Constant Amortized Time Research Articles

Loopless algorithms to generate maximum length gray cycles wrt. k-character substitutions

Given a binary word relation τ onto A⁎ and a finite language X⊆A⁎, a τ-Gray cycle over X consists in a permutation (w[i])0≤i≤|X|−1 of X such that each word w[i] is an image under τ of the previous word w[i−1]. We define the complexity measure λA,τ(n), equal to the largest cardinality of a language X having words of length at most n, and st. some τ-Gray cycle over X exists. The present paper is concerned with τ=σk, the so-called k-character substitution, st. (u,v)∈σk holds if, and only if, the Hamming distance of u and v is k. We present loopless (resp., constant amortized time) algorithms for computing specific maximum length σk-Gray cycles.

Combinatorial Generation via Permutation Languages. III. Rectangulations

A generic rectangulation is a partition of a rectangle into finitely many interior-disjoint rectangles, such that no four rectangles meet in a point. In this work we present a versatile algorithmic framework for exhaustively generating a large variety of different classes of generic rectangulations. Our algorithms work under very mild assumptions, and apply to a large number of rectangulation classes known from the literature, such as generic rectangulations, diagonal rectangulations, 1-sided/area-universal, block-aligned rectangulations, and their guillotine variants, including aspect-ratio-universal rectangulations. They also apply to classes of rectangulations that are characterized by avoiding certain patterns, and in this work we initiate a systematic investigation of pattern avoidance in rectangulations. Our generation algorithms are efficient, in some cases even loopless or constant amortized time, i.e., each new rectangulation is generated in constant time in the worst case or on average, respectively. Moreover, the Gray codes we obtain are cyclic, and sometimes provably optimal, in the sense that they correspond to a Hamilton cycle on the skeleton of an underlying polytope. These results are obtained by encoding rectangulations as permutations, and by applying our recently developed permutation language framework.

Generating 2-Gray codes for ballot sequences in constant amortized time

We present a simple algorithm that generates a cyclic 2-Gray code for ballot sequences. The algorithm generates each ballot sequence in constant amortized time using a linear amount of space. This is the first known cyclic 2-Gray code for ballot sequences that achieves this time bound. In addition, the algorithm can be easily modified to output ballot sequences in binary reflected Gray code order in constant amortized time per string using a linear amount of space.

A Dynamical System Approach to Polyominoes Generation*

We describe a method which exploits discrete dynamical systems to generate suitable classes of polyominoes. We apply the method to design an algorithm that uses O(n) space to generate in constant amortized time all polyominoes corresponding to hole-free partially directed animals consisting of n sites on the square grid. By implementing the algorithm in C++ we have obtained a new sequence that does not appear in the On-Line Encyclopedia of Integer Sequences.

A constant amortized time enumeration algorithm for independent sets in graphs with bounded clique number

In this study, we address the independent set enumeration problem. Although several efficient enumeration algorithms and careful analyses have been proposed for maximal independent sets, no fine-grained analysis has been given for the non-maximal variant. As the main result, we propose an enumeration algorithm for the non-maximal variant that runs in O(q) amortized time and linear space, where q is the clique number, i.e., the maximum size of a clique in an input graph. Note that the proposed algorithm works correctly even if the exact value of q is unknown. It is optimal for graphs with a bounded clique number, such as, triangle-free graphs, bipartite graphs, planar graphs, bounded degenerate graphs, nowhere dense graphs, and F-free graphs for any fixed graph F, where a F-free graph is a graph that has no copy of F as a subgraph. Furthermore, with a slight modification of our proposed algorithm, we can enumerate independent sets with the size at most k in the same time and space complexity. This problem is a generalization of the original problem since this is equal to the original problem if k=n.

Generating Gray codes for weak orders in constant amortized time

A weak order is a way to rank n objects where ties are allowed. In this paper, we develop a generic framework to generate Gray codes for weak orders. We then describe a simple algorithm based on the framework that generates cyclic 2-Gray codes for weak orders in constant amortized time per string. This is the first known cyclic 2-Gray codes for weak orders. The framework can easily be modified to generate other Gray codes for weak orders. We provide an example of using the framework to generate the first shift Gray codes for weak orders in constant amortized time per string, where consecutive strings differ by a shift or a symbol change.

