I am working my way through the USACO training website. I found this question which requires us to find the minimum number of swaps in a three valued sequence. Although I have solved the problem with a O(n^2) solution, I was intrigued to find out this faster O(n) approach. However, I am finding it rather difficult to understand, being a beginner.
Here is the detailed problem statement and the best approach.
In this task the possible key values are the integers 1, 2 and 3. The
required sorting order is non-decreasing. However, sorting has to be
accomplished by a sequence of exchange operations. An exchange
operation, defined by two position numbers p and q, exchanges the
elements in positions p and q.
You are given a sequence of key values. Write a program that computes
the minimal number of exchange operations that are necessary to make
the sequence sorted.
#include <fstream>
using namespace std;
int min (int a, int b) { return a < b ? a : b; }
int max (int a, int b) { return a > b ? a : b; }
int main () {
int s[1024];
int n;
int sc[4] = {0};
ifstream fin("sort3.in");
ofstream fout("sort3.out");
fin>>n;
for (int i = 0; i < n; i++) {
fin>>s[i];
sc[s[i]]++;
}
int s12 = 0, s13 = 0, s21 = 0, s31 = 0, s23 = 0, s32 = 0;
for (int i = 0; i < sc[1]; i++){
if (s[i] == 2) s12++;
if (s[i] == 3) s13++;
}
for (int i = sc[1]; i < sc[1]+sc[2]; i++){
if (s[i] == 1) s21++;
if (s[i] == 3) s23++;
}
for (int i = sc[1]+sc[2]; i < sc[1]+sc[2]+sc[3]; i++){
if (s[i] == 1) s31++;
if (s[i] == 2) s32++;
}
fout<<min(s12, s21)+min(s13, s31)+min(s23, s32) +
2*(max(s12, s21) - min(s12, s21))<<endl;
return 0;
}
I got the part where we are taking the minimum of overlaps of 1s, 2s and 3s in their respective desired ranges [0->c1, c1->c1+c2, c1+c2->n]. However, I don't really understand the formula of 2*max(s12, s21) - min(s12, s21). Also why are we not considering s13, s31, s23, and s32 in the formula. I would like you to explain this to me as I don't intend to leave any unbridged gaps in my training. Thanks in Advance!
If we approach the problem from the other side and work out how to perform the swaps in an example we can see where the formula for the number of swaps comes from.
If we start with:
22121 23331 1231
s12 is 3
s13 is 0
s21 is 1
s23 is 3
s31 is 2
s32 is 1
Then swapping 1 and 2 with 1 swap (min(s12, s21)) gives:
12121 23332 1231
There are no swaps between 1 and 3 (min(s13, s31)).
Then swapping 2 and 3 with 1 swap (min(s23, s32)) gives:
12121 22332 1331
Now we're left with no more direct swaps, we need to swap twice to get the remaining elements into the correct locations. There are 2 1s in the wrong place, 2s in the wrong place and 2 3s in the wrong place, the number for each number must always be the same. The initial number of misplaced 1s was given by max(s12, s21), we've fixed min(s12, s21) of those leaving max(s12, s21) - min(s12, s21) still to fix. We'll need to perform 2 swaps to fix each incorrect element.
First swap the 2s where 1s should be with the 1s that are where the 3s should be, this takes 2 swaps giving:
11111 22332 2332
Now we can directly swap the 2s and 3s taking another 2 swaps giving the final result:
11111 22222 3333
Adding up the total number of swaps gives:
min(s12, s21) + min(s13, s31) + min(s23, s32) + 2*(max(s12, s21) - min(s12, s21))
Related
There's a problem, which I've to solve in c++. I've written the whole code and it's working in the given test cases but when I'm submitting it, It's saying wrong answer. I can't understand that why is it showing wrong answer.
I request you to tell me an input for the given code, which will give incorrect output so I can modify my code further.
Shrink The Array
You are given an array of positive integers A[] of length L. If A[i] and A[i+1] both are equal replace them by one element with value A[i]+1. Find out the minimum possible length of the array after performing such operation any number of times.
Note:
After each such operation, the length of the array will decrease by one and elements are renumerated accordingly.
Input format:
The first line contains a single integer L, denoting the initial length of the array A.
The second line contains L space integers A[i] − elements of array A[].
Output format:
Print an integer - the minimum possible length you can get after performing the operation described above any number of times.
