Finding the Maximum Sum of a Contiguous Subvector of an Input Vector

I had a phone interview around mid-October. In the second half of the interview I was asked the obligatory programming question as follows,

Given an input vector, find the maximum sum of a contiguous sub-vector in that vector. Your algorithm should be in linear time.

This was taken verbatim from the Jon Bentley’s excellent book “Programming Pearls” (2nd. Ed.). I coud’nt solve the problem at the time (and I kicked myself because I owned and had read the book), but it got me thinking about the problem again.

I thought that even though the problem did’nt ask for it, the maximum sum is not very useful. What would be more useful is the actual range of the sub-vector. After all the problem itself was a simplified version of the (as yet open) problem for finding the largest sum of 2-D matrix. I’m not going to go into various details on how the problem can be solved, but I did think about how I could implement a generic version of the algorithm that would dove-tail nicely with various sequences in the STL. (If any reader is interested, I strongly suggest you buy the book). Here is my first cut,

template<typename SequenceIterator>
struct find_max_sum_range
    std::pair<SequenceIterator, SequenceIterator>
       operator () (typename SequenceIterator first, typename  SequenceIterator end)
        SequenceIterator::value_type max_so_far = 0;
        SequenceIterator::value_type max_ending_here = 0;
        std::pair<SequenceIterator, SequenceIterator> p;
        for (;first!=end; ++first){
            if (max_ending_here==0){
                p.first = p.second = first;
            max_ending_here =
                std::max<typename SequenceIterator::value_type>(max_ending_here + *first, 0);
            if (max_ending_here > max_so_far){
                max_so_far = max_ending_here;
                p.second  = first;
        //Have to increment STL iterator because the end always points to
        //one incrment beyond
        return p;

I was’nt very happy with this version. I thought that to keep it in the spirit of STL, I needed to use the proper STL library functions, and in this case the right algorithm to use would be the std::for_each function. The problem with this function is that there is no way to access the current iterator while the algorithm is being executed. (Actually this is a general weakness in the otherwise very powerful STL iterator concept). My second solution got around the problem by using a functor that held a reference to a local iterator that got updated whenever the functor got executed. Hence the second cut of the solution.

template<typename SequenceIterator>
struct find_max_sum_range2
    template<typename SequenceIterator>
    struct find_max_sum_range_helper
            ( typename SequenceIterator::value_type &max_so_far_
            , typename SequenceIterator::value_type &max_ending_here_
            , std::pair<SequenceIterator, SequenceIterator > &p_
            , typename SequenceIterator &current_)
            : max_so_far(max_so_far_)
            , max_ending_here(max_ending_here_)
            , p(p_)
            , current(current_)
        { /* Do nothing else */}

        void operator()(typename SequenceIterator::value_type val){
                p.first = p.second = current;

            max_ending_here =
                std::max<typename SequenceIterator::value_type>(max_ending_here + val, 0);

            if (max_ending_here > max_so_far){
                max_so_far = max_ending_here;
                p.second = current;
            //increment the current operator;

        typename SequenceIterator::value_type &max_so_far;
        typename SequenceIterator::value_type &max_ending_here;
        typename std::pair<SequenceIterator, SequenceIterator> &p;
        SequenceIterator &current;
    std::pair<SequenceIterator, SequenceIterator>
        operator() (typename SequenceIterator first, typename SequenceIterator end)
        SequenceIterator::value_type  max_so_far = 0;
        SequenceIterator::value_type  max_ending_here = 0;
        std::pair<SequenceIterator, SequenceIterator> p;
        SequenceIterator current = first;

        std::for_each(first, end,
            find_max_sum_range_helper<SequenceIterator>(max_so_far, max_ending_here, p, current));

        //increment to make sure that it points to 1 element beyond the last element.
        return p;

Comparing the various solutions, it seems the implementation using std::for_each was much more complicated, although it abstracted away the alogrithm for every element away from the iteration of the algorithm. My next step would be use to Boost::lambda to make the code somewhat more clearer while trying to stay in the STL spirit of things.

Usage is as follows

    /* TEST 1:
    * Test a sequence of ints using vectors
    int a[]={-3, 2, 4, -7, 2, 6, 8, -3, 2};
    std::vector<int> x(a, a+sizeof(a)/sizeof(int));
    typedef std::pair<std::vector<int>::iterator, std::vector<int>::iterator> vector_iterator_pair;
    vector_iterator_pair vec_pair = find_max_sum_range<std::vector<int>::iterator>()(x.begin(), x.end());
    print_sequence(vec_pair.first, vec_pair.second);
    /* Test 2:
    *   Test a sequence of longs  using lists

    long b[]={-3, 2, 4, -7, 2, 6, 8, -1, 2};
    std::list<long> list_of_longs(sizeof(b)/sizeof(long), 0);
    std::swap_ranges(list_of_longs.begin(), list_of_longs.end(), b);
    typedef std::pair<std::list<long>::iterator, std::list<long>::iterator> list_iterator_pair ;
    list_iterator_pair lp = find_max_sum_range<std::list<long>::iterator>()
        (list_of_longs.begin(), list_of_longs.end());    print_sequence(lp.first, lp.second);

    /* Test 3:
    * Test a single element sequence, using a set of flats
    float ca[]={2.365f};
    std::set<float> set_of_floats(ca, ca + sizeof(ca)/sizeof(float));
    typedef std::pair<std::set<float>::iterator, std::set<float>::iterator> float_iterator_pair;
    float_iterator_pair cp = find_max_sum_range<std::set<float>::iterator>()
        ( set_of_floats.begin(), set_of_floats.end());
    print_sequence(cp.first, cp.second);

2 Responses to Finding the Maximum Sum of a Contiguous Subvector of an Input Vector

  1. Orfeu says:

    I could not understand your code :). I made my own implementation, the result was 6, 8, -1, 2. Is it correct?

    Here is the implementation if you want to look at it:
    using namespace std;

    int main()
    int data[20], sum[20];
    int n, max, lim_inf = 0, lim_sup = 0;

    cout <>n;

    sum[0] = 0;
    for (int i = 1; i <= n; i++)
    cout <<“Number “<<i<>data[i – 1];
    sum[i] = sum[i – 1] + data[i – 1];

    max = sum[1];
    for (int i = 0; i < n; i++)
    for (int j = i + 1; j max)
    max = sum[j];
    cout <<i<<” “<<j<<” “<<max<<endl;
    lim_inf = i;
    lim_sup = j;

    for (int i = lim_inf; i < lim_sup; i++)
    cout <<data[i]<<” “;

    cout <<endl;
    return 0;

  2. What’s the input vector? Your solution runs in O(n^2) which is quadratic time, which is very inefficient. This problem is solvable in linear time.

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