Given a sentence that is spread over a linked list where each item in the list is a word, for example:
Hello -> Everybody -> How -> Are -> You -> Feeling -> |
Given that this list is sorted, eg:
Are -> Everybody -> Feeling -> Hello -> How -> You -> |
How would you write the recursion that will find the initial letter that appears the most in the sentence (in this example the letter H from Hello & How) ?
Edit: I have update the code to recursion version.
In order to run it you call
GetMostLetterRecursion(rootNode , '0', 0, '0', 0)
The code itself look like this:
public char GetMostLetterRecursion(LinkedListNode<String> node, char currentChar, int currentCount, char maxChar, int maxCount)
{
if (node == null) return maxChar;
char c = node.Value[0];
if (c == currentChar)
{
return GetMostLetterRecursion(node.Next, currentChar, currentCount++, maxChar, maxCount);
}
if(currentCount > maxCount)
{
return GetMostLetterRecursion(node.Next, c, 1, currentChar, currentCount);
}
return GetMostLetterRecursion(node.Next, c, 1, maxChar, maxCount);
}
Solution 1
Loop over the words, keeping a tally of how many words start with each letter. Return the most popular letter according to the tally (easy if you used a priority queue for the tally).
This takes O(n) time (the number of words) and O(26) memory (the number of letters in alphabet).
Solution 2
Sort the words alphabetically. Loop over the words. Keep a record of the current letter and its frequency, as well as the most popular letter so far and its frequency. At the end of the loop, that's the most popular letter over the whole list.
This takes O(n log n) time and O(1) memory.
Keep an array to store the count of occurrences and Go through the linked list once to count it. Finally loop through the array to find the highest one.
Rough sketch in C:
int count[26]={0};
While ( head->next != NULL)
{
count[head->word[0] - 'A']++; // Assuming 'word' is string in each node
head = head->next;
}
max = count[0];
for (i=0;i<26;i++)
{
if(max<a[i])
max = a[i];
}
You can modify it to use recursion and handle lower case letters.
Here is a pure recursive implementation in Python. I haven't tested it, but it should work modulo typos or syntax errors. I used a Dictionary to store counts, so it will work with Unicode words too. The problem is split into two functions: one to count the occurrences of each letter, and another to find the maximum recursively.
# returns a dictionary where dict[letter] contains the count of letter
def count_first_letters(words):
def count_first_letters_rec(words, count_so_far):
if len(words) == 0:
return count_so_far
first_letter = words[0][0]
# could use defaultdict but this is an exercise :)
try:
count_so_far[first_letter] += 1
except KeyError:
count_so_far[first_letter] = 1
# recursive call
return count_first_letters_rec(words[1:], count_so_far)
return count_first_letters(words, {})
# takes a list of (item, count) pairs and returns the item with largest count.
def argmax(item_count_pairs):
def argmax_rec(item_count_pairs, max_so_far, argmax_so_far):
if len(item_count_pairs) == 0:
return argmax_so_far
item, count = item_count_pairs[0]
if count > max_so_far:
max_so_far = count
argmax_so_far = item
# recursive call
return argmax_rec(item_count_pairs[1:], max_so_far, argmax_so_far)
return argmax_rec(item_count_pairs, 0, None)
def most_common_first_letter(words);
counts = count_first_letters(words)
# this returns a dictionary, but we need to convert to
# a list of (key, value) tuples because recursively iterating
# over a dictionary is not so easy
kvpairs = counts.items()
# counts.iteritems() for Python 2
return argmax(kvpairs)
I have an array with the length of 26 (as English letters, so index 1 is for 'a' and 2 for 'b' and so on. ). Each time a letter occurs, I increment it's value in the array. if the value becomes more than max amount, then I update the max and take that letter as most occurred one.then I call the method for the next node.
This is the code in Java:
import java.util.LinkedList;
public class MostOccurance {
char mostOccured;
int maxOccurance;
LinkedList<String> list= new LinkedList<String>();
int[] letters= new int[26];
public void start(){
findMostOccuredChar( 0, '0', 0);
}
public char findMostOccuredChar ( int node, char most, int max){
if(node>=list.size())
return most;
String string=list.get(node);
if (string.charAt(0)== most)
{max++;
letters[Character.getNumericValue(most)-10]++;
}
else{
letters[Character.getNumericValue(most)-10]++;
if (letters[Character.getNumericValue(most)-10]++>max){
max=letters[Character.getNumericValue(most)-10];
most=string.charAt(0);
}
}
findMostOccuredChar( node++, most, max);
return most;
}
}
of course, you have to add each word to your link list. I didn't do that, because I was just showing an example.
I have always wanted to do this but every time I start thinking about the problem it blows my mind because of its exponential nature.
The problem solver I want to be able to understand and code is for the countdown maths problem:
Given set of number X1 to X5 calculate how they can be combined using mathematical operations to make Y.
You can apply multiplication, division, addition and subtraction.
So how does 1,3,7,6,8,3 make 348?
Answer: (((8 * 7) + 3) -1) *6 = 348.
How to write an algorithm that can solve this problem? Where do you begin when trying to solve a problem like this? What important considerations do you have to think about when designing such an algorithm?