Example:
Input
7
3 3 4 4 4 3 3
Output
2
Sample test case explanation
3 3 4 4 4 3 3 -> 4 4 4 4 3 3 -> 4 4 4 4 4 -> 5 4 4 4 -> 5 5 4 -> 6 4.
Thus the length of the array is 2.
My code:
#include <bits/stdc++.h>
using namespace std;
int main()
{
bool end = false;
int l;
cin >> l;
int arr[l];
for(int i = 0; i < l; i++){
cin >> arr[i];
}
int len = l, i = 0;
while(i < len - 1){
if(arr[i] == arr[i + 1]){
arr[i] = arr[i] + 1;
if((i + 1) <= (len - 1)){
for(int j = i + 1; j < len - 1; j++){
arr[j] = arr[j + 1];
}
}
len--;
i = 0;
}
else{
i++;
}
}
cout << len;
return 0;
}
THANK YOU
As noted in the comments: Just picking the first two neighbours that have the same value and combining those will lead to suboptimal results.
You will need to investigate which two neighbours you should combine somehow. When you have combined two neighbours you then need to investigate which neighbours to combine on the next level. The number of combinations may become plentiful.
One way to solve this is through recursion.
If you've followed the advice in the comments, you now have all your input data in std::vector<unsigned> A(L).
You can now do std::cout << solve(A) << '\n'; where solve has the signature size_t solve(const std::vector<unsigned>& A) and is described below:
Find the indices of all neighbour pairs in A that has the same values and put the indices in a std::vector<size_t> neighbours. Example: If A contains 2 2 2 3, put 0 and 1 in neighbours.
If no neighbours are found (neighbours.empty() == true), return A.size().
Define a minimum variable and initialize it with A.size() - 1 which is the worst result you know you can get at this point. So, size_t minimum = A.size() - 1;
Loop over all indices stored in neighbours (for(size_t idx : neighbours))
Copy A into a new std::vector<unsigned>. Let's call it cpy.
Increase cpy[idx] by one and remove cpy[idx+1].
Call size_t result = solve(cpy). This is where recursion comes in.
Is result less than minimum? If so assign result to minimum.
Return minimum.
I don't think I ruined the programming exercise by providing one algorithm for solving this. It should still have plenty of things to deal with. Recursion won't be possible with big data etc.
Please when answering this question try to be as general as possible to help the wider community, rather than just specifically helping my issue (although helping my issue would be great too ;) )
I seem to be encountering this problem time and time again with the simple problems on Project Euler. Most commonly are the problems that require a computation of the prime numbers - these without fail always fail to terminate for numbers greater than about 60,000.
My most recent issue is with Problem 12:
The sequence of triangle numbers is generated by adding the natural numbers. So the 7th triangle number would be 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28. The first ten terms would be:
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...
Let us list the factors of the first seven triangle numbers:
1: 1
3: 1,3
6: 1,2,3,6
10: 1,2,5,10
15: 1,3,5,15
21: 1,3,7,21
28: 1,2,4,7,14,28
We can see that 28 is the first triangle number to have over five divisors.
What is the value of the first triangle number to have over five hundred divisors?
Here is my code:
#include <iostream>
#include <vector>
#include <cmath>
using namespace std;
int main() {
int numberOfDivisors = 500;
//I begin by looping from 1, with 1 being the 1st triangular number, 2 being the second, and so on.
for (long long int i = 1;; i++) {
long long int triangularNumber = (pow(i, 2) + i)/2
//Once I have the i-th triangular, I loop from 1 to itself, and add 1 to count each time I encounter a divisor, giving the total number of divisors for each triangular.
int count = 0;
for (long long int j = 1; j <= triangularNumber; j++) {
if (triangularNumber%j == 0) {
count++;
}
}
//If the number of divisors is 500, print out the triangular and break the code.
if (count == numberOfDivisors) {
cout << triangularNumber << endl;
break;
}
}
}
This code gives the correct answers for smaller numbers, and then either fails to terminate or takes an age to do so!
So firstly, what can I do with this specific problem to make my code more efficient?
Secondly, what are some general tips both for myself and other new C++ users for making code more efficient? (I.e. applying what we learn here in the future.)
Thanks!
The key problem is that your end condition is bad. You are supposed to stop when count > 500, but you look for an exact match of count == 500, therefore you are likely to blow past the correct answer without detecting it, and keep going ... maybe forever.
