Knuth's Algorithm Research Articles

Poisson limit theorems for the Robinson–Schensted correspondence and for the multi-line Hammersley process

We consider the Robinson–Schensted–Knuth algorithm applied to a random input and study the growth of the bottom rows of the corresponding Young diagrams. We prove a multidimensional Poisson limit theorem for the resulting Plancherel growth process. In this way we extend the result of Aldous and Diaconis to more than just one row. This result can be interpreted as convergence of the multi-line Hammersley process to its stationary distribution which is given by a collection of independent Poisson point processes.

Poisson limit of bumping routes in the Robinson\u2013Schensted correspondence

We consider the Robinson–Schensted–Knuth algorithm applied to a random input and investigate the shape of the bumping route (in the vicinity of the y-axis) when a specified number is inserted into a large Plancherel-distributed random tableau. We show that after a projective change of the coordinate system the bumping route converges in distribution to the Poisson process.

Hydrodynamic limit of the Robinson–Schensted–Knuth algorithm

AbstractWe investigate the evolution in time of the position of a fixed number in the insertion tableau when the Robinson–Schensted–Knuth algorithm is applied to a sequence of random numbers. When the length of the sequence tends to infinity, a typical trajectory after scaling converges uniformly in probability to some deterministic curve.

3D parallel tiled code implementing a modified Knuth's optimal binary search tree algorithm

This paper presents an approach to generation of 3D parallel tiled code implementing an optimal binary search tree (OBST) algorithm. We demonstrate that the features of data dependences available in the code implementing Knuth's OBST algorithm allow us to generate only 2D tiled code. We suggest a way of transformation of Knuth's OBST algorithm to a modified one exposing dependences allowing us to generate 3D parallel tiled code. The polyhedral model and the corresponding tools supporting that model are used by us to generate 3D target tiled code on the basis of the modified Knuth's OBST algorithm. Program parallelism is based on the wavefront technique and it is presented in the OpenMP C/C++ standard. Experiments carried out by us with obtained 3D tiled code demonstrate that this code considerably outperforms 2D tiled code generated on the basis of serial code implementing classic Knuth's OBST algorithm. Increased code performance is achieved due to much larger locality of 3D tiled code in comparison with that of 2D one.

An insertion algorithm on multiset partitions with applications to diagram algebras

We generalize the Robinson–Schensted–Knuth algorithm to the insertion of two row arrays of multisets. This generalization leads to new enumerative results that have representation theoretic interpretations as decompositions of centralizer algebras and the spaces they act on. In addition, restrictions on the multisets lead to further identities and representation theory analogues. For instance, we obtain a bijection between words of length k with entries in [n] and pairs of tableaux of the same shape with one being a standard Young tableau of size n and the other being a standard multiset tableau of content [k]. We also obtain an algorithm from partition diagrams to pairs of a standard tableau and a standard multiset tableau of the same shape, which has the remarkable property that it is well-behaved with respect to restricting a representation to a subalgebra. This insertion algorithm matches recent representation-theoretic results of Halverson and Jacobson [15].

Tactile Tools for Teaching: Implementing Knuth's Algorithm for Mastering Mastermind

Mordecai Meirowitz's 1970s hit Mastermind is a two-player game, codemaker against codebreaker. The codemaker first chooses a concealed sequence of four colored pegs, called a code, and the codebrea...

ULTRA: Universal Grammar as a Universal Parser

A central concern of generative grammar is the relationship between hierarchy and word order, traditionally understood as two dimensions of a single syntactic representation. A related concern is directionality in the grammar. Traditional approaches posit process-neutral grammars, embodying knowledge of language, put to use with infinite facility both for production and comprehension. This has crystallized in the view of Merge as the central property of syntax, perhaps its only novel feature. A growing number of approaches explore grammars with different directionalities, often with more direct connections to performance mechanisms. This paper describes a novel model of universal grammar as a one-directional, universal parser. Mismatch between word order and interpretation order is pervasive in comprehension; in the present model, word order is language-particular and interpretation order (i.e., hierarchy) is universal. These orders are not two dimensions of a unified abstract object (e.g., precedence and dominance in a single tree); rather, both are temporal sequences, and UG is an invariant real-time procedure (based on Knuth's stack-sorting algorithm) transforming word order into hierarchical order. This shift in perspective has several desirable consequences. It collapses linearization, displacement, and composition into a single performance process. The architecture provides a novel source of brackets (labeled unambiguously and without search), which are understood not as part-whole constituency relations, but as storage and retrieval routines in parsing. It also explains why neutral word order within single syntactic cycles avoids 213-like permutations. The model identifies cycles as extended projections of lexical heads, grounding the notion of phase. This is achieved with a universal processor, dispensing with parameters. The empirical focus is word order in noun phrases. This domain provides some of the clearest evidence for 213-avoidance as a cross-linguistic word order generalization. Importantly, recursive phrase structure “bottoms out” in noun phrases, which are typically a single cycle (though further cycles may be embedded, e.g., relative clauses). By contrast, a simple transitive clause plausibly involves two cycles (vP and CP), embedding further nominal cycles. In the present theory, recursion is fundamentally distinct from structure-building within a single cycle, and different word order restrictions might emerge in larger domains like clauses.

