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Information Processing Letters 97 (2006) 171–176

www.elsevier.com/locate/ip

Enhanced algorithms for Local Search✩

Yves F. Verhoevena,b

a Laboratoire de Recherche en Informatique, Bâtiment 490, Université Paris-Sud, 91405 Orsay Cedex, Franceb École Nationale Supérieure des Télécommunications, 46, rue Barrault, 75013 Paris, France

Received 6 June 2005; received in revised form 31 October 2005; accepted 2 November 2005

Available online 5 December 2005

Communicated by K. Iwama

Abstract

Let G = (V ,E) be a finite graph, andf :V → N be any function. The Local Search problem consists in finding alocal minimumof the functionf on G, that is a vertexv such thatf (v) is not larger than the value off on the neighbors ofv in G. In thisnote, we first prove a separation theorem slightly stronger than the one of Gilbert, Hutchinson and Tarjan for graphs ofgenus. This result allows us to enhance a previously known deterministic algorithm for Local Search with query comO(logn) · d + O(

√g ) · √

n, so that we obtain a deterministic query complexity ofd + O(√

g ) · √n, wheren is the size ofG,

d is its maximum degree, andg is its genus. We also give a quantum version of our algorithm, whose query complexityO(

√d ) + O( 4

√g ) · 4√n log logn. Our deterministic and quantum algorithms have query complexities respectively smalle

the algorithmRandomized Steepest Descent of Aldous andQuantum Steepest Descent of Aaronson for large classes of graphincluding graphs of bounded genus and planar graphs. 2005 Elsevier B.V. All rights reserved.

Keywords:Algorithms; Analysis of algorithms; Graph algorithms

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1. Introduction

The Local Search problem consists in finding alocalminimum of the functionf onG, that is a vertexv suchthatf (v) is not larger than the value off on the neigh-bors ofv in G. Obviously, such a vertex always exisas a global minimum satisfies this constraint. Anoteasy argument shows how to find such a vertex: ma walk over vertices such that at each step the next

✩ The research was supported by the EU 5th framework progRESQ IST-2001-37559, and by the ACI CR 2002-40 and ACI2003-24 grants of the French Research Ministry.

E-mail address:yves.verhoeven@normalesup.org(Y.F. Verhoeven).

0020-0190/$ – see front matter 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.ipl.2005.11.004

tex is the neighbor of the current vertex which hassmallest value; the walk will stop in a local minimumSuch a walk is called asteepest descent. Steepest descents are the basis of several approaches to efficifind a local minimum.

The Local Search problem has been previously sied and there is already a large literature on its compity. Its structural complexity, where the function and tgraph are given as an input to a Turing machine,studied in [7,10], and its query complexity, where tgraph is known but the values off are accessed througan oracle, was investigated in [9,8,2,1,11,12]. We foon the query complexity, which is obviously at most tsize of the graph. Our query model is the standard osee Section 3.2 for precise definitions.

172 Y.F. Verhoeven / Information Processing Letters 97 (2006) 171–176

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The notion ofgenusof a graph is central to our workWe recall its definition (for more details, see, forstance, [4]).

Definition 1. The genusof a graphG is the minimumnumber of handles that must be added to the sphso that the graphG can be embedded in it without ancrossings of its edges.

The deterministic query complexity of Local Searon a graphG of sizen, maximum degreed and genusg,is intimately connected to the size ofseparatorsof G:

Definition 2. A separator forG is a subset ofV whoseremoval leaves no connected component with more2n/3 vertices.

In [9], a deterministic query algorithm was exhibitewhich works using a sub-linear number of querieslarge classes of graphs: a local minimum can be foon the graphG using O(logn) ·d +O(

√g ) ·√n queries.

It is based on the recursive use of separators for smand smaller subgraphs ofG, and their complexity analysis relies on the following result:

Theorem 1 (Gilbert et al. [6]). The graphG has a sep-arator of size at most6

√gn + 2

√2n + 1.

