Peter Winkler

• Doctor of Philosophy
• Professor at Dartmouth College

We study a 2000-year-old combinatorial problem which may be the first NP-complete problem in history. The Mishna considers two versions of the problem, giving an elegant solution to the first (amounting to a pseudo-polynomial algorithm), but only a partial solution to the second. We finish the job by giving a complete solution to the second problem...

We consider an apparent paradox that arises from the following puzzle. Baseball teams A and B meet annually and play until one team has won four games and is declared the series winner. The teams are evenly matched except that each team enjoys the same slight advantage when playing at home. If, every year, the first three games are played at the ho...

We study scaling limits of random permutations (“permutons”) constrained by having fixed densities of a finite number of patterns. We show that the limit shapes are determined by maximizing entropy over permutons with those constraints. In particular, we compute (exactly or numerically) the limit shapes with fixed 12 density, with fixed 12 and 123...

Markov chains defined on the set of permutations of $1,2,\dots,n$ have been studied widely by mathematicians and theoretical computer scientists. We consider chains in which a position $i<n$ is chosen uniformly at random, and then $\sigma(i)$ and $\sigma(i{+}1)$ are swapped with probability depending on $\sigma(i)$ and $\sigma(i{+}1)$. Our objectiv...

Let $\pi$ be a permutation of $\{1,2,\ldots,n\}$. If we identify a permutation with its graph, namely the set of $n$ dots at positions $(i,\pi(i))$, it is natural to consider the minimum $L^1$ (Manhattan) distance, $d(\pi)$, between any pair of dots. The paper computes the expected value (and higher moments) of $d(\pi)$ when $n\rightarrow\infty$ an...

Let G be an infinite geometric graph; in particular, a graph whose vertices are a countable discrete set of points on the plane, with vertices u, v adjacent if their Euclidean distance is less than 1. A "fire" begins at some finite set of vertices and spreads to all neighbors in discrete steps; in the meantime f vertices can be deleted at each time...

We consider a variation of cop vs. robber on graph in which the robber is not restricted by the graph edges; instead, he picks a time-independent probability distribution on and moves according to this fixed distribution. The cop moves from vertex to adjacent vertex with the goal of minimizing expected capture time. Players move simultaneously. We...

This chapter discusses a method of computing probabilities called “coupling.” The idea of coupling is that, when comparing probabilities of two events A and B , one tries to put them into the same experiment. All one would need to do is compare Pr( A but not B ) with Pr( B but not A ). This might be quite easy, especially if most of the time either...

An abelian processor is an automaton whose output is independent of the order of its inputs. Bond and Levine have proved that a network of abelian processors performs the same computation regardless of processing order (subject only to a halting condition). We prove that any finite abelian processor can be emulated by a network of certain very simp...

We define an entropy function for scaling limits of permutations, called permutons, and prove that under appropriate circumstances, both the shape and number of large permutations with given constraints are determined by maximizing entropy over permutons with those constraints. We also describe a useful equivalent version of permutons using a recur...

Modern computers with hard-disk storage and networks with dynamic spectrum access illustrate systems having a resource RR that allows fragmented allocations. We model RR as a sequence of M>1M>1 units for which requests are made by items in a FIFO queue; each request is for a desired amount of RR, the item size, and the residence time during which i...

We show that the expected time for a smart "cop" to catch a drunk "robber" on an n-vertex graph is at most n+o(n). More precisely, let G be a simple, connected, undirected graph with distinguished points u and v among its n vertices. A cop begins at u and a robber at v; they move alternately from vertex to adjacent vertex. The robber moves randomly...

Consider two simple games played by Alice and Bob on a checkerboard or, more generally, on a graph. The games look different, but, as we know, looks can be deceiving…

Last month (May 2014) we posted three puzzles in which you were asked to sort several cards using three stacks on a table; you were allowed to move the top card of one stack to the top of another (possibly empty) stack, with the object being to get all the cards in their natural order stacked in the leftmost place. The catch was you could see only...

Last month (February 2014) we posted three games in which you were asked to pick a positive integer. The question in each was: What is the highest number you should think about picking? Here, we offer solutions to all three. How did you do? 1. Found dollar. Alice and Bob are vying for a found dollar, with lowest number the winner and a tie winning...

