d-dimensional Arrangement Revisited

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1 -Dimensional Arrangement Revisite Daniel Rotter Jens Vygen Research Institute for Discrete Mathematics University of Bonn Revise version: April 5, 013 Abstract We revisit the -imensional arrangement problem an analyze the performance ratios of previously propose algorithms base on the linear arrangement problem with -imensional cost. The two problems are relate via space-filling curves an recursive balance bipartitioning. We prove that the worst-case ratio of the optimum solutions of these problems is Θ(log n), where n is the number of vertices of the graph. This invaliates two previously publishe proofs of approximation ratios for -imensional arrangement. Furthermore, we conclue that the currently best known approximation ratio for this problem is O(log n). 1 Introuction We revisit the -imensional arrangement problem (-imap) for N: given an unirecte graph G = (V (G), E(G)) an an integer k V (G), fin an injection p : V (G) {1,..., k} minimizing p(v) p(w) 1. Throughout this paper, we write n = V (G) an m = E(G), an is a fixe constant. The case = is an interesting (though simplifie) moel of VLSI placement. Alreay the case = 1, known as the Optimal Linear Arrangement Problem, is NP-har (Garey, Johnson an Stockmeyer [1976]). The currently best known approximation guarantee is O( log n log log n), ue to Charikar et al. [010] an Feige an Lee [007] (improving an earlier result of Rao an Richa [00]). For the general case, Hansen [1989] sketche an algorithm that recursively bipartitions the vertex set using an algorithm propose by Leighton an Rao [1999]. The Leighton-Rao algorithm computes a c-balance cut (i.e., the set of eges with exactly one enpoint in U for a set U V (G) with cn U (1 c)n) that is at most O(log n) times larger than a minimum c -balance cut, for some constants 0 < c < c < 1. This can lea to 1

2 an O(log n)-approximation algorithm for -imap, although Hansen i not give a full proof. The Leighton-Rao result was improve by Arora, Rao an Vazirani [009], who obtaine O( log n) instea of O(log n). Arora, Hazan an Kale [010] obtaine the same ratio by a faster algorithm. Using this algorithm for the recursive bipartitioning improves Hansen s result by a factor of O( log n). Even et al. [000] presente an O(log n log log n)-approximation algorithm for the linear arrangement problem with -imensional cost (-LAP): given a graph G, fin a bijection p : V (G) {1,..., n} such that p(v) p(w) is minimize. Charikar, Makarychev an Makarychev [007] use the result of Arora, Rao an Vazirani [009] to obtain an O( log n)-approximation algorithm for -LAP for any. Both, Even et al. [000] an Charikar, Makarychev an Makarychev [007], claime that their approximation algorithm for -LAP implies an approximation algorithm for -imap with the same performance ratio for every fixe. The iea, propose by Even et al. [000], is to transform the linear arrangement into a -imensional arrangement accoring to a iscrete space-filling curve; this is essentially [Even et al. [000], Lemma 1] (except that they i not aress the case n < k explicitly): Lemma 1 (essentially Even et al. [000], Lemma 1) For any n,, k N with n k, there exists an injection p : {1,..., n} {1,..., k} such that p(i) p(j) 1 ( + 1) i j for all i, j {1,..., n}, an such a mapping can be compute in O(n( + log n) + log k) time. Our proof follows Even et al. [000], but contains an explicit construction of a suitable space-filling curve through the -imensional gri, also in the case n < k. Proof: Let s := log n an l := s. Consier the s-th step of the construction of the -imensional version of Hilbert s [1891] space-filling curve (see Sagan [199]), say q : {1,..., l } {1,..., l}. For any i, j {1,..., l } with i j let t = log i j ; then (t 1) < i j t an hence q(i) q(j) 1 < ( + 1) t < ( + 1) i j. Let k = min{k, l} an S = { li/k : i = 1,..., k }. Writing q(j) = (q 1 (j),..., q (j)), we finally set p(j) := ( k q 1 (j )/l,..., k q (j )/l ) for j = 1,..., n, where j = min{i : {q(1),..., q(i)} S = j}. Note that p is injective an p(i) p(j) 1 q(i ) q(j ) 1 ( + 1) i j ( + 1) i j = ( + 1) i j for any i an j. See Figure 1 for an example. Hence, for any graph, any solution of -LAP can be transforme to a solution of -imap such that the cost increases at most by a factor ( + 1). However, this transformation oes not preserve the approximation ratio, as we point out in this note. This is because the optimum value of -LAP is not boune by a constant factor times the optimum value of -imap. A factor Θ(log n) is lost because of the following theorem, our main result:

