Alternating Runs of Geometrically Distributed Random Variables *

Transcription

1 JOURNAL OF INFORMATION SCIENCE AND ENGINEERING 7, 09-0 (0) Alteratg Rus of Geometrcally Dstrbuted Radom Varables * CHIA-JUNG LEE AND SHI-CHUN TSAI + Isttute of Iformato Scece Academa Sca Nakag, 5 Tawa E-mal: leecj@sscaedutw + Departmet of Computer Scece Natoal Chao Tug Uversty Hschu, 00 Tawa E-mal: sctsa@csectuedutw Ru statstcs about a seuece of depedet geometrcally dstrbuted radom varables has attracted some atteto recetly may areas such as appled probablty, relablty, statstcal process cotrol, ad computer scece I ths paper, we frst study the mea ad varace of the umber of alteratg rus a seuece of depedet geometrcally dstrbuted radom varables The, usg the relato betwee the model of geometrcally dstrbuted radom varables ad the model of radom permutato, we ca obta the varace a radom permutato, whch s dffcult to derve drectly Moreover, usg the cetral lmt theorem for depedet radom varables, we ca obta the dstrbuto of the umber of alteratg rus a radom permutato Keywords: alteratg rus, geometrc radom varables, asymptotc propertes, radom permutato, cetral lmt theorem INTRODUCTION Let X,, X be depedet geometrcally dstrbuted radom varables satsfyg for {,,, }, Pr[X k] p k-, k N, where p + Combatorcs of depedet geometrcally dstrbuted radom varables has caught some atteto recetly may areas cludg computer scece due to the applcatos to, for example, the skp lst [-6] ad probablstc coutg [7-0] Moreover, sce takg the lmt, wth probablty o(), there are o eual values X,, X ad each relatve orderg of values s eually lkely, we have that the model of geometrcally dstrbuted radom varables approaches the model of radom permutato [, ] Therefore, varous rus of geometrcally dstrbuted radom varables have bee vestgated [, -6] For rus of cosecutve eual umbers, Louchard [5], Graber et al [], ad Erylmaz [] studed the asymptotc propertes of the umber of rus usg a Markov cha approach, probablty geeratg fuctos, ad the exstg theory for m-depedet radom varables (that s, the two sets (X,, X r ) ad (X r+m+,, X ) are depedet), respectvely Aother le of research cosdered ascedg rus, whch s a seuece of cotguous strctly creasg umbers Louchard ad Prodger vestgated the asymptotc propertes of the umber of ascedg rus ad the logest ascedg rus Receved September, 009; revsed October, 009; accepted December, 009 Commucated by Ch-Je Lu * Ths paper was supported by the Natoal Scece Coucl of Tawa, ROC No NSC 97--E MY 09

2 00 CHIA-JUNG LEE AND SHI-CHUN TSAI usg a Markov cha approach [], where they metoed the possblty of applyg ther approach to study alteratg rus propertes For a permutato, a alteratg ru s a seuece of cosecutve ascedg or descedg umbers, whch the frst ru ca be ascedg or descedg [7] For example, s cosdered to have alteratg rus: 8,9,965, ad 7 Wth permutato array, we ca use alteratg ru for formato hdg [8] For formato hdg, we wat to hde some message a object, such as a mage data or a MIDI fle, wthout chagg the appearace of the object Usg a permutato array [9], whch cotas some permutatos of a fxed legth ad satsfes the property that the dstace (uder some metrc) of ay two permutatos ths array s ot too large, oe ca corporate the capablty of error correctg to formato hdg, but we ca oly hde few bts Fortuately, wth the help of alteratg rus, we ca crease the umber of hdg bts ad preserve the ablty of error correctg, where we ca smply treat the creasg ru of legth as ad decreasg ru of legth as 0 I ths paper, we wll study the mea, varace ad dstrbuto of the umber of alteratg rus Istead of usg the approach of [], we prove wth a elemetary method ad the cetral lmt theorem for depedet radom varables [0] Sce a radom permutato (Y Y Y ), Y s are mutually depedet, t s dffcult to derve the varace of the umber of alteratg rus Hece, we frst cosder the model of geometrcally dstrbuted radom varables, ad the by takg the lmt, we ca obta the result the model of radom permutato Sce there may be some cosecutve eual umbers a seuece of depedet geometrcally dstrbuted radom varables, we modfy the defto of alteratg rus to defe modfed alteratg rus I short, a modfed alteratg ru s a seuece of alteratvely strctly ascedg ad descedg umbers Lke alteratg rus, each modfed alteratg ru ca beg wth ascedg or descedg Moreover, we also cosder turg pots a seuece of umbers I a seuece x,, x, we call x a turg pot f max{x +, x - } < x or m{x +, x - } > x Oe ca observe that the umber of alteratg rus a seuece of radom varables s exactly the sum of the umber of modfed alteratg rus ad the umber of turg pots Therefore, to obta the asymptotc propertes of the umbers of alteratg rus, we ca study the asymptotc propertes of the umbers of modfed alteratg rus ad turg pots To make the above deftos clear, we cosder the followg example Let AR, MAR, ad T deote the total umber of alteratg rus, modfed alteratg rus, ad turg pots a seuece of depedet geometrcally dstrbuted radom varables, respectvely The cosder the seuece: 5 There are 6 alteratg rus: 5,,,,,; modfed alteratg rus: 5,, ; ad turg pots: turg pots ( ad ) the modfed alteratg ru 5 ad oe turg pot () the modfed alteratg ru Hece, AR 0 6, MAR 0, ad T 0 Note that AR 0 MAR 0 + T 0 Next, we descrbe our results Frst, we derve the followg three theorems about the meas ad varaces of MAR, T, ad AR a seuece of geometrcally dstrbuted radom varables, whch have ot (to the best of our kowledge) bee dscussed before Theorem The expectato of the umber of modfed alteratg rus for s