In a recent issue of Communications, ACM President Vinton Cerf gave an excellent account of what ACM is doing to help reform K--12 education (August 2013, p. 7).

We consider the following cryptographic secret leaking problem. A group of players communicate with the goal of learning (and perhaps revealing) a secret held initially by one of them. Their conversation is monitored by a computationally unlimited eavesdropper, who wants to learn the identity of the secret-holder. Despite the unavailability of key,...

Last month (November 2013) we posted three tricky puzzles concerning coin flipping. Here, we offer solutions to all three. How did you do?

We consider a variation of a cops and robbers game in which the cop---here referred to as "hunter"---is not constrained by the graph but must play in the dark against a "mole." We characterize the graphs---which we will call "hunter-win"---on which the hunter can guarantee capture of the mole in bounded time. We also define an optimal hunter strate...

Each of these puzzles involves coin flipping. Simple stuff, right? Not necessarily…though solutions will indeed be provided in next month's column.

We consider a variation of cop vs.\ robber on graph in which the robber is not restricted by the graph edges; instead, he picks a time-independent probability distribution on $V(G)$ and moves according to this fixed distribution. The cop moves from vertex to adjacent vertex with the goal of minimizing expected capture time. Players move simultaneou...

The Communications Web site, http://cacm.acm.org, features more than a dozen bloggers in the BLOG@CACM community. In each issue of Communications, we'll publish selected posts or excerpts.twitterFollow us on ...

Last month (May 2013) we posed a trio of brainteasers concerning Ant Alice and her ant friends who always march at 1 cm/sec in whatever direction they are facing, reversing direction when they collide.

We show that the expected time for a smart "cop" to catch a drunk "robber" on an $n$-vertex graph is at most $n + {\rm o}(n)$. More precisely, let $G$ be a simple, connected, undirected graph with distinguished points $u$ and $v$ among its $n$ vertices. A cop begins at $u$ and a robber at $v$; they move alternately from vertex to adjacent vertex. T...

Two parties are said to “share a secret” if there is a question to which only they know the answer. Since possession of a shared secret allows them to communicate a bit between them over an open channel without revealing the value of the bit, shared secrets are fundamental in cryptology. We consider below the problem of when two parties with shared...

These three puzzles involve my favorite ant, Ant Alice. Like all ants on this page, Alice moves at exactly one centimeter per second in whichever direction she happens to be facing; if she meets another ant head on, both immediately reverse direction and walk away from each other, each still at speed 1 cm/sec. Figuring out how Alice and her friends...

Last month (February 2013) we posed a trio of brainteasers concerning probability and dice. Here, we offer solutions to all three. How did you do?

We describe a new concept for making photo tampering more difficult and time consuming, and for a given amount of time and effort, more amenable to detection. We record the camera preview and camera motion in the moments just prior to image capture. This information is packaged along with the full resolution image. To avoid detection, any subsequen...

We show that if a graph H is k-colorable, then (k−1)-branching walks on H exhibit long range action, in the sense that the position of a token at time 0 constrains the configuration of its descendents arbitrarily far into the future. This long range action property is one of several investigated herein; all are similar in some respects to chromatic...

Let Gλ be the graph whose vertices are points of a planar Poisson process of density λ, with vertices adjacent if they are within distance 1. A “fire” begins at some vertex and spreads to all neighbors in discrete steps; in the meantime f vertices can be deleted at each time-step. Let fλ be the least f such that, with probability 1, any fire on Gλ...

We consider irreducible Markov chains on a finite state space. We show that the mixing time of any such chain is equivalent to the maximum, over initial states $x$ and moving large sets $(A_s)_s$, of the hitting time of $(A_s)_s$ starting from $x$. We prove that in the case of the $d$-dimensional torus the maximum hitting time of moving targets is...

Last month (August 2012) we posted a trio of brainteasers concerning "magic sets." Here, we offer solutions to all three. How did you do?

Welcome to three new puzzles. Each involves a collection of items, and your job is to find a subset of them that is characterized by a particular property. Since solving the puzzles is not easy, here are a couple of hints: For the first, think about averages; for the other two, try constructing your sets sequentially, bearing in mind that if two pa...

