We all heard of bentley's beautiful proramming pearls problem
which solves maximum subsequence sum:
maxsofar = 0;
maxcur = 0;
for (i = 0; i < n; i++) {
maxcur = max(A[i] + maxcur, 0);
maxsofar = max(maxsofar, maxcur);
}
What if we add an additional condition maximum subsequence that is lesser M?
This should do this. Am I wright?
int maxsofar = 0;
for (int i = 0; i < n - 1; i++) {
int maxcur = 0;
for (int j = i; j < n; j++) {
maxcur = max(A[j] + maxcur, 0);
maxsofar = maxcur < M ? max(maxsofar, maxcur) : maxsofar;
}
}
Unfortunately this is O(n^2). You may speed it up a little bit by breaking the inner loop when maxcur >=M, but still n^2 remains.
This can be solved using dynamic programming albeit only in pseudo-polynomial time.
Define
m(i,s) := maximum sum less than s obtainable using only the first i elements
Then you can calculate max(n,M) using the following recurrence relation
m(i,s) = max(m(i-1,s), m(i-1,s-A[i]]+A[i]))
This solution is similar to the solution to the knapsack problem.
If all A[i] > 0, you can do this in O(n lg n): precompute partial sums S[i], then binary search S for S[i] + M. For instance:
def binary_search(L, x):
def _binary_search(lo, hi):
if lo >= hi: return lo
mid = lo + (hi-lo)/2
if x < L[mid]:
return _binary_search(lo, mid)
return _binary_search(mid+1, hi)
return _binary_search(0, len(L))
A = [1, 2, 3, 2, 1]
M = 4
S = [A[0]]
for a in A[1:]:
S.append(S[-1] + a)
maxsum = 0
for i, s in enumerate(S):
j = binary_search(S, s + M)
if j == len(S):
break
sum = S[j-1] - S[i]
maxsum = max(sum, maxsum)
print maxsum
EDIT: as atuls correctly points out, the binary search is overkill; since S is increasing, we can just keep track of j each iteration and advance from there.
Solveable in O(n log(n)). Using a binary search tree (balanced) to search for smallest value larger than sum-M, and then update min, and insert sum, by going from left to right. Where sum is the partial sum so far.
best = -infinity;
sum = 0;
tree.insert(0);
for(i = 0; i < n; i++) {
sum = sum + A[i];
int diff = sum - tree.find_smallest_value_larger_than(sum - M);
if (diff > best) {
best = diff;
}
tree.insert(sum);
}
print best
Related
It is the well-konw Twelvefold way:
https://en.wikipedia.org/wiki/Twelvefold_way
Where we want to find the number of solutions for following equation:
X1 + X2 + ... + XK = target
from the given array:
vector<int> vec(N);
We can assume vec[i] > 0. There are 3 cases, for example
vec = {1,2,3}, target = 5, K = 3.
Xi can be duplicate and solution can be duplicate.
6 solutions are {1,2,2}, {2,1,2}, {2,2,1}, {1,1,3}, {1,3,1}, {3,1,1}
Xi can be duplicate and solution cannot be duplicate.
2 solutions are {1,2,2}, {1,1,3}
Xi cannot be duplicate and solution cannot be duplicate.
0 solution.
The ides must be using dynamic programming:
dp[i][k], the number of solution of target = i, K = k.
And the iteration relation is :
if(i > num[n-1]) dp[i][k] += dp[i-num[n-1]][k-1];
For three cases, they depend on the runing order of i,n,k. I know the result when there is no restriction of K (sum of any number of variables):
case 1:
int KSum(vector<int>& vec, int target) {
vector<int> dp(target + 1);
dp[0] = 1;
for (int i = 1; i <= target; ++i)
for (int n = 0; n < vec.size(); n++)
if (i >= vec[n]) dp[i] += dp[i - vec[n]];
return dp.back();
}
case 2:
for (int n = 0; n < vec.size(); n++)
for (int i = 1; i <= target; ++i)
case 3:
for (int n = 0; n < vec.size(); n++)
for (int i = target; i >= 1; --i)
When there is additional variable k, do we just simply add the for loop
for(int k = 1; k <= K; k++)
at the outermost layer?
