In section2.2,a problem called"subset sum"require you to calculate in how many ways can a integer set from 1 to n be partitioned into two sets whose sums are identical.
I know the recurrence is:
f[i][j] : numbers of ways that sum up to j with 1...i
f[i][j]=f[i-1][j]+f[i-1][j-i]
if the initial condition is:
f[1][1]=1;//others are all zero,main loop start from 2
OR:
f[0][0]=1;//others are all zero,main loop start from 1
the answers are all f[n][n*(n+1)/4].Does this means the initial condition doesn't affect the answer?
but if I use a one dimension array,say f[N]:
let f[0]=1,loop from 1(so f[0] is f[0][0] in fact),the answer is f[n]/2
or f[1]=1,loop from 2(f[1] is f[1][1]),the answer is f[n]
I am so confused...
I don't know if you are still stuck on this problem, but here's a solution for anyone else who stumbles onto this problem.
Let ways[i] be the number of ways you can get a sum of i using a subset of the numbers 1...N.
Then it becomes a variant of the 0-1 knapsack algorithm:
base case: ways[0] = 1
for (int i = 1; i <= N; i++) {
for (int j = sum - i; j >= 0; --j) { //sum is n*(n+1)/2
ways[j + i] += ways[j];
}
}
Your answer is located at ways[sum/2]/2.
Related
I'm facing difficulty in understanding O(sum) complexity solution of coin changing problem.
The problem statement is:
You are given a set of coins A. In how many ways can you make sum B assuming you have infinite amount of each coin in the set.
NOTE:
Coins in set A will be unique. Expected space complexity of this problem is O(B).
The solution is:
int count( int S[], int m, int n )
{
int table[n+1];
memset(table, 0, sizeof(table));
table[0] = 1;
for(int i=0; i<m; i++)
for(int j=S[i]; j<=n; j++)
table[j] += table[j-S[i]];
return table[n];
}
can someone explain me this code.?
First, let's identify the parameters and variables used in the function:
Parameters:
S contain the denomination of all m coins. i.e. Each element contain the value of each coin.
m represents the number of coin denominations. Essentially, it's the length of array S.
n represents the sum B to be achieved.
Variables:
table: Element i in array table contains the number of ways sum i can be achieved with the given coins. table[0] = 1 because there is a single way to achieve a sum of 0 (not using any coin).
i loops through each coin.
Logic:
The number of ways to achieve a sum j = sum of the following:
number of ways to achieve a sum of j - S[0]
number of ways to achieve a sum of j - S[1]
...
number of ways to achieve a sum of j - S[m-1] (S[m-1] is the value of the mth coin)
I did not completely decipher nor validate the rest of the code, but I hope this is a step in the right direction.
Added comments to code:
#include <stdio.h>
#include <string.h>
int count( int S[], int m, int n )
{
int table[n+1];
memset(table, 0, sizeof(table));
table[0] = 1;
for(int i=0; i<m; i++) // Loop through all of the coins
for(int j=S[i]; j<=n; j++) // Achieve sum j between the value of S[i] and n.
table[j] += table[j-S[i]]; // Add to the number of ways to achieve sum j the number of ways to achieve sum j - S[i]
return table[n];
}
int main() {
int S[] = {1, 2};
int m = 2;
int n = 3;
int c = count(S, m, n);
printf("%d\n", c);
}
Notes:
The code avoids repeats: 3 = 1+1+1, 1+2 (2 ways instead of 3 if 2+1 was considered.
No dependence on the order of the coins in term of value.
I was asked this question in an interview:
Given a string (1<=|s|<=10^5), check if it is possible to partition it into three palindromes. If there are multiple answers possible, output the one where the cuts are made the earliest. If no answer is possible, print "Impossible".
**Input:**
radarnoonlevel
aabab
abcdefg
**Output:**
radar noon level
a a bab (Notice how a, aba, b is also an answer, but we will output the one with the earliest cuts)
Impossible
I was able to give a brute force solution, running two loops and checking palindrome property for every 3 substrings ( 0-i, i-j, j-end). This was obviously not optimal, but I have not been able to find a better solution since then.
I need a way of checking that if I know the palindrome property of a string, then how removing a character from the start or adding one at the end can give me the property of the new string without having to do the check for the whole string again. I am thinking of using three maps where each character key is mapped to number of occurences but that too doesn't lead me down anything.
