Unpublished Short Notes

Proof of an Abstract Algorithm for Graph Reachability.  May. 7, 2019
We present an abstract non-deterministic algorithm for identifying nodes of a directed graph that are reachable from a specific node. The algorithm encompasses all known algorithms for this problem, including breadth-first and depth-first search. Its correctness, therefore, implies that all such algorithms are correct.

A Proof of a Heavy Hitters Algorithm.  Feb. 7, 2018
This note is meant to teach program development and formal proofs in an introductory computer science course. Given a bag of items consider finding all items that occur more often than a given fraction of the bag size. The first algorithm for this problem, due to Misra and Gries, is used as the motivation for the note. First, I treat a special case, the Boyer-Moore Majority Protocol. I start with its mathematical underpinning, then show a program with abstract data structures, later a refinement by replacing the abstract data structures by concrete representations. The heavy hitters algorithm is a small generalization of the ideas inherent in this algorithm.

Knuth-Morris-Pratt string matching algorithm.  July 21, 2016
The Knuth-Morris-Pratt string matching algorithm locates all occurrences of a pattern string in a text string in linear time. It is a refined version of a naive algorithm.

Dijkstra's single-source shortest path algorithm.  June 14, 2016
Dijkstra's single-source shortest path algorithm is explained in two different ways, first as a discrete event simulation of a physical system and next in a more conventional manner.

A covergence proof.   June 6, 2016
An undirected connected finite graph has a natural number initially associated with each node. There is a distinguished node, anchor. A non-anchor node may make a move by setting its value to 1 + the minimum value over all its neighbors. The anchor node never makes a move. Show that the computation eventually converges so that no move changes any value.

Implementation of Queue with add, remove and max operations.   Oct. 08, 2014
It is required to maintain a queue that supports the traditional "add-at-the-left" and "remove-from-the-right" operations. Additionally, it should support a max operation that returns the maximum value in the queue. The inclusion of max makes it impossible to implement each operation in constant time in the worst case. What is required is an implementation with constant amortized running time.

Knaster-Tarski Theorem.   Sep 12, 2014
This note presents a proof of the famous Knaster-Tarski theorem [1]. I have opted for clarity over brevity in the proof.

Well-founded orderings over multisets.  June 09, 2014
This note gives an elementary proof of a result, due to Manna and Dershowitz, on the well-foundedness of an order relation over multisets.

A Variation of Koenig's Lemma.   Sept. 11, 2014
A variation of the famous infinite tree lemma that is useful in program semantics.

Guessing a Hat color out of N colors   Sep. 14, 2012
The following problem was communicated to me by Mike Starbird. There are N persons each of whom has a hat on his head. There are N possible hat colors. Not every color appears on someone's head. Every one can see the colors of all others' hats, but not his own. Each person guesses the color of his hat (writing it down on a piece if paper, say). Devise a protocol so that at least one person guesses his hat color correctly.

Identifying Lights with their Switches   Sep. 07, 2012
Given is a set of switches and an equal number of lights where each switch controls exactly one light and each light is controlled by exactly one switch. The wiring diagram is unavailable and the wiring itself is hidden. A step consists of selecting some number of switches and turning them on, and, presumably, noting the lights that come on as a result. It is required to determine which switch controls which light using a minimum number of steps.

Digital Cash   Aug. 24, 2012
We propose a scheme for management of digital cash that mimics the current physical management. In particular, any one can verify the authenticity of a digital bill, no one can manufacture or double-spend a bill, a transaction between two parties does not reveal the identities of the parties to others, and the bills are maintained by a distributed set of (trusted) servers that may belong to many financial institutions, much like the banks of today. Additionally, failure of any one client or server does not affect the rest of the system.

A Proof of the Infinite Ramsey Theorem   July 31, 2012

Assigning Coordinates to Events: Solving Combinatorial problems using Discrete Event Simulation.   (with David Kitchin and John Thywissen) Nov. 1, 2011
This paper is inspired by the "event list" mechanism in discrete event simulations. We argue that descriptions of many combinatorial algorithms can be simplified by casting the solution in terms of processing events according to some order. We propose generalizations of the event list mechanism, and show their applications in problems from graph theory and computational geometry.

Processing Boolean Equalities and Inequalities.   Nov. 1, 2011
Given a set of boolean variables, and a set of equalities and inequalities among those variables (called ``facts''), it is required to determine the relationship, if any, among pairs of variables (called ``queries''). We consider two versions of this problems, one when the queries are given after all the facts have been presented (off-line) and when the queries and facts are intermingled (on-line). For the off-line version we give a linear algorithm, and for the on-line version a quasi-linear one by modifying the well known union-find algorithm.