3 ALTERNATING RUNS OF GEOMETRICALLY DISTRIBUTED RANDOM VARIABLES 0 gve by EMAR [ ] ( )( ( )( The varace of the umber of modfed alteratg rus for s Var[ MAR ] ( + ) ( + + Furthermore, for, Var[MAR ] s ( + )( + + ( ( ( + )( + + ( ( Theorem The expected umber of turg pots for s + + ET [ ] ( ) ( + )( + + The varace of the umber of turg pots for s Var[ T ] ( + ) ( + + Furthermore, for, Var[T ] s ( )( ) ( )( ) ( + )( + + ( ( Theorem For, the expectato of the umber of alteratg rus satsfes EAR [ ] ( )( ( )( The varace of the umber of alteratg rus for s Var[ AR ], ( + ) ( + + whle for, Var[AR ] s

4 0 CHIA-JUNG LEE AND SHI-CHUN TSAI ( )( ) ( )( ) ( )( ) ( )( ) The proofs of these theorems are sectos,, ad, respectvely The, as metoed above, takg the lmt, we ca obta the mea ad varace a radom permutato usg the results of geometrcally dstrbuted radom varables Corollary I a radom permutato of legth, the expectato of the umber of alteratg rus satsfes, for, EAR [ ] + ( ) For, the varace of the umber of alteratg rus s /9, whle for, t s Note that the expectato s exactly the result for the case of radom permutatos [7] Moreover, sce the radom varables we use to evaluate AR are m-depedet for some teger m, we ca get the followg theorem about the dstrbuto of AR a radom permutato, whose proof s secto 5 Theorem I a radom permutato of legth, the radom varable -/ (AR E[AR ]) has a lmtg ormal dstrbuto wth mea 0 ad varace A /5 as TOTAL NUMBER OF MODIFIED ALTERNATING RUNS Throughout ths paper, let X,, X be depedet geometrcally dstrbuted radom varables satsfyg for {,,, }, Pr[X k] p k-, k N, where p + I ths secto, we prove Theorem whch s about the mea ad varace of the total umber of modfed alteratg rus X,, X Let MAR deote the total umber of modfed alteratg rus X,, X Sce each modfed alteratg ru must ed wth repeated umbers, we observe that MAR s oe more tha the umber of cosecutve detcal substrgs except for the cases of X X ad X - X Take the seueces 555 ad as examples We fd that there are modfed alteratg rus the seuece 555: 5,, whle the umber of repeated subseuece s exactly Moreover, there s modfed alteratg ru the seuece :, whle the umber of repeated subseuece s exactly Hece, to cout the umber of repeated sub-

