Recursive Structure and Bandwidth of Hales-Numbered Hypercube

Recursiv e Structur e and Bandwidt h of Hales-Num b ered Hyp erc ub e Xiaoha n W ang, Xiaol in W u, Dep artment of Ele ctric al and Computer Engine ering McMaster University Hamilton, Ontario, Canada, L8S 4K1 Abstract The Hales n u m b ered n -dimensional h yp ercub e and the corresp onding adjacency matrix exhibit in teresting recursive structures in n . These structures lead to a v ery simple pr oof of the w ell-kno wn bandw idth form ula for h yp ercub e, whose pro of w as though t to b e surprisingly difficult. A related p roblem ca lled hyp ercub e an tiband- width, for whic h Harp er prop osed an algorithm, is also reexamined in th e ligh t of the ab ov e r ec ursive structures, and a close form solution is f ound. Key wor d s: Graph band w idth, hyp ercub e. 1 In t ro duct ion The problem of g r a ph bandwidth has b een extensiv ely studied [1,2], and has found man y applications suc h as pa rallel computations, VLSI circuit design, etc. In this pap er we a re particularly in terested in the bandwidth of h yp er- cub es. The study of hypercub e bandwidth can guide the design of communi- cation co des fo r error resilien t transmission of signals o v er lossy net works suc h as the In ternet [3]. First, we restate the definitions of v ertex n um b ering and graph bandwidth, most of whic h a re adopted from [4]. Definition 1 A n umbering of a vertex se t V is any function η : V → { 1 , 2 , · · · , | V |} , (1) Email addr esses: wangx28@mcmas ter.ca (Xia ohan W ang), xwu@ece. mcmaster.ca (Xiaolin W u). Preprint su bmitted to Elsevier Science 14 No v em b er 2018 which is one- to-o n e (and ther efor e onto). A n um b ering η uniquely determines a tot al order, ≤ η , on V as: u ≤ η v if η ( u ) < η ( v ). Con v ersely , a total order defined on V uniqu ely determines a n umbering of the graph. Definition 2 The bandwidth of a n um b ering η of a gr aph G = ( V , E ) is bw ( η ) = max { u,v }∈ E | η ( u ) − η ( v ) | . (2) Definition 3 The bandwidth of a graph G is the minimum b andwidth ove r al l numb erings, η , of G , i.e. bw ( G ) = min η bw ( η ) . (3) The graph of the n-dimensional cub e, Q ( n ) , has v ertex set { 0 , 1 } n , the n -fold Cartesian pro duct of { 0 , 1 } . Th us | V ( n ) Q | = 2 n . Q ( n ) has a n edge b et w een tw o v ertices ( n -tuples of 0s and 1s) if they differ in exactly one en try . Definition 4 The Hales order , ≤ H , on V Q ( n ) , is define d by u ≤ H v if (1) w ( u ) < w ( v ) , or (2) w ( u ) = w ( v ) and u i s gr e ater than v in lexic o gr ap h ic or der r e l a tive to the right-to-left or der of the c o or dina tes, wher e w ( · ) is the Hamming weight of a vertex o f Q ( n ) . This total or der de- termines a numb ering, H ( n ) : V Q ( n ) → { 1 , 2 , · · · , 2 n } , which is c al le d Hales n umbering . Theorem 5 (Harp er, [4]) The Hale s numb ering mini m izes the b andwid th of the n -cub e, i.e. bw ( H ( n ) ) = bw ( Q ( n ) ) . (4) PR OOF. See Corollary 4.3 in [4]. ✷ Theorem 6 (Harp er, [4]) F or the n -cub e Q ( n ) , we have bw ( Q ( n ) ) = n − 1 X m =0 m ⌊ m 2 ⌋ ! . (5) Although the ab ov e result has b een kno wn for fo rt y y ear s, no pro of seemed to app ear in the literature. Ha r per p osed the pro of of Theorem 6 as an excise in his recen t b o ok [4], and noted “ it is surprisingly difficult”. In the following sec- tion w e presen t a rat her simple pro of. The pro of also rev eals some interesting effects of the Hales n umbering on h yp ercub es . 