The Structure of Forward, Reverse, and Transverse Path Graphs in The Pattern Recognition Algorithms of Sellers

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1 The Structure of Forwrd, Reverse, nd Trnsverse Pth Grhs in The Pttern Recognition Algorithms of Sellers Lewis Lsser Dertment of Mthemtics nd Comuter Science York College/CUNY Jmic, New York Louis D Alotto Dertment of Mthemtics nd Comuter Science York College/CUNY Jmic, New York dlotto@york.cuny.edu Abstrct In [3], [4], [5] Sellers develoes dynmic rogrmming ttern mtching lgorithm tht genertes forwrd, reverse, nd trnsverse th grhs tht determine the best resemblnce (lowest cost) of smller string ttern inside lrger. In this er we study the roerties nd structure of these grhs. We show tht these th grhs cn be decomosed into smll number of distinct block tyes, tht re used to nlyze grh structure. It is lso shown tht n exct ttern mtch results in disconnected trnsverse th grh. keywords: Pttern Recognition, Connected Grh, Pth Grh, Block Tye, Evolutionry Distnce. I. INTRODUCTION The ttern recognition roblem of finding n intervl i in lrge sequence, = n, tht best resembles smller sequence, b = b 1 b 2...b m, ws studied by Sellers, see [3], [4], [5]. In his work he described dynmic rogrmming ttern recognition lgorithm tht serches for best resemblence (lowest cost). String resembles ttern b if there is n intervl in tht is either equl to b or close enough to b. The term close enough deends on the choice of suitble metric. The nottion d(x, y), where x nd y re rbitrry strings, will be used to reresent such metric. The function d(x, y) is clled the evolutionry distnce. The evolutionry distnce is the length of the shortest th (lowest cost) of evolutionry stes between nd b. We ssume, s in [3], tht only certin tyes of evolutionry stes re llowed for strings (sequences) nd tht cost cn be ssocited with ech ste. We consider n evolutionry ste s n insertion, deletion, or muttion of single element in sequence. We ssign ech such ste cost of 1. More formlly, the evolutionry distnce d(, b) between ny two sequences is defined s { m+n } min d( i, b i ) i=1 Where the minimum is tken over ll irs of sequences n+m nd b 1 b 2...b n+m with m null symbols (-) inserted in sequence nd n null symbols inserted in sequence b. The null symbols reresent the deletion or insertion of term with which it is ligned in the other sequence. This roduces metric lignment of sequences nd b. Thus the evolutionry distnce is the lowest cost of evolutionry stes between nd b. The lgorithm of Sellers is described s follows. Given two finite sequences, = n nd b= b 1 b 2...b m, where n > m nd b is the secified string ttern being sought in. The lgorithm serches to identify intervls i in tht resemble b nd clcultes their evolutionry distnce from string b. We use the nottion i to sy tht i is n intervl in. Hence i= q, where 1 q n. D f = (e(i, j)) is clled the (n + 1) (m + 1) forwrd distnce mtrix nd is comuted inductively. To strt constructing the mtrix, the initil vlues of the first column re set to zero. Tht is, set e(i, 0) = 0 for i = 0, 1,..., n. Continuing, the vlues in the first row re set j e(0, j) = d(, b h ) h=1

2 for j = 1, 2,..., m. The rest of the forwrd mtrix vlues re comuted inductively by ssigning the smllest of these three vlues: (e(i 1, j) + d( i, )) (e(i 1, j 1) + d( i, b j )) (e(i, j 1) + d(, b j )) to e(i, j). These three vlues determine the forwrd neighborhood, see [1]. Note tht e(i, j) is the minimum totl cost of the muttions, insertions, nd deletions needed to trnsform the first string segment b 1 b 2...b j of b into ny intervl i of ny length tht ends t i. The reverse distnce mtrix D r = (f(k, j)) is constructed in similr mnner, however we strt with the lst terms of nd b nd end with the first. Here we note tht f(k, j) is the minimum totl cost of muttions, insertions, nd deletions necessry to convert the string segment b k b k+1...b m into n intervl i. The initil vlues of the mtrix entries in the lst column nd row re f(i, m + 1) = 0 for i = 1, 2,..., n + 1 nd f(n + 1, j) = m j h=0 d(, b m h ) for j = 1, 2,..., m. The rest of the reverse mtrix vlues re comuted