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The Work of Peter Shor Has a Strong Geometrical �Avor� Typically Coupled Doc Math J DMV The Work of Peter W Shor Ronald Graham Much of the work of Peter Shor has a strong geometrical avor typically coupled with deep ideas from probability complexity theory or combinatorics and always woven together with brilliance and insight of the rst magnitude Due to the space limitations of this note I will restrict myself to brief descriptions of just four of his remarkable achievements unfortunately omitting discussions of his seminal work on randomized incremental algorithms of fundamental imp ortance in computational geometry and his provo cative results in computational biology on selfassembling virus shells Twodimensional discrepancy minimax grid matchings and online bin packing The minimax grid matching problem is a fundamental combinatorial problem aris ing the the average case analysis of algorithms To state it we consider a square S p p of area N in the plane and a regularly spaced N N array G grid of p oints in S Let P be a set of N points selected indep endently and uniformly in S By a p erfect matching of P to G wemeanamap P G For each selection P dene LP min max dp p where ranges over all p erfect matchings pP of P to Gandd denotes Euclidean distance Theorem Shor LeightonShor With very high probability ELP logN The pro of is very intricate and ingenious and contains a wealth in new ideas which havespawned a variety of extensions and generalizations notably in the work of M Talagrand on ma jorizing measures and discrepancy A classical paradigm in the analysis of algorithms is the socalled bin packing in which a list W w w w of weights is given and problem n we are to required to pack all the w into bins with the constraint that no bin i can contain a weight total of more than Since it is NPhard to determine the minimum numb er of bins which W requires for a successful packing oreven to decide if this minimum numb er is extensive eorts have b een made for nding go o d approximation algorithms for pro ducing nearoptimal packings In the Best Fit algorithm after the rst i weights are packed the next weight w is placed into the bin in which it ts b est ie so that the unused space i DocumentaMathematica Extra Volume ICM I Ronald Graham in that bin is less than it would b e in any other bin This is actually an online algorithm In his thesis Shor proved the very surprising and deep result that when the w are chosen uniformly at random from then with very high i probability the amountofwasted space has size n log n An upright region R R f of the square S is dened as the region in S lying ab ovesomecontinuous monotonically nonincreasing function f eg S is itself upright If P isasetof N points chosen uniformly and indep endently at random in S we can dene the discrepancy R jj R P jar eaR j An old problem in mathematical statistics from the s see was the estimation of sup R over all upright regions of S This was nally answered byLeighton R and Shor in and it is nowknown that sup R N log N R The preceding results give just a hint of the numerous applications these fertile techniques have found to suchdiverse areas as pseudorandom numb er gen eration dynamic storage allo cation waferscale integration and twodimensional bin packing see DavenportSchinzel Sequences ADavenp ortSchinzel sequence DS n s is a sequence U u u u com m p osed of n distinct symb ols suchthat u u for all i and such that U con i i tains no alternating subsequence of length s ie there do not exist indices i i i such that u u u a b u u s i i i i i We dene n maxfm u u isaDS n s sequenceg S m Davenp ortSchinzel sequences have turned out to b e of central imp ortance in com putational and combinatorial geometryandhave found many applications in such Voronoi diagrams and shortest path algo areas as motion planning visibility rithms It is known that DS n ssequences provide a combinatorial character ization of the lower envelop e of n continuous univariate functions each pair of whichintersectinatmostspoints Hence n is just the maximum number of s connected comp onents of the graphs of such functions and accurate estimates of n can often b e translated into sharp b ounds for algorithms which dep end on s function minimization It is trivial to showthat nn and nn The rst surprise came when it was shown that n nn where n is dened to b e the functional inverse of the Ackermann function At ie nminft At ng Note that n is an extremely slowly growing function of n since A is dened as follows A tt t and A tA A t k t k k k t Thus A t A t is an exp onential to wer of n s and so on Then At is dened to be A t The best bounds for n s in were rather t s DocumentaMathematica Extra Volume ICM I The Work of Peter W