3 (a) Hilbert s curve q for = an s = 3. (b) The resulting injection p for =, n = 3, an k = 5. Figure 1: left: Hilbert s curve q for = an s = 3; right: the resulting injection p for =, n = 3, an k = 5. The figure shows the graph with eges {q(i), q(i + 1)}, i = 1,..., 63 on the left an the graph with eges {p(i), p(i + 1)}, i = 1,..., on the right. Note that p results from q by consiering only the points in S (which here means erasing the secon, fifth, an seventh row an column), an omitting the last k n points. Theorem Let N,. For any graph G an any injection p : V (G) {1,..., k} (where k N), there exists a bijection q : V (G) {1,..., n} such that q(v) q(w) O(log n) There are pairs (G, p) for which this boun is tight. p(v) p(w) 1. Consequently, the analysis of the algorithms of Even et al. [000] an Charikar, Makarychev an Makarychev [007] only yiels approximation ratios of O(log n log log n) an O(log n log n), respectively. However, a ifferent proof (see Section ) shows that the algorithm of Even et al. [000] oes inee achieve the claime performance ratio O(log n log log n). Moreover, from a result of Fakcharoenphol, Rao an Talwar [00] we can euce the currently best known approximation ratio of O(log n); this will be shown in Section. 3

4 Banerjee et al. [009] suggeste a similar algorithm for =. They claime an approximation ratio of O( log n m log log n) (an a weaker ratio for hypergraphs). Unfortunately, their proof contains an error, too (the complete graph is a counterexample to [Banerjee et al. [009], Lemma ]). However, the claime approximation ratio is anyway worse than the trivial O( m), which is obtaine by an arbitrary injection of the non-isolate vertices to {1,..., m }. We note that -imap is not known to be MAXSNP-har for any N (but see Ambühl, Mastrolilli an Svensson [011] an Devanur et al. [006]). The next two sections contain a proof of Theorem. Upper Boun We first consier the irection neee for proving approximation ratios for -imap via -LAP an space-filling curves. Lemma 3 For any graph G an any injection p : V (G) {1,..., k} (where k, N), there exists a bijection q : V (G) {1,..., n} such that q(v) q(w) 3 ln n p(v) p(w) 1. This essentially generalizes [Charikar, Makarychev an Makarychev [007], Theorem.1, part I] (they consier the case where p is a one-imensional bijection). The following proof is inspire by theirs, but the analysis is more involve. The basic iea is to partition the vertex set recursively. In each iteration, large vertex sets are partitione into two sets of approximately equal size (up to a constant factor) accoring to their j-th coorinates in p, where j changes in each iteration. Then all vertices in one set will precee all vertices in the other set in q. Proof: We write p(v) = (p 1 (v),..., p (v)) for v V (G). Let γ = 1. Note that 1 γ = (1 γ) 1 i=0 γi (1 γ) = 1 an 1 < γ < 1. We construct q as follows. Let i := 1 an r 1 (v) := 0 for all v V (G). Repeat the following until (r 1 (v),..., r i (v)) (r 1 (w),..., r i (w)) for all v, w V (G) with v w. At termination, the lexicographical orer of these vectors etermines q. In iteration i we will consier coorinates p ji (v) for v V (G), where j i = 1+(i mo ).