On the Importance Sampling of Self-Avoiding Walks

In a 1976 paper published in Science, Knuth presented an algorithm to sample (non-uniform) self-avoiding walks crossing a square of side k. From this sample, he constructed an estimator for the number of such walks. The quality of this estimator is directly related to the (relative) variance of a certain random variable Xk. From his experiments, Knuth suspected that this variance was extremely large (so that the estimator would not be very efficient). But how large? For the analogous Rosenbluth algorithm, which samples unconfined self-avoiding walks of length n, the variance of the corresponding estimator is believed to be exponential in n.A few years ago, Bassetti and Diaconis showed that, for a sampler à la Knuth that generates walks crossing a k × k square and consisting of North and East steps, the relative variance is only $O(\sqrt k)$. In this note we take one step further and show that, for walks consisting of North, South and East steps, the relative variance jumps to $2^{k(k+1)}/(k+1)^{2k}$. This is exponential in the average length of the walks, which is of order k2. We also obtain partial results for general self-avoiding walks crossing a square, suggesting that the relative variance could be exponential in k2 (which is again the average length of these walks).Knuth's algorithm is a basic example of a widely used technique called sequential importance sampling. The present paper, following the paper by Bassetti and Diaconis, is one of very few examples where the variance of the estimator can be found.

Parallel dynamic programming for solving the optimal search binary tree problem on CGM

This paper presents an efficient coarse grain multicomputer CGM parallel algorithm for the cost of the optimal binary search tree problem OBST problem. In the previous best CGM algorithm for this problem, the size of the local memory of each processor is not bounded. And even, in the worst case, its running time reduces to the one of Knuth's sequential algorithm for the same problem, bringing no gain by parallelising OBST. Our CGM algorithm uses Knuth's sequential algorithm for local computations to perform in time as well as the best previous CGM algorithm. Moreover, each processor can process at most two blocks and thus avoid the severe drawback raised above of the previous work.

Properties of the nonsymmetric Robinson–Schensted–Knuth algorithm

We introduce a generalization of the Robinson–Schensted–Knuth insertion algorithm for semi-standard augmented fillings whose basement is an arbitrary permutation σ∈S n . If σ is the identity, then our insertion algorithm reduces to the insertion algorithm introduced by the second author (Sémin. Lothar. Comb. 57:B57e, 2006) for semi-standard augmented fillings and if σ is the reverse of the identity, then our insertion algorithm reduces to the original Robinson–Schensted–Knuth row insertion algorithm. We use our generalized insertion algorithm to obtain new decompositions of the Schur functions into nonsymmetric elements called generalized Demazure atoms (which become Demazure atoms when σ is the identity). Other applications include Pieri rules for multiplying a generalized Demazure atom by a complete homogeneous symmetric function or an elementary symmetric function, a generalization of Knuth’s correspondence between matrices of non-negative integers and pairs of tableaux, and a version of evacuation for composition tableaux whose basement is an arbitrary permutation σ.

Knuth's Balanced Codes Revisited

In 1986, Don Knuth published a very simple algorithm for constructing sets of bipolar codewords with equal numbers of ?1?s and ?-1?s, called balanced codes. Knuth's algorithm is well suited for use with large codewords. The redundancy of Knuth's balanced codes is a factor of two larger than that of a code comprising the full set of balanced codewords. In this paper, we will present results of our attempts to improve the performance of Knuth's balanced codes.

Very Efficient Balanced Codes

The prior art construction of sets of balanced codewords by Knuth is attractive for its simplicity and absence of look-up tables, but the redundancy of the balanced codes generated by Knuth's algorithm falls a factor of two short with respect to the minimum required. We present a new construction, which is simple, does not use look-up tables, and is less redundant than Knuth's construction. In the new construction, the user word is modified in the same way as in Knuth's construction, that is by inverting a segment of user symbols. The prefix that indicates which segment has been inverted, however, is encoded in a different, more efficient, way.

Permutations Without Long Decreasing Subsequences and Random Matrices

We study the shape of the Young diagram $\lambda$ associated via the Robinson–Schensted–Knuth algorithm to a random permutation in $S_n$ such that the length of the longest decreasing subsequence is not bigger than a fixed number $d$; in other words we study the restriction of the Plancherel measure to Young diagrams with at most $d$ rows. We prove that in the limit $n\to\infty$ the rows of $\lambda$ behave like the eigenvalues of a certain random matrix (namely the traceless Gaussian Unitary Ensemble random matrix) with $d$ rows and columns. In particular, the length of the longest increasing subsequence of such a random permutation behaves asymptotically like the largest eigenvalue of the corresponding random matrix.