inductively by setting f(i, j) equl to the smllest of these three vlues in the reverse neighborhood: nd (f(i + 1, j) + d( i, )) (f(i + 1, j + 1) + d( i, b j )) (f(i, j + 1) + d(, b j )) The lgorithm determines both globl nd locl resemblnces. The bove rocedure determines ll locl resemblnces. Looking only t the forwrd mtrix determines only globl resemblnces. Given the evolutionry distnce d, n intervl i in best resembles short sequence b globlly iff d(i,b) d(j,b) for ll intervls j in. Also, given n evolutionry distnce d, n intervl i in best resembles b loclly iff d(i,b) d(h,b) nd d(i,b) d(j,b) for ll h nd j, where h i j. The forwrd th grh is constructed directly from D f by connecting ech ir of mtrix entries, tht re connected by n inductive ste, with n edge nd relcing ech mtrix entry with vertex. Similrly, the reverse th grh is constructed from D r. The common th grh is the intersection of vertices nd edges from both grhs. The trnsverse th is succession of edges tht connects the first column nd the lst column of th grh. Globl resemblnces cn be distinguished from locl by exmining the vlues of the lst column in the mtrix ssocited with the vertices of the trnsverse th grh. A miniml vlue corresonds to globl resemblnce while ny lrger vlues corresond to locl resemblnces. If the trnsverse th grh is connected then ll vertices in the lst column of the ssocited mtrix must hve the sme vlue. Throughout this er we consider grh connected if there is exctly one connected comonent fter we disregrd vertices of degree zero. A locl mtch (resemblence) only occurs when vlue lrger thn the miniml vlue is encountered. If only one vlue exists ll mtches must be globl. Hence the grh roerty of being connected corresonds to ll mtches being globl. For i 1 nd j 1, the forwrd neighborhood of vertex e(i, j) is the set of three edges e(i 1, j), e(i 1, j 1) nd e(i, j 1). For i = 0 nd j 1 it consists of the single vertex e(0, j 1) while for j = 0 nd i 1 it is just e(i 1, 0). The vertex e(0, 0) hs emty neighborhood. The forwrd neighbors of vertex e(i, j) re those vertices in its forwrd neighborhood to which it is connected. For i n nd j m, the reverse neighborhood of vertex f(i, j) is the set of three edges f(i+1, j), f(i+1, j+1) nd f(i, j+1). For i = n+1 nd j m it consists of the single vertex f(n + 1, j + 1) while for j = m + 1 nd i n it is just f(i + 1, m + 1). The vertex f(n+1, m+1) hs emty neighborhood. The reverse neighbors of vertex f(i, j) re those vertices in its reverse neighborhood to which it is connected. The lemms nd theorems resented in this er demonstrte tht the forwrd, reverse, common, nd trnsverse th grhs cn be decomosed into vrious block tyes. It is further interesting to see, tht good ttern mtch results in disconnected trnsverse th grh. b will lwys refer to the secified string nd for the longer string in which we seek substrings loclly nd/or globlly similr to b. The set Σ will consist of ll of our symbols together with the secil null symbol. We use the evolutionry distnce d(x, y) = 0 if x = y nd 1 otherwise, for ll x, y Σ. The vlue of 0 imlies direct mtch, while the vlue (cost) of 1 corresonds to muttion, deletion, or insertion of single term in the sequence. If Σ then n will men string of n consecutive s. The interior of the forwrd grh/mtrix refers to ll but the first row nd column. For the reverse grh/mtrix, the interior is ll but the lst row nd column. The roofs of the results

3 deend on the nlysis of the different block tyes of the forwrd nd reverse mtrices nd their corresonding grhs. Where no confusion rises, we will use the term block to reresent rectngulr ortions of the mtrices s well s the th grhs. In the illustrtions tht follow, it is frequently necessry to show the elements in either the long string or the short string b tht corresond to the ortion of the mtrix/grh being shown. These re indicted in one or more verticl lines long the extreme left mrgin (for the whole or the ortion of the long string) or in one or more horizontl lines long the extreme to (for the whole or the ortion of the short string). A str indictes n rbitrry symbol of the lhbet different from. The lgorithm ws imlemented to generte nd disly grhiclly the forwrd, reverse, common, nd trnsverse th grhs. The rogrm ws written in the C rogrmming lnguge nd run under Mc OS X nd Linux with the GD grhics librry instlled. II. THE STRUCTURE OF FORWARD AND REVERSE PATH GRAPHS There re eight fundmentl building blocks from which the forwrd nd reverse distnce mtrices nd their resective th grhs re constructed (four for the forwrd nd four for the reverse). The following lemms describe ech tye for the forwrd mtrix/grh cse. Ech hs counterrt for the reverse cse. Lemm 1: Suose the rectngulr ortion of the forwrd distnce mtrix between rows i 1 nd i 2 nd columns j 1 nd j 2 hs every entry in the first column equl to nd entries in the first row, from left to right, equl to, +1,..., +s, where s = j 2 j 1. Suose i = for i i i 2 nd b j for j j j 2. Then every entry in the lst column hs vlue +s nd the entries in the lst row, from left to right, re, + 1,..., + s. Furthermore every vertex in the interior of the corresonding rectngulr ortion of the forwrd th grh is connected to exctly two neighbors, the one to its left nd the one digonlly u nd left. Proof of Lemm 1: Figure 1 illustrtes this lemm for s = 4 nd i 2 i 1 = 8. The vlues in the mtrix re, s lwys, comuted in n itertive fshion. First the missing vlues in the second row, from left to right, re found. This rocedure is reeted for ech row, from to to bottom. When missing vlue e(i, j) is comuted we hve d( i, b j ) = 1 becuse i b j. Furthermore it is esily seen tht the vlues to its left nd digonlly u nd left re equl while the vlue bove is one greter. Consequently, due to Seller s construction, e(i, j) is one more thn the vlue to its left nd the forwrd th grh hs the indicted edges incident with the corresonding vertex Fig Block Tye 1 (BT1) Lemm 2: Suose the rectngulr ortion of the forwrd distnce mtrix between rows i 1 nd i 2 nd columns j 1 nd j 2 hs entries in the first column, from bottom to to, of, +1,..., +r, where r = i 2 i 1 nd entries in the first row, from left to right, equl to + r, + r + 1, + r + 2,..., + r + s, where s = j 2 j 1. Suose i = for i i i 2 nd b j for j j j 2. Then the entries in the lst row, from left to right, re, +1,..., +s nd the entries in the lst column, from bottom to to, re +s, +s+1, +s+2,..., +r+s. Furthermore every vertex in the interior of the corresonding rectngulr ortion of the forwrd th grh is connected only to the neighbor to its left. Proof of Lemm 2: Figure 2 illustrtes this lemm for r = s = 4. The roof is strightforwrd consequence of the construction Fig Block Tye 2 () Lemm 3: Suose the rectngulr ortion of the forwrd distnce mtrix between rows i 1 nd i 2 nd columns j 1 nd j 2 hs entries in the first column, from bottom to to, of, +1,..., +r, where r = i 2 i 1 nd entries in the first row, from left to right, equl to +r, +r+1, +r+2,..., +r+s, where s = j 2 j

4 Suose i = for i i i 2 nd b j = for j 1 +1 j j 2. Then the entries in the lst row, from left to right, re, + 1,..., + s nd the entries in the lst column, from bottom to to, re +s, +s+1, + s + 2,..., + r + s. Furthermore every vertex in the interior of the corresonding rectngulr ortion of the forwrd th grh is connected to exctly two neighbors: the neighbor to its left nd digonlly u nd left. Proof of Lemm 3: Figure 3 illustrtes this lemm for r = s = 4. Agin, this is strightforwrd consequence of the wy in which the mtrix nd grh re constructed Fig Block Tye 3 () Lemm 4: Suose the rectngulr ortion of the forwrd distnce mtrix between rows i 1 nd i 2 nd columns j 1 nd j 2 hs every entry in the first column equl to nd entries in the first row, from left to right, equl to, + 1,..., + s, where s = j 2 j 1. Suose i = for i i i 2 nd b j = for j j j 2. Then every entry in the lst row hs vlue nd the entries in the lst column, from bottom to to, consist of r s vlues of followed by + 1, + 2,..., + s. The edges of vertex in the interior of the corresonding rectngulr ortion of the forwrd th grh deend on the reltive row nd column numbers. If j j 1 i i 1 then the vertex is connected to exctly two neighbors: the one to its left nd the one digonlly u nd left. Otherwise the vertex is connected to the neighbor