5 v v 5 v v 5 b (0) 3 v 3 v 6 v 9 v 3 v 6 v 9 v 1 v v 1 v a (01) x b (01) a (0) = x 1 v 7 v 10 v 7 v 10 0 v (a) Iteration 1 v 8 a (00), b (00) (b) Iteration Figure : Example of the first two iterations of the algorithm efine in the proof of Lemma 3. Here, n = 10 an =, hence γ = an γ = In iteration, we have X (0,0) = 1 < γi n/ Hence, we will not partition V (0,0) = {v 7, v 8, v 10 } in iteration. 1 We write S i = {(r 1 (v),..., r i (v)) : v V (G)}. For each s S i let V s = {v V (G) : (r 1 (v),..., r i (v)) = s}, a s = min{x : {v V s : p ji (v) x + 1} γ V s }, b s = max{x : {v V s : p ji (v) x} γ V s }, X s = {a s,..., b s }. Note that b s + 1 a s because {v V s : p ji (v) a s } + {v V s : p ji (v) b s + 1} > γ V s + γ V s > V s. Let us sketch the iea behin these efinitions. Partitioning V s into the set of vertices for which the j i -th coorinate is at most x an the rest yiels sufficiently small parts if x X s. However, we will only perform such a partitioning step if X s is sufficiently large, an then we will pick a coorinate x X s that yiels the smallest cut. If V s is large, then X s will be large for at least one coorinate, an so we will make progress after at most iterations. We will now give the etails. γ i n/, then we set r i+1 (v) := 0 for all v V s. There are two cases. If X s 1 Otherwise, we split V s : for x X s an v V s let r x (v) := 0 if p ji (v) x an r x (v) := 1 otherwise. Choose x X s such that rx (v) r x (w) is minimize, an set r i+1 (v) := r x (v) for all v V s. 5

6 After oing this for each s S i, we increment i. This ens the escription of the proceure that ultimately efines q. See Figure for an illustration. To see that this proceure terminates, we prove that { } V s max 1, γ i n (1) for any s S i an any iteration i. This is trivial for i. We procee by inuction. Let i > an s S i. For 1 h < i let s h enote the prefix of s of length h, i.e., the vector resulting from s by omitting the last i h components. Case 1: V s V s i. Then the set V s resulte from splitting uring at least one of the iterations i,..., i 1. Then V s γ V s i. Since s i S i, we are one by inuction. Case : V s = V s i. Then the set V s was not split uring any of the iterations h {i,..., i 1}. This implies b s h + 1 a s h = X s h 1 γ h n/ 1 γ i n/ for h = i,..., i 1. Moreover, by the choice of a s h an b s h, we have {v V s : p jh (v) < a s h} < (1 γ ) V s an {v V s : p jh (v) > b s h + 1} < (1 γ ) V s. Combining this for h = i,..., i 1 yiels V s {v V s : a s h p jh (v) b s h + 1 for h = i,..., i 1} + an hence If 1 If 1 i 1 h=i ( {v V s : p jh (v) > b s h + 1} + {v V s : p jh (v) < a s h} ) < (1 γ ) V s + {v V s : a s h p jh (v) b s h + 1 for h = i,..., i 1} 1 V s + i 1 h=i (b s h + a s h), V s < i 1 h=i (b s h + a s h). γ i n/ < 1, we have b s h +1 = a s h for h = i,..., i 1, an conclue V s <. γ i n/ 1, we have i 1 h=i (b s h + a s h) an conclue V s < γ i n. ( γ n/) i ( γ n/) i = γ i n/, 6

7 In both cases (1) is prove. Let t enote the inex i of the last iteration; then V s = 1 for all s S t+1. From (1) we immeiately get t log 1/γ (γ n) = +log 1/γ n + ln n. Next, we compute an upper boun on the number of eges separate in one partitioning step, in iteration i for s S i with X s > 1 γ i n/: r i+1 (v) r i+1 (w) 1 X s = 1 X s 1 X s < x X s γ i n/ r x (v) r x (w) {x X s : p ji (v) x < p ji (w) or p ji (w) x < p ji (v)} p ji (v) p ji (w) p ji (v) p ji (w). () For an ege e = {v, w} let i e be the smallest i such that r i+1 (v) r i+1 (w) iffer (i.e., i e is the inex of the iteration in which e is separate). Then, both enpoints of e are in V (r1 (v),...,r ie (v)), an these vertices are place consecutively in q (see Figure 3 for an illustration). Hence, using (1), q(v) q(w) < = 1 γ = 1 γ e= e= t t < γ γ V (r1 (v),...,r ie (v)) 1 γ i e n γ i n {e E(G) : i e = i} γ i n t s S i s S i t 7 r i+1 (v) r i+1 (w) p ji (v) p ji (w) p ji (v) p ji (w)