5 ALTERNATING RUNS OF GEOMETRICALLY DISTRIBUTED RANDOM VARIABLES 0 strgs ad deal wth the case of X - X, we defe a bary radom varable x for each {,,, } by, f X X+ X+ ; ξ 0, otherwse Moreover, we defe a addtoal radom varable, f X X; ς 0, otherwse; for the case of X X The, we have MAR + ξ + ς () To evaluate the expectato of MAR, we eed to obta E[ξ ], for each ad E[ς ] For each {,, }, sce X s are depedet, the by the formula for a geometrc seres, the expectato of ξ s E[ ] Pr[ X X X ] Pr[ X X k]pr[ X k] ξ k k k k ( p ( p ( p p p p + + ( + )( + + O the other had, the expectato of ς s E X X p p k [ ς] Pr[ ] ( ) + Therefore, by the learty of expectato ad E (), we obta that the expectato of MAR s + 6 EMAR [ ] + E[ ξ] + E[ ς] + ( + )( + + ( + )( + + Next, we derve the varace of MAR Sce the varace of MAR s gve by Var[ MAR] Var[ ξ] + Var[ ς] + cov( ξ, ξ j) + cov( ξ, ς), < j () we cosder the followg values: Var[ξ ], Var[ς ], cov(ξ, ξ j ), ad cov(ξ, ς ) for ay ad j > For each {,,, }, the varace of ξ s Throughout ths paper, we use matlab to expad the polyomals

6 0 CHIA-JUNG LEE AND SHI-CHUN TSAI ( + ) ( [ ξ] [( ξ) ] ( [ ξ]) Var E E () I addto, the varace of ς s Var[ ς] E[( ς) ] ( E[ ς]) ( + ) () The, we proceed to boud the covarace Sce for, j {,,, } wth j, ξ ad ξ j are depedet, we kow cov(ξ, ξ j ) 0 Moreover, for, ξ ad ς are also depedet, that s, cov(ξ, ς ) 0 It remas to evaluate cov(ξ, ξ + ), cov(ξ, ξ + ), cov(ξ, ς ), ad cov(ξ, ς ) for ay For {,,, }, the covarace of ξ ad ξ + s cov( ξ, ξ ) E[ ξξ ] E[ ξ ] E[ ξ ] Pr[ X X X ad X X X ] ( E[ ξ ]) ( E[ ξ ]) , ( + ) ( + + (5) ad smlarly, for {,,, }, the covarace of ξ ad ξ + s cov( ξ, ξ ) + E[ ξξ ] E[ ξ ] E[ ξ ] + + Pr[ X X X ad X X X ] ( E[ ξ ]) Pr[ X X k] Pr[ X X h]pr[ X h] ( E[ ξ ]) h N; h k ( p ( p ( p ( E[ ξ ]) k k h h h k p k ( p k p p + p ( E[ ξ ]) k k ( )( ) ( )( ) (6) O the other had, the covarace of ξ ad ς s cov( ξ, ς ) E[ ξς ] E[ ξ ] E[ ς ] Pr[ X X X ad X X ] E[ ξ ] E[ ς ] Pr[ X X k]pr[ X k] E[ ξ ] E[ ς ] +, ( + ) ( + + (7)

7 ALTERNATING RUNS OF GEOMETRICALLY DISTRIBUTED RANDOM VARIABLES 05 ad the covarace of ξ ad ς s cov( ξ, ς ) E[ ξ ς ] E[ ξ ] E[ ς ] Pr[ X X X ad X X ] E[ ξ ] E[ ς ] Pr[ X X X k]pr[ X k] E[ ξ ] E[ ς ] + 5 ( + )( + + ( (8) Substtutg (), (), ad (7) to (), we ca obta the varace of MAR, ad substtutg (), (), (5), (6), (7), ad (8) to (), we ca obta the varace of MAR for Ths completes the proof of Theorem TOTAL NUMBER OF TURNING POINTS I ths secto, we prove Theorem, whch s about the propertes of the total umber of turg pots X,, X Let T be the total umber of turg pots X,, X Recall that x s a turg pot a seuece x,, x, exactly whe max{x +, x - } < x or m{x +, x - } > x Hece, for each {,,, }, defe a bary radom varable η by, f ( X < X+ ad X+ > X+ ) or ( X > X+ ad X+ < X+ ); η 0, otherwse, ad t s clear that T η (9) To smplfy the evaluato of expectato ad varace of T, we troduce the followg hady lemma frst Lemma For ay,, ad s N, Pr[X > s] s Pr[X < s] s- s ( Pr[X < s ad X + > X ] Pr[X + < s ad X > X + ] + Pr[X > s ad X + < X ] Pr[X + > s ad X < X + ] s s /( + ) Proof: By the formula for a geometrc seres ad the fact p, we have p s t s Pr[ X > s] Pr[ X t] p, t s+ t s+

