Problem Statement
Egg dropping refers to a class of problems in which it is important to find the correct response without exceeding a (low) number of certain failure states. In a toy example, there is a tower of floors, and an egg dropper with ideal eggs. The physical properties of the ideal egg is such that it will shatter if it is dropped from floor or above, and will have no damage whatsoever if it is dropped from floor or below. The problem is to find a strategy such that the egg dropper can determine the floor in as few egg drops as possible. This problem has many applications in the real world such as avoiding a call out to the slow HDD, or attempting to minimize cache misses, or running a large number of expensive queries on a database.
Problem Statement and Solution Analysis
When we have N number of eggs and K number of floors the following code finds the minimum number of drops using quadratic equation with time complexity of O(N).
(function() {
var eggs = 3, floors = 2;
function findFloor(eggs, floors) {
if (eggs === 1 || floors === 0 || floors === 1) {
return floors;
}
var minDrops = Math.ceil((-1 + Math.sqrt(1 + (8 * floors))) / 2);
return Math.min(minDrops, findFloor(eggs - 1, minDrops));
}
console.log(findFloor(eggs, floors));
})();
I have tested with some test cases but can anyone suggest, will this work for all the scenarios?
No, this will not always produce the correct results. You have used this formula:
But that formula only provides a meaningful result in case the number of eggs is two. Note how the number of eggs is not appearing in it, only the number of floors ( k ).
Counter example
Take for instance the case with 4 floors and 3 eggs. Your function returns 2, but if that were the correct answer, then which floors would you pick in those two attempts?
Let's drop from floor 3: egg breaks. Then throw from floor 1: egg does not break. Now we don't know whether the answer is floor 1 or 2. We would need to drop one more egg to be sure.
Maybe start at floor 2?: egg is OK. Then throw from floor 4: egg breaks. Now we don't know whether the answer is floor 2 or 3. We would need to drop one more egg to be sure.
So, in the worst case we need to drop at least 3 eggs.
Conclusion
Your algorithm is not correct. The article you refer two has correct implementations (although there are some typos with variable names). Here they are in JavaScript:
function getNumDropsRecursive(eggs, floors) {
if (eggs == 1 || floors == 0 || floors == 1) {
return floors
}
let minimum = Infinity;
for (let floor = 1; floor <= floors; floor++) {
minimum = Math.min(
minimum,
1 + Math.max(getNumDropsRecursive(eggs - 1, floor - 1),
getNumDropsRecursive(eggs, floors - floor))
)
}
return minimum;
}
function getNumDropsDP(eggs, floors) {
const numdrops = [
null,
[...Array(floors+1).keys()],
...Array.from(Array(eggs-1), _ => [0, 1])
];
for (let remainingEggs = 2; remainingEggs <= eggs; remainingEggs++) {
for (let choices = 2; choices <= floors; choices++) {
let minimum = Infinity;
for (let dropAt = 1; dropAt <= choices; dropAt++) {
minimum = Math.min(minimum,
1 + Math.max(numdrops[remainingEggs-1][dropAt-1],
numdrops[remainingEggs][choices-dropAt])
);
}
numdrops[remainingEggs][choices] = minimum;
}
}
return numdrops[eggs][floors];
}
Using the first one is not advised as it starts to get really slow with arguments above 20.
I would also name your function differently. The function does not find a floor, but the number of drops you need in the worst case to find the floor. So a name like getNumDrops would be more telling.
I believe the known solution is O(n log k). Here are some mismatches:
/*
W(n,k) = 1 + min{max(W(n − 1, x − 1), W(n,k − x)): x = 1, 2, ..., k }
with W(n,0) = 0 for all n > 0 and W(1,k) = k for all k.