8 i = 1 v 1,..., v 10 X s = {1,, 3}, x = 1 i = v 7, v 8, v 10 X s = {3} v 1, v, v 3, v, v 5, v 6, v 9 X s = {0, 1,, 3}, x = i = 3 v 7, v 8, v 10 X s = {0} v 1, v, v 3, v X s = {, 3}, x = v 5, v 6, v 9 X s = {3} i = v 7, v 8, v 10 X s = {3}, x = 3 v 1, v 3, v X s = {0, 1}, x = 0 v v 5, v 6, v 9 X s = {3}, x = 3 i = 5 v 7, v 8 X s = {0}, x = 0 v 10 v 1 v 3, v X s = {}, x = v v 5, v 6 X s = {3}, x = 3 v 9 i = 6 v 8 (000000) v 7 (000001) v 10 (000010) v 1 (010000) v (010010) v 3 (010011) v (010100) v 6 (011000) v 5 (011001) v 9 (011010) Figure 3: Visualization of the hierarchical ecomposition ({V s : s S i }),...,t constructe in the proof of Theorem 3 on the instance of Figure. The resulting linear orer q is the left-to-right orer inicate at the bottom of the figure. γ t p(v) p(w) 1 < ln n p(v) p(w) ln n p(v) p(w) 1. Charikar, Makarychev an Makarychev [007] calle a sequence P 0, P 1,..., P t of partitions of V (G) a hierarchical ecomposition if P 0 = {V (G)}, P t = {{v} : v V (G)}, an P i+1 is a refinement of P i for each i = 1,..., t 1. For a constant 0 < b < 1, a hierarchical ecomposition is calle b-balance if C b i n for each C P i. We remark that the sequence ({V s : s S i }),+1,+,...,t efine in the proof of Lemma 3 is a γ-balance hierarchical ecomposition (see Figure 3). 8

9 3 Lower Boun We will now show that the boun is tight up to a factor that only epens on. The graphs that we will consier are -imensional gris themselves: let G k be given by V (G k ) = {1,..., k} an E(G k ) = {{x, y} : x, y V (G k ), x y 1 = 1}. Note that the ientity function p embes V (G k ) in itself with {v,w} E(G k ) p(v) p(w) 1 = E(G k ) = (k k 1 ) < n. Therefore, the following lemma shows the lower boun, an hence, with Lemma 3, implies Theorem. Lemma Let. If q : V (G k ) {1,..., n} is any bijection, then {v,w} E(G k ) ( q(v) q(w) > 3 16 ) 1 )( ) 1/ ) n log n 3n 6. Proof: Let G = G k, an let q : V (G) {1,..., n} be any bijection. Apply the proceure in the proof of Lemma 3 to q (in the role of p, for imension = 1) to compute vectors (r 1 (v),..., r t+1 (v)) for v V (G) with r i+1 (v) r i+1 (w) < ( 3 )i n for i = 1,..., t an s S i (cf. inequality (); note that γ = 3 ). Hence, e E(G) ( 3 ie ) n = = < = t ( 3 i ) n {e E(G) : ie = i} t ( 3 ) i n s S i t ( 3 ) i n < ( 3 ) 1 ) 1 ( 3 s S i q(v) q(w) ) i n q(v) q(w) r i+1 (v) r i+1 (w) ( ( 1 3 ) ie n) q(v) q(w) i e ( ) ( 3 ) i ie 1 q(v) q(w). (3) 9

10 The last inequality hols because for e = {v, w} E(G) there is an s S ie with v, w V s an q(v) q(w) < V s ( 3 ie 1 ( ) n (cf. inequality (1), implying 3 ie ) n > 3 q(v) q(w). A subgraph of G with c vertices has at most (c c 1 1/ ) eges, an this is tight if the subgraph is inuce by a prouct of intervals of length c 1/. There are at most i 1 subgraphs G[V s ], s S i, each with at most ( 3 i 1 ) n vertices. Therefore, {e E(G) : i e i} = E(G) s S i E(G[V s ]) (n k 1 ) ( V (G[V s ]) V (G[V s ]) 1 1/) s S i = s S i V (G[V s ]) 1 1/ k 1 = n ( ( ( 3 i 1 3 ) ) i 1 1 1/ n k 1 ) n ( ( 3 ) (1 i)/ 1 ) k 1, an hence, e E(G) ( 3 ie ) n = t ( ( 3 ) ) i ( n 3 ) i+1 n {e E(G) : i e i} t ( ( 3 ) ) i ( (( n 3 i+1 ) n 3 ) ) (1 i)/ 1 k 1 ( 1 ( ) 3 1/ ( 3 ) 1/ t 1 ) i=0 ( ( 1 ( ( ) 3 1/ ( 3 ) 1/ ) t (( 1 ( ) ) 3 i/ n ) 1 ) 1/ n ) 1/ ) log n 1) ( 3 ) 1/ n. () 10