Very quick and dirty solution in Java:
public class JavaApplication1
{
public static void main(String[] args)
{
List<Integer> list = Arrays.asList(1, 3, 7, 6, 8, 3);
for (Integer integer : list) {
List<Integer> runList = new ArrayList<>(list);
runList.remove(integer);
Result result = getOperations(runList, integer, 348);
if (result.success) {
System.out.println(integer + result.output);
return;
}
}
}
public static class Result
{
public String output;
public boolean success;
}
public static Result getOperations(List<Integer> numbers, int midNumber, int target)
{
Result midResult = new Result();
if (midNumber == target) {
midResult.success = true;
midResult.output = "";
return midResult;
}
for (Integer number : numbers) {
List<Integer> newList = new ArrayList<Integer>(numbers);
newList.remove(number);
if (newList.isEmpty()) {
if (midNumber - number == target) {
midResult.success = true;
midResult.output = "-" + number;
return midResult;
}
if (midNumber + number == target) {
midResult.success = true;
midResult.output = "+" + number;
return midResult;
}
if (midNumber * number == target) {
midResult.success = true;
midResult.output = "*" + number;
return midResult;
}
if (midNumber / number == target) {
midResult.success = true;
midResult.output = "/" + number;
return midResult;
}
midResult.success = false;
midResult.output = "f" + number;
return midResult;
} else {
midResult = getOperations(newList, midNumber - number, target);
if (midResult.success) {
midResult.output = "-" + number + midResult.output;
return midResult;
}
midResult = getOperations(newList, midNumber + number, target);
if (midResult.success) {
midResult.output = "+" + number + midResult.output;
return midResult;
}
midResult = getOperations(newList, midNumber * number, target);
if (midResult.success) {
midResult.output = "*" + number + midResult.output;
return midResult;
}
midResult = getOperations(newList, midNumber / number, target);
if (midResult.success) {
midResult.output = "/" + number + midResult.output;
return midResult
}
}
}
return midResult;
}
}
UPDATE
It's basically just simple brute force algorithm with exponential complexity.
However you can gain some improvemens by leveraging some heuristic function which will help you to order sequence of numbers or(and) operations you will process in each level of getOperatiosn() function recursion.
Example of such heuristic function is for example difference between mid result and total target result.
This way however only best-case and average-case complexities get improved. Worst case complexity remains untouched.
Worst case complexity can be improved by some kind of branch cutting. I'm not sure if it's possible in this case.
Sure it's exponential but it's tiny so a good (enough) naive implementation would be a good start. I suggest you drop the usual infix notation with bracketing, and use postfix, it's easier to program. You can always prettify the outputs as a separate stage.
Start by listing and evaluating all the (valid) sequences of numbers and operators. For example (in postfix):
1 3 7 6 8 3 + + + + + -> 28
1 3 7 6 8 3 + + + + - -> 26
My Java is laughable, I don't come here to be laughed at so I'll leave coding this up to you.
To all the smart people reading this: yes, I know that for even a small problem like this there are smarter approaches which are likely to be faster, I'm just pointing OP towards an initial working solution. Someone else can write the answer with the smarter solution(s).
So, to answer your questions:
I begin with an algorithm that I think will lead me quickly to a working solution. In this case the obvious (to me) choice is exhaustive enumeration and testing of all possible calculations.
If the obvious algorithm looks unappealing for performance reasons I'll start thinking more deeply about it, recalling other algorithms that I know about which are likely to deliver better performance. I may start coding one of those first instead.
If I stick with the exhaustive algorithm and find that the run-time is, in practice, too long, then I might go back to the previous step and code again. But it has to be worth my while, there's a cost/benefit assessment to be made -- as long as my code can outperform Rachel Riley I'd be satisfied.
Important considerations include my time vs computer time, mine costs a helluva lot more.
A working solution in c++11 below.
The basic idea is to use a stack-based evaluation (see RPN) and convert the viable solutions to infix notation for display purposes only.
If we have N input digits, we'll use (N-1) operators, as each operator is binary.
First we create valid permutations of operands and operators (the selector_ array). A valid permutation is one that can be evaluated without stack underflow and which ends with exactly one value (the result) on the stack. Thus 1 1 + is valid, but 1 + 1 is not.
We test each such operand-operator permutation with every permutation of operands (the values_ array) and every combination of operators (the ops_ array). Matching results are pretty-printed.
Arguments are taken from command line as [-s] <target> <digit>[ <digit>...]. The -s switch prevents exhaustive search, only the first matching result is printed.
(use ./mathpuzzle 348 1 3 7 6 8 3 to get the answer for the original question)
This solution doesn't allow concatenating the input digits to form numbers. That could be added as an additional outer loop.
The working code can be downloaded from here. (Note: I updated that code with support for concatenating input digits to form a solution)
See code comments for additional explanation.