*/
function f(n,k){
if (k == 0 && n > 0)
return 0;
if (n == 1)
return k;
let best = Infinity;
for (let x=1; x<=k; x++)
best = Math.min(best, Math.max(f(n-1, x-1), f(n, k-x)));
return 1 + best;
}
function findFloor(eggs, floors) {
if (eggs === 1 || floors === 0 || floors === 1) {
return floors;
}
var minDrops = Math.ceil((-1 + Math.sqrt(1 + (8 * floors))) / 2);
return Math.min(minDrops, findFloor(eggs - 1, minDrops));
}
for (let i=1; i<10; i++){
for (let j=1; j<10; j++){
let a = f(i,j);
let b = findFloor(i,j);
if (a != b){
console.log(`n,k: ${i},${j}; f: ${a}; findFloors: ${b}`);
}
}
}
I am trying to learn Scala and functional programming ideology by rewriting basic exercises. Currently I have trouble with naive approach for generating primes "trial division".
The trouble described below is that I could not rewrite well-known algorithm in functional style preserving efficiency, because I have no suitable immutable data structure, like a List but with fast operations not only on head, but also on the very end.
I started with writing java code which for every odd number tests its divisibility by already found primes (limited by square root of value being tested) - and adds it to the end of the list if no divisor was found.
http://ideone.com/QE8U0I
List<Integer> primes = new ArrayList<>();
primes.add(2);
int cur = 3;
while (primes.size() < 100000) {
for (Integer x : primes) {
if (x * x > cur) {
primes.add(cur);
break;
}
if (cur % x == 0) {
break;
}
}
cur += 2;
}
Now I tried to rewrite it in "functional way" - there was no problem with using recursion instead of loops, but I stuck with immutable collections. Core idea is as following:
http://ideone.com/4DQ6mi
def primes(n: Int) = {
#tailrec
def divisibleByAny(x: Int, list: List[Int]): Boolean = {
if (list.isEmpty) false else {
val h = list.head
h * h <= x && (x % h == 0 || divisibleByAny(x, list.tail))
}
}
#tailrec
def morePrimes(from: Int, prev: List[Int]): List[Int] = {
if (prev.size == n) prev else
morePrimes(from + 2, if (divisibleByAny(from, prev)) prev else prev :+ from)
}
morePrimes(3, List(2))
}
But it is slow - if I understand correctly because operation of adding to the end of immutable list requires creation of new copy of the whole stuff.
I searched over documentation to find more suitable data structure and tried to substitute list with immutable Queue, for it is said:
Adding items to the queue always has cost O(1) ... Removing an item is on average O(1).
But it is still even slower:
http://ideone.com/v8BsuQ
def primes(n: Int) = {
#tailrec
def divisibleByAny(x: Int, list: Queue[Int]): Boolean = {
if (list.isEmpty) false else {
val (h, t) = list.dequeue
h * h <= x && (x % h == 0 || divisibleByAny(x, t))
}
}
#tailrec
def morePrimes(from: Int, prev: Queue[Int]): Queue[Int] = {
if (prev.size == n) prev else
morePrimes(from + 2, if (divisibleByAny(from, prev)) prev else prev.enqueue(from))
}
morePrimes(3, Queue(2))
}
What is going wrong or am I missing something?
P.S. I believe there are other algorithms for generating primes which are more suitable for functional style. I think I've seen some paper. But now I'm interested in this one, or more precisely in existence of suitable data structure.
According to http://docs.scala-lang.org/overviews/collections/performance-characteristics.html Vectors have an amortised constant cost for appending, prepending and seeking. Indeed, using vectors instead of lists in your solution is much faster
def primes(n: Int) = {
#tailrec
def divisibleByAny(x: Int, list: Vector[Int]): Boolean = {
if (list.isEmpty) false else {
val (h +: t) = list
h * h <= x && (x % h == 0 || divisibleByAny(x, t))
}
}
#tailrec
def morePrimes(from: Int, prev: Vector[Int]): Vector[Int] = {
if (prev.length == n) prev else
morePrimes(from + 2, if (divisibleByAny(from, prev)) prev else prev :+ from)
}
morePrimes(3, Vector(2))
}
http://ideone.com/x3k4A3
I think you have 2 main options
Use a Vector - which is better than a list for appending. It is a Bitmapped Trie data structure (http://en.wikipedia.org/wiki/Trie). It’s “effectively” O(1) for appending to (i.e. O(1) on average)
Or...possibly the answer you're not looking for
Use a mutable data structure like ListBuffer - immutability it great to try achieve, and should be your go to collections - but sometimes for efficiency reasons, you may use mutable structures . What is key it to make sure it does not “leak out” of your classes. If you look at the List.scala implementation, you’ll see ListBuffer used a lot internally. However, its coverted back to a List just before it leaves the class. If its good enough for the core Scala libraries, its probably ok for you to use under exceptional cases that warrant it.
