Programming Praxis

One of the sites that I watch from time to time is Career Cup, which publishes coding questions that have been asked in actual job interviews. These questions came through this morning:

1) Consider a sorted singly-linked list having the following nodes: 10 -> 30 -> 50 -> 70 -> NULL. You are given a pointer to node 50 and a new node having the value 40. Can you insert node 40 correctly in the list maintaining the ascending order?

2) Given a linked list 5 -> 4 -> 3 -> 2 -> 1 produce a linked list 4 -> 2 -> 0 -> 2 -> 1 by subtracting the last node of the list from the first, the next-to-last node from the second, and so on, stopping at the midpoint of the list.

3) Write a program to output the number of consecutive trailing zeros in the factorial of a number. For example, if the number is 5, then 5! = 120, and there is one trailing zero.

Your task is to write programs to answer the three interview questions posed above. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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Clustering is the process of collecting in groups all of the items from an input collection that share some common feature; for instance, the GROUP BY operator of SQL performs clustering. We will define cluster(proc, lt?, lst) as a function that takes an input list and returns a list of lists; proc computes a signature of each item in the input list, and each sub-list in the output list contains all those elements of the input list with identical signatures, with sub-lists in increasing order of signature according to lt?. The type of cluster is (α → β) × (β × β → boolean) × (list α) → (list (list α)).

Your task is to write the function cluster. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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Packed ascii is a method for compressing a useful subset of ascii characters in 6 bits, used in HART-enabled devices. The characters that may be transmitted are the space character, the 26 upper-case letters, the 10 digits, and the following punctuation characters: ! " # $ % & ' ( ) * + , - . / : ; ? @ [ \ ] ^ _. Omitted are the 26 lower-case letters, the rubout character, and the following punctuation characters: ` { | } ~.

Compression is achieved by keeping only the six low-order bits of each ascii character. Compressed characters are expanded to their original character by adding a high-order bit that is the complement of the compressed high-order bit. For instance, the string “PRAXIS” is compressed as the six 6-bit binary numbers 010000 010010 000001 011000 001001 010011.

A string of characters is compressed to 6-bit characters, then transmitted as 8-bit bytes by packing the bits from high-order to low-order, so that four characters are transmitted in three bytes, achieving a 25% compression. If the message length is not an even multiple of four, space characters are added to the end of the string as padding. Thus, the six 6-bit binary numbers representing “PRAXIS” would be padded with two space characters and transmitted as the six 8-bit binary numbers 01000001 00100000 01011000 00100101 00111000 00100000, which corresponds to the string “A X%8 “.

Your task is to write functions that compress and expand strings in packed ascii format. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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An n-bit Gray sequence is a sequence of 2ⁿ values, starting from 0, in which each differs from its predecessor by a single bit. There are always 2ⁿ⁻¹ n-bit Gray sequences; for instance, the two 2-bit Gray sequences are 0, 1, 3, 2 and 0, 2, 3, 1. It is easier to see the Gray sequences when they are written in binary; the two 2-bit Gray sequences written in binary are 00, 01, 11, 10 and 0,0 10, 11, 01, where it is clear that each element of the sequence differs from the previous one by only one bit.

Although there are many possible Gray sequences for any given number of bits, there is one Gray sequence, known as the binary reflected gray code BRGC, that is almost always the Gray sequence that is being discussed. Such sequences are generated recursively from the next-smaller sequence by reversing the sequence, prefixing the entries of the original sequence with a 0-bit, prefixing the entries of the reversed sequence with a 1-bit, then concatenating the two sequences. For instance, given the 2-bit Gray sequence 00, 01, 11, 10, its reversal is 10, 11, 01, 00, adding a 0-bit to the original gives 000, 001, 011, 010, adding a 1-bit to the reversal gives 110, 111, 101, 100, and concatenating the two gives 000, 001, 011, 010, 110, 111, 101, 100, which is 0, 1, 3, 2, 6, 7, 5, 4.

Your task is to write a function that generates an n-bit binary reflected Gray sequence. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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We have seen the Miller-Rabin and Baillie-Wagstaff primality tests in previous exercises. Today we look at a test developed by Robert M. Solovay and Volker Strassen in 1977. The test is no longer used, having been superseded by the other two tests, but was historically of great importance in the development of the RSA cryptosystem.

The test is based on Euler’s Criterion, which states that a^(p−1)/2 ≡ (a / p) (mod p) for any odd prime number p and any integer a on the range 2 to p − 1 with gcd(a, p) = 1, where (a / p) is the Legendre symbol. If a number n being tested is prime, the test will indicate that n is prime; if a number n being tested is composite, the test will indicate that it is either prime or composite with equal likelihood, a coin flip. Thus, the Solovay-Strassen test chooses k different witnesses a; if any of them indicate n is composite, then it must be composite, but if none of them indicate n is composite, it is presumed prime with odds 2^−k.

If you are willing to assume the truth of the Extended Riemann Hypothesis, the Solovay-Strassen test can be made an absolute proof of primality by testing all prime a in the range 2 to min(n − 1, log² n); if none of the a indicate the compositeness of n, then n is prime on the ERH.

The Solovay-Strassen test is no longer used, for three reasons: First, it takes more work to implement, as it involves computing the Jacobi symbol as well as modular exponentiation, whereas the Miller-Rabin test involves only modular exponentiation. Second, the Miller-Rabin test has a better error bound, k⁻⁴ compared to k⁻². Third, all Euler witnesses are also Miller-Rabin strong witnesses to the compositeness of n. The presence of the absolute proof of primality on the ERH is also not a bar to the use of the Miller-Rabin test, as there is a similar proof of primality based on Miller-Rabin strong witnesses.