#include <iostream>
#include <vector>
#include <algorithm>
#include <stack>
#include <iterator>
#include <string>
namespace {
enum class Op {
Add,
Sub,
Mul,
Div,
};
const std::size_t NumOps = static_cast<std::size_t>(Op::Div) + 1;
const Op FirstOp = Op::Add;
using Number = int;
class Evaluator {
std::vector<Number> values_; // stores our digits/number we can use
std::vector<Op> ops_; // stores the operators
std::vector<char> selector_; // used to select digit (0) or operator (1) when evaluating. should be std::vector<bool>, but that's broken
template <typename T>
using Stack = std::stack<T, std::vector<T>>;
// checks if a given number/operator order can be evaluated or not
bool isSelectorValid() const {
int numValues = 0;
for (auto s : selector_) {
if (s) {
if (--numValues <= 0) {
return false;
}
}
else {
++numValues;
}
}
return (numValues == 1);
}
// evaluates the current values_ and ops_ based on selector_
Number eval(Stack<Number> &stack) const {
auto vi = values_.cbegin();
auto oi = ops_.cbegin();
for (auto s : selector_) {
if (!s) {
stack.push(*(vi++));
continue;
}
Number top = stack.top();
stack.pop();
switch (*(oi++)) {
case Op::Add:
stack.top() += top;
break;
case Op::Sub:
stack.top() -= top;
break;
case Op::Mul:
stack.top() *= top;
break;
case Op::Div:
if (top == 0) {
return std::numeric_limits<Number>::max();
}
Number res = stack.top() / top;
if (res * top != stack.top()) {
return std::numeric_limits<Number>::max();
}
stack.top() = res;
break;
}
}
Number res = stack.top();
stack.pop();
return res;
}
bool nextValuesPermutation() {
return std::next_permutation(values_.begin(), values_.end());
}
bool nextOps() {
for (auto i = ops_.rbegin(), end = ops_.rend(); i != end; ++i) {
std::size_t next = static_cast<std::size_t>(*i) + 1;
if (next < NumOps) {
*i = static_cast<Op>(next);
return true;
}
*i = FirstOp;
}
return false;
}
bool nextSelectorPermutation() {
// the start permutation is always valid
do {
if (!std::next_permutation(selector_.begin(), selector_.end())) {
return false;
}
} while (!isSelectorValid());
return true;
}
static std::string buildExpr(const std::string& left, char op, const std::string &right) {
return std::string("(") + left + ' ' + op + ' ' + right + ')';
}
std::string toString() const {
Stack<std::string> stack;
auto vi = values_.cbegin();
auto oi = ops_.cbegin();
for (auto s : selector_) {
if (!s) {
stack.push(std::to_string(*(vi++)));
continue;
}
std::string top = stack.top();
stack.pop();
switch (*(oi++)) {
case Op::Add:
stack.top() = buildExpr(stack.top(), '+', top);
break;
case Op::Sub:
stack.top() = buildExpr(stack.top(), '-', top);
break;
case Op::Mul:
stack.top() = buildExpr(stack.top(), '*', top);
break;
case Op::Div:
stack.top() = buildExpr(stack.top(), '/', top);
break;
}
}
return stack.top();
}
public:
Evaluator(const std::vector<Number>& values) :
values_(values),
ops_(values.size() - 1, FirstOp),
selector_(2 * values.size() - 1, 0) {
std::fill(selector_.begin() + values_.size(), selector_.end(), 1);
std::sort(values_.begin(), values_.end());
}
// check for solutions
// 1) we create valid permutations of our selector_ array (eg: "1 1 + 1 +",
// "1 1 1 + +", but skip "1 + 1 1 +" as that cannot be evaluated
// 2) for each evaluation order, we permutate our values
// 3) for each value permutation we check with each combination of
// operators
//
// In the first version I used a local stack in eval() (see toString()) but
// it turned out to be a performance bottleneck, so now I use a cached
// stack. Reusing the stack gives an order of magnitude speed-up (from
// 4.3sec to 0.7sec) due to avoiding repeated allocations. Using
// std::vector as a backing store also gives a slight performance boost
// over the default std::deque.
std::size_t check(Number target, bool singleResult = false) {
Stack<Number> stack;
std::size_t res = 0;
do {
do {
do {
Number value = eval(stack);
if (value == target) {
++res;
std::cout << target << " = " << toString() << "\n";
if (singleResult) {
return res;
}
}
} while (nextOps());
} while (nextValuesPermutation());
} while (nextSelectorPermutation());
return res;
}
};
} // namespace
int main(int argc, const char **argv) {
int i = 1;
bool singleResult = false;
if (argc > 1 && std::string("-s") == argv[1]) {
singleResult = true;
++i;
}
if (argc < i + 2) {
std::cerr << argv[0] << " [-s] <target> <digit>[ <digit>]...\n";
std::exit(1);
}
Number target = std::stoi(argv[i]);
std::vector<Number> values;
while (++i < argc) {
values.push_back(std::stoi(argv[i]));
}
Evaluator evaluator{values};
std::size_t res = evaluator.check(target, singleResult);
if (!singleResult) {
std::cout << "Number of solutions: " << res << "\n";
}
return 0;
}
Input is obviously a set of digits and operators: D={1,3,3,6,7,8,3} and Op={+,-,*,/}. The most straight forward algorithm would be a brute force solver, which enumerates all possible combinations of these sets. Where the elements of set Op can be used as often as wanted, but elements from set D are used exactly once. Pseudo code:
D={1,3,3,6,7,8,3}
Op={+,-,*,/}
Solution=348
for each permutation D_ of D:
for each binary tree T with D_ as its leafs:
for each sequence of operators Op_ from Op with length |D_|-1:
label each inner tree node with operators from Op_
result = compute T using infix traversal
if result==Solution
return T
return nil
Other than that: read jedrus07's and HPM's answers.
By far the easiest approach is to intelligently brute force it. There is only a finite amount of expressions you can build out of 6 numbers and 4 operators, simply go through all of them.
How many? Since you don't have to use all numbers and may use the same operator multiple times, This problem is equivalent to "how many labeled strictly binary trees (aka full binary trees) can you make with at most 6 leaves, and four possible labels for each non-leaf node?".