8 06 CHIA-JUNG LEE AND SHI-CHUN TSAI ad Pr[X < s] Pr[X > s ] s- Moreover, sce X ad X + are depedet, we get s s t t X < s X+ > X X t X+ > t p t t Pr[ ad ] Pr[ ]Pr[ ] O the other had, we have p + s s ( ) ( ) s s t t X+ < s X > X+ X+ t X > t p t t Pr[ ad ] Pr[ ]Pr[ ] s ( Pr[ X < s ad X+ > X] + Smlarly, we obta Pr[ X > s ad X < X ] Pr[ X > s ad X < X ] t t Pr[ X t]pr[ X + < t] p ( t s+ t s+ s s s ( p + Now we derve the expectato of T For each {,, }, sce X s are depedet, the by Lemma, we obta E[η ] as Pr[ X < X ad X > X ] + Pr[ X > X ad X < X ] Pr[ X s]pr[ X < s]pr[ X < s] + Pr[ X s]pr[ X > s]pr[ X > s] p ( + p ( s s s s + + ( + )( + + ) The, by the learty of expectato ad the represetato of E (9), we have the expectato of T as + + ET [ ] E[ η ] ( ) ( + )( + + Next, we evaluate the varace of T For each {,,, }, the varace of η s

9 ALTERNATING RUNS OF GEOMETRICALLY DISTRIBUTED RANDOM VARIABLES ( + ) ( [ η] [( η) ] ( [ η]) Var E E (0) Sce for, j {,, } wth j, η ad η j are depedet, we kow cov(η, η j ) 0 I addto, for {,,, }, by Lemma, the expectato of η η + s E[η η + ] Pr[(X < X + ) ad (X + > X + ) ad (X + < X + )] + Pr[(X > X + ) ad (X + < X + ) ad (X + > X + )] Pr[ X s]pr[ X < s]pr[ X < s ad X > X ] Pr[ X s]pr[ X > s]pr[ X > s ad X < X ] s s s s ( s s s p ( + p ( , ( + )( + + (+ + + ad therefore the covarace of η ad η + s cov( η, η ) E[ ηη ] E[ η ] E[ η ] ( ( + )( + + ( ( + )( ( )( ( () Smlarly, for {,,, }, by Lemma, the expectato of η η + s E[ ηη + ] Pr[( X < X ) ad ( X > X > X ) ad ( X < X )] Pr[( X < X ) ad ( X > X ) ad ( X < X ) ad ( X > X )] Pr[( X > X ) ad ( X < X < X ) ad ( X > X )] Pr[( X > X ) ad ( X < X ) ad ( X > X ) ad ( X < X )] Pr[ X s]pr[ X > s] (Pr[ X t]pr[ X > t ad X < X ]) s N t s+ s + Pr[ X s]pr[ X < s] (Pr[ X t]pr[ X > t ad X < X ]) s N t + Pr[ X s]pr[ X > s] + t s+ (Pr[ X t]pr[ X > t ad X < X ]) Pr[ X s]pr[ X > s] (Pr[ X t]pr[ X < t ad X > X ]) s N t s+

10 08 CHIA-JUNG LEE AND SHI-CHUN TSAI t s s t t p p s N t s+ + s t s s t t + p ( p s N t + t s s t t + p p s N t s+ + p p s s t + s N t s+ t ( ) ( + )( + + ( ( whch leads to, cov(η, η + ) E[η η + ] E[η ]E[η + ] ( + )( + + ( ( () Hece, by E (0), we ca obta the varace of T, ad by Es (0)-(), we ca obta the varace of T for Ths completes the proof of Theorem TOTAL NUMBER OF ALTERNATING RUNS I ths secto, we prove Theorem, that s, we cosder the total umber of alteratg rus a seuece of depedet radom varables, whch s exactly the sum of the umber of modfed alteratg rus ad the umber of turg pots Let AR deote the umber of alteratg rus a seuece of depedet geometrcally dstrbuted radom varables X,, X Therefore, AR MAR + T By the learty of expectato alog wth Theorems ad, we have the expectato of AR as E[AR ] E[MAR ] + E[T ] ( ) ( + )( + + ( + )( + + ( + )( ( + )( + + ( + )( + + Next, we derve the varace of AR Sce for ay, j {,, } wth j, ξ are depedet of η j, ad ς s depedet of η k for ay k, we have