If you fix that, you can post it to code review. They might say something like this:
Break it down into separate functions for finding the next triangle number, and counting the factors of some number.
When you find the next triangle number, you execute pow. I perform a single addition.
For counting the number of factors in a number, a google search might help. (e.g. http://www.cut-the-knot.org/blue/NumberOfFactors.shtml ) You can build a list of prime numbers as you go, and use that to quickly find a prime factorization, from which you can compute the number of factors without actually counting them. When the numbers get big, that loop gets big.
Tldr: 76576500.
About your Euler problem, some math:
Preliminary 1:
Let's call the n-th triangle number T(n).
T(n) = 1 + 2 + 3 + ... + n = (n^2 + n)/2 (sometimes attributed to Gauss, sometimes someone else). It's not hard to figure it out:
1+2+3+4+5+6+7+8+9+10 =
(1+10) + (2+9) + (3+8) + (4+7) + (5+6) =
11 + 11 + 11 + 11 + 11 =
55 =
110 / 2 =
(10*10 + 10)/2
Because of its definition, it's trivial that T(n) + n + 1 = T(n+1), and that with a<b, T(a)<T(b) is true too.
Preliminary 2:
Let's call the divisor count D. D(1)=1, D(4)=3 (because 1 2 4).
For a n with c non-repeating prime factors (not just any divisors, but prime factors, eg. n = 42 = 2 * 3 * 7 has c = 3), D(n) is c^2: For each factor, there are two possibilites (use it or not). The 9 possibile divisors for the examples are: 1, 2, 3, 7, 6 (2*3), 14 (2*7), 21 (3*7), 42 (2*3*7).
More generally with repeating, the solution for D(n) is multiplying (Power+1) together. Example 126 = 2^1 * 3^2 * 7^1: Because it has two 3, the question is no "use 3 or not", but "use it 1 time, 2 times or not" (if one time, the "first" or "second" 3 doesn't change the result). With the powers 1 2 1, D(126) is 2*3*2=12.
Preliminary 3:
A number n and n+1 can't have any common prime factor x other than 1 (technically, 1 isn't a prime, but whatever). Because if both n/x and (n+1)/x are natural numbers, (n+1)/x - n/x has to be too, but that is 1/x.
Back to Gauss: If we know the prime factors for a certain n and n+1 (needed to calculate D(n) and D(n+1)), calculating D(T(n)) is easy. T(N) = (n^2 + n) / 2 = n * (n+1) / 2. As n and n+1 don't have common prime factors, just throwing together all factors and removing one 2 because of the "/2" is enough. Example: n is 7, factors 7 = 7^1, and n+1 = 8 = 2^3. Together it's 2^3 * 7^1, removing one 2 is 2^2 * 7^1. Powers are 2 1, D(T(7)) = 3*2 = 6. To check, T(7) = 28 = 2^2 * 7^1, the 6 possible divisors are 1 2 4 7 14 28.
What the program could do now: Loop through all n from 1 to something, always factorize n and n+1, use this to get the divisor count of the n-th triangle number, and check if it is >500.
There's just the tiny problem that there are no efficient algorithms for prime factorization. But for somewhat small numbers, todays computers are still fast enough, and keeping all found factorizations from 1 to n helps too for finding the next one (for n+1). Potential problem 2 are too large numbers for longlong, but again, this is no problem here (as can be found out with trying).
With the described process and the program below, I got
the 12375th triangle number is 76576500 and has 576 divisors
#include <iostream>
#include <vector>
#include <cstdint>
using namespace std;
const int limit = 500;
vector<uint64_t> knownPrimes; //2 3 5 7...
//eg. [14] is 1 0 0 1 ... because 14 = 2^1 * 3^0 * 5^0 * 7^1
vector<vector<uint32_t>> knownFactorizations;
void init()
{
knownPrimes.push_back(2);
knownFactorizations.push_back(vector<uint32_t>(1, 0)); //factors for 0 (dummy)
knownFactorizations.push_back(vector<uint32_t>(1, 0)); //factors for 1 (dummy)
knownFactorizations.push_back(vector<uint32_t>(1, 1)); //factors for 2
}
void addAnotherFactorization()
{
uint64_t number = knownFactorizations.size();
size_t len = knownPrimes.size();
for(size_t i = 0; i < len; i++)
{
if(!(number % knownPrimes[i]))
{
//dividing with a prime gets a already factorized number
knownFactorizations.push_back(knownFactorizations[number / knownPrimes[i]]);
knownFactorizations[number][i]++;
return;
}
}
//if this failed, number is a newly found prime
//because a) it has no known prime factors, so it must have others
//and b) if it is not a prime itself, then it's factors should've been
//found already (because they are smaller than the number itself)
knownPrimes.push_back(number);
len = knownFactorizations.size();
for(size_t s = 0; s < len; s++)
{
knownFactorizations[s].push_back(0);
}
knownFactorizations.push_back(knownFactorizations[0]);
knownFactorizations[number][knownPrimes.size() - 1]++;
}
uint64_t calculateDivisorCountOfN(uint64_t number)
{
//factors for number must be known
uint64_t res = 1;
size_t len = knownFactorizations[number].size();
for(size_t s = 0; s < len; s++)
{
if(knownFactorizations[number][s])
{
res *= (knownFactorizations[number][s] + 1);
}
}
return res;
}
uint64_t calculateDivisorCountOfTN(uint64_t number)
{
//factors for number and number+1 must be known
uint64_t res = 1;
size_t len = knownFactorizations[number].size();
vector<uint32_t> tmp(len, 0);
size_t s;
for(s = 0; s < len; s++)
{
tmp[s] = knownFactorizations[number][s]
+ knownFactorizations[number+1][s];
}
//remove /2
tmp[0]--;
for(s = 0; s < len; s++)
{
if(tmp[s])
{
res *= (tmp[s] + 1);
}
}
return res;
}
int main()
{
init();
uint64_t number = knownFactorizations.size() - 2;
uint64_t DTn = 0;
while(DTn <= limit)
{
number++;
addAnotherFactorization();
DTn = calculateDivisorCountOfTN(number);
}
uint64_t tn;
if(number % 2) tn = ((number+1)/2)*number;
else tn = (number/2)*(number+1);
cout << "the " << number << "th triangle number is "
<< tn << " and has " << DTn << " divisors" << endl;
return 0;
}
About your general question about speed:
1) Algorithms.
How to know them? For (relatively) simple problems, either reading a book/Wikipedia/etc. or figuring it out if you can. For harder stuff, learning more basic things and gaining experience is necessary before it's even possible to understand them, eg. studying CS and/or maths ... number theory helps a lot for your Euler problem. (It will help less to understand how a MP3 file is compressed ... there are many areas, it's not possible to know everything.).
2a) Automated compiler optimizations of frequently used code parts / patterns
2b) Manual timing what program parts are the slowest, and (when not replacing it with another algorithm) changing it in a way that eg. requires less data send to slow devices (HDD, hetwork...), less RAM memory access, less CPU cycles, works better together with OS scheduler and memory management strategies, uses the CPU pipeline/caches better etc.etc. ... this is both education and experience (and a big topic).
And because long variables have a limited size, sometimes it is necessary to use custom types that use eg. a byte array to store a single digit in each byte. That way, it's possible to use the whole RAM for a single number if you want to, but the downside is you/someone has to reimplement stuff like addition and so on for this kind of number storage. (Of course, libs for that exist already, without writing everything from scratch).
Btw., pow is a floating point function and may get you inaccurate results. It's not appropriate to use it in this case.
I have been given the following assignment:
Given N integers in the form of A(i) where 1≤i≤N, make each number
A(i) in the N numbers equal to M. To convert a number A(i) to M, it
will cost |M−Ai| units. Find out the minimum cost to convert all the N
numbers to M, so you should choose the best M to get the minimum cost.
Given:
1 <= N <= 10^5
1 <= A(i) <= 10^9
My approach was to calculate the sum of all numbers and find avg = sum / n and then subtract each number by avg to get the minimum cost.
But this fails in many test cases. How can I find the optimal solution for this?
You should take the median of the numbers (or either of the two numbers nearest the middle if the list has even length), not the mean.
An example where the mean fails to minimize is: [1, 2, 3, 4, 100]. The mean is 110 / 5 = 22, and the total cost is 21 + 20 + 19 + 18 + 78 = 156. Choosing the median (3) gives total cost: 2 + 1 + 0 + 1 + 97 = 101.
An example where the median lies between two items in the list is [1, 2, 3, 4, 5, 100]. Here the median is 3.5, and it's ok to either use M=3 or M=4. For M=3, the total cost is 2 + 1 + 0 + 1 + 2 + 97 = 103. For M=4, the total cost is 3 + 2 + 1 + 0 + 1 + 96 = 103.
A formal proof of correctness can be found on Mathematics SE, although you may convince yourself of the result by noting that if you nudge M a small amount delta in one direction (but not past one of the data points) -- and for example's sake let's say it's in the positive direction, the total cost increases by delta times the number of points to the left of M minus delta times the number of points to the right of M. So M is minimized when the number of points to its left and the right are equal in number, otherwise you could move it a small amount one way or the other to decrease the total cost.
#PaulHankin already provided a perfect answer. Anyway, when thinking about the problem, I didn't think of the median being the solution. But even if you don't know about the median, you can come up with a programming solution.
I made similar observations as #PaulHankin in the last paragraph of his last answer. This made me realize, that I have to eliminate outliers iteratively in order to find m. So I wrote a program that first sorts the input array (vector) A and then analyzes the minimum and maximum values.
The idea is to move the minimum values towards the second smallest values and the maximum values towards the second largest values. You always move either the minimum or maximum values, depending on whether you have less minimum values than maximum values or not. If all array items end up being the same value, then you found your m:
#include <vector>
#include <algorithm>
#include <iostream>
using namespace std;
int getMinCount(vector<int>& A);
int getMaxCount(vector<int>& A);
int main()
{
// Example as given by #PaulHankin
vector<int> A;
A.push_back(1);
A.push_back(2);
A.push_back(3);
A.push_back(4);
A.push_back(100);
sort(A.begin(), A.end());
int minCount = getMinCount(A);
int maxCount = getMaxCount(A);
while (minCount != A.size() && maxCount != A.size())
{
if(minCount <= maxCount)
{
for(int i = 0; i < minCount; i++)
A[i] = A[minCount];
// Recalculate the count of the minium value, because we changed the minimum.
minCount = getMinCount(A);
}
else
{
for(int i = 0; i < maxCount; i++)
A[A.size() - 1 - i] = A[A.size() - 1 - maxCount];
// Recalculate the count of the maximum value, because we changed the maximum.
maxCount = getMaxCount(A);
}
}
// Print out the one and only remaining value, which is m.
cout << A[0] << endl;
return 0;
}
int getMinCount(vector<int>& A)
{
// Count how often the minimum value exists.
int minCount = 1;
int pos = 1;
while (pos < A.size() && A[pos++] == A[0])
minCount++;
return minCount;
}
int getMaxCount(vector<int>& A)
{
// Count how often the maximum value exists.
int maxCount = 1;
int pos = A.size() - 2;
while (pos >= 0 && A[pos--] == A[A.size() - 1])
maxCount++;
return maxCount;
}
If you think about the algorithm, then you will come to the conclusion, that it actually calculates the median of the values in the array A. As example input I took the first example given by #PaulHankin. As expected, the code provides the correct result (3) for it.
I hope my approach helps you to understand how to tackle such kind of problems even if you don't know the correct solution. This is especially helpful when you are in an interview, for example.
So, I was solving the following question: http://www.spoj.com/problems/ROADS/en/
N cities named with numbers 1 ... N are connected with one-way roads. Each road has two parameters associated with it: the road length and the toll that needs to be paid for the road (expressed in the number of coins). Bob and Alice used to live in the city 1. After noticing that Alice was cheating in the card game they liked to play, Bob broke up with her and decided to move away - to the city N. He wants to get there as quickly as possible, but he is short on cash. We want to help Bob to find the shortest path from the city 1 to the city N that he can afford with the amount of money he has.
Input
The input begins with the number t of test cases. Then t test cases follow. The first line of the each test case contains the integer K, 0 <= K <= 10000, maximum number of coins that Bob can spend on his way. The second line contains the integer N, 2 <= N <= 100, the total number of cities. The third line contains the integer R, 1 <= R <= 10000, the total number of roads. Each of the following R lines describes one road by specifying integers S, D, L and T separated by single blank characters : S is the source city, 1 <= S <= N D is the destination city, 1 <= D <= N L is the road length, 1 <= L <= 100. T is the toll (expressed in the number of coins), 0 <= T <= 100 Notice that different roads may have the same source and destination cities.
Output
For each test case, output a single line contain the total length of the shortest path from the city 1 to the city N whose total toll is less than or equal K coins. If such path does not exist, output -1.
Now, what I did was, I tried to use the djikstra's algorithm for this which is as follows:
Instead of only having a single node as the state, I take
node and coins as one state and then apply dijkstra.
length is the weight between the states.
and I minimize the length without exceeding the total coins.
My code is as follows:
using namespace std;
#define ll long long
#define pb push_back
#define mp make_pair
class node
{
public:
int vertex;
int roadlength;
int toll;
};
int dist[101][101]; // for storing roadlength
bool visited[101][10001];
int cost[101][101]; // for storing cost
int ans[101][10001]; // actual distance being stored here
void djikstra(int totalcoins, int n);
bool operator < (node a, node b)
{
if (a.roadlength != b.roadlength)
return a.roadlength < b.roadlength;
else if (a.toll != b.toll)
return a.toll < b.toll;
return a.vertex < b.vertex;
}
int main (void)
{
int a,b,c,d;
int r,t,k,n,i,j;
cin>>t;
while (t != 0)
{
cin>>k>>n>>r;
for (i = 1; i <= 101; i++)
for (j = 1; j <= 101; j++)
dist[i][j] = INT_MAX;
for (i = 0; i <= n; i++)
for (j = 0; j <= k; j++)
ans[i][j] = INT_MAX;
for ( i = 0; i <= n; i++ )
for (j = 0; j <= k; j++ )
visited[i][j] = false;
for (i = 0; i < r; i++)
{
cin>>a>>b>>c>>d;
if (a != b)
{
dist[a][b] = c;
cost[a][b] = d;
}
}
djikstra(k,n);
int minlength = INT_MAX;
for (i = 1; i <= k; i++)
{
if (ans[n][i] < minlength)
minlength = ans[n][i];
}
if (minlength == INT_MAX)
cout<<"-1\n";
else
cout<<minlength<<"\n";
t--;
}
cout<<"\n";
return 0;
}
void djikstra(int totalcoins, int n)
{
set<node> myset;
myset.insert((node){1,0,0});
ans[1][0] = 0;
while (!myset.empty())
{
auto it = myset.begin();
myset.erase(it);
int curvertex = it->vertex;
int a = it->roadlength;
int b = it->toll;
if (visited[curvertex][b] == true)
continue;
else
{
visited[curvertex][b] = true;
for (int i = 1; i <= n; i++)
{
if (dist[curvertex][i] != INT_MAX)
{
int foo = b + cost[curvertex][i];
if (foo <= totalcoins)
{
if (ans[i][foo] >= ans[curvertex][b] + cost[curvertex][i])
{
ans[i][foo] = ans[curvertex][b] + cost[curvertex][i];
myset.insert((node){i,ans[i][foo],foo});
}
}
}
}
}
}
}
Now, I have two doubts:
Firstly, my output is not coming correct for the first given test case of the question, i.e.
Sample Input:
2
5
6
7
1 2 2 3
2 4 3 3
3 4 2 4
1 3 4 1
4 6 2 1
3 5 2 0
5 4 3 2
0
4
4
1 4 5 2
1 2 1 0
2 3 1 1
3 4 1 0
Sample Output:
11
-1
My output is coming out to be, 4 -1 which is wrong for the first test case. Where am I going wrong in this?
How do I handle the condition of having multiple edges? That is, question mentions, Notice that different roads may have the same source and destination cities. How do I handle this condition?
The simple way to store the roads is as a vector of vectors. For each origin city, you want to have a vector of all roads leading from that city.
So when you are processing a discovered "best" path to a city, you would iterate through all roads from that city to see if they might be "best" paths to some other city.
As before you have two interacting definitions of "best" than cannot be simply combined into one definition. Shortest is more important, so the main definition of "best" is shortest considering cheapest only in case of ties. But you also need the alternate definition of "best" considering only cheapest.
As I suggested for the other problem, you can sort on the main definition of "best" so you always process paths that are better in that definition before paths that are worse. Then you need to track the best seen so far for the second definition of "best" such that you only prune paths from processing when they are not better in the second definition from what you already processed prioritized by the first definition.
I haven't read your code, however I can tell you the problem cannot be solved with an unmodified version of Dijkstra's algorithm.
The problem is at least as hard as the binary knapsack problem. How? The idea is to construct the knapsack problem within the stated problem. Since the knapsack problem is known to be not solvable within polynomial time, neither is the stated problem's. Since Dijkstra's algorithm is a polynomial algorithm, it therefore could not apply.
