关于算法:CSE111C讨论

CSE-111 • Spring 2022 • Program 1 • Overloading and operators 1 of 7
$Id: asg1-dc-bigint.mm,v 1.267 2022-04-03 12:04:19-07 – – $
/afs/cats.ucsc.edu/courses/cse111-wm/Assignments/asg1-dc-bigint
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Using C++11/14/17 (g++ -std=gnu++20)
All programming in this course will be done C++ style, not C style.
Do not use : Instead, use :
char* strings <string>
C arrays <vector>
<stdio.h>, <cstdio> <iostream>, <iomanip>
pointers <shared_ptr> or <unique_ptr>
union inheritance or <variant>
<header.h> <cheader>
Include only C++ header files and use the declaration using namespace std; Include
<cheader> files only when C++ header files do not provide a necessary facility.
Include <header.h> files from C only when an appropriate <cheader> file does not
exist. Use the script cpplint.py.perl (a wrapper for cpplint.py) to check style.
The production system for all work is unix.ucsc.edu using g++. Compile with
g++ -std=gnu++20 -g -O0 -Wall -Wextra -Wpedantic -Wshadow -Wold-style-cast
Following is a description of these options :
• -std=gnu++20 Gnu dialect of C++20.
• -g produces debugging information into object files and the binary executable.
This is necessary for gdb and valgrind to use symbolic names.
• -O0 reduces compilation time and makes debugging produce more expected
results. Optimization may rearrange bugs in code in unexpected ways.
• -Wall enables all the warnngs about questionable constructions.
• -Wextra enables extra warnings that are not enabled with -Wall.
• -Wpedantic issues all warnings required by strict ISO C++ and rejects all programs
that do not conform to ISO C++.
• -Wshadow warns whenever a local variable or declaration shadows another variable,
parameter, or class member.
• -Wold-style-cast warns about the use of any old-style (C-style) cast. Instead,
use one of : static_cast, dynamic_cast, const_cast, reinterpret_cast. Better
yet, code in suchaway as to not need casts.
• -fdiagnostics-color=never prevents the compiler from using those silly annoying
colors in diagnostics.
The particular g++ compiler we will be using is
-bash-1$ which g++
/opt/rh/devtoolset-11/root/usr/bin/g++
-bash-2$ g++ –version | grep -i g++
g++ (GCC) 11.2.1 20210728 (Red Hat 11.2.1-1)
-bash-3$ uname -npo
unix1.lt.ucsc.edu x86_64 GNU/Linux
If you develop on your personal system, be sure to port and test your code on the
Linux timeshares. If it compiles and runs on your system, but not on the timeshares,
then it does not wor k.
CSE-111 • Spring 2022 • Program 1 • Overloading and operators 2 of 7
Overview
This assignment will involve overloading basic integer operators to perform arbitrary
precision integer arithmetic in the style of dc(1). Your class bigint will intermix
arbitrarily with simple integer arithmetic.
To begin read the man(1) page for the command dc(1) :
man -s 1 dc
A copy of that page is also in this directory. Your program will use the standard dc
as a reference implemention and must produce exactly the same output for the
commands you have to implement :
+-*/%^cdfpq
If you have any questions as to the exact format of your output, just run dc(1) and
make sure that, for the operators specified above, your program produces exactly
the same output. A useful program to compare output from your program with that
of dc(1) is diff(1), which compares the contents of two files and prints only the differences.
Also look in the subdirectory misc/ for some examples.
See also :
• dc (computer program)
https://en.wikipedia.org/wiki…(computer_program)
• dc, an arbitrary precision calculator
https://www.gnu.org/software/…
Implementation strategy
As before, you have been given starter code.
(a) Makefile, debug, and util If you find you need a function which does not properly
belong to a given module, you may add it to util.
(b) The module scanner reads in tokens, namely a NUMBER, an OPERATOR, or SCANEOF.
Each token returns a token_t, which indicates what kind of token it is (the
terminal_symbol symbol), and the string lexinfo associated with the token.
Only in the case of a number is there more than one character. Note that on
input, an underscore (_) indicates a negative number. The minus sign (-) is
reserved only as a binary operator. The scanner also has defined a couple of
operator<< for printing out scanner results in debug mode. This is strictly for
debugging.
(c) The main program main.cpp, has been implemented for you. For the six binary
arithmetic functions, the right operand is popped from the stack, then the left
operand, then the result is pushed onto the stack.
(d) The module iterstack can not just be the STL stack, since we want to iterate
from top to bottom, and the STL stack does not have an iterator. A stack
depends on the operations back(), push_back(), and pop_back() in the underlying
container. We could use a vector, a deque, or just a list, as long as the requisite
operations are available.