A planar set that contains a unit segment in every direction is called a Kakeya set. We relate these sets to a game of pursuit on a cycle ℤ_n. A hunter and a rabbit move on the nodes of ℤ_n without seeing each other. At each step, the hunter moves to a neighbouring vertex or stays in place, while the rabbit is free to jump to any node. Adler et al....

Last month (May 2012) we posted a trio of brainteasers concerning designs on square grids. Here, we offer solutions to all three. How did you do?

Welcome to, as usual, three new puzzles. However, unlike previous columns, where solutions to two were known (and included in the related Solutions and Sources in the next issue), this time expect to see solutions to all three in June.

Last month (Feb. 2012) we posted a trio of brainteasers concerning where sets meet, or Venn diagrams. Here, we offer solutions to two of them. How did you do?

Welcome to three new puzzles. Solutions to the first two will be published next month; the third is as yet unsolved.

To improve efficiency, we allow a carrier in a TDMA (Time Division Multiple Access) network to be shared by adjacent cells. This sharing of time slots is seriously hampered by the lack of synchronization in distinct cells. We study packing algorithms that overcome this obstacle by clustering calls. The results suggest that even simple greedy algori...

Last month (Nov. 2011, p. 120) we posted a trio of brainteasers, including one as yet famously unsolved, concerning distances between points on the plane. Here, we offer solutions to two of them. How did you do?

We examine the question of whether a collection of random walks on a graph can be coupled so that they never collide. In particular, we show that on the complete graph on n vertices, with or without loops, there is a Markovian coupling keeping apart Omega(n/log n) random walks, taking turns to move in discrete time.

We consider Glauber dynamics (starting from an extremal configuration) in a monotone spin system, and show that interjecting extra updates cannot increase the expected Hamming distance or the total variation distance to the stationary distribution. We deduce that for monotone Markov random fields, when block dynamics contracts a Hamming metric, sin...

We solve the game of Babylon when played with chips of two colors, giving a winning strategy for the second player in all previously unsolved cases.

Welcome to three new puzzles. Solutions to the first two will be published next month; the third is as yet (famously) unsolved.

We show that the expected time for a random walk on a (multi-)graph $G$ to traverse all $m$ edges of $G$, and return to its starting point, is at most $2m^2$; if each edge must be traversed in both directions, the bound is $3m^2$. Both bounds are tight and may be applied to graphs with arbitrary edge lengths, with implications for Brownian motion o...

Last month (Aug. 2011, p. 120) we posted a trio of brainteasers, including one as yet unsolved, concerning divisibility of numbers. Here, we offer solutions to two of them and a remark about the third. How did you do?

Welcome to three new puzzles. Solutions to the first two will be published next month; the third is (as yet) unsolved. In fact, this time itﾒs a famously unsolved problem.

A dyadic tile of order n is any rectangle obtained from the unit square by n successive bisections by horizontal or vertical cuts. Let each dyadic tile of order n be available with probability p, independently of the others. We prove that for p sufficiently close to 1, there exists a set of pairwise disjoint available tiles whose union is the unit...

Last month (May 2011, p. 120) we posted a trio of brainteasers, including one as yet unsolved, concerning games and their roles and turns, with randomness either removed or inserted. Here, we offer solutions to two of them. How did you do?

Welcome to three new puzzles. Solutions to the first two will be published next month; the third is (as yet) unsolved. In each, the issue is how your intuition matches up with the mathematics.

Welcome to three new puzzles. Solutions to the first two will be published next month; the third is (as yet) unsolved. In each, the issue is how your intuition matches up with the mathematics.

It's amazing how little we know about good old plane geometry. Last month (August 2010, p. 128) we posted a trio of brainteasers, including one as yet unsolved, concerning figures on a plane. Here, we offer solutions to two of them.

Last month (August 2014), we presented three puzzles concerning the Path Game and the Match Game, each of which can be played on any finite graph. To start, Alice marks a vertex; Bob and Alice then alternate marking vertices until one (the loser) is unable to mark any more. In the Path Game, each vertex thus marked, following the first one, must be...

It's amazing how little we know about the simple, ordinary, axis-aligned rectangle. Last month (p. 112) we posted a trio of brainteasers, including one as yet unsolved, concerning rectangles galore. Here, we offer solutions to at least two of them. How did you do?