The only known sub-linear query algorithm for geeral graphs is a randomized algorithm, which we cRandomized Steepest Descent, was exhibited by Al-dous [2], and has a query complexity�(

√nd ). The idea

of this algorithm is to choose√

nd vertices at randomquery their values, start a steepest descent from thetex with smallest value for at most

√nd steps, and to

return the last visited vertex. This idea was later refiby Aaronson [1] to give a sub-linear quantum qualgorithm for general graphs, which we callQuantumSteepest Descent, using�(n1/3d1/6) queries.

On the side of lower-bounds, it follows from [9that the minimal size of a separator is a lower-bouon the deterministic complexity of Local Search. Alsfrom [1], we know thatd is a lower-bound on the deterministic query complexity,�(d) a lower-bound forthe randomized query complexity, and�(

√d ) a lower-

bound for the quantum query complexity.

2. Results

In this note, we first improve Theorem 1 in Setion 4.1, to obtain the following slightly stronger searation theorem:

,

-

Theorem 2 (Strong separation for graphs of genusg).Assumen � 3. There exists a separatorC for G suchthatC contains no more than(6+2

√2−12/n+6

√g+

4g/n+1/√

n ) ·√n vertices, and the subgraph induconG byV \ C has maximal degree at most

√n.

As a result it allows us to enhance, in Section 5,deterministic algorithm of Llewellyn et al. [9] whoscomplexity is of O(logn) · d + O(

√g ) · √

n. We alsoderive a quantum algorithm from it. More precisely,obtain the following result:

Theorem 3. There exist a deterministic and a quantuquery algorithms that find a local minimum off onG using respectivelyd + O(

√g ) · √

n and O(√

d ) +O( 4

√g ) · 4

√n log logn queries.

We analyze how our algorithms compare toRan-domized Steepest Descent of Aldous [2] andQuantumSteepest Descent of Aaronson [1] in detail in Section 6

Independently from this work, Zhang [12] hascently given an algorithm which finds a local miimum on the planar grid over{1, . . . ,

√n}2 using

O( 4√

n(log logn)2) queries. Our quantum algorithm cbe viewed as a strongly generalized, and slightlyhanced version of this algorithm.

3. Preliminaries

3.1. Notations

We denote by logn the natural logarithm ofn, andfor every positive real numberb we denote by logb n

the logarithm ofn in baseb. If G is a graph andv is anyvertex ofG, we denote by∂G(v) the set of neighbors ov in G.

3.2. Query complexity

In the query model of computation we count onqueries made by the algorithm, and all other comptions are free.

We next describe a general setting using the Dnotation for query complexity, which allows us to easderive the definition of quantum, randomized and deministic query complexities from it. However, our deinitions of randomized and deterministic models mathe usual ones (our randomized model is thebounded-error model).

The state of the computation is represented by thregisters, the query registeri ∈ {1, . . . , n}, the answeregistera ∈ Σ , and the work registerz ∈ W , whereΣ

Y.F. Verhoeven / Information Processing Letters 97 (2006) 171–176 173

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andW are finite sets. The computation takes placethe vector space spanned by all basis states|i〉|a〉|z〉. Inthequantum query modelintroduced by Beals et al. [3the state of the computation is a complex combinaof all basis states which has unit length for the norm�2,and the allowed non-query operations on the state ocomputation are all isometric operators for the�2 normacting over the computation space. In therandomizedmodel, the state of the computation is a non-negareal combination of all basis states of unit length fornorm�1, and the allowed operations on the state ofcomputation are all isometric operators for the norm�1

acting over the computation space. In thedeterministicmodel, the state of the computation is always one ofbasis states, and the allowed operations are all opermapping a basis state to another basis state.

Assume thatx ∈ Σn is the input of the problem which can be accessed only through the oraThe query operationOx is the permutation whichmaps the basis state|i〉|a〉|z〉 into the state|i〉|(a +xi) mod |Σ |〉|z〉 (here we identifyΣ with the residueclasses mod|Σ |). Non-query operations are indepedent ofx. A k-query algorithmis a sequence of(k + 1)

operations(U0,U1, . . . ,Uk) where Ui is an allowedoperation in the chosen model of computation. Itially the state of the computation is set to some fixvalue|0〉|0〉|0〉, and then the sequence of operationsU0,

Ox,U1,Ox, . . . ,Uk−1,Ox,Uk is applied. The finastate is denoted byΦ.