Queues, stores, and tableaux

Consider the single-server queue with an infinite buffer and a first-in–first-out discipline, either of type M/M/1 or Geom/Geom/1. Denote by 𝒜 the arrival process and by s the services. Assume the stability condition to be satisfied. Denote by 𝒟 the departure process in equilibrium and by r the time spent by the customers at the very back of the queue. We prove that (𝒟, r) has the same law as (𝒜, s), which is an extension of the classical Burke theorem. In fact, r can be viewed as the sequence of departures from a dual storage model. This duality between the two models also appears when studying the transient behaviour of a tandem by means of the Robinson–Schensted–Knuth algorithm: the first and last rows of the resulting semistandard Young tableau are respectively the last instant of departure from the queue and the total number of departures from the store.

Queues, stores, and tableaux

Consider the single-server queue with an infinite buffer and a first-in–first-out discipline, either of type M/M/1 or Geom/Geom/1. Denote by 𝒜 the arrival process and by s the services. Assume the stability condition to be satisfied. Denote by 𝒟 the departure process in equilibrium and by r the time spent by the customers at the very back of the queue. We prove that (𝒟, r) has the same law as (𝒜, s), which is an extension of the classical Burke theorem. In fact, r can be viewed as the sequence of departures from a dual storage model. This duality between the two models also appears when studying the transient behaviour of a tandem by means of the Robinson–Schensted–Knuth algorithm: the first and last rows of the resulting semistandard Young tableau are respectively the last instant of departure from the queue and the total number of departures from the store.

Refined sign-balance on 321-avoiding permutations

The number of even 321-avoiding permutations of length n is equal to the number of odd ones if n is even, and exceeds it by the n − 1 2 th Catalan number otherwise. We present an involution that proves a refinement of this sign-balance property respecting the length of the longest increasing subsequence of the permutation. In addition, this yields a combinatorial proof of a recent analogous result of Adin and Roichman dealing with the last descent. In particular, we answer the question of how to obtain the sign of a 321-avoiding permutation from the pair of tableaux resulting from the Robinson–Schensted–Knuth algorithm. The proof of the simple solution is based on a matching method given by Elizalde and Pak.

Magnetic interpretation of the Robinson–Schensted–Knuth algorithm

Combinatorial aspects of the Robinson–Schensted–Knuth (RSK) algorithm have been discused in the context of a Heisenberg magnetic ring with N nodes, each with the spin s. Each magnetic configuration acquires a natural interpretation as a word of the length N in the alphabet of spins, consisting of n = 2s+1 letters. We demonstrate that the construction of n-tuple cover of the ring, with a separate copy for each letter of the alphabet of spins, allows for a transparent determination of maximal length of non-decreasing subwords. Moreover, it yields completeness of the RSK correspondence in classification of the irreducible basis of the Weyl duality between actions of unitary and symmetric groups in the space spanned on all magnetic configurations.

Kinematics of the Heisenberg chain and irreducible bases of the Weyl duality

Quantum kinematics of the linear Heisenberg ring, consisting of N crystal nodes, each with spin s, is presented in terms of the Weyl duality between actions of the symmetric and unitary groups in the space of quantum states of the magnet. This space is spanned on magnetic configurations which gives rise to an application of the combinatorial Robinson–Schensted–Knuth algorithm for a unique classification of irreducible basis of the duality of Weyl in terms of a pair of tableaux: a standard Young tableau in the alphabet of nodes, accompanied by a semistandard Weyl tableau in the alphabet of spins. Similarities and distinctions between various group-theoretic and combinatorial objects are discussed within the context of the magnetic interpretation. In particular, the role of the spectrum of Jucys–Murphy operators in the clasification and construction of magnetic eigenstates corresponding to Young tableaux is illustrated.

Weighted Deductive Parsing and Knuth's Algorithm

We discuss weighted deductive parsing and consider the problem of finding the derivation with the lowest weight. We show that Knuth's generalization of Dijkstra's algorithm for the shortest-path problem offers a general method to solve this problem. Our approach is modular in the sense that Knuth's algorithm is formulated independently from the weighted deduction system.

Attributed tree grammars

This paper develops a notion of attributes on tree grammars. We study the case when graphs of attributes are associated with terminal symbols of tree grammars. An algebraic formalism is given in this case for attributed tree grammars. We also describe algorithms for the test of noncircularity. These algorithms generalize Knuth's algorithm to attributed tree grammars. The main idea behind this approach is to represent graphics (attributed graphs) by trees whose nodes are enriched with attributes.