optimal time of weak le is shown to be , if either or is known to the nodes .more precisely , we give two algorithms , one working for the class of networks of given diameter and the other for the class of networks of given size , that elect a leader in time on networks with diameter and level of symmetry , whenever election is possible .moreover , we prove that this complexity can not be improved .we show , for any values and , a network of diameter and level of symmetry on which leader election is possible but takes time at least , even when , and are known .if neither nor is known , then even weak le is impossible .cccccc & & & & & + & &  & + & & &  & & & & + & & & & +  +  for strong le , we show that knowing only is insufficient to perform it .then we prove that , if only is known , then optimal time is .we give an algorithm working for the class of networks of given size , which performs strong le in time and we show , for arbitrarily large , a network of size , diameter and level of symmetry 0 , such that any algorithm performing correctly strong le on all networks of size must take time on .finally , if both and are known , then optimal time is .here we give an algorithm , working for the class of networks of given size and given diameter which performs strong le in time on networks with level of symmetry . in this casethe matching lower bound carries over from our result for weak le .table [ summary ] gives a summary of our results .the main difficulty of this study is to prove lower bounds on the time of leader election , showing that the complexity of the proposed algorithms can not be improved for any of the considered scenarios .the comparison of our results for various scenarios shows two exponential gaps .the first is between the time of strong and weak le .when only the size is known , strong le takes time on some graphs of logarithmic diameter and level of symmetry 0 , while weak le is accomplished in time for such graphs .the second exponential gap is for the time of strong le , depending on the knowledge provided .while knowledge of the diameter alone does not help to accomplish strong le , when this knowledge is added to the knowledge of the size , it may exponentially decrease the time of strong le .indeed , strong le with the knowledge of alone takes time on some graphs of logarithmic diameter and level of symmetry 0 , while strong le with the knowledge of and is accomplished in time , for such graphs .leader election was first studied for rings , under the assumption that all labels are distinct . a synchronous algorithm , based on comparisons of labels , and using messages was given in .it was proved in that this complexity is optimal for comparison - based algorithms . on the other hand, the authors showed an algorithm using a linear number of messages but requiring very large running time .an asynchronous algorithm using messages was given , e.g. , in and the optimality of this message complexity was shown in .deterministic leader election in radio networks has been studied , e.g. , in and randomized leader election , e.g. , in .many authors studied various computing problems in anonymous networks .in particular , characterize networks in which leader election can be achieved when nodes are anonymous . in authors study the problem of leader election in general networks , under the assumption that labels are not unique .they characterize networks in which this can be done and give an algorithm which performs election when it is feasible .they assume that the number of nodes of the network is known to all nodes .in the authors study feasibility and message complexity of sorting and leader election in rings with nonunique labels , while in the authors provide algorithms for the generalized leader election problem in rings with arbitrary labels , unknown ( and arbitrary ) size of the ring and for both synchronous and asynchronous communication . in the leader election problem is approached in a model based on mobile agents . characterizations of feasible instances for leader election and naming problems have been provided in  .memory needed for leader election in unlabeled networks has been studied in .to the best of our knowledge , the problem of time of leader election in arbitrary unlabeled networks has never been studied before .we say that leader election is _ possible _ for a given graph , or that this graph is a _ solvable graph _ , if there exists an algorithm which performs le for this graph .we consider two versions of the leader election task for a class of graphs : + weak le .let be any graph in class .if leader election is possible for the graph , then it is accomplished .+ strong le .let be any graph in class .if leader election is possible for the graph , then it is accomplished .otherwise , all nodes eventually declare that the graph is not solvable and stop .hence weak le differs from strong le in that , in the case of impossibility of leader election , no restriction on the behavior of nodes is imposed : they can , e.g. , elect different leaders , or no leader at all , or the algorithm may never stop .we will use the following notion from .let be a graph and a node of .  the _ view_ from is the infinite rooted tree with labeled ports , defined recursively as follows . has the root corresponding to . for every node , , adjacent to , there is a neighbor in such that the port number at corresponding to edge is the same as the port number at corresponding to edge , and the port number at corresponding to edge is the same as the port number at corresponding to edge .node , for , is now the root of the view from .see fig .