EDIT:
I tried case 1，just add for loop of K most inside:
int KSum(vector<int> vec, int target, int K) {
vector<vector<int>> dp(K+1,vector<int>(target + 1,0));
dp[0][0] = 1;
for (int n = 0; n < vec.size(); n++)
for (int i = 1; i <= target; ++i)
for (int k = 1; k <= K; k++)
{
if (i >= vec[n]) dp[k][i] += dp[k - 1][i - vec[n]];
}
return dp[K][target];
}
Is it true for case 2 and case 3？
In your solution without variable K dp[i] represents how many solutions are there to achieve sum i.
Including the variable K means that we added another dimension to our subproblem. This dimension doesn't necessarily have to be on a specific axis. Your dp array could look like dp[i][k] or dp[k][i].
dp[i][k] means how many solutions to accumulate sum i using k numbers (duplicate or unique)
dp[k][i] means using k numbers how many solutions to accumulate sum i
Both are the same things. Meaning that you can add the loop outside or inside.
Let L be a list of positive integers.
We are allowed to merge two elements of L if they have adjacent indices.
The cost of this operation is the sum of both elements.
For example: [1,2,3,4] -> [3,3,4] with a cost of 3.
We are looking for the minimum cost to merge L into one integer.
Is there a fast way of doing this? I came up with this naive recursive approach but that should
be O(n!).
I have noticed that it benefits a lot from memoization so I think there must be a way to avoid trying all possible permutations which will always result in O(n!).
def solveR(l):
if len(l) <= 2:
return sum(l)
else:
return sum(l) + min(solveR(l[1:]), solveR(l[:-1]),
solveR(l[len(l) // 2:]) + solveR(l[:len(l) // 2]))
This is much like this LeetCode problem, but with K = 2. The comments suggest that the time complexity is O(n^3). Here is some C++ code that implements the algorithm:
class Solution {
public:
int mergeStones(vector<int>& stones, int K) {
K = 2;
int N = stones.size();
if((N-1)%(K-1) > 0) return -1;
int sum[N+1] = {0};
for(int i = 1; i <= N; i++)
sum[i] = sum[i-1] + stones[i-1];
vector<vector<int>> dp(N, vector<int>(N,0));
for(int L=K; L<= N; L++)
for(int i=0, j=i+L-1; j<N; i++,j++) {
dp[i][j] = INT_MAX;
for (int k = i; k < j; k += (K-1))
dp[i][j] = min(dp[i][j], dp[i][k] + dp[k+1][j]);
if ((L-1)%(K-1) == 0)
dp[i][j] += (sum[j+1] - sum[i]); // add sum in [i,j]
}
return dp[0][N-1];
}
};
I was wondering how could I get the longest positive-sum subsequence in a sequence:
For example I have -6 3 -4 4 -5, so the longest positive subsequence is 3 -4 4. In fact the sum is positive (3), and we couldn't add -6 neither -5 or it would have become negative.
It could be easily solvable in O(N^2), I think could exist something much more faster, like in O(NlogN)
Do you have any idea?
EDIT: the order must be preserved, and you can skip any number from the substring
EDIT2: I'm sorry if I caused confusion using the term "sebsequence", as #beaker pointed out I meant substring
O(n) space and time solution, will start with the code (sorry, Java ;-) and try to explain it later:
public static int[] longestSubarray(int[] inp) {
// array containing prefix sums up to a certain index i
int[] p = new int[inp.length];
p[0] = inp[0];
for (int i = 1; i < inp.length; i++) {
p[i] = p[i - 1] + inp[i];
}
// array Q from the description below
int[] q = new int[inp.length];
q[inp.length - 1] = p[inp.length - 1];
for (int i = inp.length - 2; i >= 0; i--) {
q[i] = Math.max(q[i + 1], p[i]);
}
int a = 0;
int b = 0;
int maxLen = 0;
int curr;
int[] res = new int[] {-1,-1};
while (b < inp.length) {
curr = a > 0 ? q[b] - p[a-1] : q[b];
if (curr >= 0) {
if(b-a > maxLen) {
maxLen = b-a;
res = new int[] {a,b};
}
b++;
} else {
a++;
}
}
return res;
}
we are operating on input array A of size n
Let's define array P as the array containing the prefix sum until index i so P[i] = sum(0,i) where `i = 0,1,...,n-1'
let's notice that if u < v and P[u] <= P[v] then u will never be our ending point
because of the above we can define an array Q which has Q[n-1] = P[n-1] and Q[i] = max(P[i], Q[i+1])
now let's consider M_{a,b} which shows us the maximum sum subarray starting at a and ending at b or beyond. We know that M_{0,b} = Q[b] and that M_{a,b} = Q[b] - P[a-1]
with the above information we can now initialise our a, b = 0 and start moving them. If the current value of M is bigger or equal to 0 then we know we will find (or already found) a subarray with sum >= 0, we then just need to compare b-a with the previously found length. Otherwise there's no subarray that starts at a and adheres to our constraints so we need to increment a.