The output in the quantum model is an elemz ∈ W that appears with probability equal to the squof the �2 norm of the orthogonal projection ofΦover the vector spaceV spanned by{|i〉|a〉|z〉 | i ∈ {1,

. . . , n}, a ∈ Σ}. The output in the randomized modelan elementz ∈ W that appears with probability equalthe�1 norm of the orthogonal projection ofΦ over thevector spaceV . The output in the deterministic modis the elementz ∈ W such that there existi anda withΦ = |i〉|a〉|z〉.

Assume thatR ⊆ Σn × W is a total relation (i.e. foreveryx ∈ Σn there existsz ∈ W such that(x, z) ∈ R)that we want to compute. A quantum or randomizedgorithm computes (with two-sided error)R if its outputyield somez ∈ W such that(x, z) ∈ R with probabilityat least 2/3. A deterministic algorithm computesR if itsoutput yield somez ∈ W such that(x, z) ∈ R.

Then the query complexity of a relationR in a modelof computation (deterministic, randomized or quantuis the smallestk for which there exists ak-query algo-rithm, in that model of computation, which computesR.

s

4. Tools

In this section, we recall and prove the results thatneed in order to design the algorithms of Section 5.

4.1. Separation in graphs of higher genus

We first recall the following well-known theorem fographs of higher genus (see, for instance, [4]):

Theorem 4. Any n-vertex graphG of genusg withn � 3 contains no more than3n − 6+ 2g edges.

This result, together with Theorem 1, allows usprove Theorem 2.

Proof of Theorem 2. For every vertexv of G, denoteby d(v) the degree ofv in G. Theorem 1 shows thatis possible to find a separatorC′ for G such thatC′ hassize at most 6

√gn + 2

√2n + 1. LetB be the set of al

verticesv of G such that d(v) >√

n. Using Theorem 4we have

|B| · √n �∑v∈V

d(v) = 2 · |E| � 6n − 12+ 4g.

The setC = C′ ∪ B being a superset of a separatorG is also a separator ofG. From the definition ofB, thesubgraph induced onG byV \C obviously has maximadegree at most

√n. �

The genus of a graph being in O(n2), the asymp-totic inequalityg/n = O(

√g ) holds and therefore The

orem 2 can be interpreted as stating the existenceparticular O(

√g ) · √n separatorG.

4.2. Minimum-finding algorithms

In this paragraph, we recall results about the qucomplexity of finding the minimum value of a functioon a set.

Let n be a positive integer,S be a set of cardinalityn,g :S → N be a function andOg be an oracle forg. Wedenote by argmin{g(s) | s ∈ S} the set of all elements′ ∈ S such thatg(s′) = min{g(s) | s ∈ S}.

Definition 3. Let ε < 1 be any positive real number.A is a randomized algorithm that outputs an elemenargmin{g(s) | s ∈ S} with probability at least 1− ε > 0,then we denote by MINAg (S) the random variable equto its output.

It is obvious that, for every deterministic algrithm A, outputting an element in argminA{g(s) |

174 Y.F. Verhoeven / Information Processing Letters 97 (2006) 171–176

are

in

l-ingin-

ned

ster

oofeill

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atorpo-

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rd in

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on-

out-t atec-

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itsna-do itess.

l-

um

s ∈ S} requires querying alln values ofg to Og . It isnatural that, for every randomized algorithmA, com-puting MINA

g (S) requires querying�(n) values ofgto Og . It is more surprising that much less queriesneeded when quantum queries are allowed:

Theorem 5 (Dürr and Høyer [5]). There exists aquantum algorithmA which outputs an elementargminA{g(s) | s ∈ S} with probability at least1/2, us-ing O(

√n ) quantum queries toOg .