The amount of full binary trees with n leaves is equal to catalan(n-1). You can see this as follows:
Every full binary tree with n leaves has n-1 internal nodes and corresponds to a non-full binary tree with n-1 nodes in a unique way (just delete all the leaves from the full one to get it). There happen to be catalan(n) possible binary trees with n nodes, so we can say that a strictly binary tree with n leaves has catalan(n-1) possible different structures.
There are 4 possible operators for each non-leaf node: 4^(n-1) possibilities
The leaves can be numbered in n! * (6 choose (n-1)) different ways. (Divide this by k! for each number that occurs k times, or just make sure all numbers are different)
So for 6 different numbers and 4 possible operators you get Sum(n=1...6) [ Catalan(n-1) * 6!/(6-n)! * 4^(n-1) ] possible expressions for a total of 33,665,406. Not a lot.
How do you enumerate these trees?
Given a collection of all trees with n-1 or less nodes, you can create all trees with n nodes by systematically pairing all of the n-1 trees with the empty tree, all n-2 trees with the 1 node tree, all n-3 trees with all 2 node tree etc. and using them as the left and right sub trees of a newly formed tree.
So starting with an empty set you first generate the tree that has just a root node, then from a new root you can use that either as a left or right sub tree which yields the two trees that look like this: / and . And so on.
You can turn them into a set of expressions on the fly (just loop over the operators and numbers) and evaluate them as you go until one yields the target number.
I've written my own countdown solver, in Python.
Here's the code; it is also available on GitHub:
#!/usr/bin/env python3
import sys
from itertools import combinations, product, zip_longest
from functools import lru_cache
assert sys.version_info >= (3, 6)
class Solutions:
def __init__(self, numbers):
self.all_numbers = numbers
self.size = len(numbers)
self.all_groups = self.unique_groups()
def unique_groups(self):
all_groups = {}
all_numbers, size = self.all_numbers, self.size
for m in range(1, size+1):
for numbers in combinations(all_numbers, m):
if numbers in all_groups:
continue
all_groups[numbers] = Group(numbers, all_groups)
return all_groups
def walk(self):
for group in self.all_groups.values():
yield from group.calculations
class Group:
def __init__(self, numbers, all_groups):
self.numbers = numbers
self.size = len(numbers)
self.partitions = list(self.partition_into_unique_pairs(all_groups))
self.calculations = list(self.perform_calculations())
def __repr__(self):
return str(self.numbers)
def partition_into_unique_pairs(self, all_groups):
# The pairs are unordered: a pair (a, b) is equivalent to (b, a).
# Therefore, for pairs of equal length only half of all combinations
# need to be generated to obtain all pairs; this is set by the limit.
if self.size == 1:
return
numbers, size = self.numbers, self.size
limits = (self.halfbinom(size, size//2), )
unique_numbers = set()
for m, limit in zip_longest(range((size+1)//2, size), limits):
for numbers1, numbers2 in self.paired_combinations(numbers, m, limit):
if numbers1 in unique_numbers:
continue
unique_numbers.add(numbers1)
group1, group2 = all_groups[numbers1], all_groups[numbers2]
yield (group1, group2)
def perform_calculations(self):
if self.size == 1:
yield Calculation.singleton(self.numbers[0])
return
for group1, group2 in self.partitions:
for calc1, calc2 in product(group1.calculations, group2.calculations):
yield from Calculation.generate(calc1, calc2)
#classmethod
def paired_combinations(cls, numbers, m, limit):
for cnt, numbers1 in enumerate(combinations(numbers, m), 1):
numbers2 = tuple(cls.filtering(numbers, numbers1))
yield (numbers1, numbers2)
if cnt == limit:
return
#staticmethod
def filtering(iterable, elements):
# filter elements out of an iterable, return the remaining elements
elems = iter(elements)
k = next(elems, None)
for n in iterable:
if n == k:
k = next(elems, None)
else:
yield n
#staticmethod
#lru_cache()
def halfbinom(n, k):
if n % 2 == 1:
return None
prod = 1
for m, l in zip(reversed(range(n+1-k, n+1)), range(1, k+1)):
prod = (prod*m)//l
return prod//2
class Calculation:
def __init__(self, expression, result, is_singleton=False):
self.expr = expression
self.result = result
self.is_singleton = is_singleton
def __repr__(self):
return self.expr
#classmethod
def singleton(cls, n):
return cls(f"{n}", n, is_singleton=True)
#classmethod
def generate(cls, calca, calcb):
if calca.result < calcb.result:
calca, calcb = calcb, calca
for result, op in cls.operations(calca.result, calcb.result):
expr1 = f"{calca.expr}" if calca.is_singleton else f"({calca.expr})"
expr2 = f"{calcb.expr}" if calcb.is_singleton else f"({calcb.expr})"
yield cls(f"{expr1} {op} {expr2}", result)
#staticmethod
def operations(x, y):
yield (x + y, '+')
if x > y: # exclude non-positive results
yield (x - y, '-')
if y > 1 and x > 1: # exclude trivial results
yield (x * y, 'x')
if y > 1 and x % y == 0: # exclude trivial and non-integer results
yield (x // y, '/')
def countdown_solver():
# input: target and numbers. If you want to play with more or less than
# 6 numbers, use the second version of 'unsorted_numbers'.
try:
target = int(sys.argv[1])
unsorted_numbers = (int(sys.argv[n+2]) for n in range(6)) # for 6 numbers
# unsorted_numbers = (int(n) for n in sys.argv[2:]) # for any numbers
numbers = tuple(sorted(unsorted_numbers, reverse=True))
except (IndexError, ValueError):
print("You must provide a target and numbers!")