CSE-111 • Spring 2022 • Program 1 • Overloading and operators 3 of 7
Class bigint
Then we come to the most complex part of the assignment, namely the class bigint.
Operators in this class are heavily overloaded.
(a) Most of the functions take a arguments of type const bigint&, i.e., a constant
reference, for the sake of efficiency. But they have to return the result by
value.
(b) The operator<< can’t be a member since its left operand is an ostream, so we
make it a friend, so that it can see the innards of a bigint. Note now dc prints
really big numbers. operator<< is used by debugging functions.
(c) The function print (suitably modified) is used for actual output.
(d) The relational operators == and < are coded individually as member functions.
The others, !=, <=, >, and >= are defined in terms of the essential two.
(e) All of the functions of bigint only work with the sign, using ubigint to do the
actual computations. So bigint::operator+ and bigint::operator- will check
the signs of the two operands then call ubigint::operator+ or ubigint::operator-,
as appropriate, depending on the relative signs and magnitudes. The
multiplication and division operators just call the corresponding ubigint operators,
then adjust the resulting sign according to the rule of signs.
Class ubigint
Class ubigint implements unsigned large integers and is where the computational
work takes place. Class bigint takes care of the sign. Now we turn to the representation
of a ubigint, which will be represented by vector of bytes.
(a) Replace the declaration
using ubigvalue_t = unsigned long;
with
using ubigvalue_t = vector<uint8_t>;
in the header file <ubigint.h>. The type uint8_t is an unsigned 8-bit integer
defined in <cstdint>.
(b) In storing the big integer, each digit is kept as an integer in the range 0 to 9 in
an element of the vector. Since the arithmetic operators add and subtract
work from least significant digit to most significant digit, store the elements of
the vector in the same order. That means, for example, that the number 4629
would be stored in a vector v as : v[3]==4, v[2]==6, v[1]==2, v[0]==9. In other
words, if v[k]==d, then the digit’s place value is d*pow(10,k). In mathematical
notation, the value of a radix 10 (base 10) number v with n digits is :
n−1
i=0
Σ vi10i
= vn−110n−1
vn−210n−2
… + v2102
v1101
v0100
(c) In order for the comparisons to work correctly, always store numbers in a
canonical form : After computing a value from any one of the six arithmetic
operators, always trim the vector by removing all high-order zeros :
while (size() > 0 and back() == 0) pop_back();
CSE-111 • Spring 2022 • Program 1 • Overloading and operators 4 of 7
(d) Canonical form.
• Zero is represented as a vector of size zero and a positive sign.
• High-order zeros are suppressed.
• All digits are stored as uint8_t values in the range 0…9, not as characters in
the range’0’…’9’.
• To print a digit, cast it to an integer : cout << static_cast<int> (digit).
• This can be done more easily by : cout << int (digit), which looks like a
ctor call.
(e) The scanner will produce numbers as strings, so scan each string from the end
of the string, using a const_reverse_iterator (or other means) from the end of
the string (least significant digit) to the beginning of the string (most signifi-
cant digit) using push_back to append them to the vector.
Implementation of operators
(a) For bigint::operator+, check the signs.
(1) If the signs are the same :
• Call ubigint::operator+ with the unsigned numbers.
• The sign of the result is the sign of either number.
(2) If the signs are different :
• Call ubigint::operator- with the larger number as its left number.
• The sign of the result is the sign of the larger number.
(b) The operator bigint::operator-, check the signs.
(1) If the signs are different :
• Call ubigint::operator+ with the unsigned numbers.
• The sign of the result is the sign of the left number.
(2) If the signs are the same :
• Call ubigint::operator- with the larger number as its left number.
• If the left number is larger, the sign of the result is its sign.
• Else the the result has the opposite of the sign of the right number.
(c) For the above bigint::operator+ and bigint::operator-, to find the‘‘larger’’
number, make use of ubigint::operator<. Since the numbers are kept in
canonical form (see above), to compare them :
(1) Check the size() of each vector. If different, the larger number has the
greater size.
(2) If the sizes are the same, write a loop iterating from the highest-order
digit toward the lowest-order digit, comparing digit by digit.
• As soon as a difference is found, return true or false, as appriate.
• If all digits are the same, then return false.
(d) To implement ubigint::operator+, create a new ubigint and proceed from the
low order end to the high order end, adding digits pairwise. For any sum >=
10, take the remainder and add the carry to the next digit. Use push_back to
append the new digits to the ubigint. When you run out of digits in the
shorter number, continue, matching the longer vector with zeros, until it is
done. Make sure the sign of 0 is positive.