Amplification of the probability of success of the agorithm of Dürr and Høyer can be obtained by runnthe algorithm several times, and then taking the mimum value of all the values that have been returby each repetition of the algorithm. Afterk repetitions,the probability of having found the minimum is at lea1 − 2−k . In particular, for every positive real numbε < 1, there exists a quantum algorithmA outputtingan element in argminA{g(s) | s ∈ S} with probability atleast 1− ε using O(

√n log(1/ε)) quantum queries.

5. Enhanced algorithms for Local Search

In this paragraph, we prove Theorem 3. The prwill be in two steps: in Theorem 6 we will prove thcorrection of our algorithms, and in Theorem 7 we wprove their complexity.

The basic procedure of our algorithms is a simfied reformulation of the algorithms of Llewellyn et aof [9] and Santha and Szegedy [11]. It is given in Algrithm 1. The main idea is to adopt a divide-and-conqapproach: the graph is split into connected componof small size by removing a separator; then, query

the values of the vertices in and close to that separmake it possible to find one of these connected comnents in which there is a local minimum off onG.

Notice that neither the way the separators are csen, or how the minimum-finding algorithmsAi workfor integersi � 1, are specified in Algorithm 1. Oualgorithms consist in using the procedure describeAlgorithm 1 with the following specific choices:

• a separatorC of a graphG′ will be chosen according to Theorem 2 ifG′ has more than two verticeandC contains all vertices ofG′ otherwise,

• the minimum-finding algorithmAi will behave asfollows when requested to minimize the functif over a setS ⊆ V of vertices: in the deterministic case, the local minimum off is found byexhaustive search. In the quantum cases, theput is the one found by the last measurementhe end of the quantum procedure described in Stion 4.2; moreover, we request that the minimufinding algorithmA1 has error probability 1/12 us-ing O(

√|S| ) queries, andAi has error probability1/(12 log3/2 n) using O(

√|S| log logn) queries, fori �= 1.

Although the structure of our algorithm is not new,use in a non-deterministic setting is. We therefore alyze its correctness in the quantum setting, and alsoin the deterministic setting for the sake of completen

Theorem 6. With our choice of minimum-finding agorithm, Algorithm1 always returns a local minimumin the deterministic case, and returns a local minimwith probability at least2/3 in the quantum case.

i := 0, G(0) := G, v(0) := any vertex ofG, output := ∅.while output = ∅ do

i := i + 1.Create a separatorC(i) for G(i−1).

m(i) := MINAig (C(i)).

z(i) := MINAig (∂G(i−1) (m

(i))).v(i) := any element in argmin{f (v) | v ∈ {v(i−1),m(i), z(i)}}.if v(i) = m(i) then

output := {v(i)}.else

V (i) := the connected component ofV (i−1) \ C(i) that containsv(i).G(i) := G[V (i)].

end ifend whileReturnoutput.

Algorithm 1. Procedure for finding a local minimum of a functionF :V → N on a graphG = (V ,E), using separators.

Y.F. Verhoeven / Information Processing Letters 97 (2006) 171–176 175

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o-st

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Proof. Let j be the largest value of the variablei fora run of the algorithm. First, an easy inductions shothat for every iterationi � j of the main loop, and everv ∈ V (i) we have∂G(v) ⊆ ∂G(i) (v) ∪ C(1) ∪ C(2) ∪ · · · ∪C(i). So, to prove thatf is minimized onv(j), one mustonly prove thatf (v(j)) is not larger than min{f (v) | v ∈∂G(j) (v) ∪ C(1) ∪ C(2) ∪ · · · ∪ C(j)}.

If, during the run of the algorithm, the calls to thalgorithmsAi for 1 � i � j have always successfulreturned elements minimizingf , then for every positiveintegeri � j we havef (v(i)) � min{f (v(i−1)), f (m(i)),

f (z(i))}. Therefore, an easy induction shows thaf (v(i)) � min{f (v) | v ∈ C(k)}, for every positive in-tegersk � i � j . Moreover, the equalityv(j) = m(j)

impliesf (v(j)) � min{f (v) | v ∈ ∂G(j) (v(j))}. So, ifAi

never failed to find a minimizing element, then the cterion given in the previous paragraph shows thatv(j) isa local minimum.