return
solutions = Solutions(numbers)
smallest_difference = target
bestresults = []
for calculation in solutions.walk():
diff = abs(calculation.result - target)
if diff <= smallest_difference:
if diff < smallest_difference:
bestresults = [calculation]
smallest_difference = diff
else:
bestresults.append(calculation)
output(target, smallest_difference, bestresults)
def output(target, diff, results):
print(f"\nThe closest results differ from {target} by {diff}. They are:\n")
for calculation in results:
print(f"{calculation.result} = {calculation.expr}")
if __name__ == "__main__":
countdown_solver()
The algorithm works as follows:
The numbers are put into a tuple of length 6 in descending order. Then, all unique subgroups of lengths 1 to 6 are created, the smallest groups first.
Example: (75, 50, 5, 9, 1, 1) -> {(75), (50), (9), (5), (1), (75, 50), (75, 9), (75, 5), ..., (75, 50, 9, 5, 1, 1)}.
Next, the groups are organised into a hierarchical tree: every group is partitioned into all unique unordered pairs of its non-empty subgroups.
Example: (9, 5, 1, 1) -> [(9, 5, 1) + (1), (9, 1, 1) + (5), (5, 1, 1) + (9), (9, 5) + (1, 1), (9, 1) + (5, 1)].
Within each group of numbers, the calculations are performed and the results are stored. For groups of length 1, the result is simply the number itself. For larger groups, the calculations are carried out on every pair of subgroups: in each pair, all results of the first subgroup are combined with all results of the second subgroup using +, -, x and /, and the valid outcomes are stored.
Example: (75, 5) consists of the pair ((75), (5)). The result of (75) is 75; the result of (5) is 5; the results of (75, 5) are [75+5=80, 75-5=70, 75*5=375, 75/5=15].
In this manner, all results are generated, from the smallest groups to the largest. Finally, the algorithm iterates through all results and selects the ones that are the closest match to the target number.
For a group of m numbers, the maximum number of arithmetic computations is
comps[m] = 4*sum(binom(m, k)*comps[k]*comps[m-k]//(1 + (2*k)//m) for k in range(1, m//2+1))
For all groups of length 1 to 6, the maximum total number of computations is then
total = sum(binom(n, m)*comps[m] for m in range(1, n+1))
which is 1144386. In practice, it will be much less, because the algorithm reuses the results of duplicate groups, ignores trivial operations (adding 0, multiplying by 1, etc), and because the rules of the game dictate that intermediate results must be positive integers (which limits the use of the division operator).
I think, you need to strictly define the problem first. What you are allowed to do and what you are not. You can start by making it simple and only allowing multiplication, division, substraction and addition.
Now you know your problem space- set of inputs, set of available operations and desired input. If you have only 4 operations and x inputs, the number of combinations is less than:
The number of order in which you can carry out operations (x!) times the possible choices of operations on every step: 4^x. As you can see for 6 numbers it gives reasonable 2949120 operations. This means that this may be your limit for brute force algorithm.
Once you have brute force and you know it works, you can start improving your algorithm with some sort of A* algorithm which would require you to define heuristic functions.
In my opinion the best way to think about it is as the search problem. The main difficulty will be finding good heuristics, or ways to reduce your problem space (if you have numbers that do not add up to the answer, you will need at least one multiplication etc.). Start small, build on that and ask follow up questions once you have some code.
I wrote a terminal application to do this:
https://github.com/pg328/CountdownNumbersGame/tree/main
Inside, I've included an illustration of the calculation of the size of the solution space (it's n*((n-1)!^2)*(2^n-1), so: n=6 -> 2,764,800. I know, gross), and more importantly why that is. My implementation is there if you care to check it out, but in case you don't I'll explain here.
Essentially, at worst it is brute force because as far as I know it's impossible to determine whether any specific branch will result in a valid answer without explicitly checking. Having said that, the average case is some fraction of that; it's {that number} divided by the number of valid solutions (I tend to see around 1000 on my program, where 10 or so are unique and the rest are permutations fo those 10). If I handwaved a number, I'd say roughly 2,765 branches to check which takes like no time. (Yes, even in Python.)
TL;DR: Even though the solution space is huge and it takes a couple million operations to find all solutions, only one answer is needed. Best route is brute force til you find one and spit it out.