Amortized efficiency of constructing multiple independent spanning trees on bubble-sort networks

A set of spanning trees in a graph G is called independent spanning trees (ISTs for short) if they are rooted at the same vertex, say r, and for each vertex $$v(\ne r)$$ in G, the two paths from v to r in any two trees share no common edge and no common vertex except for v and r. Constructing ISTs has applications on fault-tolerant broadcasting and secure message distribution in reliable communication networks. Since Cayley graphs have been used extensively to design the topologies of interconnection networks, construction of ISTs on Cayley graphs is significative. It is well-known that star networks $$S_n$$ and bubble-sort network $$B_n$$ are two of the most attractive subclasses of Cayley graphs. Although it has been dealt with about two decades for the construction of ISTs on $$S_n$$ (which has been pointed out that there is a flaw and has been corrected recently), so far the problem of constructing ISTs on $$B_n$$ is not dealt with yet. In this paper, we present an algorithm to construct $$n-1$$ ISTs of $$B_n$$ . Moreover, we show that our algorithm has amortized efficiency for multiple trees construction. In particular, every vertex can determine its parent in each spanning tree in a constant amortized time. Accordingly, except for the star networks, it seems that our work is the latest breakthrough on the problem of ISTs for all subfamilies of Cayley graphs.

From Tetris to polyominoes generation

The behaviour of a bad Tetris player suggests a class of polyominoes that we call prefix-closed. Such a class contains all polyominoes P such that for any integer i>0 the first i columns of P form a polyomino. We provide a simple discrete dynamical system that allows us to define an algorithm for generating all prefix-closed polyominoes of size n in constant amortized time using O(n) space.

On the exhaustive generation of k-convex polyominoes

The degree of convexity of a convex polyomino P is the smallest integer k such that any two cells of P can be joined by a monotone path inside P with at most k changes of direction. In this paper we present a simple algorithm for computing the degree of convexity of a convex polyomino and we show how it can be used to design an algorithm that generates, given an integer k, all k-convex polyominoes of area n in constant amortized time, using space O(n). Furthermore, by applying few changes, we are able to generate all convex polyominoes whose degree of convexity is exactly k.

A recursive structure of sand pile model and its applications

Abstract The Sand Pile Model (SPM) and its generalization, the Ice Pile Model (IPM), originate from physics and have various applications in the description of the evolution of granular systems. In this article, we deal with the enumeration and the exhaustive generation of the accessible configuration of the system. Our work is based on a new recursive decomposition theorem for SPM configurations using the notion of staircase bases. Based on this theorem, we provide a recursive formula for the enumeration of SPM(n) and a constant amortized time (CAT) algorithm for the generation of all SPM(n) configurations. The extension of the same approach to the Ice Pile Model is also discussed.

The lexicographically smallest universal cycle for binary strings with minimum specified weight

H. Fredricksen, I.J. Kessler and J. Maiorana discovered a simple but elegant construction of a universal cycle for binary strings of length n: Concatenate the aperiodic prefixes of length n binary necklaces in lexicographic order. We generalize their construction to binary strings of length n whose weights are in the range c,c+1,…,n by simply omitting the necklaces with weight less than c. We also provide an efficient algorithm that generates the universal cycles in constant amortized time per bit using O(n) space. Our universal cycles have the property of being the lexicographically smallest universal cycle for the set of binary strings of length n with weight ≥c.

On the generation of convex polyominoes

A convex polyomino can be easily represented by a pair of suitable integer vectors. This lets us design a CAT (Constant Amortized Time) algorithm for generating all convex polyominoes of size n using O(n) space.