11 The inequalities (3) an () imply q(v) q(w) > ( 3 ) 1/ ( ) 1 3 > 3 16 n n ( ( ( ( ) 1 )( ) 1 )( ) ) ( 1/ log n ) ) ) 1/ log n 1. ) 1 ) ) Approximation Algorithms After showing him the above proofs, Guy Even [personal communication, 011] sent us a sketch of a revise proof of the performance ratio of the -imensional arrangement algorithm of Even et al. [000]. This algorithm begins by solving the following linear program (cf. [Even et al. [000], page 606]): min s.t. l(v, w) (5) l(u, v) u U ( U 1)1+1/ U V (G), v U (6) l(u, v) + l(v, w) l(u, w) u, v, w V (G) (7) l(v, w) 0 v, w V (G) (8) An optimum solution l : V (G) V (G) R 0 of this LP can be foun in polynomial time [Even et al. [000], Section 6.1]. The following lemma strengthens [Even et al. [000], Lemma 1] by showing that the LP (5) (8) constitutes a lower boun for the cost of any -imensional arrangement, up to a constant factor. Lemma 5 (Guy Even, personal communication 011) Let q be an optimum solution to -imap. Then l(v, w) := ( + 1)( 1)! q (v) q (w) 1 for v, w V (G) efines a feasible solution to the LP (5) (8). Proof: Since (7) an (8) hol eviently, we prove (6). Let = U V (G) an v U, w.l.o.g. q (v) = 0. If U 6, then, q (u) 1 U 1 > 1 6 ( U 1)1+1/ u U 1 +1 ( + 1)( 1)! ( U 1)1+1/. 11

12 If U > 6, then let R := U / 1 1 U 1 an S(, r) := {x Z : x 1 = r} for r N. Observe that {x S(, r) : x 0} = ( ) r+ 1 1, an thus, Since we have (r + 1) 1 ( 1)! ( ) r R S(, r) r=1 ( ) r + 1 S(, r) 1 (r + 1) 1. R (r + 1) 1 R(R + 1) 1 U 1, r=1 q (u) 1 u U = R r=1 u S(,r) R r S(, r) r=1 R r=1 R 0 r ( 1)! u 1 x ( 1)! x R +1 ( + 1)( 1)! 1 +1 ( + 1)( 1)! ( U 1)1+1/ The algorithm of Even et al. [000] computes a solution within an O(log n log log n) factor of the cost of an optimum solution to LP (5) (8) an hence, of the cost of q (Lemma 5). Therefore, the result in that paper (though not its original proof) is correct. Now we show that we can even get an O(log n)-approximation algorithm. To this en, we use a result of Fakcharoenphol, Rao an Talwar [00], who showe how to approximate an arbitrary metric by a special kin of tree metric: We call a tree T together with a vertex r V (T ) an a weight function c : E(T ) R 0 -hierarchically well separate if there exists a constant γ > 0 such that c(e) = γ h, where h is the number of eges in the unique path starting in r an ening with e. This inuces a metric l : V (T ) R 0, where l (v, w) is the weight of the v-w-path in (T, c). See Figure. 1