I wrote a slightly simpler version:
for every combination of 2 (distinct) elements from the list and combine them using +,-,*,/ (note that since a>b then only a-b is needed and only a/b if a%b=0)
if combination is target then record solution
recursively call on the reduced lists
import sys
def driver():
try:
target = int(sys.argv[1])
nums = list((int(sys.argv[i+2]) for i in range(6)))
except (IndexError, ValueError):
print("Provide a list of 7 numbers")
return
solutions = list()
solve(target, nums, list(), solutions)
unique = set()
final = list()
for s in solutions:
a = '-'.join(sorted(s))
if not a in unique:
unique.add(a)
final.append(s)
for s in final: #print them out
print(s)
def solve(target, nums, path, solutions):
if len(nums) == 1:
return
distinct = sorted(list(set(nums)), reverse = True)
rem1 = list(distinct)
for n1 in distinct: #reduce list by combining a pair
rem1.remove(n1)
for n2 in rem1:
rem2 = list(nums) # in case of duplicates we need to start with full list and take out the n1,n2 pair of elements
rem2.remove(n1)
rem2.remove(n2)
combine(target, solutions, path, rem2, n1, n2, '+')
combine(target, solutions, path, rem2, n1, n2, '-')
if n2 > 1:
combine(target, solutions, path, rem2, n1, n2, '*')
if not n1 % n2:
combine(target, solutions, path, rem2, n1, n2, '//')
def combine(target, solutions, path, rem2, n1, n2, symb):
lst = list(rem2)
ans = eval("{0}{2}{1}".format(n1, n2, symb))
newpath = path + ["{0}{3}{1}={2}".format(n1, n2, ans, symb[0])]
if ans == target:
solutions += [newpath]
else:
lst.append(ans)
solve(target, lst, newpath, solutions)
if __name__ == "__main__":
driver()
I know it is simple and possible to convert any base to any base. First, convert any base to decimal and then decimal to any other base. However, I had done this before for range 2 to 36 but never done for 2 to 46.
I don't understand what I will put after 36, because 36 means 'z' (1-10 are decimal numbers then the 26 characters of the alphabet).
Please explains what happens after 36.
Every base has a purpose. Usually we do base conversion to make complex computations simpler.
Here are some most popular bases used and their representation.
2-binary numeral system
used internally by nearly all computers, is base two. The two digits are 0 and 1, expressed from switches displaying OFF and ON respectively.
8-octal system
is occasionally used in computing. The eight digits are 0–7.
10-decimal system
the most used system of numbers in the world, is used in arithmetic. Its ten digits are 0–9.
12-duodecimal (dozenal) system
is often used due to divisibility by 2, 3, 4 and 6. It was traditionally used as part of quantities expressed in dozens and grosses.
16-hexadecimal system
is often used in computing. The sixteen digits are 0–9 followed by A–F.
60-sexagesimal system
originated in ancient Sumeria and passed to the Babylonians. It is still used as the basis of our modern circular coordinate system (degrees, minutes, and seconds) and time measuring (minutes and hours).
64-Base 64
is also occasionally used in computing, using as digits A–Z, a–z, 0–9, plus two more characters, often + and /.
256-bytes
is used internally by computers, actually grouping eight binary digits together. For reading by humans, bytes are usually shown in hexadecimal.
The octal, hexadecimal and base-64 systems are often used in computing because of their ease as shorthand for binary. For example, every hexadecimal digit has an equivalent 4 digit binary number.
Radices are usually natural numbers. However, other positional systems are possible, e.g. golden ratio base (whose radix is a non-integer algebraic number), and negative base (whose radix is negative).
Your doubt is whether we can convert any base to any other base after base exceeds 36
( # of Alphabets + # of digits = 26+ 10= 36)
Taking example of 64-Base
It uses A–Z(Upper case)(26), a–z(lower case)(26), 0–9(10), plus 2 more characters. This way the constraint of 36 is resolved.
As we have (26+26+10+2)64 symbols in 64-base for representation, we can represent any number in 64 base. Similarly for more base they use different symbols for representation.
Source: http://en.wikipedia.org/wiki/Radix
The symbols you use for digits are arbitrary. For example base64 encoding uses 'A' to represent the zero valued digit and '0' represents the digit with the value 52. In base64 the digits go through the alphabet A-Z, then the lower case alphabet a-z, then the traditional digits 0-9, and then usually '+' and '/'.
One base 60 system used these symbols:
So the symbols used are arbitrary. There's nothing that 'happens' after 36 except what you say happens for your system.
With number systems, you are allowed to play god.
Playing god
What you need to understand is, that symbols are completely arbitrary. There is no god-given rule for "what comes after 36". You are free to define whatever you like.
To encode numbers with a certain base, all you need is the following:
base-many distinct symbols
a total order on the symbols
An arbitrary example
Naturally, there's an infinite amount of possibilities to create such a symbol table for a certain base:
Θ
ェ
す
)
０
・
＿
ｏ
や
ι
You could use this, to encode numbers with base 10. Θ being the zero-element, ェ being the one, etc.
Conventions
Of course, your peers would not be too happy if you started using the above symbol table. Because the symbols are arbitrary, we need conventions. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 is a convention, as are the symbols we use for hexadecimal, binary, etc. It is generally agreed upon what symbol table we use for what basis, that is why we can read the numbers someone else writes down.
The important thing to remember is that all numbers are symbolic of a value. Thus if you wanted to do that, you could just make a list containing the values at each position. After base 36, you simply run out of characters you can make a logical sequence out of. For example, if you used the Cambodian Alphabet with 70 odd characters, you could do base 80.