CSE-111 • Spring 2022 • Program 1 • Overloading and operators 5 of 7
(e) To implement ubigint::operator-, also create a new empty vector, starting
from the low order end and continuing until the high end. If the left digit is
smaller than the right digit, the subtraction will be less than zero. In that
case, add 10 to the digit, and set the borrow to the next digit to −1. After the
algorithm is done, pop_back all high order zeros from the vector before returning
it. Make sure the sign of 0 is positive.
(f) To implement bigint::operator==, check to see if the signs are the same and
ubigint::operator== returns true.
(g) To implement ubigint::operator==, just use the vector::operator== comparison
function.
(h) To implement bigint::operator<, remember that a negative number is less
than a positive number. If the signs are the same, use ubigint::operator< for
a comparison. For positive numbers, the smaller one is less. and for negative
nubmers, the larger one is less.
(i) To implement ubigint::operator<, check the size() of each vector. The
shorter one is less than the longer one. If the size() are the same, scan the
vectors in parallel from the most significant digit to the last significant digit
until a difference is found.
(j) Implement function bigint::operator*, which uses the rule of signs to determine
the result. The number crunching is delegated to ubigint::operator*,
which produces the unsigned result.
(k) Multiplication in ubigint::operator* proceeds by allocating a new vector
whose size is equal to the sum of the sizes of the other two operands. If u is a
vector of size m and v is a vector of size n, then in O(mn) speed, perform an outer
loop over one argument and an inner loop over the other argument, adding the
new partial products to the product p as you would by hand. The algorithm
can be described as follows :
p=all zeros
for i in interval [0,m):
carry = 0
for j in interval [0,n):
digit = p[i+j] + u[i] * v[j] + carry
p[i+j] = digit % 10
carry = digit / 10
p[i+n] = carry
Note that the interval [a,b) refers to the half-open interval including a but
excluding b. This is the set {x| a<=x && x<b}. In the same way,apair of iterators
in C++ is used to bound an interval (begin and end pair).
(l) Long division is complicated if done correctly. See a paper by P. Brinch
Hansen,‘‘Multiple-length division revisited : A tour of the minefield’’, Software
— Practice and Experience 24, (June 1994), 579–601. Algorithms 1 to 12 are
on pages 13–23, Note that in Pascal, array bounds are part of the type, which
is not true for vectors in C++.
CSE-111 • Spring 2022 • Program 1 • Overloading and operators 6 of 7
• multiple-length-division.pdf
• http://brinch-hansen.net/pape…
• http://citeseerx.ist.psu.edu/…
(m) The function divide as implemented uses the ancient Egyptian division algorithm,
which is slower than Hansen’s Pascal program, but is easier to understand.
Replace the long values in it by vector<digit_t>. The logic is shown
also in misc/divisioncpp.cpp. The algorithm is rather slow, but the big-O
analysis is reasonable.
(n) The unsigned division function that is provided depends on two private functions,
multiply_by_2 and divide_by_2, which are in-lace non-constant functions.
They both perform without creating a new object.
(1) To implement multiply_by_2, iterate from the low order digit, and double
each digit (remainder 10), carrying to the next higher digit. At the end, if
the carry is 1, use push_back.
(2) To implement divide_by_2, iterate from the low order digit, and divide
each digit by 2. Then, if the next higher digit is odd, add 5 to the current
digit. Be careful of the end, and pop_back any remaining high order
zeros.
(o) Modify operator<<, first just to print out the number all in one line. You will
need this to debug your program.
(p) The function print will print numbers in the same way as dc(1) does.
(q) The pow function uses other operations to raise a number to a power. If the
exponent does not fit into a single long print an error message, otherwise do
the computation. The power function is not a member of either bigint or ubigint,
and is just considered a library function that is implemented using more
primitive operations.
Memory leak and other problems
Make sure that you test your program completely so that it does not crash on a Segmentation
Fault or any other unexpected error. Since you are not using pointers,
and all values are inline, there should be no memory leak. Use valgrind(1) to check
for and eliminate uninitialized variables and memory leak.
If your program is producing strange output or segmentation faults, use gdb(1) and
the debug macros in the debug module of the code/ subdirectory.
What to submit
Submit source files and only source files : Makefile, README, and all of the header
and implementation files necessary to build the target executable. If gmake does not
build ydc your program can not be tested and you lose 1/2 of the points for the
assignment. Use checksource on your code to verify basic formatting.
Look in the grader’s score subdirectory for instructions to graders. Read Syllabus/
pair-programming/ and also submit PARTNER if you are doing pair programming.
Either way submit the README described therein.
CSE-111 • Spring 2022 • Program 1 • Overloading and operators 7 of 7
Et cetera (και τα‘´ετερα).
The accuracy of the Unix utility dc(1) can be checked by :
echo’82 43/25 43+65P80P82P73P76P32P70P79P79P76P10P’| dc

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