13 r u w T u x v y Figure : Illustration of a tree metric efine by a -hierarchically well separate tree (T, r, c). Here, γ = 8, l (v, w) = 5 an l (x, y) = 6. The function q efine in the proof of Theorem 7 orers the leaves from left to right. Lemma 6 (Fakcharoenphol, Rao an Talwar [00]) Let G be a graph with n vertices an l : V (G) V (G) R 0 a metric. Then one can compute in polynomial time a -hierarchically well separate tree (T, r, c) such that V (G) is the set of leaves of T an the inuce tree metric l satisfies the following properties: (a) l (v, w) l(v, w) for all v, w V (G); an (b) l (v, w) O(log n) l(v, w). We conclue: Theorem 7 There is an O(log n)-approximation algorithm for -imap. Proof: Let l be an optimum solution to the LP (5) (8). Let (T, r, c) an l be as efine in Lemma 6. For u V (T ) let T u enote the set of leaves v V (G) such that the r-v-path in T contains u. Define a bijection q : V (G) {1,..., n} such that for all u V (T ) the elements of T u are numbere consecutively. Let {v, w} E(G) an u be the unique vertex with v, w T u an T u maximal (see Figure for an illustration). Due to the spreaing constraints (6), there are x, y T u such that l(x, y) 1 Tu 1. Note that l (x, y) l (v, w) since (T, r, c) is -hierarchically well separate. Therefore, q(v) q(w) T u 1 l(x, y) l (x, y) 8 l (v, w), 13

14 an hence, q(v) q(w) 8 The result now follows from Lemma 1 an 5. l (v, w) O(log n) l(v, w). We consiere the unweighte version of -imap in this paper, but only to simplify the exposition. It is straightforwar that all results also hol for the weighte version (where nonnegative ege weights are given an the weighte sum is minimize). Acknowlegement We thank Guy Even for Lemma 5 an the anonymous reviewers for careful reaing an their suggestions. References Ambühl, C., Mastrolilli, M., an Svensson, O. [011]: Inapproximability results for maximum ege biclique, minimum linear arrangement, an sparsest cut. SIAM Journal on Computing 0 (011), Arora, S., Hazan, E., an Kale, S. [010]: O( log n) approximation to Sparsest Cut in Õ(n ) time. SIAM Journal on Computing 39 (010), Arora, S., Rao, S., an Vazirani, U. [009]: Expaner flows, geometric embeings an graph partitioning. Journal of the ACM 56 (009), Article 5 Banerjee, P., Sur-Kolay, S., Bishnu, A., Das, S., Nany, S.C., an Bhattacharjee, S. [009]: FPGA placement using space-filling curves: theory meets practice. ACM Transactions on Embee Computing Systems 9 (009), Article 1 Charikar, M., Hajiaghayi, M.T., Karloff, H. an Rao, S. [010]: l spreaing metrics for vertex orering problems. Algorithmica 56 (010), Charikar, M., Makarychev, K., an Makarychev, Y. [007]: A ivie an conquer algorithm for -imenisonal arrangement. Proceeings of the 18th ACM-SIAM Symposium on Discrete Algorithms (007), Devanur, N, Khot, S., Saket, R., an Vishnoi, N. [006]: On the harness of minimum linear arrangement. Manuscript, 006 Even, G., Naor, J., Rao, S., an Schieber, B. [000]: Divie-an-conquer approximation algorithms via spreaing metrics. Journal of the ACM 7 (000),

15 Fakcharoenphol, J., Rao, S., an Talwar, K. [00]: A tight boun on approximating arbitrary metrics by tree metrics. Journal of Computer an System Sciences 69 (00), Feige, U., an Lee, J.R. [007]: An improve approximation ratio for the minimum linear arrangement problem. Information Processing Letters 101 (007), 6 9 Garey, M.R., Johnson, D.S., an Stockmeyer, L. [1976]: Some simplifie NP-complete graph problems. Theoretical Computer Science 1 (1976), Hansen, M.D. [1989]: Approximation algorithms for geometric embeings in the plane with applications to parallel processing problems. Proceeings of the 30th Annual IEEE Symposium on Founations of Computer Science (1989), Hilbert, D. [1891]: Über ie stetige Abbilung einer Linie auf ein Flächenstück. Mathematische Annalen 38 (1891), Leighton, T., an Rao, S. [1999]: Multicommoity max-flow min-cut theorems an their use in esigning approximation algorithms. Journal of the ACM 6 (1999), Rao, S., an Richa, A.W. [00]: New approximation techniques for some linear orering problems. SIAM Journal on Computing 3 (00), Sagan, H. [199]: Space-Filling Curves. Springer, New York