2 2 Pro of of the Bandwidth F orm ula for Hyp ercub es T o pro v e Theorem 6, w e first need a lemma and some definitions. Lemma 7 We define a 2 n × n (0 , 1) -m atrix S ( n ) as S ( n ) =           A ( n ) 0 A ( n ) 1 . . . A ( n ) n           , (6) wher e A ( n ) k , k = 0 , 1 , · · · , n , is an  n k  × n (0 , 1) -ma trix satisfying the fol lowing r e cursive formula A ( n ) k =    A ( n − 1) k − 1 1 A ( n − 1) k 0    , k = 1 , 2 , · · · , n − 1 , (7) wher e 0 and 1 ar e c olumn ve ctors c o ntaining only 0 s and 1 s r esp e ctively. As the b ase c ase, we have A ( n ) 0 = 0 T and A ( n ) n = 1 T . Then the r ow ve ctors of S ( n ) , fr om top to b o ttom, ar e al l vertic es of Q ( n ) in the incr e a s ing Hales or d e r. PR OOF. F rom Definition 4, it is sufficien t to sho w tha t the ro w v ectors of A ( n ) k , k = 0 , 1 , · · · , n, are all distinct v ectors with Hamming w eight k , whic h are sorted, from top to b ottom, in the decreasing lexicographic order. W e pro v e by induction on n . The abov e assertion is t r ivially true for n = 1. Assume the assertion holds for n − 1 ≥ 1. Now for n , A ( n ) 0 is a v ector o f Hamming w eight 0 and A ( n ) n a ve ctor of Hamming w eigh t n , so the assertion trivially holds. F or 1 ≤ k ≤ n − 1 , the first  n − 1 k − 1  v ectors of A ( n ) k are all distinct and ha ve Hamming w eigh t k b y the induction assumption that all ro w v ectors in A ( n − 1) k − 1 are distinct and ha v e Hamming w eigh t k − 1. F urther, these vectors are in the dec reasing lexicographic order b ecause they share the same rightmos t bit and all ve ctors in A ( n − 1) k are sorted. By the same arg ument the next  n − 1 k  v ectors of A ( n ) k are distinct, o f Hamming w eight k , and sorted in the decreasing lexic ographic order as w ell. Com bining the ab ov e facts and (7) conclude s that the row vec tors of A ( n ) k are dis tinct, of Hamming w eigh t k , and in the decreasing lexicographic order. ✷ Definition 8 Given a gr aph G = ( V , E ) , for two vertex subsets V 1 ⊆ V and V 2 ⊆ V numb er e d by numb erings η 1 and η 2 r esp e ctively, the adjacency matrix 3 of V 1 and V 2 is a | V 1 | × | V 2 | ma trix M such that fo r any u ∈ V 1 and v ∈ V 2 M ( η 1 ( u ) , η 2 ( v )) =      1 if { u, v } ∈ E ; 0 otherwise. (8) Definition 9 The bandwidth of an s × t matrix M is the maximum absolute value of the differ enc e b etwe en the r ow and c olumn indic es of a nonzer o ele ment of that matrix, i.e. bw ( M ) = max 1 ≤ i ≤ s, 1 ≤ j ≤ t {| i − j | | M ( i, j ) 6 = 0 } . (9) Remark 10 The b andwidth of a numb ering η of a gr aph G is e qual to the b andwidth of the adjac ency ma trix of G numb er e d by η . The bandwidth o f a square matrix is obvious ly the maxim um Manhattan distance from a nonzero elemen t to the main diagonal of the matrix. Definition 11 F or an s × t matrix M , its Manhattan r adius r ( M ) is define d by r ( M ) = max 1 ≤ i ≤ s, 1 ≤ j ≤ t { s − i + j | M ( i, j ) 6 = 0 } , (10) which is the maximum Manhattan distanc e fr om a nonzer o e l e ment of M to the p osition imme diately to the left of the b ottom-le f t c orner of m atrix M (an imaginary matrix element M ( s, 0) ), as shown in Fig. 1. This imaginary matrix element M ( s , 0) is c al le d the anchor of M . 