Let's make a naive implementation and then improve it.
We move from the left to the right calculating partial sums and for each position we find the most-left partial sum such as the current partial sum is greater than that.
input a
int partialSums[len(a)]
for i in range(len(a)):
partialSums[i] = (i == 0 ? 0 : partialSums[i - 1]) + a[i]
if partialSums[i] > 0:
answer = max(answer, i + 1)
else:
for j in range(i):
if partialSums[i] - partialSums[j] > 0:
answer = max(answer, i - j)
break
This is O(n2). Now the part of finding the left-most "good" sum could be actually maintained via BST, where each node would be represented as a pair (partial sum, index) with a comparison by partial sum. Also each node should support a special field min that would be the minimum of indices in this subtree.
Now instead of the straightforward search of an appropriate partial sum we could descend the BST using the current partial sum as a key following the next three rules (assuming C is the current node, L and R are the roots of the left and the right subtrees respectively):
Maintain the current minimal index of "good" partial sums found in curMin, initially +∞.
If C.partial_sum is "good" then update curMin with C.index.
If we go to R then update curMin with L.min.
And then update the answer with i - curMin, also add the current partial sum to the BST.
That would give us O(n * log n).
We can easily have a O(n log n) solution for longest subsequence.
First, sort the array, remember their indexes.
Pick all the largest numbers, stop when their sum are negative, and you have your answer.
Recover their original order.
Pseudo code
sort(data);
int length = 0;
long sum = 0;
boolean[] result = new boolean[n];
for(int i = n ; i >= 1; i--){
if(sum + data[i] <= 0)
break;
sum += data[i];
result[data[i].index] = true;
length++;
}
for(int i = 1; i <= n; i++)
if(result[i])
print i;
So, rather than waiting, I will propose a O(n log n) solution for longest positive substring.
First, we create an array prefix which is the prefix sum of the array.
Second, we using binary search to look for the longest length that has positive sum
Pseudocode
int[]prefix = new int[n];
for(int i = 1; i <= n; i++)
prefix[i] = data[i];
if(i - 1 >= 1)
prefix[i] += prefix[i - 1];
int min = 0;
int max = n;
int result = 0;
while(min <= max){
int mid = (min + max)/2;
boolean ok = false;
for(int i = 1; i <= n; i++){
if(i > mid && pre[i] - pre[i - mid] > 0){//How we can find sum of segment with mid length, and end at index i
ok = true;
break;
}
}
if(ok){
result = max(result, mid)
min = mid + 1;
}else{
max = mid - 1;
}
}
Ok, so the above algorithm is wrong, as pointed out by piotrekg2 what we need to do is
create an array prefix which is the prefix sum of the array.
Sort the prefix array, and we need to remember the index of the prefix array.
Iterate through the prefix array, storing the minimum index we meet so far, the maximum different between the index is the answer.
Note: when we comparing value in prefix, if two indexes have equivalent values, so which has smaller index will be considered larger, this will avoid the case when the sum is 0.
Pseudo code:
class Node{
int val, index;
}
Node[]prefix = new Node[n];
for(int i = 1; i <= n; i++)
prefix[i] = new Node(data[i],i);
if(i - 1 >= 1)
prefix[i].val += prefix[i - 1].val;
sort(prefix);
int min = prefix[1].index;
int result = 0;
for(int i = 2; i <= n; i ++)
if(prefix[i].index > min)
result = max(prefix[i].index - min + 1, result)
min = min(min, prefix[i].index);
There's an array A containing (positive and negative) integers. Find a (contiguous) subarray whose elements' absolute sum is minimal, e.g.:
A = [2, -4, 6, -3, 9]
|(−4) + 6 + (−3)| = 1 <- minimal absolute sum
I've started by implementing a brute-force algorithm which was O(N^2) or O(N^3), though it produced correct results. But the task specifies:
complexity:
- expected worst-case time complexity is O(N*log(N))
- expected worst-case space complexity is O(N)
After some searching I thought that maybe Kadane's algorithm can be modified to fit this problem but I failed to do it.
My question is - is Kadane's algorithm the right way to go? If not, could you point me in the right direction (or name an algorithm that could help me here)? I don't want a ready-made code, I just need help in finding the right algorithm.