Monty Hall game  Aug. 10, 2011
Four short proofs for the famous Monty Hall game.

Chameleons  Sept. 8, 2009, Revised 01/05/2014
There are 3 piles of chips. A step consists of removing one chip each from two different piles and adding both chips to the third pile. The game terminates (reaches a final state) when no more steps can be taken, i.e., there are at least two empty piles. It is required to devise a strategy to arrive at a terminating state, or prove that no such strategy exists for the given initial state. This puzzle first appeared in the Comm. of ACM in the form of Chameleons of 3 different colors. I have made it less colorful by converting it to piles of chips.

A puzzle from Peter Winkler  July 19, 2006
A secret is a triple where each component is a natural number below n, for some given n. A guess is a triple of the same form. A guess has an outcome which is revealed to the guesser: it succeeds if it matches at least two corresponding components of the secret, and fails otherwise. What is the minimum number of guesses required to succeed for n=8?

Unique Prime Factorization Theorem.   Feb 4, 2006
For every positive integer there is a unique bag of primes whose product equals that integer. The fact that there is a bag of primes corresponding to every positive integer is readily proven using induction. We prove the uniqueness part: distinct bags of primes have distinct products.

(With William Cook) Some Facts about String Interleaving  February 17, 2005
We prove commutativity, associativity and a distributivity property of string interleaving.

Pairing Integers so that their sums are primes.   January 31, 2005
For every even positive integer n, pair the integers up to n so that the sum of each pair is prime.

A Consensus Protocol in a Prison.   February 2, 2004
A set of prisoners --assume there are at least 2-- are asked to play the following game by the warden. There is a room in the prison which has two switches; initially, the switches are in arbitrary positions. The warden will bring one prisoner at a time to the room, and the prisoner must flip one of the switches. The prisoners do not know the order in which they will be taken to the room, but they know that every prisoner will visit the room over and over until the end of the game. The game ends when some prisoner announces, ``every prisoner has been in this room at least once''. If the announcement is correct, all prisoners go free; if incorrect, they all are executed. The game continues until the announcement is made.

The prisoners are allowed to confer and decide on a protocol prior to the start of the game. Once the game starts, they are not allowed to communicate, nor can they find out who is being taken to the room. The problem is to devise a protocol for the prisoners.

A Note on EWD 1312  Sep 14, 2001
Integers h and k are disjoint provided in their binary representations no position has 1s in both h and k. (In comparing two integers, append leading zeroes to the binary representation of the smaller number to make their lengths identical.) Dijkstra proved that for positive h and k, among (h, k), (h, k-1), and (h-1, k), an even number of pairs are disjoint. In this note we prove this result using some elementary algebraic properties of disjointness.

Tree Isomorphism: An Exercise in Functional Programming  Sep 10, 2001
The problem is to decide if two unordered trees are the same.

Parities of Binomial Coefficients  May 16, 2001
We give a formula for the parity of any binomial coefficient.

A Problem due to J Moore.   Mar 28, 2001
The following problem was posed by J Moore during the faculty lunch today. Let there be two machines with two registers each, which can read/write a shared counter. Initially the counter holds the value 1 and all registers are empty. There are two atomic actions:

  • (read) A machine may read the counter value into one of its empty registers (and then the register becomes nonempty).
  • (write) A machine may write the sum of its register values provided that both are nonempty into the counter and then empty both registers. We show that for any positive integer there is an execution in which the counter holds that value.

    Computing Spans.   Jan 22, 2001
    Given is a sequence of integers. For any element, e, of the sequence, define its "span" to be the length of the longest segment ending at e where each element of the segment is at most e in value. This note contains development of a linear algorithm for computing all the spans.

    Enumerating the Strings of a Regular Expression.   Aug 29, 2000
    We develop an algorithm to enumerate the strings of a regular expression in increasing order.

    Locating the Center of a Set of Points on a Curve.   June 15, 2000
    Given are a finite number of points on a simple closed curve; call these points anchors. It is required to find a point, called the center, so that the sum of distances between the center and the anchors is the minimum over all points.

    A proof by Erdos.   Oct 1, 1999
    A sequence of numbers whose length exceeds n2 contains either an ascending or a descending subsequence longer than n.

    Permutations Generated by a Stack Machine.   Sep 28, 1999
    An output string is computed as follows from an input string using an infinite stack. In each step either the next input symbol is added to the stack, or the top of the stack is moved to the output. We characterize the outputs for a given input.

    Russell Paradox, Cantor Diagonalization.   Sep 28, 1999
    Formal Proofs are simpler than their informal counterparts, for Russell Paradox and Cantor's Diagonalization argument.