We will explore the computational complexity of satisfying certain sets of neighborhood conditions in graphs with various properties. More precisely, fix a radius ρ and let N(G) be the set of isomorphism classes of ρ-neighborhoods of vertices of G where G is a graph whose vertices are colored (not necessarily properly) by colors from a fixed finite...

Welcome to three new puzzles. Solutions to the first two will be published next month; the third is (as yet) unsolved. In each, the issue is how your intuition matches up with the mathematics.

Last month (November 2012) we posted a trio of brainteasers concerning the use of a balance scale to determine the weight of various numbers of coins. Here, we offer solutions to all three. How did you do?

Welcome to three new puzzles. Solutions to the first two will be published next month; the third is (as yet) unsolved. In each, the issue is how your intuition matches up with the mathematics.

Last month (May 2010, p. 120) we posted a trio of brainteasers, including one as yet unsolved, concerning variations on the Ham Sandwich Theorem.

Solutions to puzzles concerning circular food shapes presented in a previous issue are presented.

Welcome to three new puzzles. Solutions to the first two will be published next month; the third is (as yet) unsolved. In each, the issue is how your intuition matches up with the mathematics.

Last month (February 2010, p. 120) we posted a trio of brainteasers, including one as yet unsolved, concerning the breaking of a bar of chocolate.

Welcome to three new puzzles. Solutions to the first two will be published next month; the third is (as yet) unsolved. In each, the issue is how your intuition matches up with the mathematics.

Ebenso wie guter Wein können auch Rätsel mit der Zeit reifen und neue, aufregende Varianten entwickeln, oder es finden sich elegantere Lösungen zu alten Varianten. Dieses Kapitel enthält einige Knobeleien, die Ihnen vielleicht vertraut vorkommen; doch selbst, wenn Sie eines von ihnen in guter Erinnerung zu haben glauben, sind Sie vermutlich von den...

Die meisten von uns sind in der Schule im Rahmen der euklidischen ebenen Geometrie zum ersten Mal den Konzepten von Sätzen und Beweisen begegnet. Doch die Rätsel, um die es im Folgenden geht, haben nicht viel mit Euklids Elementen zu tun. Sie testen eher Ihre Vertrautheit mit der zwei- und dreidimensionalen Welt.

Welcome to three new puzzles. Solutions to the first two will be published next month; the third is (as yet) unsolved. In each, the issue is how your intuition matches up with the mathematics.

Last month (August 2009, p. 104) we posted a trio of brainteasers, including one as yet unsolved, concerning probability and intuition.

Welcome to three new puzzles. Solutions to the first two will be published next month; the third is (as yet) unsolved. In each puzzle, the issue is how your intuition matches up with the mathematics.

Last month (November 2009, p. 112) we posted a trio of brain teasers, including one as yet unsolved, concerning the covering of a plane.

Welcome to three new challenging mathematical puzzles. Solutions to the first two will be published next month; the third is as yet (famously) unsolved. In each puzzle, the issue is how numbers interact with one another.

Last month (November 2009, p. 112) we posted a trio of brain teasers, including one as yet unsolved, concerning the covering of a plane.

In Mathematical Mind-Benders, Peter Winkler. Wellesley, MA: A.K. Peters, 2007. Reprinted by permission of A.K. Peters, Ltd.

Let f : L -> R be a submodular function on a modular lattice L; we show that there is a maximal chain C in L on which the sequence of values of f is minimal among all paths from 0 to 1 in the Hasse diagram of L, in a certain well-behaved partial order on sequences of reals. One consequence is that the maximum value of f on C is minimized over all s...

Last month (November 2008, p. 112) we posed a trio of brain teasers concerning circular food shapes. Here, we offer some possible solutions. How did you do?

Let a i , b i , i = 0, 1, 2, . . . be drawn uniformly and independently from the unit interval, and let t be a fixed real number. Let a site ( i , j ) ∈ $\N^2$ be open if a i + b j ≤ t , and closed otherwise. We obtain a simple, exact expression for the probability Θ( t ) that there is an infinite path (oriented or not) of open sites, containing th...

