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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];
}
};
How can we develop a dynamic programming algorithm that calculates the minimum number of different primes that sum to x?
Assume the dynamic programming calculates the minimum number of different primes amongst which the largest is p for each couple of x and p. Can someone help?
If we assume the Goldbach conjecture is true, then every even integer > 2 is the sum of two primes.
So we know the answer if x is even (1 if x==2, or 2 otherwise).
If x is odd, then there are 3 cases:
x is prime (answer is 1)
x-2 is prime (answer is 2)
otherwise x-3 is an even number bigger than 2 (answer is 3)
First of all, you need a list of primes up to x. Let's call this array of integers primes.
Now we want to populate the array answer[x][p], where x is the sum of primes and p is maximum for each prime in the set (possibly including, but not necessarily including p).
There are 3 possibilities for answer[x][p] after all calculations:
1) if p=x and p is prime => answer[x][p] contains 1
2) if it's not possible to solve problem for given x, p => answer[x][p] contains -1
3) if it's possible to solve problem for given x, p => answer[x][p] contains number of primes
There is one more possible value for answer[x][p] during calculations:
4) we did not yet solve the problem for given x, p => answer[x][p] contains 0
It's quite obvious that 0 is not the answer for anything but x=0, so we are safe initializing array with 0 (and making special treatment for x=0).
To calculate answer[x][p] we can iterate (let q is prime value we are iterating on) through all primes up to (including) p and find minimum over 1+answer[x-q][q-1] (do not consider all answer[x-q][q-1]=-1 cases though). Here 1 comes for q and answer[x-q][q-1] should be calculated in recursive call or before this calculation.
Now there's small optimization: iterate primes from higher to lower and if x/q is bigger than current answer, we can stop, because to make sum x we will need at least x/q primes anyway. For example, we will not even consider q=2 for x=10, as we'd already have answer=3 (actually, it includes 2 as one of 3 primes - 2+3+5, but we've already got it through recursive call answer(10-5, 4)), since 10/2=5, that is we'd get 5 as answer at best (in fact it does not exist for q=2).
package ru.tieto.test;
import java.util.ArrayList;
public class Primers {
static final int MAX_P = 10;
static final int MAX_X = 10;
public ArrayList<Integer> primes= new ArrayList<>();
public int answer[][] = new int[MAX_X+1][MAX_P+1];
public int answer(int x, int p) {
if (x < 0)
return -1;
if (x == 0)
return 0;
if (answer[x][p] != 0)
return answer[x][p];
int max_prime_idx = -1;
for (int i = 0;
i < primes.size() && primes.get(i) <= p && primes.get(i) <= x;
i++)
max_prime_idx = i;
if (max_prime_idx < 0) {
answer[x][p] = -1;
return -1;
}
int cur_answer = x+1;
for (int i = max_prime_idx; i >= 0; i--) {
int q = primes.get(i);
if (x / q >= cur_answer)
break;
if (x == q) {
cur_answer = 1;
break;
}
int candidate = answer(x-q, q-1);
if (candidate == -1)
continue;
if (candidate+1 < cur_answer)
cur_answer = candidate+1;
}
if (cur_answer > x)
answer[x][p] = -1;
else
answer[x][p] = cur_answer;
return answer[x][p];
}
private void make_primes() {
primes.add(2);
for (int p = 3; p <= MAX_P; p=p+2) {
boolean isPrime = true;
for (Integer q : primes) {
if (q*q > p)
break;
if (p % q == 0) {
isPrime = false;
break;
}
}
if (isPrime)
primes.add(p);
}
// for (Integer q : primes)
// System.out.print(q+",");
// System.out.println("<<");
}
private void init() {
make_primes();
for (int p = 0; p <= MAX_P; p++) {
answer[0][p] = 0;
answer[1][p] = -1;
}
for (int x = 2; x <= MAX_X; x++) {
for (int p = 0; p <= MAX_P; p++)
answer[x][p] = 0;
}
for (Integer p: primes)
answer[p][p] = 1;
}
void run() {
init();
for (int x = 0; x <= MAX_X; x++)
for (int p = 0; p <= MAX_P; p++)
answer(x, p);
}
public static void main(String[] args) {
Primers me = new Primers();
me.run();
// for (int x = 0; x <= MAX_X; x++) {
// System.out.print("x="+x+": {");
// for (int p = 0; p <= MAX_P; p++) {
// System.out.print(String.format("%2d=%-3d,",p, me.answer[x][p]));
// }
// System.out.println("}");
// }
}
}
Start with a list of all primes lower than x.