    A property of Identity Function: An Exercise in Induction.  Sep 16, 1999
    Let f be a function from naturals to naturals. It is given that for all n, f2(n) < f(n+1). Prove that f is the identity function. We will actually prove a generalization of this result.

    Minimum Spanning Tree.  Dec 12, 1998
    This note develops two well-known algorithms for finding the minimum spanning tree.

    Rules for Division by 3 and 11.  Dec 2, 1998
    Show that for any natural number n, n mod 3 = r.n mod 3, where r.n is the sum of decimal digits in n. The rule for division by 11 is to start from the lowest digit of the number, and add and subtract the digits alternately.

    The Muddy Children Puzzle  Oct 8, 1998
    There is a finite group of children where each child is clean or dirty. No child knows if it is clean or dirty, but it can see if every other child is clean or dirty. It is common knowledge that there is at least one dirty child. In a round, (1) the children are asked: do you know if you are dirty, and (2) each of them responds with ``NO'', ``YES, I am dirty'', or ``YES, I am clean''. Responses are heard by all children. Rounds are repeated ad infinitum starting at round 0. Prove that a child who sees n dirty children, n \ge 0, will answer YES in round n, but no earlier.

    Arithmetic Mean is greater than or equal to Geometric Mean  Sept. 23, 1998
    A simple proof of: the arithmetic mean of a finite non-empty bag of positive real numbers is greater than or equal to its geometric mean.

    A Proof about the Harmonic Series.   Aug 13, 1998
    The following problem and its solution was shown to me by Carroll Morgan.

    Knowledge, Product and Sum   Jan 30, 1998
    There are two parties, Product and Sum, who are given numbers p and s, respectively, where p = m x n and s = m+n, for some unknown integers m and n, each between 2 and 100. The parties deduce the values of m and n after a certain exchange. This note explains a solution due to Dijkstra (EWD-666).

    Dijkstra's Proof of Hall's Theorem  Dec 22, 1997
    This note contains my interpretation of a proof of Hall's theorem by Dijkstra.
    Remarks on Hall's Theorem of Distinct Representatives  Dec 30, 1997.
    This is an observation by me on Hall's theorem.

    Common Meeting Time  Dec 5, 1997
    We develop the appropriate conditions on the functions which are used in the common meeting time problem.

    Making Multiple Copies under possibility of Failure  April 4, 1997
    It is required to copy a file N times; call each copy a clone of the original. The file consists of a sequence of symbols each of which is independently copied. Therefore, we consider the problem when the file consists of a single symbol.

    There are Infinitely Many Primes  Mar 21, 1997
    The following proof of the classical theorem is due to Dijkstra.

    Coloring Grid Points, without Rabbits and Snakes  Dec 18, 1996
    For any finite set of grid points in the plane, we can colour each of the points either red or blue such that on each grid line the number of red points differs by at most 1 from the number of blue points.

    A Useful Recurrence for Division  June 20, 1996
    We show a recurrence that is useful for division.

    Reducing Satisfiability to Quadratic Programming  May 15, 1996
    We show that the boolean satisfiability problem can be reduced to the quadratic programming problem.

    A puzzle on infinite sequences: An application of Koenig's Lemma  April 12, 1996
    Define a word to be any non-empty finite sequence of symbols. Each word is either good or bad. Given an infinite sequence of symbols, show that beyond some point, the sequence can be broken into words that are all good or that are all bad.

    A puzzle on Termination  April 9, 1996
    Given is a 10 X 10 square grid. A cell is a square in the grid. Two cells are neighbors if they share a side. Initially 9 cells are chosen and colored red. Next, any cell may be colored red provided it has two red neighbors. Prove or disprove that the entire region can be colored red.

    A Counting Problem Communicated by Dijkstra  April 9, 1996
    Given that there are 25 boys and 25 girls. A party has 12 tables, each of which seats 2 boys and 2 girls. Thus, a party is attended by 24 boys and 24 girls. A boy sees 2 girls at his table in a party, and so do the girls. A set of parties, P, is feasible if each boy/girl sees different girls/boys in different parties in P. What is the size of the largest feasible set?

    Random Number Generation without Repetition  Mar 27, 1996
    We are given a function from naturals to naturals, which is applied repeatedly starting at a fixed seed. Floyd has shown how repetitions can be detected in this sequence. We give a proof of this result.

    Sorting The Rows of a Matrix Preserves the Sorted Columns  Feb 19, 1996
    Given is a matrix whose columns are sorted. Show that if each row is sorted individually then the columns remain sorted.

    On the union of well-founded relations: An application of Koenig's Lemma  Feb 16, 1996
    We are given two well-founded relations on the same set, and that their union is transitive. We show that the union is well-founded.

    A position paper on fairness.   1988
    A rebuttal of Dijkstra's position on fairness.