digonlly u nd left. Proof of Lemm 4: Figure 4 illustrtes this lemm for r = 8 nd s = 4. Although more comlex thn lemms 1 through 3 it is roved in the sme fshion. For future reference we refer to both the ortion of the mtrix nd the ortion of the grh in lemms 1 through 4 s being of block tye one through four, resectively. For grhs of block tye four there is n re in the uer right corner tht is connected with number of disjoint ths running on digonls below. The vertex where the tringulr region intersects the lst column hs mtrix vlue of. We cll this vertex Fig. 4. Block Tye 4 (BT4) the trnsition vertex for grh of block tye four. All mtrix vlues below the trnsition vertex hve vlue nd those bove, from bottom to to, hve vlues + 1, + 2,..., + s. We now describe ll ossible reltions between the four block tyes tht cn occur in forwrd or reverse th grh. The next lemm shows tht to the right of every block tye one is block tye four. This is shown in Figure 5. Lemm 5: Suose ortion P 1 of the forwrd mtrix/grh is of block tye one. Suose the vlues of the mtrix in the first row, from left to right, re, + 1,..., + s nd tht this row of the mtrix extends to the right with vlues +s+1, +s+2,..., +s+t. Suose every element of the short sequence b bove these extended mtrix vlues is. Then there is ortion P 2 of the forwrd mtrix/grh of block tye four tht shres every mtrix-vlue/vertex of the rightmost column of P 1. Proof of Lemm 5: Since P 1 is of block tye one, the vlues in the mtrix for P 1 in the rightmost column, from bottom to to, re ll + s. We now hve ortion of the mtrix (to the right of P 1 ) whose first column contins only the vlue + s nd whose first row, from left to right, contins the vlues + s, +s+1,..., +s+t. Since ll elements of the short sequence s bove ll but the first column re equl to, lemm 4 identifies this ortion of the forwrd mtrix/grh to be of block tye four. The next lemm shows tht to the right of every block tye four re blocks of tye two nd one. This is shown in figure 6. Lemm 6: Suose ortion P 1 of the forwrd mtrix/grh is of block tye four. Suose the vlues of the mtrix in the first row, from left to right, re, + 1,..., + s nd tht this row of the mtrix

5 +s +s+t +s +s+t BT1 BT4 BT4 +t BT1 +s Fig. 5. BT1 with BT4 to its right Fig. 6. BT4 with nd BT1 to its right extends to the right with vlues + s + 1, + s + 2,..., + s + t. Suose every element of the short sequence b bove these extended mtrix vlues differs from. Then there is ortion P 2 of the forwrd mtrix/grh of block tye two tht shres the mtrixvlues/vertices of the rightmost column of P 1 from the first vertex down to n including the trnsition vertex of P 1 nd ortion P 3 of the forwrd mtrix/grh of block tye one tht shres the mtrix-vlues/vertices of the rightmost column of P 1 from the trnsition vertex of P 1 down to the lst vertex. Proof of Lemm 6: Lemm 5 rovides the vlues of the mtrix in the lst column of P 1. The trnsition vertex of P 1 divides these vlues u into two rts. The vlues on nd bove this vertex, together with those in the extended row yield ortion P 2 of block tye two. Lemm 2 rovides the vlues of the mtrix in the lst row of P 2. Similrly the vlues on nd below this vertex, together with those in the lst row of P 2 yield ortion P 3 block tye one. The next lemm shows tht to the right of every block tye two is block tye three. See figure 7. Lemm 7: Suose ortion P 1 of the forwrd mtrix/grh is of block tye two. Suose the vlues of the mtrix in the first row, from left to right, re +r, +r +1,..., +r+s nd tht this row of the mtrix extends to the right with vlues + r + s + 1, + r + s + 2,..., + r + s + t. (The vlues in the first column, from bottom to to, re, + 1,..., + r.) Suose every element of the short sequence s bove these extended mtrix vlues is. Then there is ortion P 2 of the forwrd mtrix/grh of block tye three tht shres every mtrix-vlue/vertex of the rightmost column of P 1. Proof of Lemm 7: This lemm is roved in wy similr to lemm 5. +r Fig. 7. +r+s +s with to its right +r+s+t The next lemm shows tht to the right of every block tye three is block tye two. See figure 8. Lemm 8: Suose ortion P 1 of the forwrd mtrix/grh is of block