Take the largest. Now we need to solve the problem for (x - pmax). At this stage, that will be easy, x - pmax will be low. Mark all the primes as "used" and store solution 1. Now take the largest prime still in the list and repeat until all the primes are either used or rejected. If (x - pmax) is high, the problem gets more complex.
That's your first pass, brute force algorithm. Get that working first before considering how to speed things up.
Assuming you're not using goldbach conjecture, otherwise see Peter de Rivaz excellent answer, :
dynamic programming generally takes advantage of overlapping subproblems. Usually you go top down, but in this case bottom up may be simpler
I suggest you sum various combinations of primes.
lookup = {}
for r in range(1, 3):
for primes in combinations_with_replacement(all_primes, r):
s = sum(primes)
lookup[s] = lookup.get(s, r) //r is increasing, so only set it if it's not already there
this will start getting slow very quickly if you have a large number of primes, in that case, change max r to something like 1 or 2, whatever the max that is fast enough for you, and then you will be left with some numbers that aren't found, to solve for a number that doesnt have a solution in lookup, try break that number into sums of numbers that are found in lookup (you may need to store the prime combos in lookup and dedupe those combinations).
Most of us are familiar with the maximum sum subarray problem. I came across a variant of this problem which asks the programmer to output the maximum of all subarray sums modulo some number M.
The naive approach to solve this variant would be to find all possible subarray sums (which would be of the order of N^2 where N is the size of the array). Of course, this is not good enough. The question is - how can we do better?
Example: Let us consider the following array:
6 6 11 15 12 1
Let M = 13. In this case, subarray 6 6 (or 12 or 6 6 11 15 or 11 15 12) will yield maximum sum ( = 12 ).
We can do this as follow:
Maintaining an array sum which at index ith, it contains the modulus sum from 0 to ith.
For each index ith, we need to find the maximum sub sum that end at this index:
For each subarray (start + 1 , i ), we know that the mod sum of this sub array is
int a = (sum[i] - sum[start] + M) % M
So, we can only achieve a sub-sum larger than sum[i] if sum[start] is larger than sum[i] and as close to sum[i] as possible.
This can be done easily if you using a binary search tree.
Pseudo code:
int[] sum;
sum[0] = A[0];
Tree tree;
tree.add(sum[0]);
int result = sum[0];
for(int i = 1; i < n; i++){
sum[i] = sum[i - 1] + A[i];
sum[i] %= M;
int a = tree.getMinimumValueLargerThan(sum[i]);
result = max((sum[i] - a + M) % M, result);
tree.add(sum[i]);
}
print result;
Time complexity :O(n log n)
Let A be our input array with zero-based indexing. We can reduce A modulo M without changing the result.
First of all, let's reduce the problem to a slightly easier one by computing an array P representing the prefix sums of A, modulo M:
A = 6 6 11 2 12 1
P = 6 12 10 12 11 12
Now let's process the possible left borders of our solution subarrays in decreasing order. This means that we will first determine the optimal solution that starts at index n - 1, then the one that starts at index n - 2 etc.
In our example, if we chose i = 3 as our left border, the possible subarray sums are represented by the suffix P[3..n-1] plus a constant a = A[i] - P[i]:
a = A[3] - P[3] = 2 - 12 = 3 (mod 13)
P + a = * * * 2 1 2
The global maximum will occur at one point too. Since we can insert the suffix values from right to left, we have now reduced the problem to the following:
Given a set of values S and integers x and M, find the maximum of S + x modulo M
This one is easy: Just use a balanced binary search tree to manage the elements of S. Given a query x, we want to find the largest value in S that is smaller than M - x (that is the case where no overflow occurs when adding x). If there is no such value, just use the largest value of S. Both can be done in O(log |S|) time.