[ fig.view ] for an example of a view .the following proposition directly follows from and expresses the feasibility of leader election in terms of views .[ possible ] let be a graph .the following conditions are equivalent : + 1 .leader election is possible in ; + 2 .views of all nodesare different ; + 3 .there exists a node with a unique view .in particular , the impossibility of deterministic leader election , when there are two nodes with identical views , is implied by proposition 4 from  , while corollary 1 from  implies that one node has a unique view if and only if all views of the nodes are different .  by denote the view truncated to depth .we call it the view of at depth .in particular , consists of the node , together with its degree .the following proposition was proved in .[ trunc ] for a -node graph , , if and only if .  for graphs of sublinear diametera better bound than that of proposition [ trunc ] has been proved by hendrickx  after the publication of the conference version of this paper ; for an -node graph of diameter , , if and only if , where .define the following equivalence relations on the set of nodes of a graph . if and only if , and if and only if .let be the partition of all nodes into equivalence classes of , and the corresponding partition for .it was proved in ( corollary 1 ) that all equivalence classes in are of equal size . in view of proposition [ trunc] this is also the case for , where . on the other hand , for smaller , equivalence classes in may be of different sizes .every equivalence class in is a union of some equivalence classes in , for .the following result was proved in ( lemma 1 ) .it says that if the sequence of partitions stops changing at some point , it will never change again .  [ stop ] if , then .  for a set ,let denote its size . by definition of partitions , if then .hence in order to see when partitions stabilize , it is enough to check when their sizes stabilize .this allows us to modify proposition [ stop ] and obtain proposition [ stopbysize ] below , which we will use in some of our algorithms .[ stopbysize ] if , then .  for any graph define its _ level of symmetry _ as the smallest integer , for which there exists a node satisfying the condition .  by proposition[ possible ] , for solvable graphs , the level of symmetry is the smallest for which there is a node with a unique view at depth . in general, the level of symmetry is the smallest integer for which some equivalence class in has size .define to be the smallest integer for which .we have  [ lambda ] .it is enough to show that if , for some nodes and , then .let be a node for which .consider nodes and for which .let be a shortest path ( coded as a sequence of ports ) from to .let be the node at the end of the same path starting at . is included in and is included in .it follows that and hence .suppose for contradiction that .let and be paths in views and witnessing to their difference .let be the reverse of path .then the concatenations and are paths witnessing to the difference of and , which gives a contradiction.  we fix a canonical linear order on all finite rooted trees with unlabeled nodes and labeled ports , e.g. , as the lexicographic order of dfs traversals of these trees , starting from the root and exploring children of a node in increasing order of ports . for any subset of this class, the term `` smallest '' refers to this order .since views at a depth are such trees , we will elect as leader a node whose view at some depth is the smallest in some class of views .the difficulty is to establish when views at some depth are already unique for a solvable graph , and to decide fast if the graph is solvable , in the case of strong le .all our algorithms are written for a node of the graph .in this section we show that the optimal time of weak leader election is , if either the diameter of the graph or its size is known to the nodes .we first give two algorithms , one working for the class of graphs of given diameter and the other for the class of graphs of given size , that elect a leader in time on graphs with diameter and level of symmetry , whenever election is possible .our algorithms use the subroutine to exchange views at different depths with their neighbors .this subroutine is detailed in algorithm [ alg : com ] .* let * be the view of node at depth + * send * to all neighbors ; + * foreach * neighbor of + from   algorithm [ alg : knowndiameter ] works for the class of graphs of given diameter .  * for * * to * * do * + compute ; + * repeat * + ; ; compute + * until * + the set of nodes in having the smallest + elect as leader the node in having the lexicographically smallest path from   [ weak - known - diam ] algorithm [ alg : knowndiameter ] - wle - known - diameter( ) - elects a leader in every solvable graph of diameter , in time , where is the level of symmetry of the graph .all nodes of the graph find ( i.e. , the number of different node degrees in the graph ) after rounds , and then they find sizes of consecutive partitions , for ( by counting the number of distinct views at depth ) . at this timethe exit condition of the `` repeat '' loop is satisfied .all nodes stop simultaneously and elect a leader . since the graph is solvable , by the definition of and in view of propositions [ possible ] and [ stopbysize ] , all elements of the partition are singletons ( recall that equivalence classes in , and thus in , are of equal size , and in solvable graphs ) .hence all nodes in correspond to the same node in the graph and consequently all nodes elect as leader the same node .all nodes stop in round , which is at most by proposition [ lambda ] .algorithm [ alg : knownsize ] works for the class of graphs of given size . ; ; + * while * * do * + ; + * to * * do * + the set of nodes in at distance at most from ( including ) + the number of nodes in with distinct views at depth + max + compute and + the set of nodes in having the smallest + elect as leader the node in having the lexicographically smallest path from + * while * * do * + ; //allow late nodes to terminate their protocol !  [ weak - known - size ] algorithm [ alg : knownsize ] - wle - known - size - elects a leader in every solvable graph of size , in time , where is the level of symmetry of the graph .  consider a node .after rounds , node gets a view that contains views , for all nodes . since the graph is solvable , all views different , and hence exits the first `` while '' loop after seeing different views at this depth . at this point reconstruct an isomorphic copy of the graph and hence compute and .the election rule is as in algorithm [ alg : knowndiameter ] .the second `` while '' loop guarantees that every node will be able to get and hence will exit the first while loop and elect the same leader .  in order to show that algorithm [ alg : knowndiameter ] is optimal for the class of graphs of diameter and algorithm[ alg : knownsize ] is optimal for the class of graphs of size we prove the following theorem .it shows a stronger property : both above algorithms have optimal complexity even among weak le algorithms working only when all three parameters , and are known .[ lower - weak ] for any and any , with , there exists an integer and a solvable graph of size , diameter and level of symmetry , such that every algorithm for weak le working for the class of graphs of size , diameter and level of symmetry takes time at least on the graph and , because the latter condition , for solvable graphs , means that there is a node of a unique degree , contradicting the requirement , i.e. , having a clique of at least 2 nodes . ]  before proving theorem [ lower - weak ] we present a construction of a family of complete graphs ( cliques ) that will be used in the proof of our lower bound .the construction consists in assigning port numbers . in order to facilitate subsequent analysiswe also assign labels to nodes of the constructed cliques .we will use induction to describe the construction of .cc ( a ) &   +  +  + ( b ) & +  +  + ( c ) & +  +  + ( d ) &   is the single node graph . is a 4-node clique with port numbers defined as follows . for a node , we say that edge is _ incident _ to , if the edge corresponding to port at node corresponds to port at some node .nodes in are uniquely identifiable by the set of their incident edges . belowwe assign distinct labels to the four nodes of the clique depending on the sets of their incident edges ( see fig .[ fig.q2q3].a ) :  * set of edges , ,  corresponding to label ; * set of edges , ,  corresponding to label ; * set of edges , ,  corresponding to label ; * set of edges , ,  corresponding to label .we additionally assign colors 0 , 1 , and 2 to edges of as follows : edges and get color 0 , edges and get color 1 , and edges and get color 2 . is constructed starting from two disjoint copies of clique as follows .denote by one of the copies and by the other one .  for each node in , we denote by its corresponding node in .ports 3 , 4 , 5 , and 6 are used to connect each node in to all nodes in to construct an 8-node clique .we will use two types of edges ._ monochromatic edges_  will have port 3 or 4 at both endpoints , while _ skew edges _ will have port 5 at one endpoint and port 6 at the other endpoint .for monochromatic edges we call _ color _ the port number at both their endpoints .we will also consider as monochromatic the edges in .more precisely , edges of color 3 connect nodes , , , and . edges of color 4 connect nodes , , , and .  notice that until now , this construction results in a graph for which each node has exactly one other node with the same view , e.g. , and have the same view .uniqueness of all views at depth 2 is guaranteed in clique by the definitions of the skew edges .in particular , skew edges connect nodes :  * , , , and with port 6 at nodes , and and port 5 at nodes , and ; * , , , and with port 6 at nodes , and and port 5 at nodes , and .this concludes the construction of ( see fig . [fig.q2q3].b ) .  a node in whose outgoing port 6 leads to a node receives as its label the concatenation of the labels of nodes and in their respective copies of ( removing all overlines ) .the complete labeling of nodes in is : , , , , , , , and .  