Here is the complete code I have written, hope this will help.
import java.util.Scanner;
/*
* author : roottraveller, nov 4th 2017
*/
public class BaseXtoBaseYConversion {
BaseXtoBaseYConversion() {
}
public static String convertBaseXtoBaseY(String inputNumber, final int inputBase, final int outputBase) {
int decimal = baseXToDecimal(inputNumber, inputBase);
return decimalToBaseY(decimal, outputBase);
}
private static int baseXNumeric(char input) {
if (input >= '0' && input <= '9') {
return Integer.parseInt(input + "");
} else if (input >= 'a' && input <= 'z') {
return (input - 'a') + 10;
} else if (input >= 'A' && input <= 'Z') {
return (input - 'A') + 10;
} else {
return Integer.MIN_VALUE;
}
}
public static int baseXToDecimal(String input, final int base) {
if(input.length() <= 0) {
return Integer.MIN_VALUE;
}
int decimalValue = 0;
int placeValue = 0;
for (int index = input.length() - 1; index >= 0; index--) {
decimalValue += baseXNumeric(input.charAt(index)) * (Math.pow(base, placeValue));
placeValue++;
}
return decimalValue;
}
private static char baseYCharacter(int input) {
if (input >= 0 && input <= 9) {
String str = String.valueOf(input);
return str.charAt(0);
} else {
return (char) ('a' + (input - 10));
//return ('A' + (input - 10));
}
}
public static String decimalToBaseY(int input, int base) {
String result = "";
while (input > 0) {
int remainder = input % base;
input = input / base;
result = baseYCharacter(remainder) + result; // Important, Notice the reverse order here
}
return result;
}
public static void main(String[] args) {
Scanner scanner = new Scanner(System.in);
System.out.println("Enter : number baseX baseY");
while(true) {
String inputNumber = scanner.next();
int inputBase = scanner.nextInt();
int outputBase = scanner.nextInt();
String outputNumber = convertBaseXtoBaseY(inputNumber, inputBase, outputBase);
System.out.println("Result = " + outputNumber);
}
}
}
In this problem we consider only strings of lower-case English letters (a-z).
A string is a palindrome if it has exactly the same sequence of characters when traversed left-to-right as right-to-left. For example, the following strings are palindromes:
"kayak"
"codilitytilidoc"
"neveroddoreven"
A string A is an anagram of a string B if it consists of exactly the same characters, but possibly in another order. For example, the following strings are each other's anagrams:
A="mary" B="army" A="rocketboys" B="octobersky" A="codility" B="codility"
Write a function
int isAnagramOfPalindrome(String S);
which returns 1 if the string s is a anagram of some palindrome, or returns 0 otherwise.
For example your function should return 1 for the argument "dooernedeevrvn", because it is an anagram of a palindrome "neveroddoreven". For argument "aabcba", your function should return 0.
'Algorithm' would be too big word for it.
You can construct a palindrome from the given character set if each character occurs in that set even number of times (with possible exception of one character).
For any other set, you can easily show that no palindrome exists.
Proof is simple in both cases, but let me know if that wasn't clear.
In a palindrome, every character must have a copy of itself, a "twin", on the other side of the string, except in the case of the middle letter, which can act as its own twin.
The algorithm you seek would create a length-26 array, one for each lowercase letter, and start counting the characters in the string, placing the quantity of character n at index n of the array. Then, it would pass through the array and count the number of characters with an odd quantity (because one letter there does not have a twin). If this number is 0 or 1, place that single odd letter in the center, and a palindrome is easily generated. Else, it's impossible to generate one, because two or more letters with no twins exist, and they can't both be in the center.
I came up with this solution for Javascript.
This solution is based on the premise that a string is an anagram of a palindrome if and only if at most one character appears an odd number of times in it.
function solution(S) {
var retval = 0;
var sorted = S.split('').sort(); // sort the input characters and store in
// a char array
var array = new Array();
for (var i = 0; i < sorted.length; i++) {
// check if the 2 chars are the same, if so copy the 2 chars to the new
// array
// and additionally increment the counter to account for the second char
// position in the loop.
if ((sorted[i] === sorted[i + 1]) && (sorted[i + 1] != undefined)) {
array.push.apply(array, sorted.slice(i, i + 2));
i = i + 1;
}
}
// if the original string array's length is 1 or more than the length of the
// new array's length
if (sorted.length <= array.length + 1) {
retval = 1;
}
//console.log("new array-> " + array);
//console.log("sorted array-> " + sorted);
return retval;
}
i wrote this code in java. i don't think if its gonna be a good one ^^,
public static int isAnagramOfPalindrome(String str){
ArrayList<Character> a = new ArrayList<Character>();
for(int i = 0; i < str.length(); i++){
if(a.contains(str.charAt(i))){
a.remove((Object)str.charAt(i));
}
else{
a.add(str.charAt(i));
}
}
if(a.size() > 1)
return 0;
return 1;
}
Algorithm:
Count the number of occurrence of each character.
Only one character with odd occurrence is allowed since in a palindrome the maximum number of character with odd occurrence can be '1'.
All other characters should occur in an even number of times.
If (2) and (3) fail, then the given string is not a palindrome.
This adds to the other answers given. We want to keep track of the count of each letter seen. If we have more than one odd count for a letter then we will not be able to form a palindrome. The odd count would go in the middle, but only one odd count can do so.