Welcome to three new challenging mathematical puzzles. Solutions to the first two will be published next month; the third is as yet unsolved, so you may need extra luck with that one. Here, I concentrate on circular food, so you might want to eat something before jumping in.

Ants, even in a one-dimensional environment, are a source of fascination for amateur puzzlists and mathematicians. Presented here are ten puzzles (devised by myself, except where noted) involving our “favorite ant” Alice. Each puzzle is intended to illustrate some mathematical idea.

In sorting situations where the final destination of each item is known, it is natural to repeatedly choose items and place them where they belong, allowing the intervening items to shift by one to make room. (In fact, a special case of this algorithm is commonly used to hand-sort files.) However, it is not obvious that this algorithm necessarily t...

Welcome to the new puzzle column. Each column will present three puzzles. The first two will have known (and usually elegant) solutions that will appear in the next issue of Communications. The third will be an open problem; good luck with that one. Readers are encouraged to submit prospective puzzles for future columns to puzzled@cacm.acm.org. We...

A branched polymer is a connected configuration of non-overlapping unit balls in space. Building on and from the work of D. Brydges and J. Z. Imbrie [Ann. Math. (2) 158, No. 3, 1019–1039 (2003; Zbl 1140.82314)], this article presents an elementary calculation of the volume of the space of branched polymers of order n in the plane and in 3-space. Ou...

Recent legislation in the US regarding gambling over the web has led to renewed interest in the question of which games are games of skill. We take a statistical approach to the problem, defining the skill index of a game to be the average amount of playing time after which variance due to chance and variance due to skill differences are equal. We...

A 150-year-old problem asks how many identical rectangular blocks of length 1, balanced at the edge of a table, are needed to achieve some given overhang D beyond the table edge. A well-known construction based on the harmonic series requires a number of blocks which grows exponentially with D. Many people had assumed that this was optimal until a...

A k-majority tournament T on a finite vertex set V is defined by a set of 2 k - 1 linear orderings of V, with u → v if and only if u lies above v in at least k of the orders. Motivated in part by the phenomenon of "non-transitive dice", we let F ( k ) be the maximum over all k-majority tournaments T of the size of a minimum dominating set of T. We...

We prove that a system of particles in the plane, interacting only with a certain hard-core constraint, undergoes a fluid-solid phase transition.

We prove that uniformly random packings of copies of a certain simply-connected figure in the plane exhibit global connectedness at all sufficiently high densities, but not at low densities.

In contrast to a typical single source of data updates in Internet applications, data files in a networked information system are often distributed, replicated, accessed and updated by multiple nodes. Due to concurrent updates, replicated data files must be synchronized. For certain applications, stringent concurrency control must be employed to en...

We prove that a system of particles in the plane, interacting only with a certain hard-core constraint, undergoes a fluid/solid phase transition.

The main activities supported under this grant are research and support for C. Chekuri, B. Shepherd and P. Winkler. Funds also supported one summer intern, Andrew McGregory from U-Penn, who worked with Shepherd on recognizing Hilbert Bases and other theoretical topics in Math Programming. Visits from scientists include a 2-week visit from Gianpaolo...

We determine the mixing time (up to a constant factor) of the Markov chain whose state space consists of n "dots" on the unit interval, wherein a dot is selected uniformly at random and moved to a uniformly random point between its two neighbors. The method involves a two-step coupling for the upper bound, and an unusual probabilistic second- momen...

We determine, up to a log factor, the mixing time of a Markov chain whose state space consists of the successive distances between n labeled “dots” on a circle, in which one dot is selected uniformly at random and moved to a uniformly random point between its two neighbors. The method involves novel use of auxiliary discrete Markov chains to keep t...

We show that the problem of counting the number of Eulerian circuits in an undirected graph is complete for the class #P. The method employed is mod-p reduction from counting Eulerian orientations.

We prove the existence of, and describe, a (random) process which builds subtrees of a rooted d-branching tree one node at a time, in such a way that the subtree created at stage n is precisely a uniformly random subtree of size n. The union of these subtrees is a “uniformly random” infinite subtree, which we describe and generate in several ways....

We show that the problem of counting the number of Eulerian circuits in an undirected graph is complete for the class #P.