0 1 0 1 0 1 0 0 0 0 1 0 0 1 1 0 1 1 0 0 0 1 1 0 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 0 1 0 1 0 Anc hor Fig. 1. Ma nhattan radius and anc hor of a m atrix. Let M ( n ) b e the 2 n × 2 n adjacency matrix of Q ( n ) n umbered by H ( n ) . Recall from Lemma 7 that matrix S ( n ) has as ro ws a ll v ertices of Q ( n ) sorted b y H ( n ) . Consider the submatrices A ( n ) k and A ( n ) k ′ in S ( n ) , and let the  n k  ×  n k ′  matrix M ( n ) k ,k ′ b e the adjacency matr ix b et we en A ( n ) k and A ( n ) k ′ . Then M ( n ) k ,k ′ , 0 ≤ k , k ′ ≤ n , f orm the 2 n × 2 n adjacency matrix of the Hales n um b ered h yp ercub e: M ( n ) = [ M ( n ) k ,k ′ ]. Obviously , M ( n ) k ,k ′ is an all-zero matrix if | k − k ′ | 6 = 1. 4 Therefore, w e hav e M ( n ) =                  0 M ( n ) 0 , 1 0 · · · 0 0 M ( n ) 1 , 0 0 M ( n ) 1 , 2 · · · 0 0 0 M ( n ) 2 , 1 0 · · · 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 · · · 0 M ( n ) n − 1 ,n 0 0 0 · · · M ( n ) n,n − 1 0                  . (11) The bandwidth of M ( n ) equals to the ma ximum Manhattan distance from a nonzero elemen t of M ( n ) to the ma in diagonal of M ( n ) . Because of the sym- metry of M ( n ) , the bandwidth of M ( n ) is equal to the maximum Manhattan distance from a nonzero elemen t of M ( n ) k ,k +1 , k = 0 , 1 , · · · , n − 1, to the main diagonal of M ( n ) . Not e that the anc hors of M ( n ) k ,k +1 are all on the main diag o nal. Therefore, b y Definition 11 the bandwidth of M ( n ) can be expressed in terms of Manhattan radii of M ( n ) k ,k +1 : bw ( M ( n ) ) = max k =0 , ··· ,n − 1 r ( M ( n ) k ,k +1 ) . (12) A pleasing recurrence structure of the Manhatta n radius r ( M ( n ) k ,k +1 ) affords us the follo wing pro of of Theorem 6. PR OOF. [Pro of of Theorem 6] Becaus e of T heorem 5 and Remark 10, w e only need to sho w that the bandwidth of the adjacency matrix of Q ( n ) with the Hales n umbering H ( n ) satisfies (5). Rewrite (7) as, A ( n ) k =    A ( n − 1) k − 1 1 A ( n − 1) k 0    and A ( n ) k +1 =    A ( n − 1) k 1 A ( n − 1) k +1 0    , k = 1 , 2 , · · · , n − 1 . (13) Then M ( n ) k ,k +1 , the adjacency matrix b etw een A ( n ) k and A ( n ) k +1 , can b e divided into four submatrices. The top-left one is the adjacency matrix betw een [ A ( n − 1) k − 1 1 ] and [ A ( n − 1) k 1 ], whic h equals to the adjacency matrix b et w een A ( n − 1) k − 1 and A ( n − 1) k , i.e. M ( n − 1) k − 1 ,k . Similarly , t he bot t om-righ t one is M ( n − 1) k ,k +1 . Because there is no pair of Hamming distance one betw een A ( n − 1) k − 1 and A ( n − 1) k +1 , the top-right submatrix is an all-zero matrix. The bo ttom-left submatrix is the a djacenc y matrix b et ween [ A ( n − 1) k 0 ] and [ A ( n − 1) k 1 ], whic h is an iden tity matrix I ( n − 1 k ) of 5 dimension  n − 1 k  . Namely , M ( n ) k ,k +1 =    M ( n − 1) k − 1 ,k 0 I ( n − 1 k ) M ( n − 1) k ,k +1    , k = 1 , 2 , · · · , n − 1 . (14) Because A ( n ) 0 is the all zero v ector and A ( n ) 1 con ta ins n v ectors of Hamming w eight 1, w e hav e M ( n ) 0 , 1 = 1 T . 