Generating fixed density bracelets of arbitrary base

Bracelets are lexicographically minimal k-ary strings symmetric under rotation and reversal. In this paper, we present an algorithm for lexicographic listing of bracelets with fixed density. Our algorithm works for arbitrarily large alphabet size and each successive bracelet is generated in constant amortized time.

Generating bracelets with fixed content

We present an algorithm to generate bracelets with fixed content. An analysis shows that the algorithm runs in constant amortized time. The algorithm can be applied to efficiently list all non-isomorphic unicyclic graphs with n vertices.

On the exhaustive generation of plane partitions

We present a CAT (Constant Amortized Time) algorithm for generating all plane partitions of an integer n, that is, all integer matrices with non-increasing rows and columns having sum n.

A Gray code for the shelling types of the boundary of a hypercube

We consider two shellings of the boundary of the hypercube equivalent if one can be transformed into the other by an isometry of the cube. We observe that a class of indecomposable permutations, bijectively equivalent to standard double occurrence words, may be used to encode one representative from each equivalence class of the shellings of the boundary of the hypercube. These permutations thus encode the shelling types of the boundary of the hypercube. We construct an adjacent transposition Gray code for this class of permutations. This construction can be implemented in constant amortized time. Our result is a signed variant of King’s result showing that there is a transposition Gray code for indecomposable permutations.

Stamp Foldings, Semi-Meanders, and Open Meanders: Fast Generation Algorithms

By considering a permutation representation for stamp-foldings and semi-meanders we construct tree-like data structures that will allow us to generate these objects in constant amortized time. Additionally, by maintaining the wind-factor and applying an additional optimization, the algorithm for semi-meanders can be modified to produce the fastest known algorithm to generate open meanders.

A Gray code for fixed-density necklaces and Lyndon words in constant amortized time

This paper develops a constant amortized time algorithm to produce a cyclic cool-lex Gray code for fixed-density binary necklaces, Lyndon words, and pseudo-necklaces. It is the first Gray code for these objects that achieves this time bound. The algorithm is applied: (i) to develop a constant amortized time cyclic Gray code for necklaces, Lyndon words, and pseudo-necklaces ordered by density and (ii) to obtain a fixed-density de Bruijn sequence using constant time per n bits on average. In addition to Gray code order, the algorithms can be easily modified to output the strings in co-lex order.

De Bruijn Sequences for Fixed-Weight Binary Strings

De Bruijn sequences are circular strings of length $2^n$ whose length n substrings are the binary strings of length n. Our focus is on creating circular strings of length $\binom{n}{w}$ for the binary strings of length n with weight (number of 1s) equal to w. In this case, each fixed-weight string can be encoded by its first $n{-}1$ bits since the final bit is redundant. For this reason, we construct circular strings of length $\binom{n-1}{w}+\binom{n-1}{w-1}$ whose length $n{-}1$ substrings are the binary strings of length $n{-}1$ with weight w or $w{-}1$. Our construction is reminiscent of the construction for the lexicographically least de Bruijn sequence, except the underlying algorithm is applied to cool-lex order instead of lexicographic order. The construction can be efficiently implemented so that successive blocks of n bits are generated in constant amortized time while using $O(n \log n)$-space. This article's results were also used to create de Bruijn sequences for binary strings of length n with a specified maximum weight.

An Analysis of Random-Walk Cuckoo Hashing

In this paper, we provide a polylogarithmic bound that holds with high probability on the insertion time for cuckoo hashing under the random-walk insertion method. Cuckoo hashing provides a useful methodology for building practical, high-performance hash tables. The essential idea of cuckoo hashing is to combine the power of schemes that allow multiple hash locations for an item with the power to dynamically change the location of an item among its possible locations. Previous work on the case where the number of choices is larger than two has analyzed breadth-first search, which is both inefficient in practice and currently has only a polynomial upper bound on the insertion time that holds with high probability. On the other hand, it does have expected constant amortized insertion time. Here we significantly advance the state of the art by proving a polylogarithmic bound that holds with high probability on the more efficient random-walk method, where items repeatedly kick out random blocking items until a free location for an item is found.