for , the clique is produced starting from disjoint copies and of clique as follows .the set of nodes of the clique is the union of the sets of nodes of and .the skew edges connecting nodes of and will have port numbers , while the monochromatic edges will use the same port , for  ] , any node is connected to some node by a monochromatic edge of color .the label of a node in is given by the concatenation of the labels of the node in ( respectively in ) corresponding to and its neighbor in ( respectively ) connected by the skew edge incident to .while rule 1 is constructive , rule 2 is not .however , a port assignment for monochromatic edges of that fulfills rule 2 , can be obtained , for , by exploiting the already defined edges of as follows .if there is a monochromatic edge with color between nodes in , then nodes and nodes are connected by a monochromatic edge with color . if are connected by the skew edge incident to , then nodes and nodes are connected by monochromatic edges of color .now consider the skew edges from previous steps of the construction i.e. , skew edges with port numbers of graph , with ) .let be four nodes connected by these edges , where are nodes from copy in the construction of , and nodes are the corresponding nodes in copy .more precisely , let :  * edge incident to connect nodes and ; * edge incident to connect nodes and ; * edge incident to connect nodes and ; * edge incident to connect nodes and .finally , let and from correspond to nodes and , respectively , in copy of the construction of .then nodes , , , are connected by monochromatic edges of color ; nodes , , , are connected by monochromatic edges of color .this concludes the definition of monochromatic edges of , and thus completes its construction .extracts from graph are depicted in fig .[ fig.q2q3].c and [ fig.q2q3].d .a node of the clique is said to be of _ type _ , , , or if it is obtained from a node with label , , , or ( respectively ) in a copy of in the construction of .notice that the type of a node corresponds to the first letter of its label .consider a path defined as a sequence of consecutive monochromatic and skew edges ( i.e. , is uniquely defined by the sequence of outgoing port numbers ) .if a skew edge is traversed from node to by , then the _ color _ of in is given by its port number at .we call a _ distinguishing _ path for nodes and in , if it yields two different sequences of node types , , , and traversed proceeding along , depending on whether the origin of is or .[ clique ] the clique has level of symmetry .  we will prove a stronger statement . we will show that , for , if  ] and consider two nodes and whose labels have a common prefix but no common prefix .let  ] can not appear in any distinguishing path of minimal length between two nodes and whose labels have a common prefix of length up to , and therefore the depth at which views of such nodes differ in is the same as that of their corresponding nodes in .it remains to consider the case of nodes and , sharing a label prefix in . for any such pair of nodes , if $ ] is the color of a monochromatic edge connecting to , then another monochromatic edge of color connects to .any pair of such nodes have the same view ( at any depth ) in . on the other hand ,the skew edge incident to and that incident to lead , by rule 1 , to two nodes having a common label prefix of length , but no common prefix ( the same reasoning applies to the skew edges incident to and ) .these nodes have identical views up to depth in , but distinct at depth , by the inductive hypothesis .hence and have the same view up to depth and distinct views at depth in , which concludes the proof .after the publication of the conference version of this paper a more general version of lemma [ clique ] has been proved in : for any there exists a -node graph of diameter with level of symmetry .we will also use the following family of cliques . is the clique on 2 nodes , with port number 0 . is a clique on 4 nodes , where all nodes have the same set of incident edges . for , is a clique obtained from two disjoint copies of .the construction of mimics the construction of for all edges but the skew edges between nodes in and , that are replaced by monochromatic edges with port number at both endpoints , connecting nodes and , and by monochromatic edges with port number at both endpoints , connecting nodes and and nodes and .notice that , in graph , nodes and have identical views . nevertheless , in order to describe our construction , we artificially assign to nodes and the label they would respectively receive in the construction of graph .we finally define a family of graphs that allow us to prove theorem [ lower - weak ] . for any pair of integers , with and , the graph obtained using one copy of graph and copies of graph . the construction of graph proceeds as follows .arrange disjoint copies of and one copy of in a cyclic order .connect each node in a clique with all nodes in the subsequent clique .let be two nodes in and let be the color that would be assigned to edge edge in the construction of .assign port numbers to the edge connecting node in some clique to node in the subsequent clique , where has label and has label .a distinguishing path in is defined in the same way as in , which is possible , since in both graphs each node has type , , , or .  [ ring ]let and .let and be two nodes in and let be the maximum depth at which views of and in are identical .then the views at depth of nodes and belonging to the copy of in are identical .it follows from the definition that the length of a shortest distinguishing path for nodes and in is . by construction of , nodes and have labels with an identical prefix of length .suppose , for contradiction , that views of and at depth are different in .hence a shortest distinguishing path for these nodes has length in .let be a distinguishing path of length in for nodes and .we will show how to construct a distinguishing path of length at most in , which will give a contradiction .edge colors inside each clique are assigned according to the rules used for cliques .monochromatic edges with port number at both endpoints get color . for edges connecting nodes from different cliques in we assign as color the smaller of their port numbers .