Your task is to write two versions of the Solovay-Strassen primality test, one probabilistic and one on the ERH. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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We studied a simple assembler for a hypothetical computer in the two previous exercises. Today we finish our look at the assembler by writing a program that produces neat listings, like this one for our sample program:

# print sum of input numbers (terminated by zero)

000:  03010        ld    zero # initialize sum to zero          
001:  04011        st    sum                                    
002:  01000   loop get        # read a number                   
003:  08007        jz    done # no more input if number is zero 
004:  05011        add   sum  # add input to accumulated sum    
005:  04011        st    sum  # store new value back in sum     
006:  09002        j     loop # go back and read another number 

007:  03011   done ld    sum  # print sum                       
008:  02000        put                                          
009:  10000        halt                                         

010:  00000   zero const 0                                      
011:  00000   sum  const

The listing shows the memory address in the first column, the contents of memory at the beginning of the program in the second column, the mnemonic label in the third column, the opcode in the fourth column, the object of the operation in the fifth column, and the comment in the sixth column.

Your task is to write a program that produces assembler listings. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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In the previous exercise we built an assembler for a hypothetical computer. Today we write a simulator for that hypothetical computer. The basic idea is simple: Start with a program counter at location 0, either perform the action at the location and advance the program counter to the next location, or set the program counter to the object of a jump instruction.

Your task is to write the simulator. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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In this exercise and the next one we will write a simple assembler for a hypothetical computer; we are following the assembler in the book The Awk Programming Langauge by Alfred V. Aho, Brian W. Kernighan and Peter J. Weinberger. Our computer has a memory of a thousand five-digit words, a single accumulator, and eleven opcodes:

An assembly-language program is a series of blank lines and statements consisting of up to four fields: The first field, if it exists, is a label; it must start at the first position on the line and may not start with a digit. The second field, which is mandatory, is the opcode; it follows the optional label and mandatory spaces. The third field, which is used only with some opcodes, is the object; if it is present, it follows the opcode and mandatory spaces. The fourth field, which is optional, is a comment; everything on the line following a hash-sign is ignored. Here is a sample assembly-language program that prints the sum of a series of integers entered by the user, with the end of input marked by a 0:

# print sum of input numbers (terminated by zero)

     ld    zero   # initialize sum to zero
     st    sum
loop get          # read a number
     jz    done   # no more input if number is zero
     add   sum    # add input to accumulated sum
     st    sum    # store new value back in sum
     j     loop   # go back and read another number

done ld    sum    # print sum
     put
     halt

zero const 0
sum  const

The contents of memory after loading the sample program are shown on the next page.

Your task is to write a program that assembles a program written in our simple assembly language and loads the program into memory; we’ll write a simulator for our hypothetical computer in the next exercise. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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Programs that create graphs and charts are an important tool in many commercial and technical endeavors; for instance, a programmer might look at a graph of memory usage during a debugging run of his program to identify hot spots that need to be trimmed.

A simple type of graph is a scatter graph, sometimes called an x/y plot. The input is a list of points on an x,y-grid; the output is a picture with the points plotted on a grid. A simple input file appears on the next page.

Your task is to write a program to plot scatter graphs. When you are finished, you are welcome to read or run a suggested solution, or to post your own program or discuss the exercise in the comments below.

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The only output functions provided by standard Scheme are display, for plain output, and write, for system-formatted output. That’s rather limiting. At the other extreme, Lisp provides the format function, which has more options than you can dream about (before I discarded it in favor of the HyperSpec, the spine of my printed copy of CLtL2 was broken in two places, format and loop). Many languages provide some kind of formatted output — does anyone remember PIC in COBOL? In today’s exercise we will implement the printf function popularized by C and used in several languages.

There are three functions in the printf family: (sprintf fmt expr …) returns a string formatted according to the specification given by format and containing the values expr …; (printf fmt expr …) displays a string formatted similarly to sprintf to the current output port, and (fprintf port fmt expr …) displays a string formatted similarly to sprintf to the indicated port. The fmt and port arguments are always required.

The fmt argument is a string that contains literal text, escape sequences, and specifications of how the expressions should be formatted. A specification consists of a literal percent-sign %, zero or more modifiers, and a single-character specifier. The single-character specifiers that we will support are:

c ascii character
d decimal integer
f floating-point number
o octal integer
s string
x hexadecimal integer
% literal percent sign

There should be as many expressions as there are format specifiers in the fmt string, except that a % literal percent sign specifier does not consume an expression. As many as four modifiers may appear between the literal percent sign % that starts a specifier and the single-character specifier that ends it:

- left-justify the expression in its field; if not given, the expression is right-justified
0 left-pad with leading zeros instead of spaces
width pad field to this width using spaces (or zeros)
.prec digits to right of decimal point, or maximum string length

The modifiers must appear in the order shown above. The width and prec arguments are unsigned decimal integers.

Escape sequences are introduced by a backslash; the following escape sequences are supported:

\b backspace
\f formfeed
\n newline
\r carriage return
\t horizontal tab
\ddd character with octal value ddd, where ddd is 1 to 3 digits between 0 and 7 inclusive
\c any other character c literally, for instance \\ for backslash or \" for quotation mark

Depending on the environment, a newline may (or may not) imply a carriage return, or vice-versa. Several examples are given on the next page.

Your task is to write a function that provides formatted output for your language, using the definition of printf given above or some other specification that is suitable for your language and your aspirations. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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