Except using Vector, also consider using higher-order functions instead of recursion. That's also a completely valid functional style. On my machine the following implementation of divisibleByAny is about 8x faster, than #Pyetras tailrec implementation when running primes(1000000):
def divisibleByAny(x: Int, list: Vector[Int]): Boolean =
list.view.takeWhile(el => el * el <= x).exists(x % _ == 0)
Imagine you have 3 buckets, but each of them has a hole in it. I'm trying to fill a bath tub. The bath tub has a minimum level of water it needs and a maximum level of water it can contain. By the time you reach the tub with the bucket it is not clear how much water will be in the bucket, but you have a range of possible values.
Is it possible to adequately fill the tub with water?
Pretty much you have 3 ranges (min,max), is there some sum of them that will fall within a 4th range?
For example:
Bucket 1 : 5-10L
Bucket 2 : 15-25L
Bucket 3 : 10-50L
Bathtub 100-150L
Is there some guaranteed combination of 1 2 and 3 that will fill the bathtub within the requisite range? Multiples of each bucket can be used.
EDIT: Now imagine there are 50 different buckets?
If the capacity of the tub is not very large ( not greater than 10^6 for an example), we can solve it using dynamic programming.
Approach:
Initialization: memo[X][Y] is an array to memorize the result. X = number of buckets, Y = maximum capacity of the tub. Initialize memo[][] with -1.
Code:
bool dp(int bucketNum, int curVolume){
if(curVolume > maxCap)return false; // pruning extra branches
if(curVolume>=minCap && curVolume<=maxCap){ // base case on success
return true;
}
int &ret = memo[bucketNum][curVolume];
if(ret != -1){ // this state has been visited earlier
return false;
}
ret = false;
for(int i = minC[bucketNum]; i < = maxC[bucketNum]; i++){
int newVolume = curVolume + i;
for(int j = bucketNum; j <= 3; j++){
ret|=dp(j,newVolume);
if(ret == true)return ret;
}
}
return ret;
}
Warning: Code not tested
Here's a naïve recursive solution in python that works just fine (although it doesn't find an optimal solution):
def match_helper(lower, upper, units, least_difference, fail = dict()):
if upper < lower + least_difference:
return None
if fail.get((lower,upper)):
return None
exact_match = [ u for u in units if u['lower'] >= lower and u['upper'] <= upper ]
if exact_match:
return [ exact_match[0] ]
for unit in units:
if unit['upper'] > upper:
continue
recursive_match = match_helper(lower - unit['lower'], upper - unit['upper'], units, least_difference)
if recursive_match:
return [unit] + recursive_match
else:
fail[(lower,upper)] = 1
return None
def match(lower, upper):
units = [
{ 'name': 'Bucket 1', 'lower': 5, 'upper': 10 },
{ 'name': 'Bucket 2', 'lower': 15, 'upper': 25 },
{ 'name': 'Bucket 3', 'lower': 10, 'upper': 50 }
]
least_difference = min([ u['upper'] - u['lower'] for u in units ])
return match_helper(
lower = lower,
upper = upper,
units = sorted(units, key = lambda u: u['upper']),
least_difference = min([ u['upper'] - u['lower'] for u in units ]),
)
result = match(100, 175)
if result:
lower = sum([ u['lower'] for u in result ])
upper = sum([ u['upper'] for u in result ])
names = [ u['name'] for u in result ]
print lower, "-", upper
print names
else:
print "No solution"
It prints "No solution" for 100-150, but for 100-175 it comes up with a solution of 5x bucket 1, 5x bucket 2.
