# Minimum difficulty job schedule

## Problem statement

Write a function that takes as an input a list of tasks $$I$$ and an integer $$d$$. The elements in $$I$$ are dependent on each other and to schedule a certain task $$I_i$$ all the tasks $$I_j \: : j < i$$ have to be completed. The function should return the minimum complexity among all possible schedules of length exactly $$d$$ days. The complexity of a job is calculated as the sum of the complexity of every single day of the schedule. The complexity of a day of the schedule is defined as the maximum complexity of the tasks planned for that day.

As an additional constraint, you have to make sure that there is at least one task scheduled for each day.

Given:

• $$I = \{6,5,4,3,2,1\}$$

• $$d = 2$$

the function returns $$7$$. You can schedule tasks $$0$$ to $$4$$ during the first day and the last task during the second day. You cannot just schedule all tasks during the first day because then you would have a day in the schedule without planned tasks which is not permitted.

Given:

• $$I = \{10,10,10\}$$

• $$d = 4$$

the function returns $$-1$$. There is no way to schedule tasks for $$4$$ days when there are only $$3$$ tasks available for scheduling.

Given:

• $$I = \{7,1,7,1,7,1\}$$

• $$d = 3$$

the function returns $$15$$. You can schedule the first $$4$$ tasks the first day for a total complexity of $$7$$. Tasks at index $$4$$ and $$5$$ can be scheduled for days $$2$$ and $$3$$ respectively.

Notice that in this case if $$d = 2$$ then the function would return $$8$$.

Given:

• $$I = \{11,111,22,222,33,333,44,444\}$$

• $$d = 6$$

the function returns $$843$$. You can schedule tasks $$0,1,2,3,4$$ in the first $$5$$ days and the rest during the .

## Clarification Questions

What should the function return in the case where it is not possible to make a valid schedule? For instance when $$|I| < d$$?

You can return $$-1$$ in that case.

Is it guaranteed for the complexity values to be positive ($$\geq 0$$)?

Yes you can assume complexities are always positive.

## Discussion

This is a classic example of a problem that can be easily solved via dynamic programming but can be very challenging if you try to approach it differently. Fortunately, the statement is full of hints that this problem can be solved using DP. For instance:

it is an optimization problem, and,

you are not asked to find an actual schedule, but only the value of the best possible one.

. Very often those are the two most common ingredients in a DP problem. It’s important, therefore, to be able to quickly identify the clues within the statement that point to a DP based solution.

### Brute-force

If you do not immediately think about DP one of the possible approaches to this problem would be to try out all possible schedules, and for each of them calculate its cost, and return the smallest. The problem explicitly mentions a case where a solution does not exist. This is an easy case as there is only one scenario where you cannot schedule jobs for $$d$$ days: when the number of jobs to be scheduled is strictly less than $$d$$. The core of the problem is really about the case where $$|I| \geq d$$. You can think about a schedule as a way of splitting $$I$$ into $$d$$ non-empty sub-arrays. You can split an array into $$d$$ parts by placing $$d-1$$ splitting-points in $$I$$ at different locations. A different placing of the splitting-points leads univocally to a different schedule. There is, therefore, a one-to-one correspondence between a subset of size $$d-1$$ of $$\{0,1,2, \ldots, |I|-2\}$$ (the splitting point locations) and schedules (see Equation [eq:min_difficulty_job_scheduler:cost_combination]). We can therefore generate all possible schedules by generating all possible combinations of $$d-1$$ elements from $$\{0,1,2, \ldots, |I|-2\}$$ where each number of a combination $$\{e_0, \ldots, e_{d-1}\}$$ represents a splitting point in $$I$$ and $$e_i$$ identifies the following subarray of $$I$$: $$\{A_{e_i-1+1}, A_{e_i-1+1}, \ldots , A_{e_i}\}$$.

In order to solve this problem we can calculate the costs for each of the schedules represented by a combination of $$d-1$$ elements of $$\{0,1,2, \ldots, |I|-1\}$$, and return the cost of the best (the one having minimum cost overall). The cost of a schedule - as shown in the problem statement - is the sum of the costs for each of the $$d$$ day where the cost of a single day is the cost of the most expensive job scheduled for that particular day. So given a schedule represented by the combination $$e = \{e_1, \ldots, \_{d-1}\}$$ we can easily calculate its cost, $$C(e)$$, by using: $C(e) = \underbrace{\max(A_0, A_1, \ldots, A_{e_1})}_{\text{cost for the } 1^{st} \text{day}} + \underbrace{\max(A_{e_1+1}, A_{e_1+2}, \ldots, A_{e_2})}_{\text{cost for the } 2^{nd} \text{day}} + \ldots + \underbrace{\max(A_{e_{d-1}+1}, A_{e_{d-1}+2}, \ldots, A_{|I|-1})}_{\text{cost for the } d^{th} \text{day}} \label{eq:min_difficulty_job_scheduler:cost_combination}$