In the deterministic case, the algorithmsAi , for 1�i � j , always return an element minimizingf , andtherefore Algorithm 1 always returns a local minimu

In the quantum case, a call toAi returns an elemenminimizing f with error probability at most 1/12 fori = 1, and at most 1− 1/(12 log3/2 n) for 1 < i � j .The setC(i) being a separator ofG(i−1) for every pos-itive integeri � j , we have|V (i)| � 2|V (i−1)|/3. Thisimplies thatj � log3/2 n, and sinceAi is called twice ineach iteration of the main loop the probability thatAi

did not return an element minimizingf at some point isat most 2· 1/12+ 2 · log3/2 n/(12 log3/2 n) = 1/3. �Theorem 7. With our choices of separators, Algrithm 1 has a deterministic query complexity at mod + O(

√g ) · √

n, and a quantum query complexitymostO(

√d ) + O( 4

√g ) · 4

√n log logn.

Proof. Again, letj be the largest value of the variabi for a run of the algorithm. Let us denote byCAi

(s) thenumber of queries made by the minimum-finding algrithm Ai on a set of sizes, and byLi(n, d) the numberof queries that are made in theith iteration of the mainloop of our algorithm on a graphG′ that hasn verticesand is of maximum degreed . We denote also bydi themaximum degree of|G(i)|, for a non-negative integei � j . Analysis of the main loop of Algorithm 1 givesfor every positive integeri � j ,

Li(n, d) � CAi

(|C(i)|) + CAi(di−1) + 3.

Let us denote byT iγ (α,β) the number of queries mad

by our algorithm in the main loops between itsith it-eration and the end of the algorithm ifi < j , and 0 ifi � j , on an input graph which hasα vertices, is of

maximum degreeβ and has genus at mostγ . Theo-rem 2 ensures that for every positive integeri � j wehave|V (i)| � 2|V (i−1)|/3, anddi �

√|V (i−1)|. More-over, the genus of beG(i) is not larger than the genusG(i−1). So, by induction we have|V (i)| � (2/3)in, andthe genus of|G(i)| is at mostg. Therefore, for every integer 1� i � j we have|C(i)| � O(

√g ) · √(2/3)i−1n,

d0 = d anddi �√

(2/3)i−1n. This leads to the following equations:

T 1g (n, d) � L1(n, d) + T 2

g (n, d)

� CA1

(O(

√g ) · √n

)

+ CA1(d) + 3+ T 2g (n, d),

and for everyi ∈ {2, . . . , log3/2 n� − 1},T i

g (n, d) � Li(n, d) + T i+1g (n, d)

� CAi

(O(

√g ) ·

√(2/3)i−1n

)

+ CAi

(√(2/3)i−2n

)+ 3+ T i+1

g (n, d).

In the deterministic case we haveCAi(k) = k for all pos-

itive integersk andi, and in the quantum case we hafor all positive integerk, CAi

(k) = O(√

k ) wheni = 1,and CAi

(k) = O(√

k log logn) when i �= 1. So, in thedeterministic case, summing all the previous inequties gives

T 1g (n, d) � O(

√g ) · √n ·

∞∑i=0

√2/3

i + d

+ √n ·

∞∑i=0

√2/3

i + 3 log3/2 n

+ Tlog3/2 n�g (n, d),

which shows T 1g (n, d) = d + O(

√g ) · √

n, as

Tlog3/2 n�g (n, d) = O(1), and the query complexity o

our deterministic algorithm isT 1g (n, d) = d + O(

√g ) ·√

n. In the quantum case, it gives

T 1g (n, d) � O( 4

√g ) · 4

√n log logn ·

∞∑i=0

4√

2/3i + O(√

d )