Assuming you are saying that the "range" for each bucket is the amount of water that it may have when it reaches the tub, and all you care about is if they could possibly fill the tub...
Just take the "max" of each bucket and sum them. If that is in the range of what you consider the tub to be "filled" then it can.
Updated:
Given that buckets can be used multiple times, this seems to me like we're looking for solutions to a pair of equations.
Given buckets x, y and z we want to find a, b and c:
a*x.min + b*y.min + c*z.min >= bathtub.min
and
a*x.max + b*y.max + c*z.max <= bathtub.max
Re: http://en.wikipedia.org/wiki/Diophantine_equation
If bathtub.min and bathtub.max are both multiples of the greatest common divisor of a,b and c, then there are infinitely many solutions (i.e. we can fill the tub), otherwise there are no solutions (i.e. we can never fill the tub).
This can be solved with multiple applications of the change making problem.
Each Bucket.Min value is a currency denomination, and Bathtub.Min is the target value.
When you find a solution via a change-making algorithm, then apply one more constraint:
sum(each Bucket.Max in your solution) <= Bathtub.max
If this constraint is not met, throw out this solution and look for another. This will probably require a change to a standard change-making algorithm that allows you to try other solutions when one is found to not be suitable.
Initially, your target range is Bathtub.Range.
Each time you add an instance of a bucket to the solution, you reduce the target range for the remaining buckets.
For example, using your example buckets and tub:
Target Range = 100..150
Let's say we want to add a Bucket1 to the candidate solution. That then gives us
Target Range = 95..140
because if the rest of the buckets in the solution total < 95, then this Bucket1 might not be sufficient to fill the tub to 100, and if the rest of the buckets in the solution total > 140, then this Bucket1 might fill the tub over 150.
So, this gives you a quick way to check if a candidate solution is valid:
TargetRange = Bathtub.Range
foreach Bucket in CandidateSolution
TargetRange.Min -= Bucket.Min
TargetRange.Max -= Bucket.Max
if TargetRange.Min == 0 AND TargetRange.Max >= 0 then solution found
if TargetRange.Min < 0 or TargetRange.Max < 0 then solution is invalid
This still leaves the question - How do you come up with the set of candidate solutions?
Brute force would try all possible combinations of buckets.
Here is my solution for finding the optimal solution (least number of buckets). It compares the ratio of the maximums to the ratio of the minimums, to figure out the optimal number of buckets to fill the tub.
private static void BucketProblem()
{
Range bathTub = new Range(100, 175);
List<Range> buckets = new List<Range> {new Range(5, 10), new Range(15, 25), new Range(10, 50)};
Dictionary<Range, int> result;
bool canBeFilled = SolveBuckets(bathTub, buckets, out result);
}
private static bool BucketHelper(Range tub, List<Range> buckets, Dictionary<Range, int> results)
{
Range bucket;
int startBucket = -1;
int fills = -1;
for (int i = buckets.Count - 1; i >=0 ; i--)
{
bucket = buckets[i];
double maxRatio = (double)tub.Maximum / bucket.Maximum;
double minRatio = (double)tub.Minimum / bucket.Minimum;
if (maxRatio >= minRatio)
{
startBucket = i;
if (maxRatio - minRatio > 1)
fills = (int) minRatio + 1;
else
fills = (int) maxRatio;
break;
}
}
if (startBucket < 0)
return false;
bucket = buckets[startBucket];
tub.Maximum -= bucket.Maximum * fills;
tub.Minimum -= bucket.Minimum * fills;
results.Add(bucket, fills);
return tub.Maximum == 0 || tub.Minimum <= 0 || startBucket == 0 || BucketHelper(tub, buckets.GetRange(0, startBucket), results);
}
public static bool SolveBuckets(Range tub, List<Range> buckets, out Dictionary<Range, int> results)
{
results = new Dictionary<Range, int>();
buckets = buckets.OrderBy(b => b.Minimum).ToList();
return BucketHelper(new Range(tub.Minimum, tub.Maximum), buckets, results);
}
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()