We can use a hashmap to keep track of the counts. The lookup for a hashmap is O(1) so it is fast. We are able to run the whole algorithm in O(n). Here's it is in code:
if __name__ == '__main__':
line = input()
dic = {}
for i in range(len(line)):
ch = line[i]
if ch in dic:
dic[ch] += 1
else:
dic[ch] = 1
chars_whose_count_is_odd = 0
for key, value in dic.items():
if value % 2 == 1:
chars_whose_count_is_odd += 1
if chars_whose_count_is_odd > 1:
print ("NO")
else:
print ("YES")
I have a neat solution in PHP posted in this question about complexities.
class Solution {
// Function to determine if the input string can make a palindrome by rearranging it
static public function isAnagramOfPalindrome($S) {
// here I am counting how many characters have odd number of occurrences
$odds = count(array_filter(count_chars($S, 1), function($var) {
return($var & 1);
}));
// If the string length is odd, then a palindrome would have 1 character with odd number occurrences
// If the string length is even, all characters should have even number of occurrences
return (int)($odds == (strlen($S) & 1));
}
}
echo Solution :: isAnagramOfPalindrome($_POST['input']);
It uses built-in PHP functions (why not), but you can make it yourself, as those functions are quite simple. First, the count_chars function generates a named array (dictionary in python) with all characters that appear in the string, and their number of occurrences. It can be substituted with a custom function like this:
$count_chars = array();
foreach($S as $char) {
if array_key_exists($char, $count_chars) {
$count_chars[$char]++;
else {
$count_chars[$char] = 1;
}
}
Then, an array_filter with a count function is applied to count how many chars have odd number of occurrences:
$odds = 0;
foreach($count_chars as $char) {
$odds += $char % 2;
}
And then you just apply the comparison in return (explained in the comments of the original function).
return ($odds == strlen($char) % 2)
This runs in O(n). For all chars but one, must be even. the optional odd character can be any odd number.
e.g.
abababa
def anagram_of_pali(str):
char_list = list(str)
map = {}
nb_of_odds = 0
for char in char_list:
if char in map:
map[char] += 1
else:
map[char] = 1
for char in map:
if map[char] % 2 != 0:
nb_of_odds += 1
return True if nb_of_odds <= 1 else False
You just have to count all the letters and check if there are letters with odd counts. If there are more than one letter with odd counts the string does not satisfy the above palindrome condition.
Furthermore, since a string with an even number letters must not have a letter with an odd count it is not necessary to check whether string length is even or not. It will take O(n) time complexity:
Here's the implementation in javascript:
function canRearrangeToPalindrome(str)
{
var letterCounts = {};
var letter;
var palindromeSum = 0;
for (var i = 0; i < str.length; i++) {
letter = str[i];
letterCounts[letter] = letterCounts[letter] || 0;
letterCounts[letter]++;
}
for (var letterCount in letterCounts) {
palindromeSum += letterCounts[letterCount] % 2;
}
return palindromeSum < 2;
}
All right - it's been a while, but as I was asked such a question in a job interview I needed to give it a try in a few lines of Python. The basic idea is that if there is an anagram that is a palindrome for even number of letters each character occurs twice (or something like 2n times, i.e. count%2==0). In addition, for an odd number of characters one character (the one in the middle) may occur only once (or an uneven number - count%2==1).
I used a set in python to get the unique characters and then simply count and break the loop once the condition cannot be fulfilled. Example code (Python3):
def is_palindrome(s):
letters = set(s)
oddc=0
fail=False
for c in letters:
if s.count(c)%2==1:
oddc = oddc+1
if oddc>0 and len(s)%2==0:
fail=True
break
elif oddc>1:
fail=True
break
return(not fail)
def is_anagram_of_palindrome(S):
L = [ 0 for _ in range(26) ]
a = ord('a')
length = 0
for s in S:
length += 1
i = ord(s) - a
L[i] = abs(L[i] - 1)
return length > 0 and sum(L) < 2 and 1 or 0
While you can detect that the given string "S" is a candidate palindrome using the given techniques, it is still not very useful. According to the implementations given,
isAnagramOfPalindrome("rrss") would return true but there is no actual palindrome because:
A palindrome is a word, phrase, number, or other sequence of symbols or elements, whose meaning may be interpreted the same way in either forward or reverse direction. (Wikipedia)
And Rssr or Srrs is not an actual word or phrase that is interpretable. Same with it's anagram. Aarrdd is not an anagram of radar because it is not interpretable.
So, the solutions given must be augmented with a heuristic check against the input to see if it's even a word, and then a verification (via the implementations given), that it is palindrome-able at all. Then there is a heuristic search through the collected buckets with n/2! permutations to search if those are ACTUALLY palindromes and not garbage. The search is only n/2! and not n! because you calculate all permutations of each repeated letter, and then you mirror those over (in addition to possibly adding the singular pivot letter) to create all possible palindromes.
I disagree that algorithm is too big of a word, because this search can be done pure recursively, or using dynamic programming (in the case of words with letters with occurrences greater than 2) and is non trivial.
Here's some code: This is same as the top answer that describes algorithm.
1 #include<iostream>
2 #include<string>
3 #include<vector>
4 #include<stack>
5
6 using namespace std;
7
8 bool fun(string in)
9 {
10 int len=in.size();
11 int myints[len ];
12
13 for(int i=0; i<len; i++)
14 {
15 myints[i]= in.at(i);
16 }
17 vector<char> input(myints, myints+len);
18 sort(input.begin(), input.end());
19
20 stack<int> ret;
21
22 for(int i=0; i<len; i++)
23 {
24 if(!ret.empty() && ret.top()==input.at(i))
25 {
26 ret.pop();
27 }
28 else{
29 ret.push(input.at(i));
30 }
31 }
32
33 return ret.size()<=1;
34
35 }
36
37 int main()
38 {
39 string input;
40 cout<<"Enter word/number"<<endl;
41 cin>>input;
42 cout<<fun(input)<<endl;
43
44 return 0;
45 }