1 1 1 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 1 0 0 1 1 r ( M ( n − 1) k − 1 , k )  n − 1 k  r ( I ( n − 1 k ) ) r ( M ( n − 1) k,k +1 )  n − 1 k  Fig. 2. The recurs iv e structure of r ( M ( n ) k ,k +1 ), wh ere n = 5 and k = 2. It follow s from (1 4 ) that the Manhattan radius r ( M ( n ) k ,k +1 ) equals to the max- im um Manhattan distance from a nonzero elemen t in submatrices M ( n − 1) k − 1 ,k , M ( n − 1) k ,k +1 or I ( n − 1 k ) to the anc hor o f M ( n ) k ,k +1 , as illustrated in F ig ure 2. F rom the prop erty of Manhattan distance and the f act that r ( · ) > 0, w e hav e for k = 1 , 2 , · · · , n − 1, r ( M ( n ) k ,k +1 ) = max ( n − 1 k ! + r ( M ( n − 1) k − 1 ,k ) , n − 1 k ! + r ( M ( n − 1) k ,k +1 ) , r ( M ( n − 1) k ,k +1 ) ) = n − 1 k ! + max ( r ( M ( n − 1) k − 1 ,k ) , r ( M ( n − 1) k ,k +1 ) ) , (15) and r ( M ( n ) 0 , 1 ) = 1 b ecaus e M ( n ) 0 , 1 = 1 T . No w we pr ov e Theorem 6 by induction on n . It is trivial that when n = 1, r ( M (1) 0 , 1 ) = 1 =  0 0  , and 0 = ⌊ 1 − 1 2 ⌋ . Assume that r ( M ( n − 1) k ,k +1 ) ≤ n − 2 X m =0 m ⌊ m 2 ⌋ ! , k = 0 , 1 , · · · , n − 2 , (16) 6 where equalit y holds if k = ⌊ n − 1 2 ⌋ . Then w e hav e r ( M ( n ) k ,k +1 ) = n − 1 k ! + max ( r ( M ( n − 1) k − 1 ,k ) , r ( M ( n − 1) k ,k +1 ) ) ≤ n − 1 ⌊ n 2 ⌋ ! + n − 2 X m =0 m ⌊ m 2 ⌋ ! = n − 1 X m =0 m ⌊ m 2 ⌋ ! , (17) in which the equalit y holds if k = ⌊ n 2 ⌋ , b ecause when n is eve n, ⌊ n 2 ⌋− 1 = ⌊ n − 1 2 ⌋ , so r ( M ( n − 1) k − 1 ,k ) achiev es equalit y in (16); when n is o dd, ⌊ n 2 ⌋ = ⌊ n − 1 2 ⌋ , r ( M ( n − 1) k ,k +1 ) also achiev es equalit y in (1 6 ). F ro m (4), (12) and (17) , Theorem 6 f ollo ws. ✷ 3 An t iban dwidth problem Another v ertex num b ering problem related to graph bandwidth is what we call an tibandwidth problem. It is po se d b y rev ersing the ob jectiv e o f v ertex n umbering in that w e now w ant to maximize the minim um distance b et we en an y adjacen t pair of v ertices. Definition 12 The antib andwidth pr o b l e m of a gr aph G = ( V , E ) is define d as f ( G ) = max η min { v,w }∈ E | η ( v ) − η ( w ) | , (18) wher e η is a numb e ri n g of G . The antibandwid th problem has applications in co de design for comm unica- tions [3]. On h yp ercub es the antibandwidth problem has a v ery simple solution due to Harp er [5]. Corollary 13 (Harp er, [5]) F or the n -cub e, first numb er the vertic es with even Hammi n g w e ights and then numb er the vertic es with o dd Hamming weights, in the Hales or der. The r esulting numb ering achieves f ( G ) . PR OOF. See [5]. ✷ In t his section, w e pro vide a close form form ula for the solution of the an- tibandwidth problem on n -cub es, whic h is a new result. Theorem 14 F or the n -cub e Q ( n ) , we have f ( Q ( n ) ) = 2 n − 1 − n − 2 X m =0 m ⌊ m 2 ⌋ ! . (19) 7 PR OOF. The n um b ering described in Corollary 13 determines a new order- ing of v ertices ˜ S ( n ) =                          A ( n ) 0 A ( n ) 2 . . . A ( n ) 2 ⌊ n 2 ⌋ A ( n ) 1 A ( n ) 3 . . . A ( n ) 2 ⌊ n − 1 2 ⌋ +1                          . (20) Similar to the pro of of Theorem 6, w e ha ve the adjacency matrix of Q ( n ) with the v ertices n um b ered in the order of (20) ˜ M ( n ) =                          0 0 · · · 0 M ( n ) 0 , 1 0 · · · 0 0 0 · · · 0 M ( n ) 2 , 1 M ( n ) 2 , 3 · · · 0 . . . . . . . . . . . . . . . . . . . . . . . . 