Still O(n^2) solution, but you can store the result of palindrome substrings in a table and use that to get to the answer.
vector<string> threePalindromicSubstrings(string word) {
int n = word.size();
vector<vector<bool>> dp (n,vector<bool>(n,false));
for(int i = 0 ; i < n ; ++i)
dp[i][i] = 1;
for(int l = 2 ; l <= n ; ++l){
for(int i = 0 ; i < n - l +1 ; ++i){
int j = i + l - 1;
if(l == 2)
dp[i][j] = (word[i] == word[j]);
else
dp[i][j] = (word[i] == word[j]) && (dp[i+1][j-1]);
}
}
vector<string> ans;
for(int i = 0 ; i < n - 2 ; ++i){
if(dp[0][i]) {
for(int j = i+1 ; j < n - 1 ; ++j){
if(dp[i+1][j] && dp[j+1][n-1]){
ans.push_back(word.substr(0,i + 1));
ans.push_back(word.substr(i+1,j-i));
ans.push_back(word.substr(j+1,n-j));
return ans;
}
}
}
}
if(ans.empty())
ans.push_back("Impossible");
return ans;
}
I am given a limit, and I have to return the smallest value for n to make it true: 1+2+3+4+...+n >= limit. I feel like there's one thing missing, but I can't tell.
public int whenToReachLimit(int limit) {
int sum = 0;
for (int i = 1; sum < limit; i++) {
sum = sum + i;
}
return sum;
}
The output would be:
1 : 1
4 : 3
10 : 4
You get avoid the loop to compute the sum of the n first integers, using:
Thus the inequality becomes:
Notice that the left-hand side is positive (if n is negative, the sum is empty) and strictly increasing. Notice also that you are looking for the first integer satisfying the inequality. The idea here is first to replace the inequality by an equality which will allow us to solve the equation for n. In a second step, the possibly non-integer solution will be rounder to the closest integer.
Solving this equation for n should give you two solutions. The negative one can be discarded (remember n is positive). That is:
Finally, let's round this solution to the closest integer that will also satisfy the inequality:
NB: it can be overkilled for small inputs
I'm not sure if I know exactly what you want to do. But I would recommend to make a "practice run".
If Limit = 0 the function returns 0
If Limit = 1 the function returns 1
If Limit = 2 the function return 3
If Limit = 3 the function return 3
If Limit = 4 the function return 6
If Limit = 5 the function return 6
Now you decide by your own if the functions does what you're expecting.
I've found the answer. Turns out it doesn't work with a for loop which I find odd. But this is the answer to my own question.
public int whenToReachLimit(int limit) {
int n = 0;
int sum = 0;
while (sum < limit) {
sum += n;
n++;
}
return n-1;
}
You don't want to return sum, you want to return n (smallest possible value satisfying the given requirement).
return i-1 instead of sum.
Given a binary string s, we need to find the number of its substrings, containing exactly k characters that are '1'.
For example: s = "1010" and k = 1, answer = 6.
Now, I solved it using binary search technique over the cumulative sum array.
I also used another approach to solve it. The approach is as follows:
For each position i, find the total substrings that end at i containing
exactly k characters that are '1'.
To find the total substrings that end at i containing exactly k characters that are 1, it can be represented as the set of indices j such that substring j to i contains exactly k '1's. The answer would be the size of the set. Now, to find all such j for the given position i, we can rephrase the problem as finding all j such that
number of ones from [1] to [j - 1] = the total number of ones from 1 to i - [the total number of ones from j to i = k].
i.e. number of ones from [1] to [j - 1] = C[i] - k
which is equal to
C[j - 1] = C[i] - k,
where C is the cumulative sum array, where
C[i] = sum of characters of string from 1 to i.
Now, the problem is easy because, we can find all the possible values of j's using the equation by counting all the prefixes that sum to C[i] - k.
But I found this solution,
int main() {
cin >> k >> S;
C[0] = 1;
for (int i = 0; S[i]; ++i) {
s += S[i] == '1';
++C[s];
}
for (int i = k; i <= s; ++i) {
if (k == 0) {
a += (C[i] - 1) * C[i] / 2;
} else {
a += C[i] * C[i - k];
}
}
cout << a << endl;
return 0;
}
In the code, S is the given string and K as described above, C is the cumulative sum array and a is the answer.
What is the code exactly doing by using multiplication, I don't know.
Could anybody explain the algorithm?
If you see the way C[i] is calculated, C[i] represents the number of characters between ith 1 and i+1st 1.