Total runtime of this solution: O(n log n)
Here's some C++ code to compute the maximum sum. It would need some minor adaptions to also return the borders of the optimal subarray:
#include <bits/stdc++.h>
using namespace std;
int max_mod_sum(const vector<int>& A, int M) {
vector<int> P(A.size());
for (int i = 0; i < A.size(); ++i)
P[i] = (A[i] + (i > 0 ? P[i-1] : 0)) % M;
set<int> S;
int res = 0;
for (int i = A.size() - 1; i >= 0; --i) {
S.insert(P[i]);
int a = (A[i] - P[i] + M) % M;
auto it = S.lower_bound(M - a);
if (it != begin(S))
res = max(res, *prev(it) + a);
res = max(res, (*prev(end(S)) + a) % M);
}
return res;
}
int main() {
// random testing to the rescue
for (int i = 0; i < 1000; ++i) {
int M = rand() % 1000 + 1, n = rand() % 1000 + 1;
vector<int> A(n);
for (int i = 0; i< n; ++i)
A[i] = rand() % M;
int should_be = 0;
for (int i = 0; i < n; ++i) {
int sum = 0;
for (int j = i; j < n; ++j) {
sum = (sum + A[j]) % M;
should_be = max(should_be, sum);
}
}
assert(should_be == max_mod_sum(A, M));
}
}
For me, all explanations here were awful, since I didn't get the searching/sorting part. How do we search/sort, was unclear.
We all know that we need to build prefixSum, meaning sum of all elems from 0 to i with modulo m
I guess, what we are looking for is clear.
Knowing that subarray[i][j] = (prefix[i] - prefix[j] + m) % m (indicating the modulo sum from index i to j), our maxima when given prefix[i] is always that prefix[j] which is as close as possible to prefix[i], but slightly bigger.
E.g. for m = 8, prefix[i] being 5, we are looking for the next value after 5, which is in our prefixArray.
For efficient search (binary search) we sort the prefixes.
What we can not do is, build the prefixSum first, then iterate again from 0 to n and look for index in the sorted prefix array, because we can find and endIndex which is smaller than our startIndex, which is no good.
Therefore, what we do is we iterate from 0 to n indicating the endIndex of our potential max subarray sum and then look in our sorted prefix array, (which is empty at the beginning) which contains sorted prefixes between 0 and endIndex.
def maximumSum(coll, m):
n = len(coll)
maxSum, prefixSum = 0, 0
sortedPrefixes = []
for endIndex in range(n):
prefixSum = (prefixSum + coll[endIndex]) % m
maxSum = max(maxSum, prefixSum)
startIndex = bisect.bisect_right(sortedPrefixes, prefixSum)
if startIndex < len(sortedPrefixes):
maxSum = max(maxSum, prefixSum - sortedPrefixes[startIndex] + m)
bisect.insort(sortedPrefixes, prefixSum)
return maxSum
From your question, it seems that you have created an array to store the cumulative sums (Prefix Sum Array), and are calculating the sum of the sub-array arr[i:j] as (sum[j] - sum[i] + M) % M. (arr and sum denote the given array and the prefix sum array respectively)
Calculating the sum of every sub-array results in a O(n*n) algorithm.
The question that arises is -
Do we really need to consider the sum of every sub-array to reach the desired maximum?
No!
For a value of j the value (sum[j] - sum[i] + M) % M will be maximum when sum[i] is just greater than sum[j] or the difference is M - 1.
This would reduce the algorithm to O(nlogn).
You can take a look at this explanation! https://www.youtube.com/watch?v=u_ft5jCDZXk
There are already a bunch of great solutions listed here, but I wanted to add one that has O(nlogn) runtime without using a balanced binary tree, which isn't in the Python standard library. This solution isn't my idea, but I had to think a bit as to why it worked. Here's the code, explanation below:
def maximumSum(a, m):
prefixSums = [(0, -1)]
for idx, el in enumerate(a):
prefixSums.append(((prefixSums[-1][0] + el) % m, idx))
prefixSums = sorted(prefixSums)
maxSeen = prefixSums[-1][0]
for (a, a_idx), (b, b_idx) in zip(prefixSums[:-1], prefixSums[1:]):
if a_idx > b_idx and b > a:
maxSeen = max((a-b) % m, maxSeen)
return maxSeen
As with the other solutions, we first calculate the prefix sums, but this time we also keep track of the index of the prefix sum. We then sort the prefix sums, as we want to find the smallest difference between prefix sums modulo m - sorting lets us just look at adjacent elements as they have the smallest difference.
