On universal partial words

A universal word for a finite alphabet $A$ and some integer $n\geq 1$ is a word over $A$ such that every word in $A^n$ appears exactly once as a subword (cyclically or linearly). It is well-known and easy to prove that universal words exist for any $A$ and $n$. In this work we initiate the systematic study of universal partial words. These are words that in addition to the letters from $A$ may contain an arbitrary number of occurrences of a special `joker' symbol $\Diamond\notin A$, which can be substituted by any symbol from $A$. For example, $u=0\Diamond 011100$ is a linear partial word for the binary alphabet $A=\{0,1\}$ and for $n=3$ (e.g., the first three letters of $u$ yield the subwords $000$ and $010$). We present results on the existence and non-existence of linear and cyclic universal partial words in different situations (depending on the number of $\Diamond$s and their positions), including various explicit constructions. We also provide numerous examples of universal partial words that we found with the help of a computer

On the Hardness of Gray Code Problems for Combinatorial Objects

Can a list of binary strings be ordered so that consecutive strings differ in a single bit? Can a list of permutations be ordered so that consecutive permutations differ by a swap? Can a list of non-crossing set partitions be ordered so that consecutive partitions differ by refinement? These are examples of Gray coding problems: Can a list of combinatorial objects (of a particular type and size) be ordered so that consecutive objects differ by a flip (of a particular type)? For example, 000, 001, 010, 100 is a no instance of the first question, while 1234, 1324, 1243 is a yes instance of the second question due to the order 1243, 1234, 1324. We prove that a variety of Gray coding problems are NP-complete using a new tool we call a Gray code reduction.Comment: 15 pages, 5 figures, WALCOM 202

Trimming and Gluing Gray Codes

We consider the algorithmic problem of generating each subset of [n]:={1,2,...,n} whose size is in some interval [k,l], 0 <= k <= l <= n, exactly once (cyclically) by repeatedly adding or removing a single element, or by exchanging a single element. For k=0 and l=n this is the classical problem of generating all 2^n subsets of [n] by element additions/removals, and for k=l this is the classical problem of generating all n over k subsets of [n] by element exchanges. We prove the existence of such cyclic minimum-change enumerations for a large range of values n, k, and l, improving upon and generalizing several previous results. For all these existential results we provide optimal algorithms to compute the corresponding Gray codes in constant time O(1) per generated set and space O(n). Rephrased in terms of graph theory, our results establish the existence of (almost) Hamilton cycles in the subgraph of the n-dimensional cube Q_n induced by all levels [k,l]. We reduce all remaining open cases to a generalized version of the middle levels conjecture, which asserts that the subgraph of Q_(2k+1) induced by all levels [k-c,k+1+c], c in {0, 1, ...k}, has a Hamilton cycle. We also prove an approximate version of this conjecture, showing that this graph has a cycle that visits a (1-o(1))-fraction of all vertices

All your bases are belong to us : listing all bases of a matroid by greedy exchanges

You provide us with a matroid and an initial base. We say that a subset of the bases "belongs to us" if we can visit each one via a sequence of base exchanges starting from the initial base. It is well-known that "All your base are belong to us". We refine this classic result by showing that it can be done by a simple greedy algorithm. For example, the spanning trees of a graph can be generated by edge exchanges using the following greedy rule: Minimize the larger label of an edge that enters or exits the current spanning tree and which creates a spanning tree that is new (i.e., hasn't been visited already). Amazingly, this works for any graph, for any labeling of its edges, for any initial spanning tree, and regardless of how you choose the edge with the smaller label in each exchange. Furthermore, by maintaining a small amount of information, we can generate each successive spanning tree without storing the previous trees. In general, for any matroid, we can greedily compute a listing of all its bases matroid such that consecutive bases differ by a base exchange. Our base exchange Gray codes apply a prefix-exchange on a prefix-minor of the matroid, and we can generate these orders using "history-free" iterative algorithms. More specifically, we store O(m) bits of data, and use O(m) time per base assuming O(1) time independence and coindependence oracles. Our work generalizes and extends a number of previous results. For example, the bases of the uniform matroid are combinations, and they belong to us using homogeneous transpositions via an Eades-McKay style order. Similarly, the spanning trees of fan graphs belong to us via face pivot Gray codes, which extends recent results of Cameron, Grubb, and Sawada [Pivot Gray Codes for the Spanning Trees of a Graph ft. the Fan, COCOON 2021]

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum