The Data:
A list of integers increasing in order (0,1,2,3,4,5.......)
A list of values that belong to those integers. As an example, 0 = 33, 1 = 45, 2 = 21, ....etc.
And an incrementing variable x which represent a minimum jump value.
x is the value of each jump. For example if x = 2, if 1 is chosen you cannot choose 2.
I need to determine the best way to choose integers, given some (x), that produce the highest total value from the value list.
EXAMPLE:
A = a set of 1 foot intervals (0,1,2,3,4,5,6,7,8,9)
B = the amount of money at each interval (9,5,7,3,2,7,8,10,21,12)
Distance = the minimum distance you can cover
- i.e. if the minimum distance is 3, you must skip 2 feet and leave the money, then you can
pick up the amount at the 3rd interval.
if you pick up at 0, the next one you can pick up is 3, if you choose 3 you can
next pick up 6 (after skipping 4 and 5). BUT, you dont have to pick up 6, you
could pick up 7 if it is worth more. You just can't pick up early.
So, how can I programmatically make the best jumps and end with the most money at the end?
So I am using the below equation for computing the opt value in the dynamic programming:
Here d is distance.
if (i -d) >= 0
opt(i) = max (opt(i-1), B[i] + OPT(i-d));
else
opt(i) = max (opt(i-1), B[i]);
Psuedo-code for computing the OPT value:
int A[] = {integers list}; // This is redundant if the integers are consecutive and are always from 0..n.
int B[] = {values list};
int i = 0;
int d = distance; // minimum distance between two picks.
int numIntegers = sizeof(A)/sizeof(int);
int opt[numIntegers];
opt[0] = B[0]; // For the first one Optimal value is picking itself.
for (i=1; i < numIntegers; i++) {
if ((i-d) < 0) {
opt[i] = max (opt[i-1], B[i]);
} else {
opt[i] = max (opt[i-1], B[i] + opt[i-d]);
}
}
EDIT based on OP's requirement about getting the selected integers from B:
for (i=numIntegres - 1; i >= 0;) {
if ((i == 0) && (opt[i] > 0)) {
printf ("%d ", i);
break;
}
if (opt[i] > opt[i-1]) {
printf ("%d ", i);
i = i -d;
} else {
i = i - 1;
}
}
If A[] does not have consecutive integers from 0 to n.
int A[] = {integers list}; // Here the integers may not be consecutive
int B[] = {values list};
int i = 0, j = 0;
int d = distance; // minimum distance between two picks.
int numAs = sizeof(A)/sizeof(int);
int numIntegers = A[numAs-1]
int opt[numIntegers];
opt[0] = 0;
if (A[0] == 0) {
opt[0] = B[0]; // For the first one Optimal value is picking itself.
j = 1;
}
for (i=1; i < numIntegers && j < numAs; i++, j++) {
if (i < A[j]) {
while (i < A[j]) {
opt[i] = opt[i -1];
i = i + 1:
}
}
if ((i-d) < 0) {
opt[i] = max (opt[i-1], B[j]);
} else {
opt[i] = max (opt[i-1], B[j] + opt[i-d]);
}
}
Related
Consider the sequence of numbers from 1 to ๐. For example, for ๐ = 9,
we have 1, 2, 3, 4, 5, 6, 7, 8, 9.
Now, place among the numbers one of the three following operators:
"+" sum
"-" subtraction
"#" Paste Operator --> paste the previous and the next operands.
For example, 1#2 = 12
How can I calculate the number of possible sequences that yield zero ?
Example for N = 7:
1+2-3+4-5-6+7
1+2-3-4+5+6-7
1-2#3+4+5+6+7
1-2#3-4#5+6#7
1-2+3+4-5+6-7
1-2-3-4-5+6+7
See the fourth sequence, it is same as 1-23-45+67 and the result is 0.
All of the above sequences evaluate to zero.
Here is my recursion based solution just to build your intuition so that you can approach and improve this solution using dynamic programming on your own (implemented in c++):
// N is the input
// index_count is the index count in the given sequence
// sum is the total sum of a given sequence
int isEvaluteToZero(int N, int index_count, int sum){
// if N==1, then the sequence only contains 1 which is not 0, so return 0
if(N==1){
return 0;
}
// Base case
// if index_count is equal to N and total sum is 0, return 1, else 0
if(index_count==N){
if(sum==0){
return 1;
}
return 0;
}
// recursively call by considering '+' between index_count and index_count+1
// increase index_count by 1
int placeAdd = isEvaluteToZero(N, index_count+1, sum+index_count+1);
// recursively call by considering '-' between index_count and index_count+1
// increase index_count by 1
int placeMinus = isEvaluteToZero(N, index_count+1, sum-index_count-1);
// place '#'
int placePaste;
if(index_count+2<=N){
// paste the previous and the next operands
// For e.g., (8#9) = 8*(10^1)+9 = 89
// (9#10) = 9*(10^2)+10 = 910
// (99#100) = 99*(10^3)+100 = 99100
// (999#1000) = 999*(10^4)+1000 = 9991000
int num1 = index_count+1;
int num2 = index_count+2;
int concat_num = num1*(int)(pow(10, (int)num2/10 + 1) + 0.5)+num2;
placePaste = isEvaluteToZero(N, index_count+2, sum+concat_num) + isEvaluteToZero(N, index_count+2, sum-concat_num);
}else{
// in case index_count+2>N
placePaste = 0;
}
return (placeAdd+placeMinus+placePaste);
}
int main(){
int N, res=1, index_count=1;
cout<<"Enter N:";
cin>>N;
cout<<isEvaluteToZero(N, index_count, res)<<endl;
return 0;
}
output:
N=1 output=0
N=2 output=0
N=3 output=1
N=4 output=1
N=7 output=6
We need to find the maximum element in an array which is also equal to product of two elements in the same array. For example [2,3,6,8] , here 6=2*3 so answer is 6.
My approach was to sort the array and followed by a two pointer method which checked whether the product exist for each element. This is o(nlog(n)) + O(n^2) = O(n^2) approach. Is there a faster way to this ?
There is a slight better solution with O(n * sqrt(n)) if you are allowed to use O(M) memory M = max number in A[i]
Use an array of size M to mark every number while you traverse them from smaller to bigger number.
For each number try all its factors and see if those were already present in the array map.
Here is a pseudo code for that:
#define M 1000000
int array_map[M+2];
int ans = -1;
sort(A,A+n);
for(i=0;i<n;i++) {
for(j=1;j<=sqrt(A[i]);j++) {
int num1 = j;
if(A[i]%num1==0) {
int num2 = A[i]/num1;
if(array_map[num1] && array_map[num2]) {
if(num1==num2) {
if(array_map[num1]>=2) ans = A[i];
} else {
ans = A[i];
}
}
}
}
array_map[A[i]]++;
}
There is an ever better approach if you know how to find all possible factors in log(M) this just becomes O(n*logM). You have to use sieve and backtracking for that
#JerryGoyal 's solution is correct. However, I think it can be optimized even further if instead of using B pointer, we use binary search to find the other factor of product if arr[c] is divisible by arr[a]. Here's the modification for his code:
for(c=n-1;(c>1)&& (max==-1);c--){ // loop through C
for(a=0;(a<c-1)&&(max==-1);a++){ // loop through A
if(arr[c]%arr[a]==0) // If arr[c] is divisible by arr[a]
{
if(binary_search(a+1, c-1, (arr[c]/arr[a]))) //#include<algorithm>
{
max = arr[c]; // if the other factor x of arr[c] is also in the array such that arr[c] = arr[a] * x
break;
}
}
}
}
I would have commented this on his solution, unfortunately I lack the reputation to do so.
Try this.
Written in c++
#include <vector>
#include <algorithm>
using namespace std;
int MaxElement(vector< int > Input)
{
sort(Input.begin(), Input.end());
int LargestElementOfInput = 0;
int i = 0;
while (i < Input.size() - 1)
{
if (LargestElementOfInput == Input[Input.size() - (i + 1)])
{
i++;
continue;
}
else
{
if (Input[i] != 0)
{
LargestElementOfInput = Input[Input.size() - (i + 1)];
int AllowedValue = LargestElementOfInput / Input[i];
int j = 0;
while (j < Input.size())
{
if (Input[j] > AllowedValue)
break;
else if (j == i)
{
j++;
continue;
}
else
{
int Product = Input[i] * Input[j++];
if (Product == LargestElementOfInput)
return Product;
}
}
}
i++;
}
}
return -1;
}
Once you have sorted the array, then you can use it to your advantage as below.
One improvement I can see - since you want to find the max element that meets the criteria,
Start from the right most element of the array. (8)
Divide that with the first element of the array. (8/2 = 4).
Now continue with the double pointer approach, till the element at second pointer is less than the value from the step 2 above or the match is found. (i.e., till second pointer value is < 4 or match is found).
If the match is found, then you got the max element.
Else, continue the loop with next highest element from the array. (6).
Efficient solution:
2 3 8 6
Sort the array
keep 3 pointers C, B and A.
Keeping C at the last and A at 0 index and B at 1st index.
traverse the array using pointers A and B till C and check if A*B=C exists or not.
If it exists then C is your answer.
Else, Move C a position back and traverse again keeping A at 0 and B at 1st index.
Keep repeating this till you get the sum or C reaches at 1st index.
Here's the complete solution:
int arr[] = new int[]{2, 3, 8, 6};
Arrays.sort(arr);
int n=arr.length;
int a,b,c,prod,max=-1;
for(c=n-1;(c>1)&& (max==-1);c--){ // loop through C
for(a=0;(a<c-1)&&(max==-1);a++){ // loop through A
for(b=a+1;b<c;b++){ // loop through B
prod=arr[a]*arr[b];
if(prod==arr[c]){
System.out.println("A: "+arr[a]+" B: "+arr[b]);
max=arr[c];
break;
}
if(prod>arr[c]){ // no need to go further
break;
}
}
}
}
System.out.println(max);
I came up with below solution where i am using one array list, and following one formula:
divisor(a or b) X quotient(b or a) = dividend(c)
Sort the array.
Put array into Collection Col.(ex. which has faster lookup, and maintains insertion order)
Have 2 pointer a,c.
keep c at last, and a at 0.
try to follow (divisor(a or b) X quotient(b or a) = dividend(c)).
Check if a is divisor of c, if yes then check for b in col.(a
If a is divisor and list has b, then c is the answer.
else increase a by 1, follow step 5, 6 till c-1.
if max not found then decrease c index, and follow the steps 4 and 5.
Check this C# solution:
-Loop through each element,
-loop and multiply each element with other elements,
-verify if the product exists in the array and is the max
private static int GetGreatest(int[] input)
{
int max = 0;
int p = 0; //product of pairs
//loop through the input array
for (int i = 0; i < input.Length; i++)
{
for (int j = i + 1; j < input.Length; j++)
{
p = input[i] * input[j];
if (p > max && Array.IndexOf(input, p) != -1)
{
max = p;
}
}
}
return max;
}
Time complexity O(n^2)
Given a value N, if we want to make change for N cents, and we have infinite supply of each of S = { S1, S2, .. , Sm} valued coins, how many ways can we make the change? The order of coins doesnโt matter.There is additional restriction though: you can only give change with exactly K coins.
For example, for N = 4, k = 2 and S = {1,2,3}, there are two solutions: {2,2},{1,3}. So output should be 2.
Solution:
int getways(int coins, int target, int total_coins, int *denomination, int size, int idx)
{
int sum = 0, i;
if (coins > target || total_coins < 0)
return 0;
if (target == coins && total_coins == 0)
return 1;
if (target == coins && total_coins < 0)
return 0;
for (i=idx;i<size;i++) {
sum += getways(coins+denomination[i], target, total_coins-1, denomination, size, i);
}
return sum;
}
int main()
{
int target = 49;
int total_coins = 15;
int denomination[] = {1, 2, 3, 4, 5};
int size = sizeof(denomination)/sizeof(denomination[0]);
printf("%d\n", getways(0, target, total_coins, denomination, size, 0));
}
Above is recursive solution. However i need help with my dynamic programming solution:
Let dp[i][j][k] represent sum up to i with j elements and k coins.
So,
dp[i][j][k] = dp[i][j-1][k] + dp[i-a[j]][j][k-1]
Is my recurrence relation right?
I don't really understand your recurrence relation:
Let dp[i][j][k] represent sum up to i with j elements and k coins.
I think you're on the right track, but I suggest simply dropping the middle dimension [j], and use dp[sum][coinsLeft] as follows:
dp[0][0] = 1 // coins: 0, desired sum: 0 => 1 solution
dp[i][0] = 0 // coins: 0, desired sum: i => 0 solutions
dp[sum][coinsLeft] = dp[sum - S1][coinsLeft-1]
+ dp[sum - S2][coinsLeft-1]
+ ...
+ dp[sum - SM][coinsLeft-1]
The answer is then to be found at dp[N][K] (= number of ways to add K coins to get N cents)
Here's some sample code (I advice you to not look until you've tried to solve it yourself. It's a good exercise):
public static int combinations(int numCoinsToUse, int targetSum, int[] denom) {
// dp[numCoins][sum] == ways to get sum using numCoins
int[][] dp = new int[numCoinsToUse+1][targetSum];
// Any sum (except 0) is impossible with 0 coins
for (int sum = 0; sum < targetSum; sum++) {
dp[0][sum] = sum == 0 ? 1 : 0;
}
// Gradually increase number of coins
for (int c = 1; c <= numCoinsToUse; c++)
for (int sum = 0; sum < targetSum; sum++)
for (int d : denom)
if (sum >= d)
dp[c][sum] += dp[c-1][sum - d];
return dp[numCoinsToUse][targetSum-1];
}
Using your example input:
combinations(2, 4, new int[] {1, 2, 3} ) // gives 2
I was going through suffix array and its use to compute longest common prefix of two suffixes.
The source says:
"The lcp between two suffixes is the minimum of the lcp's of all pairs of adjacent suffixes between them on the array"
i.e. lcp(x,y)=min{ lcp(x,x+1),lcp(x+1,x+2),.....,lcp(y-1,y) }
where x and y are two index of the string from where the two suffix of the string starts.
I am not convinced with the statement as in example of string "abca".
lcp(1,4)=1 (considering 1 based indexing)
but if I apply the above equation then
lcp(1,4)=min{lcp(1,2),lcp(2,3),lcp(3,4)}
and I think lcp(1,2)=0.
so the answer must be 0 according to the equation.
Am i getting it wrong somewhere?
I think the index referred by the source is not the index of the string itself, but index of the sorted suffixes.
a
abca
bca
ca
Hence
lcp(1,2) = lcp(a, abca) = 1
lcp(1,4) = min(lcp(1,2), lcp(2,3), lcp(3,4)) = 0
You can't find LCP of any two suffixes by simply calculating the minimum of the lcp's of all pairs of adjacent suffixes between them on the array.
We can calculate the LCPs of any suffixes (i,j)
with the Help of Following :
LCP(suffix i,suffix j)=LCP[RMQ(i + 1; j)]
Also Note (i<j) as LCP (suff i,suff j) may not necessarly equal LCP (Suff j,suff i).
RMQ is Range Minimum Query .
Page 3 of this paper.
Details:
Step 1:
First Calculate LCP of Adjacents /consecutive Suffix Pairs .
n= Length of string.
suffixArray[] is Suffix array.
void calculateadjacentsuffixes(int n)
{
for (int i=0; i<n; ++i) Rank[suffixArray[i]] = i;
Height[0] = 0;
for (int i=0, h=0; i<n; ++i)
{
if (Rank[i] > 0)
{
int j = suffixArray[Rank[i]-1];
while (i + h < n && j + h < n && str[i+h] == str[j+h])
{
h++;
}
Height[Rank[i]] = h;
if (h > 0) h--;
}
}
}
Note: Height[i]=LCPs of (Suffix i-1 ,suffix i) ie. Height array contains LCP of adjacent suffix.
Step 2:
Calculate LCP of Any two suffixes i,j using RMQ concept.
RMQ pre-compute function:
void preprocesses(int N)
{
int i, j;
//initialize M for the intervals with length 1
for (i = 0; i < N; i++)
M[i][0] = i;
//compute values from smaller to bigger intervals
for (j = 1; 1 << j <= N; j++)
{
for (i = 0; i + (1 << j) - 1 < N; i++)
{
if (Height[M[i][j - 1]] < Height[M[i + (1 << (j - 1))][j - 1]])
{
M[i][j] = M[i][j - 1];
}
else
{
M[i][j] = M[i + (1 << (j - 1))][j - 1];
}
}
}
}
Step 3: Calculate LCP between any two Suffixes i,j
int LCP(int i,int j)
{
/*Make sure we send i<j always */
/* By doing this ,it resolve following
suppose ,we send LCP(5,4) then it converts it to LCP(4,5)
*/
if(i>j)
swap(i,j);
/*conformation over*/
if(i==j)
{
return (Length_of_str-suffixArray[i]);
}
else
{
return Height[RMQ(i+1,j)];
//LCP(suffix i,suffix j)=LCPadj[RMQ(i + 1; j)]
//LCPadj=LCP of adjacent suffix =Height.
}
}
Where RMQ function is:
int RMQ(int i,int j)
{
int k=log((double)(j-i+1))/log((double)2);
int vv= j-(1<<k)+1 ;
if(Height[M[i][k]]<=Height[ M[vv][ k] ])
return M[i][k];
else
return M[ vv ][ k];
}
Refer Topcoder tutorials for RMQ.
You can check the complete implementation in C++ at my blog.
I can't figure out how to generate all compositions (http://en.wikipedia.org/wiki/Composition_%28number_theory%29) of an integer N into K parts, but only doing it one at a time. That is, I need a function that given the previous composition generated, returns the next one in the sequence. The reason is that memory is limited for my application. This would be much easier if I could use Python and its generator functionality, but I'm stuck with C++.
This is similar to Next Composition of n into k parts - does anyone have a working algorithm?
Any assistance would be greatly appreciated.
Preliminary remarks
First start from the observation that [1,1,...,1,n-k+1] is the first composition (in lexicographic order) of n over k parts, and [n-k+1,1,1,...,1] is the last one.
Now consider an exemple: the composition [2,4,3,1,1], here n = 11 and k=5. Which is the next one in lexicographic order? Obviously the rightmost part to be incremented is 4, because [3,1,1] is the last composition of 5 over 3 parts.
4 is at the left of 3, the rightmost part different from 1.
So turn 4 into 5, and replace [3,1,1] by [1,1,2], the first composition of the remainder (3+1+1)-1 , giving [2,5,1,1,2]
Generation program (in C)
The following C program shows how to compute such compositions on demand in lexicographic order
#include <stdio.h>
#include <stdbool.h>
bool get_first_composition(int n, int k, int composition[k])
{
if (n < k) {
return false;
}
for (int i = 0; i < k - 1; i++) {
composition[i] = 1;
}
composition[k - 1] = n - k + 1;
return true;
}
bool get_next_composition(int n, int k, int composition[k])
{
if (composition[0] == n - k + 1) {
return false;
}
// there'a an i with composition[i] > 1, and it is not 0.
// find the last one
int last = k - 1;
while (composition[last] == 1) {
last--;
}
// turn a b ... y z 1 1 ... 1
// ^ last
// into a b ... (y+1) 1 1 1 ... (z-1)
// be careful, there may be no 1's at the end
int z = composition[last];
composition[last - 1] += 1;
composition[last] = 1;
composition[k - 1] = z - 1;
return true;
}
void display_composition(int k, int composition[k])
{
char *separator = "[";
for (int i = 0; i < k; i++) {
printf("%s%d", separator, composition[i]);
separator = ",";
}
printf("]\n");
}
void display_all_compositions(int n, int k)
{
int composition[k]; // VLA. Please don't use silly values for k
for (bool exists = get_first_composition(n, k, composition);
exists;
exists = get_next_composition(n, k, composition)) {
display_composition(k, composition);
}
}
int main()
{
display_all_compositions(5, 3);
}
Results
[1,1,3]
[1,2,2]
[1,3,1]
[2,1,2]
[2,2,1]
[3,1,1]
Weak compositions
A similar algorithm works for weak compositions (where 0 is allowed).
bool get_first_weak_composition(int n, int k, int composition[k])
{
if (n < k) {
return false;
}
for (int i = 0; i < k - 1; i++) {
composition[i] = 0;
}
composition[k - 1] = n;
return true;
}
bool get_next_weak_composition(int n, int k, int composition[k])
{
if (composition[0] == n) {
return false;
}
// there'a an i with composition[i] > 0, and it is not 0.
// find the last one
int last = k - 1;
while (composition[last] == 0) {
last--;
}
// turn a b ... y z 0 0 ... 0
// ^ last
// into a b ... (y+1) 0 0 0 ... (z-1)
// be careful, there may be no 0's at the end
int z = composition[last];
composition[last - 1] += 1;
composition[last] = 0;
composition[k - 1] = z - 1;
return true;
}
Results for n=5 k=3
[0,0,5]
[0,1,4]
[0,2,3]
[0,3,2]
[0,4,1]
[0,5,0]
[1,0,4]
[1,1,3]
[1,2,2]
[1,3,1]
[1,4,0]
[2,0,3]
[2,1,2]
[2,2,1]
[2,3,0]
[3,0,2]
[3,1,1]
[3,2,0]
[4,0,1]
[4,1,0]
[5,0,0]
Similar algorithms can be written for compositions of n into k parts greater than some fixed value.
You could try something like this:
start with the array [1,1,...,1,N-k+1] of (K-1) ones and 1 entry with the remainder. The next composition can be created by incrementing the (K-1)th element and decreasing the last element. Do this trick as long as the last element is bigger than the second to last.
When the last element becomes smaller, increment the (K-2)th element, set the (K-1)th element to the same value and set the last element to the remainder again. Repeat the process and apply the same principle for the other elements when necessary.
You end up with a constantly sorted array that avoids duplicate compositions