At this point you might think we're neglecting an essential part of the problem - we want the smallest difference between prefix sums, but the larger prefix sum needs to appear before the smaller prefix sum (meaning it has a smaller index). In the solutions using trees, we ensure that by adding prefix sums one by one and recalculating the best solution.
However, it turns out that we can look at adjacent elements and just ignore ones that don't satisfy our index requirement. This confused me for some time, but the key realization is that the optimal solution will always come from two adjacent elements. I'll prove this via a contradiction. Let's say that the optimal solution comes from two non-adjacent prefix sums x and z with indices i and k, where z > x (it's sorted!) and k > i:
x ... z
k ... i
Let's consider one of the numbers between x and z, and let's call it y with index j. Since the list is sorted, x < y < z.
x ... y ... z
k ... j ... i
The prefix sum y must have index j < i, otherwise it would be part of a better solution with z. But if j < i, then j < k and y and x form a better solution than z and x! So any elements between x and z must form a better solution with one of the two, which contradicts our original assumption. Therefore the optimal solution must come from adjacent prefix sums in the sorted list.
Here is Java code for maximum sub array sum modulo. We handle the case we can not find least element in the tree strictly greater than s[i]
public static long maxModulo(long[] a, final long k) {
long[] s = new long[a.length];
TreeSet<Long> tree = new TreeSet<>();
s[0] = a[0] % k;
tree.add(s[0]);
long result = s[0];
for (int i = 1; i < a.length; i++) {
s[i] = (s[i - 1] + a[i]) % k;
// find least element in the tree strictly greater than s[i]
Long v = tree.higher(s[i]);
if (v == null) {
// can't find v, then compare v and s[i]
result = Math.max(s[i], result);
} else {
result = Math.max((s[i] - v + k) % k, result);
}
tree.add(s[i]);
}
return result;
}
Few points from my side that might hopefully help someone understand the problem better.
You do not need to add +M to the modulo calculation, as mentioned, % operator handles negative numbers well, so a % M = (a + M) % M
As mentioned, the trick is to build the proxy sum table such that
proxy[n] = (a[1] + ... a[n]) % M
This then allows one to represent the maxSubarraySum[i, j] as
maxSubarraySum[i, j] = (proxy[j] - proxy[j]) % M
The implementation trick is to build the proxy table as we iterate through the elements, instead of first pre-building it and then using. This is because for each new element in the array a[i] we want to compute proxy[i] and find proxy[j] that is bigger than but as close as possible to proxy[i] (ideally bigger by 1 because this results in a reminder of M - 1). For this we need to use a clever data structure for building proxy table while keeping it sorted and
being able to quickly find a closest bigger element to proxy[i]. bisect.bisect_right is a good choice in Python.