If you compute the partial sums
such as
2, 2 +(-4), 2 + (-4) + 6, 2 + (-4) + 6 + (-3)...
Then the sum of any contiguous subarray is the difference of two of the partial sums. So to find the contiguous subarray whose absolute value is minimal, I suggest that you sort the partial sums and then find the two values which are closest together, and use the positions of these two partial sums in the original sequence to find the start and end of the sub-array with smallest absolute value.
The expensive bit here is the sort, so I think this runs in time O(n * log(n)).
This is C++ implementation of Saksow's algorithm.
int solution(vector<int> &A) {
vector<int> P;
int min = 20000 ;
int dif = 0 ;
P.resize(A.size()+1);
P[0] = 0;
for(int i = 1 ; i < P.size(); i ++)
{
P[i] = P[i-1]+A[i-1];
}
sort(P.begin(),P.end());
for(int i = 1 ; i < P.size(); i++)
{
dif = P[i]-P[i-1];
if(dif<min)
{
min = dif;
}
}
return min;
}
I was doing this test on Codility and I found mcdowella answer quite helpful, but not enough I have to say: so here is a 2015 answer guys!
We need to build the prefix sums of array A (called P here) like: P[0] = 0, P[1] = P[0] + A[0], P[2] = P[1] + A[1], ..., P[N] = P[N-1] + A[N-1]
The "min abs sum" of A will be the minimum absolute difference between 2 elements in P. So we just have to .sort() P and loop through it taking every time 2 successive elements. This way we have O(N + Nlog(N) + N) which equals to O(Nlog(N)).
That's it!
The answer is yes, Kadane's algorithm is definitely the way to go for solving your problem.
http://en.wikipedia.org/wiki/Maximum_subarray_problem
Source - I've closely worked with a PhD student who's entire PhD thesis was devoted to the maximum subarray problem.
def min_abs_subarray(a):
s = [a[0]]
for e in a[1:]:
s.append(s[-1] + e)
s = sorted(s)
min = abs(s[0])
t = s[0]
for x in s[1:]:
cur = abs(x)
min = cur if cur < min else min
cur = abs(t-x)
min = cur if cur < min else min
t = x
return min
You can run Kadane's algorithmtwice(or do it in one go) to find minimum and maximum sum where finding minimum works in same way as maximum with reversed signs and then calculate new maximum by comparing their absolute value.
Source-Someone's(dont remember who) comment in this site.
Here is an Iterative solution in python. It's 100% correct.
def solution(A):
memo = []
if not len(A):
return 0
for ind, val in enumerate(A):
if ind == 0:
memo.append([val, -1*val])
else:
newElem = []
for i in memo[ind - 1]:
newElem.append(i+val)
newElem.append(i-val)
memo.append(newElem)
return min(abs(n) for n in memo.pop())
Short Sweet and work like a charm. JavaScript / NodeJs solution
function solution(A, i=0, sum =0 ) {
//Edge case if Array is empty
if(A.length == 0) return 0;
// Base case. For last Array element , add and substart from sum
// and find min of their absolute value
if(A.length -1 === i){
return Math.min( Math.abs(sum + A[i]), Math.abs(sum - A[i])) ;
}
// Absolute value by adding the elem with the sum.
// And recusrively move to next elem
let plus = Math.abs(solution(A, i+1, sum+A[i]));
// Absolute value by substracting the elem from the sum
let minus = Math.abs(solution(A, i+1, sum-A[i]));
return Math.min(plus, minus);
}
console.log(solution([-100, 3, 2, 4]))
Here is a C solution based on Kadane's algorithm.
Hopefully its helpful.