#### Generate all combinations

The real challenge at this point concerns the generation of combinations in groups of $$d-1$$ elements. We can generate all the combinations one at a time by using a backtracking algorithm where we try to construct one combination of elements at a time. A possible recursive implementation of such an algorithm is shown in Listing [list:min_difficulty_job_scheduler:combinations]. The function takes as a input two integers $$k$$ and $$l$$. $$k$$ represents the size of the combination and $$l$$ identifies the elements of the combinations i.e. $$0,1,\ldots, l-1$$. If you had to write a generic function for generating combinations you would also most likely have a parameter containing the list of elements from which to generate the combinations. In this case, such a list is implicit as we need to generate combinations of splitting points and can be uniquely identified by a single integer. For instance generates all combinations of three elements from $$\{0,1,2,3,4,5,6,7,8,9\}$$ and generates all combinations of $$2$$ elements from $$\{0,1,2,3\}$$. is a recursive function which enumerates all combinations. It takes the following parameters:

• : the output list of all generated combination,

• : the work-in-progress combination,

• : the size of the combinations

• : the last number we can add to the work-in-progress combination

• : the first number we can add to the combination

Each call tries to place a number in the work-in-progress at a location specified by $$curr_el$$. Initially $$curr_el = 0$$ and each recursive call increases it by $$1$$. Eventually $$curr_el = d$$ and we can stop the recursion and return. At that point the combination is ready and saved into . After each recursive call, the last inserted element is removed from the work-in-progress combination and another number is pushed. The process repeats until there are no more numbers to be pushed.

Listing 1: Function that generates all the combinations of size $k$ from the elements ${0,1,\ldots,l}$

using Combination = std::vector<int>;
using CombinationList = std::vector<Combination>;

void all_combinations_helper(CombinationList& all_combinations, Combination& combination, const unsigned d, const int curr_idx, const unsigned limit)
{
if(combination.size() == d){
all_combinations.push_back(combination);
return;
}

for(size_t i = curr_idx; i < limit ; i++)
{
combination.push_back(i);
all_combinations_helper(all_combinations,combination, d, i+1, limit);
combination.pop_back();
}

}

auto all_combinations(const unsigned d, const unsigned limit)
{
Combination work_in_progress;
work_in_progress.reserve(d);
CombinationList all_combinations;
//limit -1 because the last cut cannot be empty
all_combinations_helper(all_combinations,work_in_progress,d,0,limit-1 );
return all_combinations;
}

#### Wrapping-up

Once we are able to generate all the possible schedules we are going to evaluate the cost associated with each of them, and pick the one with the smallest difficulty overall. All that is left to do at this point is to come up with a way to evaluate a given schedule $$c$$. We have already seen in Equation [eq:min_difficulty_job_scheduler:cost_combination] how a certain combination of $$d-1$$ splitting points maps directly to subarrays of $$I$$. The function in Listing [list:min_difficulty_job_scheduler:combinations] uses this idea to evaluate a schedule and calculate its difficulty by summing up the difficulties of each of the tasks scheduled each day. Note that and identify the elements of $$I$$ in the following range: $$[start, finish]$$ (the element pointed by is included). The function is the driver that is responsible for keeping track of the minimum difficulty among all the processed schedules.

Listing 2: Brute-force solution that works by evaluating all the possible schedules generated using Listing \ref{list:min_difficulty_job_scheduler:combinations}

int calculate_cost_schedule(const std::vector<int>& I,
const std::vector<int>& cutpoints_combo)
{
int ans    = 0;
auto start = std::begin(I);
for (const auto& cutpoint : cutpoints_combo)
{
const auto finish = std::begin(I) + cutpoint + 1;
ans += *std::max_element(start, finish);
start = finish;
}
ans += *std::max_element(start, std::end(I));
return ans;
}

int min_difficulty_scheduler_combinations(const std::vector<int>& I,
const unsigned d)
{
if (I.size() < d)
return -1;

auto all_combinations_cutpoints = all_combinations(d - 1, I.size());
int ans                         = std::numeric_limits<int>::max();
for (const auto& cutpoints_combo : all_combinations_cutpoints)
{
ans = std::min(ans, calculate_cost_schedule(I, cutpoints_combo));
}
return ans;
}

### Dynamic Programming

The key insight needed to solve this problem with DP is that given that you have decided on a set of tasks that are scheduled in the first day, say the first $$i$$ tasks, than the minimum difficulty of a schedule across $$d$$ days having the first $$i$$ elements scheduled the first day is the sum of

the largest difficulty among the first $$i$$ tasks, and

the cost of the best possible schedule of the last $$|I|- i$$ tasks across $$d-1$$ days.