tye three. Suose the vlues of the mtrix in the first row, from left to right, re +r, +r +1,..., +r +s nd tht this row of the mtrix extends to the right with vlues + r + s + 1, + r + s + 2,..., + r + s + t. (The vlues in the first column, from bottom to to, re, + 1,..., + r.) Suose every element of the short sequence b bove these extended mtrix vlues differs from. Then there is ortion P 2 of the forwrd mtrix/grh of block tye two tht shres every mtrix-vlue/vertex of the rightmost column of P 1. Proof of Lemm 8: This roof is similr to lemm 5. The long sequence consists of n rbitrry but finite number of s. The short sequence b cn contin rbitrry chrcters. A rtition of sequence v is sequence of subsequences such tht every element of v lies in some subsequence, every subsequence consists

6 +r Fig. 8. +r+s +s with to its right +r+s+t either entirely of s or entirely of symbols different from, nd ech subsequence is s long s ossible. If s is rtitioned into subsequences s i nd s 1 consists entirely of s then s i consists entirely of s for odd i nd entirely of symbols different from for i even. If the s i s re conctented in order from left to right then the originl sequence s is obtined. We now define the min digonl (min bnd) of mtrix/grh. The uer bound of this bnd consists of ll entries whose row number equls its column number. The lower bound is defined similrly, excet we strt t the element in the lst row nd column nd move u nd to the left long digonl. All elements on nd between these digonls constitute the min digonl. Theorem 1: The forwrd th grh cn be decomosed into rts ech of which is of one of the four block tyes. Proof of Theorem 1: It is imortnt to note tht the rts overl long their common borders. For exmle, if block tye two hs block tye three to its right then the entries in the lst column of the mtrix/grh of the block on the left re identicl to those in the first column of the block on the right. A decomosition in the sense of this theorem is not the sme s rtition. Prtition the short sequence b into subsequences s 1, s 2,...s z. It mkes no difference whether s 1 consists only of s or only of symbols different from. Without loss of generlity ssume the former, see Figure 9. The first column of the mtrix contins only zeros. The first row strts with zero nd increments by one until the lst column is reched. By lemm 4 there is block of tye four tht stretches verticlly the entire height of the mtrix/grh nd stretches horizontlly from the first column u to nd including the column corresonding to the lst symbol in s 1. We lso know the numericl vlues of the mtrix for this column by the lemm. Since s 1 contins only s, no symbol in s 2 is n. By lemm 6 we know the ortion of the mtrix/grh under s 2 strts with block tye two with block tye one below. Note tht we hve defined verticl stri within the mtrix/grh nd tht the block tye two hs height equl to its width while block tye one extends ll the wy to the bottom row. Since s 2 contins no s, s 3 contins only s. By lemm 7 we hve block tye three to the right of the block tye two in the second verticl stri. Similrly, by lemm 5 we hve block tye four to the right of the block tye one in the second verticl stri. The two blocks we hve just described form the third verticl stri. No element of s 4 is n. By lemm 8 there is block tye two to the right of the block tye three in verticl stri three. By lemm 6 we hve block tyes two nd one to the right of block tye four in verticl stri three. These three blocks (in order from to to bottom: two, two, nd one) define the fourth verticl stri. From the four verticl stris two fcts my be derived. First, for those s i s comosed only of s there cn only be block tyes three nd four. All block tye threes re bove the min digonl nd the single block tye four begins bove the min digonl nd continues until the lst row. Second, for the remining s i s there cn only be blocks tyes two nd one. All block tye twos re bove the min digonl nd the single block tye one begins bove the min digonl nd continues until the lst row. Due to the lternting nture of the s i s nd lemms 5 through 8 this ttern will continue for s long s there re s i s. For