#include <stdio.h>
int min(int a, int b)
{
return (a >= b)? b: a;
}
int min_slice(int A[], int N) {
if (N==0 || N>1000000)
return 0;
int minTillHere = A[0];
int minSoFar = A[0];
int i;
for(i = 1; i < N; i++){
minTillHere = min(A[i], minTillHere + A[i]);
minSoFar = min(minSoFar, minTillHere);
}
return minSoFar;
}
int main(){
int A[]={3, 2, -6, 4, 0}, N = 5;
//int A[]={3, 2, 6, 4, 0}, N = 5;
//int A[]={-4, -8, -3, -2, -4, -10}, N = 6;
printf("Minimum slice = %d \n", min_slice(A,N));
return 0;
}
public static int solution(int[] A) {
int minTillHere = A[0];
int absMinTillHere = A[0];
int minSoFar = A[0];
int i;
for(i = 1; i < A.length; i++){
absMinTillHere = Math.min(Math.abs(A[i]),Math.abs(minTillHere + A[i]));
minTillHere = Math.min(A[i], minTillHere + A[i]);
minSoFar = Math.min(Math.abs(minSoFar), absMinTillHere);
}
return minSoFar;
}
int main()
{
int n; cin >> n;
vector<int>a(n);
for(int i = 0; i < n; i++) cin >> a[i];
long long local_min = 0, global_min = LLONG_MAX;
for(int i = 0; i < n; i++)
{
if(abs(local_min + a[i]) > abs(a[i]))
{
local_min = a[i];
}
else local_min += a[i];
global_min = min(global_min, abs(local_min));
}
cout << global_min << endl;
}
I have this problem , where given an array of positive numbers i have to find the maximum sum of elements such that no two adjacent elements are picked. The maximum has to be less than a certain given K. I tried thinking on the lines of the similar problem without the k , but i have failed so far.I have the following dp-ish soln for the latter problem
int sum1,sum2 = 0;
int sum = sum1 = a[0];
for(int i=1; i<n; i++)
{
sum = max(sum2 + a[i], sum1);
sum2 = sum1;
sum1 = sum;
}
Could someone give me tips on how to proceed with my present problem??
The best I can think of off the top of my head is an O(n*K) dp:
int sums[n][K+1] = {{0}};
int i, j;
for(j = a[0]; j <= K; ++j) {
sums[0][j] = a[0];
}
if (a[1] > a[0]) {
for(j = a[0]; j < a[1]; ++j) {
sums[1][j] = a[0];
}
for(j = a[1]; j <= K; ++j) {
sums[1][j] = a[1];
}
} else {
for(j = a[1]; j < a[0]; ++j) {
sums[1][j] = a[1];
}
for(j = a[0]; j <= K; ++j) {
sums[1][j] = a[0];
}
}
for(i = 2; i < n; ++i) {
for(j = 0; j <= K && j < a[i]; ++j) {
sums[i][j] = max(sums[i-1][j],sums[i-2][j]);
}
for(j = a[i]; j <= K; ++j) {
sums[i][j] = max(sums[i-1][j],a[i] + sums[i-2][j-a[i]]);
}
}
sums[i][j] contains the maximal sum of non-adjacent elements of a[0..i] not exceeding j. The solution is then sums[n-1][K] at the end.
Make a copy (A2) of the original array (A1).
Find largest value in array (A2).
Extract all values before the it's preceeding neighbour and the values after it's next neighbour into a new array (A3).
Find largest value in the new array (A3).
Check if sum is larger that k. If sum passes the check you are done.
If not you will need to go back to the copied array (A2), remove the second larges value (found in step 3) and start over with step 3.
Once there are no combinations of numbers that can be used with the largest number (i.e. number found in step 1 + any other number in array is larger than k) you remove it from the original array (A1) and start over with step 0.
If for some reason there are no valid combinations (e.g. array is only three numbers or no combination of numbers are lower than k) then throw an exception or you return null if that seems more appropriate.
First idea: Brute force
Iterate all legal combination of indexes and build the sum on the fly.
Stop with one sequence when you get over K.
keep the sequence until you find a larger one, that is still smaller then K
Second idea: maybe one can force this into a divide and conquer thing ...
Here is a solution to the problem without the "k" constraint which you set out to do as the first step: https://stackoverflow.com/a/13022021/1110808
The above solution can in my view be easily extended to have the k constraint by simply amending the if condition in the following for loop to include the constraint: possibleMax < k
// Subproblem solutions, DP
for (int i = start; i <= end; i++) {
int possibleMaxSub1 = maxSum(a, i + 2, end);
int possibleMaxSub2 = maxSum(a, start, i - 2);
int possibleMax = possibleMaxSub1 + possibleMaxSub2 + a[i];
/*
if (possibleMax > maxSum) {
maxSum = possibleMax;
}
*/
if (possibleMax > maxSum && possibleMax < k) {
maxSum = possibleMax;
}
}
As posted in the original link, this approach can be improved by adding memorization so that solutions to repeating sub problems are not recomputed. Or can be improved by using a bottom up dynamic programming approach (current approach is a recursive top down approach)
You can refer to a bottom up approach here: https://stackoverflow.com/a/4487594/1110808