Shor weak This was remedied in where Shor and his coauthors managed to showby n extremely delicate and clever techniques that nn Thus DS n sequences can b e much longer than DS n sequences but are still only slightly nonlinear In addition they also obtained almost tight bounds on all other ns s n Tiling R with cubes In Minkowski made the conjecture in connection with his work on extremal n lattices that in any lattice tiling of R with unit ncub es there must b e two cub es having a complete facet n face in common This was generalized by n O Keller in to the conjecture that any tiling of R by unit ncub es must have this prop erty This was conrmed by Perron in for n and shortly thereafter Ha jos proved Minkowskis original conjecture for all n However in spite of rep eated eorts no further progress was made in proving Kellers conjecture for the next years Then in Shor struck He showed with his colleague J Lagarias that in fact Kellers conjecture is false for all dimensions n They managed to do this with an very ingenious argument n showing that certain sp ecial graphs suggested by Corradi and Szabo of size n must always have cliques of size contrary to the prevailing opinions then from n which it followed that Kellers conjecture must fail for R The reader is referred to for the details of this combinatorial gem and to for a fascinating history of this problem I mightpoint out that this is another example of an old conjecture in geometry b eing shattered by a subtle combinatorial construction an earlier one b eing the recent dispro of of the Borsuk conjecture by Kahn and Kalai It is still not known what the truth for Kellers conjecture is when n or Quantum computation It has b een generally b elieved that a digital computer or more abstractlya Turing machine can simulate any physically realizable computational device This in fact is the thrust of the celebrated ChurchTuring thesis Moreover it was also assumed that this could always b e done in an ecientway ie involving at most a p olynomial expansion in the time required However it was rst p ointed out by Feynman that certain quantum mechanical systems seemed to b e extremely on Neumann dicult in fact imp ossible to simulate eciently on a standard v computer This led him to suggest that it might b e p ossible to take advantage of the quantum mechanical b ehavior of nature itself in designing a computer which overcame these diculties In fact in doing so such a quantum computer might be able to solvesomeofthe classical dicult problems much more eciently as well These ideas were pursued by Benio Deutsch Bennett and others and slowly a mo del of quantum computation b egan to evolve However the rst bombshell in this embryonic eld o ccurred when Peter Shor in announced the rst signicant algorithm for such a hyp othetical quantum DocumentaMathematica Extra Volume ICM I Ronald Graham computer namely a metho d for factoring an arbitrary comp osite integer N in clog N log log N log log log N steps This should be contrasted with the b est current algorithm on classical digital computers whose b est running time estimates growlike expcN log N Of course no one has yet ruled out the p ossibility that a p olynomialtime factoring algorithm exists for classical computers cf the infamous P vs NP problem but it is felt by most knowledgeable people that this is extremely unlikely In the same pap er Shor also gives a p olynomialtime algorithm for a quantum computer for computing discrete logarithms another apparently intractable problem for classical computers There is not space here to describ e these algorithms in any detail but a few remarks may b e in order In a classical computer information is represented by n binary symbols and bits An n bit memory can exist in any of logical states Such computers also manipulate this binary data using functions likethe Bo olean AND and NOT By contrast a quantum bit or qubit is typically a microscopic system such as an electron with its spin or a p olarized photon The Bo olean states and are represented by reliably distinguishable states of the and ji spin However according to the laws of qubit eg ji spin quantum mechanics the qubit can also exist in a continuum of intermediate states or sup erp ositions ji ji where and are complex numb ers satisfying jj j j More generally a string of n qubits can exist in any state of the form X jxi x x P where the are complex numbers suchthat j j In other words a quan x x x n tum state of n qubits is represented byaunitvector in a dimensional complex Hilb ert space dened as the tensor pro duct of the n copies of the dimensional Hilb ert space representing the state of a single qubit It is the exp onentially large dimensionality of this space which
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