the forwrd distnce mtrix the vlues in the first column re set to zero nd those of the first row re, from left to right, 0, 1, 2,..., n, where n is the length of the short sequence b. Vlues in the interior of the mtrix re found by looking t vlues bove, to the left, nd digonlly u nd left from the current element. For the reverse distnce mtrix the vlues in the lst column re set to zero nd those of the lst row re, from right to left, 0, 1, 2,..., n. Vlues in the interior of the mtrix re found by looking t vlues below, to the right, nd digonlly down nd right from the current element. The rocedure to determine vlue in the interior of the mtrix is identicl in the forwrd nd reverse cses when these orienttion issues re tken into ccount. Consequently, everything we hve done so fr for the forwrd cse crries over to the reverse cse rovided orienttion issues re hndled roerly. Geometriclly, ech of the four block tyes must

7 - - BT4 Fig. 9. BT1 BT4 BT1 BT4 Block structure of forwrd mtrix/grh BT1 be rotted bout their centers by 180 degrees. If block tye i is trnsformed this wy we will refer to it s rotted block tye i. Lemms 1 through 8 cn now be rehrsed for the reverse cse. For exmle, lemm 6 stted tht to the right of every block tye four re blocks of tye two nd one (with tye two bove tye one nd tye one extending to the bottom of the mtrix/grh). Its counterrt sttes tht to the left of every rotted block tye four re rotted block tyes two nd one (with tye two below tye one nd tye one extending to the to of the mtrix/grh). With these conventions we hve the counterrt for theorem 1 for the cse of the reverse mtrix/grh. Theorem 2: The reverse th grh cn be decomosed into rts ech of which is of one of the four rotted block tyes. Proof of Theorem 2: Similr in nture to tht of Theorem 1, nd hence omitted. In the roof of Theorem 1 the blocks re orgnized into verticl stris. The blocks re lso orgnized into verticl stris for the reverse mtrix/grh. The lbeling of the vertices in the forwrd nd reverse mtrices nd grhs re shifts of one nother, i.e. the dsh ws moved to the other side of the short nd long sequences nd their symbols were shifted ccordingly. It is esy to see tht these shifts force the lignments of the verticl stris we hve defined for the forwrd nd reverse cses. (Although Sellers did not utilize such stris in his ers, the notion of lignments of this sort were used in his roofs tht the lgorithms for finding locl nd globl mtches were correct.) Since the stris re common to the forwrd nd reverse mtrices/grhs we my refer to them in the common nd trnsverse grhs s well. III. CONNECTEDNESS IN THE COMMON AND TRANSVERSE PATH GRAPHS Lemm 9: In ech verticl stri of the common grh for which the corresonding ortion of the short sequence b consists only of s, ech interior vertex is connected only to its neighbor u nd to the left. Proof of Lemm 9: Figure 10 illustrtes the structure of the verticl stri. We will refer to this s block tye five (BT5 for short). The common grh contins exctly those edges common to the forwrd nd reverse th grhs. In the forwrd th grh we will hve one or more block tye threes bove single block tye four tht extends downwrds for the reminder of the verticl height. In the reverse th grh we will hve one or more rotted block tye threes below single rotted block tye four tht extends uwrds for the reminder of the verticl height. Every one of these four block tyes hve ll ossible digonl edges. A block tye four will hve some horizontl edges but they re ll bove the min digonl. Similrly rotted block tye four will hve some horizontl edges s well but they re ll below the min digonl. Consequently no horizontl edge cn er in both the forwrd nd reverse th grhs, nd hence cn not er in the common grh. Since ll digonl edges er in both the forwrd nd reverse grhs they must er in the common grh. Fig. 10. Block Tye 5 (BT5) Lemm 10: Suose P is verticl stri of the Common Grh for which the corresonding ortion of the short sequence s contins no s. Then P cn be divided into three horizontl stris P 1, P 2, nd P 3, to be described below, where P 2 is lwys resent nd