See my Python implementation below (hope this helps but I am aware this might not necessarily be as concise as others' solutions):
def maximumSum(a, m):
prefix_sum = [a[0] % m]
prefix_sum_sorted = [a[0] % m]
current_max = prefix_sum_sorted[0]
for elem in a[1:]:
prefix_sum_next = (prefix_sum[-1] + elem) % m
prefix_sum.append(prefix_sum_next)
idx_closest_bigger = bisect.bisect_right(prefix_sum_sorted, prefix_sum_next)
if idx_closest_bigger >= len(prefix_sum_sorted):
current_max = max(current_max, prefix_sum_next)
bisect.insort_right(prefix_sum_sorted, prefix_sum_next)
continue
if prefix_sum_sorted[idx_closest_bigger] > prefix_sum_next:
current_max = max(current_max, (prefix_sum_next - prefix_sum_sorted[idx_closest_bigger]) % m)
bisect.insort_right(prefix_sum_sorted, prefix_sum_next)
return current_max
Total java implementation with O(n*log(n))
import java.io.BufferedReader;
import java.io.InputStreamReader;
import java.util.TreeSet;
import java.util.stream.Stream;
public class MaximizeSumMod {
public static void main(String[] args) throws Exception{
BufferedReader in = new BufferedReader(new InputStreamReader(System.in));
Long times = Long.valueOf(in.readLine());
while(times --> 0){
long[] pair = Stream.of(in.readLine().split(" ")).mapToLong(Long::parseLong).toArray();
long mod = pair[1];
long[] numbers = Stream.of(in.readLine().split(" ")).mapToLong(Long::parseLong).toArray();
printMaxMod(numbers,mod);
}
}
private static void printMaxMod(long[] numbers, Long mod) {
Long maxSoFar = (numbers[numbers.length-1] + numbers[numbers.length-2])%mod;
maxSoFar = (maxSoFar > (numbers[0]%mod)) ? maxSoFar : numbers[0]%mod;
numbers[0] %=mod;
for (Long i = 1L; i < numbers.length; i++) {
long currentNumber = numbers[i.intValue()]%mod;
maxSoFar = maxSoFar > currentNumber ? maxSoFar : currentNumber;
numbers[i.intValue()] = (currentNumber + numbers[i.intValue()-1])%mod;
maxSoFar = maxSoFar > numbers[i.intValue()] ? maxSoFar : numbers[i.intValue()];
}
if(mod.equals(maxSoFar+1) || numbers.length == 2){
System.out.println(maxSoFar);
return;
}
long previousNumber = numbers[0];
TreeSet<Long> set = new TreeSet<>();
set.add(previousNumber);
for (Long i = 2L; i < numbers.length; i++) {
Long currentNumber = numbers[i.intValue()];
Long ceiling = set.ceiling(currentNumber);
if(ceiling == null){
set.add(numbers[i.intValue()-1]);
continue;
}
if(ceiling.equals(currentNumber)){
set.remove(ceiling);
Long greaterCeiling = set.ceiling(currentNumber);
if(greaterCeiling == null){
set.add(ceiling);
set.add(numbers[i.intValue()-1]);
continue;
}
set.add(ceiling);
ceiling = greaterCeiling;
}
Long newMax = (currentNumber - ceiling + mod);
maxSoFar = maxSoFar > newMax ? maxSoFar :newMax;
set.add(numbers[i.intValue()-1]);
}
System.out.println(maxSoFar);
}
}
Adding STL C++11 code based on the solution suggested by #Pham Trung. Might be handy.
#include <iostream>
#include <set>
int main() {
int N;
std::cin>>N;
for (int nn=0;nn<N;nn++){
long long n,m;
std::set<long long> mSet;
long long maxVal = 0; //positive input values
long long sumVal = 0;
std::cin>>n>>m;
mSet.insert(m);
for (long long q=0;q<n;q++){
long long tmp;
std::cin>>tmp;
sumVal = (sumVal + tmp)%m;
auto itSub = mSet.upper_bound(sumVal);
maxVal = std::max(maxVal,(m + sumVal - *itSub)%m);
mSet.insert(sumVal);
}
std::cout<<maxVal<<"\n";
}
}
As you can read in Wikipedia exists a solution called Kadane's algorithm, which compute the maximum subarray sum watching ate the maximum subarray ending at position i for all positions i by iterating once over the array. Then this solve the problem with with runtime complexity O(n).
Unfortunately, I think that Kadane's algorithm isn't able to find all possible solution when more than one solution exists.