Consider a binary knapsack problem with a set of D many values X and a maximum value m = max(X). Now construct the proposed problem as such:
Let there be D + 1 cities where city n is connected to city n + 1 by two roads. Let cities 1 through D uniquely correspond to a value v in X. Let only two roads from such a city n go only to city n + 1, one costing v with distance m - v + 1, and the other costing 0 with a distance of m + 1.
In essence, "you get exactly what you pay for" -- for every coin you spend, your trip will be one unit of distance shorter.
This reframes the problem to be "what's the maximum Bob can spend by only spending money either no or one time on each toll?" And that's the same as the binary knapsack problem we started with.
Hence, if we solve the stated problem, we also can solve the binary knapsack problem, and therefore the stated problem cannot be any more "efficient" to solve than the binary knapsack problem -- with Dijkstra's algorithm is.
I am having writing an algorithm to generate all possible permutations of an array of this kind:
n = length
k = number of 1's in array
So this means if we have k 1's we will have n-k 0's in the array.
For example:
n = 5;
k = 3;
So obviously there are 5 choose 3 possible permutations for this array because
n!/(k!(n-k)!
5!/(3!2!) = (5*4)/2 = 10
possible values for the array
Here are all the values:
11100
11010
11001
10110
10101
10011
01110
01101
01011
00111
I am guessing i should use a recursive algorithms but i am just not seeing it. I am writing this algorithm in C++.
Any help would be appreciated!
Just start with 00111 and then use std::next_permutation to generate the rest:
#include <algorithm>
#include <iostream>
#include <string>
int main()
{
std::string s = "00111";
do
{
std::cout << s << '\n';
}
while (std::next_permutation(s.begin(), s.end()));
}
output:
00111
01011
01101
01110
10011
10101
10110
11001
11010
11100
You can split up the combinations into those starting with 1 (n-1, k-1) and those starting with 0 (n-1, k).
This is essentially the recursive formula for the choose function.
What you want is actually a combination since the 1's and 0's are indistinguishable and therefore their order doesn't matter (e.g. 1 1 1 vs 1 1 1).
I recently had to rewrite a combination function myself because my initial version was written recursively in a very straightforward way (pick an element, get all the combinations of the remaining array, insert the element in various places) and did not perform very well.
I searched StackOverflow and just seeing the pictures in this answer lit up the iconic lightbulb over my head.
If you want to do this recursively, observe that the set of permutations you want is equal to all the ones that start with "1", together with all the ones that start with "0". So in calculating (n,k), you will recurse on (n-1,k-1) and (n-1,k), with special cases where k = 0 and k = n.
This recursion is the reason that the binomial coefficients appear as the values in Pascal's triangle.
Homework and recursive algorithm? OK, here you go:
Base case:
You have two elements, name them "a" and "b" and produce the concatenations ab, then ba.
Step: If your second Element is longer than 1, split it up in first field/letter and the other part, and pass that recursively as parameters to the function itself.
That will work for any strings and arrays.
Its about 0-1 permutations, so probably its much more eficient to iteratively increment an integer and in case it has desired bits count, print out its binary representation.
Here a sketch:
void printAllBinaryPermutations(int k, int n)
{
int max = pow(2, n) - 1;
for(int i=0; i<=max;i++)
{
if(hasBitCountOf(i, k)) // i has k 1's?
{
printAsBinary(i, n);
}
}
}
bool hasBitCountOf(int v, int expectedBitCount)
{
int count = 0;
while(v>0 && count<= expectedBitCount)
{
int half = v >> 1;
if(half<<1 != v)
{
// v is odd
count++;
}
v = half;
}
return count==expectedBitCount;
}
void printAsBinary(int number, int strLen)
{
for(int i=strLen-1; i>=0; i--)
{
bool is0 = (number & pow(2,i)) == 0;
if (is0)
{
cout<<'0';
}
else
{
cout<<'1';
}
}
cout<<endl;
}
I am not sure this helps, plus it is just some weird pseudocode, but this should give you the desired ouput.
permutation (prefix, ones, zeros, cur) {
if (ones + zeros == 0) output(cur);
else {
if (cur != -1) prefix = concat(prefix,cur);
if (ones > 0) permutation(prefix, ones - 1, zeros, 1);
if (zeros > 0) permutation(prefix, ones, zeros - 1, 0);
}
}
permutation(empty, 3, 2, -1);
greetz
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