8 P 1 nd P 3 my or my not be resent. Ech vertex in P 1, excet for those in the first column, is connected only to its neighbor to the left. The sme holds for P 3. Ech vertex in P 2, excet for those in the first row or first column, is connected to exctly two vertices: tht to the left nd the one digonlly u nd to the left. Proof of Lemm 10: Figure 11 illustrtes the structure of the verticl stri. We will refer to this s block tye six, or BT6. The roof is done in the sme wy s in Lemm 9, by intersecting BT1 s nd s in the forwrd th grh nd rotted BT1 nd of the reverse th grh. Fig. 11. Block Tye 6 (BT6) Theorem 3: The common grh cn be decomosed into n lternting series of block tyes five nd six. Proof of Theorem 3: As usul, rtition the short sequence b into subsequences s i. Ech subsequence contining only s give rise to block tye five. Otherwise we hve block tye six. Since the s i s lternte by definition, the block tyes must lternte. We now stte nd rove our min result. Theorem 4: Suose the long sequence contins only the symbol nd the short sequence b is rbitrry. The trnsverse grh is connected if nd only if b contins t lest one symbol different from. Proof of Theorem 4: We need only eliminte from the common grh ny edges tht do not lie on some th beginning t the first column nd ending t the lst column. Every block tye five contins every ossible digonl th. Every block tye six contins every ossible horizontl th (s well s some digonl edges, which we will ignore for now). These block tyes lternte. If we were to choose n rbitrry digonl edge in block tye five or n rbitrry horizontl edge in block tye six, then this edge will be rt of th of mximl length formed by combining edges from the vrious verticl stris. Suose s contins t lest one symbol different from. Then there will be t lest one block tye six. Choose n rbitrry horizontl edge from block tye six tht is common to the blocks P 1 nd P 2 from the sttement of lemm 10. If no such edge exists then P 1 does not exist nd we choose ny edge in the first row of P 2. In either cse we hve n edge which will belong to some th s described in the revious rgrh. Cll this th the uer criticl th. We cn define lower criticl th by choosing n edge between P 2 nd P 3, if one exists, or n edge from the lst row of P 2 otherwise. Any th bove the criticl th will fil to rech the first column. Similrly ny th below the criticl th will fil to rech the lst column. Now consider the digonl edges of block tyes six tht we hve ignored until now. These edges will link the two criticl ths nd ny ths between them nd will not link ny th bove the uer criticl th nor ny th below the lower criticl th. Consequently ll ths bove the uer criticl th nd ll ths below the lower criticl th will fil to rech either the first or lst column nd will hve no edges connecting them to the centrl region tht reches both the first nd lst column. These ths will be bsent from the trnsverse grh. The remining ths, together with the digonl edges of block tyes six, will form single comonent tht will be resent in the trnsverse grh. Hence the trnsverse grh will be connected. We now consider the cse when b is mde u of only s. In this cse there re no block tyes six. In fct, the entire grh is single block tye five nd the Common Grh consists only of ths like those in Figure 10. It is cler tht the Trnsverse Grh contins t lest two ths tht re connected by ny dditionl edges, i.e. we hve t lest two comonents. Hence the trnsverse grh will be disconnected. REFERENCES [1] D Alotto, L., Lsser, L., Disconnectedness in Forwrd nd Reverse Pth Grhs for Pttern Mtching, Proceedings of the Interntionl Conference on Artificil Intelligence, Vol. I, , Ls Vegs, NV, (2004). [2] Snkoff, D., Kruskl, J., Time Wrs, String Edits, nd Mcromolecules, The Theory nd Prctice of Sequence Comrison, CSLI Publictions, (1999). [3] Sellers, P. H., The Theory nd Comuttion of Evolutionry Distnces; Pttern Recognition, J. Algorithms 1, , (1980). [4] Sellers, P. H., Pttern recognition in genetic sequences, Proc. Ntl. Acd. Sci. USA 76, 3041, (1979). [5] Sellers, P. H., On the Theory nd Comuttion of Evolutionry Distnces, SIAM J. Al. Mth. 26, , (1974). [6] West, D. B., Introduction to Grh Theory, 2nd ed.. Prentice Hll, NJ (2001).