An implementation in Java, I didn't tested it:
public int[] kadanesAlgorithm (int[] array) {
int start_old = 0;
int start = 0;
int end = 0;
int found_max = 0;
int max = array[0];
for(int i = 0; i<array.length; i++) {
max = Math.max(array[i], max + array[i]);
found_max = Math.max(found_max, max);
if(max < 0)
start = i+1;
else if(max == found_max) {
start_old=start;
end = i;
}
}
return Arrays.copyOfRange(array, start_old, end+1);
}
I feel my thoughts are aligned with what have been posted already, but just in case - Kotlin O(NlogN) solution:
val seen = sortedSetOf(0L)
var prev = 0L
return max(a.map { x ->
val z = (prev + x) % m
prev = z
seen.add(z)
seen.higher(z)?.let{ y ->
(z - y + m) % m
} ?: z
})
Implementation in java using treeset...
import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;
import java.util.TreeSet;
public class Main {
public static void main(String[] args) throws IOException {
BufferedReader read = new BufferedReader(new InputStreamReader(System.in)) ;
String[] str = read.readLine().trim().split(" ") ;
int n = Integer.parseInt(str[0]) ;
long m = Long.parseLong(str[1]) ;
str = read.readLine().trim().split(" ") ;
long[] arr = new long[n] ;
for(int i=0; i<n; i++) {
arr[i] = Long.parseLong(str[i]) ;
}
long maxCount = 0L ;
TreeSet<Long> tree = new TreeSet<>() ;
tree.add(0L) ;
long prefix = 0L ;
for(int i=0; i<n; i++) {
prefix = (prefix + arr[i]) % m ;
maxCount = Math.max(prefix, maxCount) ;
Long temp = tree.higher(prefix) ;
System.out.println(temp);
if(temp != null) {
maxCount = Math.max((prefix-temp+m)%m, maxCount) ;
}
//System.out.println(maxCount);
tree.add(prefix) ;
}
System.out.println(maxCount);
}
}
Here is one implementation of solution in java for this problem which works using TreeSet in java for optimized solution !
public static long maximumSum2(long[] arr, long n, long m)
{
long x = 0;
long prefix = 0;
long maxim = 0;
TreeSet<Long> S = new TreeSet<Long>();
S.add((long)0);
// Traversing the array.
for (int i = 0; i < n; i++)
{
// Finding prefix sum.
prefix = (prefix + arr[i]) % m;
// Finding maximum of prefix sum.
maxim = Math.max(maxim, prefix);
// Finding iterator poing to the first
// element that is not less than value
// "prefix + 1", i.e., greater than or
// equal to this value.
long it = S.higher(prefix)!=null?S.higher(prefix):0;
// boolean isFound = false;
// for (long j : S)
// {
// if (j >= prefix + 1)
// if(isFound == false) {
// it = j;
// isFound = true;
// }
// else {
// if(j < it) {
// it = j;
// }
// }
// }
if (it != 0)
{
maxim = Math.max(maxim, prefix - it + m);
}
// adding prefix in the set.
S.add(prefix);
}
return maxim;
}
public static int MaxSequence(int[] arr)
{
int maxSum = 0;
int partialSum = 0;
int negative = 0;
for (int i = 0; i < arr.Length; i++)
{
if (arr[i] < 0)
{
negative++;
}
}
if (negative == arr.Length)
{
return 0;
}
foreach (int item in arr)
{
partialSum += item;
maxSum = Math.Max(maxSum, partialSum);
if (partialSum < 0)
{
partialSum = 0;
}
}
return maxSum;
}
Modify Kadane algorithm to keep track of #occurrence. Below is the code.
#python3
#source: https://github.com/harishvc/challenges/blob/master/dp-largest-sum-sublist-modulo.py
#Time complexity: O(n)
#Space complexity: O(n)
def maxContiguousSum(a,K):
sum_so_far =0
max_sum = 0
count = {} #keep track of occurrence
for i in range(0,len(a)):
sum_so_far += a[i]
sum_so_far = sum_so_far%K
if sum_so_far > 0:
max_sum = max(max_sum,sum_so_far)
if sum_so_far in count.keys():
count[sum_so_far] += 1
else:
count[sum_so_far] = 1
else:
assert sum_so_far < 0 , "Logic error"
#IMPORTANT: reset sum_so_far
sum_so_far = 0
return max_sum,count[max_sum]
a = [6, 6, 11, 15, 12, 1]
K = 13
max_sum,count = maxContiguousSum(a,K)
print("input >>> %s max sum=%d #occurrence=%d" % (a,max_sum,count))
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;
}
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