11 ALTERNATING RUNS OF GEOMETRICALLY DISTRIBUTED RANDOM VARIABLES 09 Var[AR ] Var + ξ + ς + η [ ] + [ ] + [ ] + (, ) + (, ) + Var ξ Var ς Var η cov ξ ξ+ cov ξ ξ+ cov( ξ, ς) + cov( ξ, ς) + cov( η, η+ ) + cov( η, η+ ) + cov( η cov( η, ς) + cov( ξ, η) + cov( ξ, η+ ) + cov( ξ, η+ ) + cov( ξ+, η) + cov( ξ+, η) Var[ MAR] Var[ T] cov( η, ς) cov( η, ς) cov( ξ, η) cov ξ η+ cov ξ η+ cov ξ+ η cov ξ+ η ς (, ) + (, ) + (, ) + (, ), ) + () Hece, by Theorems ad, t remas to evaluate cov(η, ς ), cov(η, ς ), cov(ξ, η ), cov(ξ, η + ), cov(ξ, η + ), cov(ξ +, η ), ad cov(ξ +, η ) for ay The covarace of η ad ς s cov( η, ς) E[ ης ] E[ η] E[ ς] Pr[(( X < X ad X > X) or ( X > X ad X < X)) ad X X] E[ η] E[ ς] E[ η] E[ ς] +, () ( + ) ( + + ad by Lemma, the covarace of η ad ς s cov( η, ς ) E[ ης] E[ η] E[ ς] Pr[(( X < X ad X > X ) or ( X > X ad X < X )) ad X X ] E[ η ] E[ ς ] Pr[ X X < X ad X > X ] Pr[ X X > X ad X < X ] E[ η ] E[ ς ] Pr[ X X s] (Pr[ s< X ad X > X ] + Pr[ s> X ( + ) s s s s ( p E[ η] E[ ς] ( + )( + + ( ad X < X ]) E[ η ] E[ ς ] (5)

12 00 CHIA-JUNG LEE AND SHI-CHUN TSAI O the other had, for {,, }, the covarace of ξ ad η s cov( ξ, η ) E[ ξη ] E[ ξ ] E[ η ] Pr[ X X X ad (( X < X ad X > X ) or ( X > X ad X < X ))] E[ ξ ] E[ η ] E[ ξ ] E[ η ] 6 + +, ( + ) ( + + (6) ad smlarly, for {,, }, the covarace of ξ + ad η s cov(ξ +, η ) E[ξ + η ] E[ξ + ]E[η ] Pr[ X+ X+ X+ ad (( X < X+ ad X+ > X+ ) or ( X > X+ ad X+ < X+ ))] E[ ξ+ ] E[ η] E[ ξ+ ] E[ η] ( + ) ( + + (7) The, by Lemma ad E (6), for {,, }, the covarace of ξ ad η + s cov( ξ, η ) + E[ ξη ] E[ ξ ] E[ η ] + + Pr[ X X X ad (( X < X ad X > X ) or ( X > X ad X < X ))] E[ ξ ] E[ η ] Pr[ X X s](pr[ s< X ad X > X ] Pr[ s > X ad X < X ]) E[ ξ ] E[ η ] ( + ) + s s s s ( p E[ ξ] E[ η+ ] ( )( )( ) (8) Smlarly, for {,, }, the covarace of ξ ad η + s cov( ξ, η ) + E[ ξη ] E[ ξ ] E[ η ] + + Pr[ X X X ad (( X < X ad X > X ) or ( X > X ad X < X ))] E[ ξ ] E[ η ]