If you take an example S = 1001000
C[0] = 1
C[1] = 3 // length of 100
C[2] = 4 // length of 1000
So coming to your doubt, Why multiplication
Say your K=1, then you want to find out the substring which have only one 1, now you know that after first 1 there are two zeros since C[1] = 3. So number of of substrings will be 3, because you have to include this 1.
{1,10,100}
But when you come to the second part: C[2] =4
now if you see 1000 and you know that you can make 4 substrings (which is equal to C[2])
{1,10,100,1000}
and also you should notice that there are C[1]-1 zeroes before this 1.
So by including those zeroes you can make more substring, in this case by including 0 once
0{1,10,100,1000}
=> {01,010,0100,01000}
and 00 once
00{1,10,100,1000}
=> {001,0010,00100,001000}
so essentially you are making C[i] substrings starting with 1 and you can append i number of zeroes before this one and make another C[i] * C[i-k]-1 substrings. i varies from 1 to C[i-k]-1 (-1 because we want to leave that last one).
((C[i-k]-1)* C[i]) +C[i]
=> C[i-k]*C[i]
Given two bit strings, x and y, with x longer than y, I'd like to compute a kind of asymmetric variant of the Levensthein distance between them. Starting with x, I'd like to know the minimum number of deletions and substitutions it takes to turn x into y.
Can I just use the usual Levensthein distance for this, or do I need I need to modify the algorithm somehow? In other words, with the usual set of edits of deletion, substitution, and addition, is it ever beneficial to delete more than the difference in lengths between the two strings and then add some bits back? I suspect the answer is no, but I'm not sure. If I'm wrong, and I do need to modify the definition of Levenshtein distance to disallow deletions, how do I do so?
Finally, I would expect intuitively that I'd get the same distance if I started with y (the shorter string) and only allowed additions and substitutions. Is this right? I've got a sense for what these answers are, I just can't prove them.
If i understand you correctly, I think the answer is yes, the Levenshtein edit distance could be different than an algorithm that only allows deletions and substitutions to the larger string. Because of this, you would need to modify, or create a different algorithm to get your limited version.
Consider the two strings "ABCD" and "ACDEF". The Levenshtein distance is 3 (ABCD->ACD->ACDE->ACDEF). If we start with the longer string, and limit ourselves to deletions and substitutions we must use 4 edits (1 deletion and 3 substitutions. The reason is that strings where deletions are applied to the smaller string to efficiently get to the larger string can't be achieved when starting with the longer string, because it does not have the complimentary insertion operation (since you're disallowing that).
Your last paragraph is true. If the path from shorter to longer uses only insertions and substitutions, then any allowed path can simply be reversed from the longer to the shorter. Substitutions are the same regardless of direction, but the inserts when going from small to large become deletions when reversed.
I haven't tested this thoroughly, but this modification shows the direction I would take, and appears to work with the values I've tested with it. It's written in c#, and follows the psuedo code in the wikipedia entry for Levenshtein distance. There are obvious optimizations that can be made, but I refrained from doing that so it was more obvious what changes I've made from the standard algorithm. An important observation is that (using your constraints) if the strings are the same length, then substitution is the only operation allowed.
static int LevenshteinDistance(string s, string t) {
int i, j;
int m = s.Length;
int n = t.Length;
// for all i and j, d[i,j] will hold the Levenshtein distance between
// the first i characters of s and the first j characters of t;
// note that d has (m+1)*(n+1) values
var d = new int[m + 1, n + 1];
// set each element to zero
// c# creates array already initialized to zero
// source prefixes can be transformed into empty string by
// dropping all characters
for (i = 0; i <= m; i++) d[i, 0] = i;
// target prefixes can be reached from empty source prefix
// by inserting every character
for (j = 0; j <= n; j++) d[0, j] = j;
for (j = 1; j <= n; j++) {
for (i = 1; i <= m; i++) {
if (s[i - 1] == t[j - 1])
d[i, j] = d[i - 1, j - 1]; // no operation required
else {
int del = d[i - 1, j] + 1; // a deletion
int ins = d[i, j - 1] + 1; // an insertion
int sub = d[i - 1, j - 1] + 1; // a substitution
// the next two lines are the modification I've made
//int insDel = (i < j) ? ins : del;
//d[i, j] = (i == j) ? sub : Math.Min(insDel, sub);
// the following 8 lines are a clearer version of the above 2 lines
if (i == j) {
d[i, j] = sub;
} else {
int insDel;
if (i < j) insDel = ins; else insDel = del;
// assign the smaller of insDel or sub
d[i, j] = Math.Min(insDel, sub);
}
}
}
}
return d[m, n];
}