+ O( 4√

n log logn) ·∞∑i=0

4√

2/3i

+ 3 log3/2 n + Tlog3/2 n�g (n, d),

leading to a quantum query complexityT 1g (n, d) =

O(√

d ) + O( 4√

g ) · 4√

n log logn. �

176 Y.F. Verhoeven / Information Processing Letters 97 (2006) 171–176

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6. Comparison with generic algorithms

Let us first compare the query complexity of our dterministic algorithm with the query complexity of thalgorithmRandomized Steepest Descent of Aldous [2].The complexity of our algorithm isd + O(

√g ) · √

n,and the complexity ofRandomized Steepest Descentis �(

√nd ). As d � n, our algorithm performs as we

as Randomized Steepest Descent (up to a constanspeedup factor) as soon asg = O(d), and performs asymptotically better wheng = o(d). In particular, our deterministic algorithm has lower query complexity thRandomized Steepest Descent on classes of graphwith bounded genus, which includes the class of plagraphs.

Let us now compare the query complexity of oquantum algorithm with the query complexity of thegorithm Quantum Steepest Descent of Aaronson [1].The complexity of our algorithm is O(

√d ) + O( 4

√g ) ·

4√

n log logn, and the complexity ofQuantum Steep-est Descent is �(n1/3d1/6). As d � n, we have

√d �

n1/3d1/6, and our algorithm performs as well asQuan-tum Steepest Descent (up to a constant speedup fator) as soon asg1/4 · n1/4 log logn = O(n1/3d1/6), thatis to sayg = O(n1/3d2/3/(log logn)4). This holds ifg = O(d/(log logn)4). Also, our quantum algorithmperforms asymptotically better wheng = o(n1/3d2/3/

(log logn)2), and therefore wheng = o(d/(log logn)4).In particular, our quantum algorithm has lower qucomplexity thanQuantum Steepest Descent on classesof graphs with bounded genus, which includes the cof planar graphs.

In conclusion, the algorithms we have designed pform better than the known generic algorithms for so

classes of graphs, in particular planar graphs and grof constant genus, both for classical (deterministicrandomized) computation, and for quantum comption.

References

[1] S. Aaronson, Lower bounds for local search by quantum aments, in: Proc. 36th Annual ACM Symp. on Theory of Coputing, ACM Press, New York, 2004, pp. 465–474.

[2] D. Aldous, Minimization algorithms and random walk on thed-cube, Ann. Probab. 11 (2) (1983) 403–413.

[3] R. Beals, H. Buhrman, R. Cleve, M. Mosca, R. de Wolf, Qutum lower bounds by polynomials, J. ACM 48 (4) (2001) 77797.

[4] B. Bollob’as, Modern Graph Theory, Graduate Texts in Matmatics, vol. 184, Springer-Verlag, Berlin, 1998.

[5] C. Dürr, P. Høyer, A quantum algorithm for finding the minmum, 1996, quant-ph/9607014.

[6] J.R. Gilbert, J.P. Hutchinson, R.E. Tarjan, A separator theofor graphs of bounded genus, J. Algorithms 5 (3) (1984) 3407.

[7] D.S. Johnson, C.H. Papadimitriou, M. Yannakakis, How easlocal search?, J. Comput. System Sci. 37 (1) (1988) 79–100

[8] D.C. Llewellyn, C.A. Tovey, Dividing and conquering thsquare, Discrete Appl. Math. 43 (2) (1993) 131–153.

[9] D.C. Llewellyn, C.A. Tovey, M. Trick, Local optimization ographs, Discrete Appl. Math. 23 (2) (1989) 157–178.

[10] C.H. Papadimitriou, A.A. Schaëffer, M. Yannakakis, Oncomplexity of local search, in: Proc. 22nd Annual ACM Symon Theory of Computing, ACM Press, New York, 1990, pp. 43445.

[11] M. Santha, M. Szegedy, Quantum and classical query compities of local search are polynomially related, in: Proc. 36th Anual ACM Symp. on Theory of Computing, ACM Press, NYork, 2004, pp. 494–501.

[12] S. Zhang, (Almost) tight bounds for randomized and quanlocal search on hypercubes and grids, 2005, quant-ph/0504