13 ALTERNATING RUNS OF GEOMETRICALLY DISTRIBUTED RANDOM VARIABLES 0 Pr[ X X k] Pr[ X s] (Pr[ s< X ad X > X ] , s k Pr[ s > X ad X < X ]) E[ ξ ] E[ η ] ( + ) s s k s s ( p p E[ ξ] E[ η+ ], s k + + k ( p k k + + k k ( p [ ] [ + E ξ E η+ ] (+ )( ( ) ( ) ( )( ) (9) Fally, for {,, }, the covarace of ξ + ad η s cov( ξ, η ) + E[ ξ η ] E[ ξ ] E[ η ] + + Pr[ X X X ad (( X < X ad X > X ) or ( X > X ad X < X ))] E[ ξ ] E[ η ] k Pr[ X k]pr[ X < k] Pr[ X s]pr[ X s]pr[ X s] k N s + Pr[ X k N s k+ E[ ξ ] E[ η ] + k k k s s k N s k]pr[ X < k] Pr[ X s]pr[ X s]pr[ X s] p ( ( p ( p + k k s s p p p E ξ+ E k N s k+ ( ) ( ) [ ] [ η ] k k k k p( p ( p ( ) p p k k k k p E ξ + E k N [ ] [ η ] ( ) ( ) ( )( ) + (0) Now, substtutg Theorems ad alog wth Es () ad (6) to E (), we ca obta the varace of AR, ad substtutg Theorems ad alog wth Es ()-(0) to E (), we ca obta the varace of AR for Ths completes the proof of Theorem

14 0 CHIA-JUNG LEE AND SHI-CHUN TSAI 5 THE DISTRIBUTION OF THE NUMBER OF ALTERNATING RUNS I ths secto, we prove Theorem Note that AR MAR T ς ( ξ η ) We observe that the seuece of radom varables ς, ξ, η, ξ, η, s 5-depedet, where a seuece of radom varables {Y } s called m-depedet, f the two sets (Y, Y,, Y r ) ad (Y r+m+, Y r+m+,, Y ) are depedet for ay r Hece, we ca apply the cetral lmt theorem for depedet radom varables [0] to obta the dstrbuto of AR The followg lemma s mmedately from Theorem [0] Lemma Let Y, Y, be a m-depedet seuece of radom varables such that E[ Y E[Y ] ] < for ay ad lm A+ k A exsts, uformly for all, where k m A Var[ Y ] + cov( Y, Y ) + m + m j + m j / The as, the radom varables ( Y E[ Y]) has a lmtg ormal dstrbuto wth mea 0 ad varace A For the seuece of radom varables ς, ξ, η, ξ, η,, we have that for ay, A Var[ ξ+ ] + { cov( η+, ξ+ ) + cov( ξ+, ξ+ ) + cov( η, ξ+ ) + cov( ξ, ξ+ )} A, A Var[ η+ ] + { cov( ξ+, η+ ) + cov( η+, η+ ) + cov( ξ+, η+ ) + cov( η, η+ ) + cov( ξ, η )} A + The, by Es (), (5), (6), (0)-(), (6)-(0), we obta that for ay, lm A+ k ( A+ A) k ( + )( + + ( ( Hece, accordg to Lemma ad takg the lmt, we complete the proof of Theorem ACKNOWLEDGMENT We would lke to thaks Phlppe Flajolet, ad Helmut Prodger for helpful formato about the covergece of the model of geometrcally dstrbuted radom varables, ad the aoymous referees for very useful commets

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16 0 CHIA-JUNG LEE AND SHI-CHUN TSAI 0 W Hoeffdg ad H Robbs, The cetral lmt theorem for depedet radom varables, Duke Mathematcal Joural, Vol 5, 98, pp Cha-Jug Lee ( ) receved the BS degree from the Natoal Tawa Normal Uversty, Tape, Tawa, 000; ad PhD degree Computer Scece from the Natoal Chao Tug Uversty, Hschu, Tawa, 00 She s ow dog postdoctoral research at the Isttute of Iformato Scece, Academa Sca, Tape, Tawa Her research terests are radomess computato, cryptography, ad theoretcal computer scece Sh-Chu Tsa ( ) was bor Chay, Tawa, ROC He receved the BS ad MS degrees from Natoal Tawa Uversty, Tape, Tawa, 98 ad 988, respectvely, ad the PhD degree from the Uversty of Chcago, Chcago, IL, 996, all computer scece He s curretly a Professor the Computer Scece Departmet, Natoal Chao Tug Uversty, Hschu, Tawa Hs research terests clude computatoal complexty, algorthms, cryptography, radomzed computato ad dscrete mathematcs Dr Tsa s a member of ACM, IEEE ad SIAM