( notice that edges connecting nodes with the same label in their cliques will get the same color but these edges will be subsequently deleted in the construction of the distinguishing path . ) for the remaining edges ( i.e. , skew edges in the construction of a graph , for some ) , the color is defined by the outgoing port number , according to path .the distinguishing path is constructed as follows . consider all edges in , in reverse order .if has color then we remove it ; if has color such that then we replace it with an edge with color .this corresponds to replacing edges going from one copy of or from to another copy of or to with internal edges of some clique or of .deleted edges are those going to corresponding nodes of different cliques , hence their deletion corresponds to removing self loops .we show that path is distinguishing for nodes and in .  indeed , the only edge replacements that could modify the sequence of , , , and types yielded by paths and , when starting from nodes and , are those of edges with color and , as these edges may be replaced by skew edges of the last step of the construction of the clique , that are defined differently in cliques . for each such edge leading to a node with label ,the corresponding skew edge in leads to a node with label .hence , views in of nodes and are identical up to depth , as shown in the proof of lemma [ clique ] , and thus the path , of length at most , is distinguishing for and in , contradiction .* proof of theorem [ lower - weak ] : * lemma [ small ] below proves the theorem if either or are less than 2 . herewe give the general argument for .consider the clique antipodal to the clique in graph .consider nodes and from this clique .any distinguishing path for nodes and in must contain a node from .let be a minimum length distinguishing path for nodes and and assume without loss of generality that and are the first nodes from found along path , if starting from and , respectively . by lemmas [ clique ] and [ ring ] ,nodes and have the same views at depth and different views at depth in the graph .thus the minimum length distinguishing path in for and has length . since nodes and are at distance from and , respectively , the views at depth of and are identical .the following lemma proves theorem [ lower - weak ] in the case when either or are small , thus concluding the proof of theorem [ lower - weak ] .[ small ] for and any , and for any and , there exists an integer and a solvable graph of size , diameter and level of symmetry , such that every algorithm for weak le , working for the class of graphs of size , diameter and level of symmetry , takes time at least on the graph .consider the following cases . and .the required graph is depicted in fig . [ fig.lambda0 ] .the lower bound on the time of weak le is straightforward . and .the construction of the graph is analogous to the general construction of .we take one copy of clique and copies of clique in circular order . in copies of cliques we arbitrarily assign labels , , , and to nodes .the argument remains the same as in the general case . and .consider the clique constructed from cliques and by connecting each node with label in to each node with label in by an edge having the color that would be assigned to edge in the construction of .the obtained clique requires time for weak le .for strong leader election more knowledge is required to accomplish it , and even more knowledge is needed to perform it fast .we first prove that knowledge of the diameter is not sufficient for this task .the idea is to show , for sufficiently large , one solvable and one non - solvable graph of diameter , such that both graphs have the same sets of views .[ impos ] for any there is no strong le algorithm working for all graphs of diameter  .cc ( a ) & +  +  + ( b ) &  let .we will use the following family of graphs ( see also fig .[ fig.t - k].a ) .consider a torus .let and be the two rings of size in .for each node , let be its unique neighbor from in .the graph is obtained by connecting each node with nodes and in , where and are the two neighbors of in .an additional node of degree 2 , connected to a pair of nodes , , completes the construction of .the assignment of port numbers can be performed arbitrarily and is thus omitted in the construction .consider the following family of graphs ( see also fig .[ fig.t - k].b ) .the graph is obtained from two copies of the graph .let us call one copy , and , the other .similarly we call the copy in of node .the graph is obtained by removing all edges and from and , and replacing them with edges and , maintaining the same port numbers .  since the graph has exactly one node of degree 2 , it is solvable for any assignment of port numbers . by the construction of graph , each node from one copy of has the same view as the corresponding node from the other copy .it follows that graph is not solvable . to prove the theorem ,it is thus enough to show that graphs and have the same diameter .indeed , this will imply that no algorithm knowing only the diameter can tell apart some solvable graph from an unsolvable one .the diameter of graph is .indeed , is the distance from the unique node of degree two to the two nodes in rings and that are antipodal to the neighbors of .all other pairs of nodes are at distance at most .consider the graph .since no edge from the original torus in the copies of is modified in the construction of , we have that all pairs of nodes from the copy , as well as all pairs from the copy are within distance from each other . by using one edge node can reach any node in within steps ( similarly for ) .nodes and are at distance 3 in , for any .the two copies of the node of degree two are at distance 4 in , for any .hence , for any , all pairs of nodes in are within distance from each other . on the other hand , the distance in from a node in and a node in , where is the antipodal node of in , is equal to .hence the diameter of is , which concludes the proof .by contrast , knowledge of the size alone is enough to accomplish strong leader election , but ( unlike for weak le ) , it may be slow .we will show that optimal time for strong le is in this case .we first show algorithm [ alg : sleknownsize ] , working in time .  * for * * to * * do * + the set of nodes in at distance at most from ( including ) + the number of nodes in with distinct views at depth + * if * * then * report `` le impossible '' + * else * + the set of nodes in having the smallest + elect as leader the node in having the lexicographically smallest path from   [ strong - size ] algorithm [ alg : sleknownsize ] - sle - known - size - performs strong le in the class of graphs of size , in time .after rounds of communication every node has view , which contains views of all nodes .a copy of each node appears in the set . by propositions[ possible ] and [ trunc ] the graph is solvable if and only if there are exactly different views of nodes in .hence the report `` le impossible '' is correct when . if le is possible , the election rule is the same as in the weak le algorithms .hence all nodes elect the same leader in this case .our next result shows that algorithm [ alg : sleknownsize ] is optimal if only is known .compared to theorem [ weak - known - size ] it shows that the optimal time of strong le with known size can be exponentially slower than that of weak le with known size .indeed , it shows that strong le may take time on some graphs of diameter logarithmic in their size and having level of symmetry 0 , while weak le takes time , on any solvable graph of diameter and level of symmetry 0 .the high - level idea of proving that algorithm [ alg : sleknownsize ] is optimal if only is known is the following .for arbitrarily large , we show one solvable and one non - solvable graph of size , such that there are nodes in one graph having the same view at depth as some nodes of the other .[ lower - strong ] for arbitrarily large there exist graphs of size , level of symmetry 0 and diameter , such that every strong le algorithm working for the class of graphs of size takes time on graph .consider the following family of graphs , for .the construction of graph starts from a ring with nodes , where all edges have port numbers 0 and 1 at the endpoints .all nodes in will have degree 3 at the end of the construction .hence all edges that do not belong to the original ring will have port number 2 at both endpoints .pick an edge of the ring and call its first node .consider the whole ring as a segment of edges having edge as its first and last edge ( hence nodes and appear twice in ) .cc ( a ) & +  +  + ( b ) & +  let be the central edge of segment , where precedes in .connect the second endpoint of the first edge of ( i.e. , node ) to .connect the first endpoint of the last edge of ( i.e. , node ) to .let be the segment obtained from by removing the first edge , edge and all subsequent edges .let be the segment obtained from by removing all edges preceding edge ( included ) and the last edge .if and have length larger than 2 , proceed recursively on both segments .if and have length 2 , connect their central nodes .this concludes the construction of graph ( see fig .[ fig.g2].a for an example ) .for we define .since the length of a segment is given by and , we get the formula .hence every segment has a central edge if . the diameter of graph is at most . indeed , starting from node it is possible to reach any other node in using at most edges .views of all nodes in are identical , hence leader election in graph is impossible .consider the family of graphs , obtained , for any , from a ring with nodes , where all edges have port numbers 0 and 1 at the endpoints , as follows .add a new node and connect it to some node in the ring ; divide the remaining nodes of the ring into groups of 4 consecutive nodes and connect the first node of each group with the third one of the same group , as well as the second with the fourth . again , new edges joining nodes of the original ring have port number 2 at both endpoints .leave the remaining 2 consecutive nodes unchanged ( see fig .[ fig.g2].b for an example ) .suppose , for contradiction , that some algorithm correctly solves strong le for all graphs of known size , where , in time at most .in particular , for any , algorithm must report `` le impossible '' in graph in time at most . however , any node at distance at least from the unique node of degree 1 in graph has the same view , up to depth , as nodes in .thus node would report `` le impossible '' in graph and algorithm would fail to elect a leader in the solvable graph , contradicting the assumption that solves strong le for all graphs of known size , for .this contradiction implies that every algorithm solving strong le with known size must take time on graphs which have diameter and level of symmetry 0 .  * for * * to * * do * + compute ; + * repeat * + ; ; construct + * until * + * if * * then * report `` le impossible '' + * else * + the set of nodes in having the smallest + elect as leader the node in having the lexicographically smallest path from   we finally show that if both and are known , then the optimal time of strong le is , for graphs with level of symmetry .the upper bound is given by algorithm [ alg : sleknownsizeanddiameter ] - sle - known - size - and - diameter .the algorithm is a variation of algorithm [ alg : knowndiameter ] - wle - known - diameter - with an added test on the size of the partition after exiting the `` repeat '' loop .. the following result says that algorithm sle - known - size - and - diameter is fast .in fact , compared to theorems [ impos ] and [ lower - strong ] , it shows that while knowledge of the diameter alone does not help to accomplish strong le , when this knowledge is added to the knowledge of the size , it may exponentially decrease the time of strong le .[ all ] algorithm sle - known - size - and - diameter performs strong le in the class of graphs of size and diameter , in time , for graphs with level of symmetry .the time complexity of algorithm sle - known - size - and - diameter on solvable graphs follows directly from the analysis of algorithm [ alg : knowndiameter ] .the test on the number of sets in partition correctly identifies solvable and non - solvable graphs .indeed , in view of proposition [ stop ] , and it is a direct consequence of proposition [ possible ] that leader election is possible on an -node graph , if and only if , contains sets .proposition [ lambda ] guarantees completion time on non - solvable graphs , while uniqueness of the elected leader on solvable graphs follows from the analysis of algorithm [ alg : knowndiameter ] .this concludes the proof .since the lower bound in theorem [ lower - weak ] was formulated for known , and , it implies a matching lower bound for the optimal time of strong le with known and , showing that this time is indeed for graphs with level of symmetry .we established the optimal time of weak and strong leader election , depending on the knowledge of the size and of the diameter of the graph . for each scenariothe upper bounds were shown by proposing an algorithm , and matching lower bounds were proved .the optimal time turned out to depend on the knowledge of one or both of the parameters and .notice that the comparison of assumptions for our matching upper and lower bounds in various scenarios of weak and strong leader election shows that , while the level of symmetry may significantly influence optimal election time , the _ knowledge _ of this level is not important .indeed , for each task and for various combinations of knowledge of the size and/or the diameter , adding the knowledge of the symmetry level does not help .more precisely , weak le is impossible knowing neither nor , even if knowledge of , e.g. , is added , cf . . on the other hand , theorems [ weak - known - diam ] , [ weak - known - size ] , and [ lower - weak ]show that with known or , optimal time of weak le is , regardless of whether is known or not ( but the time itself depends on both and ) .the same is true for strong le .indeed , the impossibility result in theorem [ impos ] and the lower bound from theorem [ lower - strong ] hold even for known , while the upper bound from theorem [ strong - size ] does not require the knowledge of . forknown and , the upper bound holds without knowing by theorem [ all ] , while the lower bound carried over from theorem [ lower - weak ] holds even when is known.  it would be interesting to investigate other complexity measures of the leader election problem , such as the bit or message complexity of communication needed to accomplish it , in relation to the knowledge of the three parameters of the network used in our present study .p. boldi , s. shammah , s. vigna , b. codenotti , p. gemmell , and j. simon , symmetry breaking in anonymous networks : characterizations .4th israel symposium on theory of computing and systems , ( istcs 1996 _ ) , 1626 .             m.a .haddar , a.h .kacem , y. mtivier , m. mosbah , and m. jmaiel , electing a leader in the local computation model using mobile agents .6th acs / ieee international conference on computer systems and applications ( aiccsa 2008 _ ) , 473480 .

Abstract: we study the time needed for deterministic leader election in the model , where in every round a node can exchange any messages with its neighbors and perform any local computations . the topology of the network is unknown and nodes are unlabeled , but ports at each node have arbitrary fixed labelings which , together with the topology of the network , can create asymmetries to be exploited in leader election . we consider two versions of the leader election problem : strong le in which exactly one leader has to be elected , if this is possible , while all nodes must terminate declaring that leader election is impossible otherwise , and weak le , which differs from strong le in that no requirement on the behavior of nodes is imposed , if leader election is impossible . we show that the time of leader election depends on three parameters of the network : its diameter , its size , and its _ level of symmetry _  , which , when leader election is feasible , is the smallest depth at which some node has a unique view of the network . it also depends on the knowledge by the nodes , or lack of it , of parameters and . * keywords : * leader election , anonymous network , level of symmetry  [ section ] [ section ] [ section ] [ section ] [ section ] [ section ] [ section ] [ section ]  [ section ]