0 0 · · · 0 0 0 · · · M ( n ) 2 ⌊ n 2 ⌋ , 2 ⌊ n − 1 2 ⌋ +1 M ( n ) 1 , 0 M ( n ) 1 , 2 · · · 0 0 0 · · · 0 0 M ( n ) 3 , 2 · · · 0 0 0 · · · 0 . . . . . . . . . . . . . . . . . . . . . . . . 0 0 · · · M ( n ) 2 ⌊ n − 1 2 ⌋ +1 , 2 ⌊ n 2 ⌋ 0 0 · · · 0                          . (21) F rom the symmetric structure of ˜ M ( n ) , w e only tak e in to accoun t the lo w er part of the matrix ˜ M ( n ) . Then w e hav e f ( ˜ H ( n ) ) = min ( min k =1 , 3 , ··· , 2 ⌊ n − 1 2 ⌋ +1 δ k ,k − 1 , min k =1 , 3 , ··· , 2 ⌊ n 2 ⌋− 1 δ k ,k +1 ) , (22) where δ k ,k ′ is the minim um Manhattan distance from a nonzero elemen t in the submatrix M ( n ) k ,k ′ to the main diagonal of ˜ M ( n ) . T ak e in t o account a ro w of submatrices M ( n ) k ,κ , κ = k + 1 , k + 3 , · · · , 2 ⌊ n 2 ⌋ , 1 , 3 , · · · , k a nd apply the prop ert y of Manhattan distance, w e hav e δ k ,k +1 = X κ = k +1 ,k +3 , · ·· , 2 ⌊ n 2 ⌋ , 1 , 3 , ··· , k W ( M ( n ) κ,k ) − r ( M ( n ) k ,k +1 ) , k = 1 , 3 , · · · , 2 ⌊ n − 1 2 ⌋ +1 (23) 8 where W ( · ) is the width of a matrix and hence W ( M ( n ) k ,κ ) =  n κ  . Therefore δ k ,k +1 = n 1 ! + n 3 ! + · · · + n k ! + n k + 1 ! + n k + 3 ! + · · · + n 2 ⌊ n 2 ⌋ ! − r ( M ( n ) k ,k +1 ) . (24) Similarly , w e hav e for k = 1 , 3 , · · · , 2 ⌊ n 2 ⌋ − 1, δ k ,k − 1 = n 1 ! + n 3 ! + · · · + n k ! + n k − 1 ! + n k + 1 ! + · · · + n 2 ⌊ n 2 ⌋ ! − r ( M ( n ) k ,k − 1 ) . (25) F rom (7) and similar to the analysis of (14), w e derive the recursion form M ( n ) k ,k − 1 =    M ( n − 1) k − 1 ,k − 2 I ( n − 1 k − 1 ) 0 M ( n − 1) k ,k − 1 .    , k = 2 , · · · , n, (26) where as the base case M ( n ) 1 , 0 = 1 . The zero matrix at the b ottom-left corner has dimension  n − 1 k  ×  n − 1 k − 2  . Therefore, r ( M ( n ) k ,k − 1 ) = n − 1 k ! + n − 1 k − 2 ! + r ( I ( n − 1 k − 1 ) ) = n − 1 k ! + n − 1 k − 2 ! + n − 1 k − 1 ! . (27) Substituting r ( M ( n ) k ,k − 1 ) of (27) in to (2 5), w e hav e δ k ,k − 1 = n 1 ! + n 3 ! + · · · + n k ! + n k − 1 ! + n k + 1 ! + · · · + n 2 ⌊ n 2 ⌋ ! − n − 1 k ! − n − 1 k − 2 ! − n − 1 k − 1 ! =2 n − 1 , (28) whic h can be easily establis hed by considering the parit y of n and using the binomial coefficien ts relations  n k  =  n − 1 k − 1  +  n − 1 k  and P k =0 , ··· ,n − 1  n − 1 k  = 2 n − 1 . 9 Using the same justification and substituting r ( M ( n ) k ,k +1 ) of (1 7) into ( 2 4), w e ha v e δ k ,k +1 = n 1 ! + n 3 ! + · · · + n k ! + n k + 1 ! + n k + 3 ! + · · · + n 2 ⌊ n 2 ⌋ ! − n − 1 k ! − max ( r ( M ( n − 1) k − 1 ,k ) , r ( M ( n − 1) k ,k +1 ) ) =2 n − 1 − max ( r ( M ( n − 1) k − 1 ,k ) , r ( M ( n − 1) k ,k +1 ) ) ≥ 2 n − 1 − n − 2 X m =0 m ⌊ m 2 ⌋ ! , (29) where equalit y holds when k = ⌊ n − 1 2 ⌋ + 1 or ⌊ n − 1 2 ⌋ , whic hev er b eing o dd. Com bining (22), (28) and (29) completes the pro of. ✷ References [1] P . Z. Chinn , J. 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