More formally, if $$C(I,d)$$ is a function returning the minimum cost of a schedule of the tasks in $$I$$ across $$d$$ days and can be defined as follows: $\begin{cases} C(\emptyset, 0) = 0 \; : \text{the cost of scheduling 0 task in 0 days is 0}\\ C(\emptyset, d > 0) = +\infty \; : \text{it is impossible to schedule 0 tasks in 1 or more days}\\ C(|I|, 0) = +\infty \: :\text{it is impossible to schedule 1 or more tasks in 0 days}\\\\ C(|I|, d) = \underbrace{\min_{\forall j \in \{0,1,\ldots,|I-1|\}}}_{\text{ \forall schedule of the d^{th} day}} \Bigg( \max I_j + \underbrace{C\Big(I - \{0,1,\ldots,j\}, d-1\Big)}_{\text{optimal solution to a subproblem}}\Bigg)\\ \end{cases} \label{eq:min_difficulty_job_scheduler:dpformula}$ $$C(I,d)$$ has a recursive definition and we can quickly see that the problem has both the properties any DP problem has:

optimal substructure:

can be solved by solving and combining together various optimal solutions to smaller subproblems.

overlapping subproblems:

the same problems are solved over and over again. (try to draw the recursion tree for $$C$$ if you are not entirely convinced)

### Top-down

Without any optimization, the function that we obtain by translating the recursive definition of Equation [eq:min_difficulty_job_scheduler:dpformula] is extremely inefficient due to the fact that problems are recalculated over and over (See Appendix [sect:appendix:DP]). In order to make good use of DP, we can therefore use memoization to avoid unnecessary recomputation. Listing [list:min_difficulty_job_scheduler:solutiondp] shows a possible implementation of this idea where a is used to remember the calls to (the equivalent of the function $$C$$). Note that given $$I$$ and $$d$$ the function can be invoked in $$|I| \times d$$ ways. Therefore, in the worst case scenario, by using memoization we will never make more than $$|I| \times d$$ calls to . Because the cost of a single call to is linear in $$|I|$$ the complexity of the whole algorithm is $$O(|I|^2 d)$$

Listing 3: Dynamic programming top-down solution.

struct KeyHash
{
std::size_t operator()(const std::tuple<int, int>& key) const
{
return std::get<0>(key) ^ std::get<1>(key);
}
};

using Cache = std::unordered_map<std::tuple<int, int>, int, KeyHash>;

long min_difficulty_scheduler_DP_topdown_helper(const std::vector<int>& I,
const size_t start,
const int d,
Cache& cache)
{
if (start >= I.size() && d == 0)
return 0;

const size_t remaining = I.size() - start;
if (remaining < d)
return std::numeric_limits<int>::max();

auto t = std::make_tuple(start, d);
if (auto it = cache.find(t); it != cache.end())
return it->second;

int M    = I[start];
long ans = std::numeric_limits<int>::max();
for (size_t i = start; i < I.size(); i++)
{
M   = std::max(M, I[i]);
ans = std::min(
ans,
M + min_difficulty_scheduler_DP_topdown_helper(I, i + 1, d - 1, cache));
}
cache[t] = ans;
return ans;
}

int min_difficulty_scheduler_DP_topdown(const vector<int>& I, int d)
{
Cache cache;
auto ans = min_difficulty_scheduler_DP_topdown_helper(I, 0, d, cache);
if (ans >= std::numeric_limits<int>::max())
return -1;
return ans;
}

### Bottom-up

In this section, we are going to have a look at how we can implement the DP idea in a bottom-up fashion. Like many bottom-up solutions, we have to come up with a way of filling out a table of values of some sort. For this problem we can use a table $$T$$ of size $$d \times |I|$$ where each element of the table $$T[i][j]$$ will eventually contain the solution to the problem of scheduling the elements of $$|I|$$ up to and including the task at index $$j$$ in exactly $$i$$ days. Clearly, filling some cells of $$T$$ is easier than others. For instance, all the values of the first column of $$T$$ are all filled with a value indicating that the problem has no solution because you cannot schedule any task in $$0$$ days. An exception should be made for $$T[0][0]$$ that is filled with $$0$$ as the cost of scheduling $$0$$ tasks in $$0$$ days is equivalent to the cost of doing nothing. To mark that a subproblem is impossible we can use a large value, or perhaps the largest value a cell of $$T$$ can hold.

Additionally, the values of the first row are also relatively easy to fill in as they contain values that are symmetrically equal to the cells in the first column. Each value of the first row represents a solution to the problem of scheduling some task in $$0$$ days and there is clearly no way that can be done (except for the case when you have $$0$$ task to schedule). An element of the first row $$T[1][j]$$ corresponds to the subproblem of scheduling the first $$j$$ of $$I$$ in exactly one day. Its cost is clearly the maximum difficulty among the elements of $$I$$ from $$0$$ to $$j$$ as there is only one way of scheduling all the $$j+1$$ tasks in a single day.

Things get a bit more interesting when looking at the second row. When we have two days at our disposal to schedule $$j$$ tasks we have more freedom over which task to schedule on the first day and which on the second. The values we just filled for the first column and row can be helpful in making the best decision for the elements of the second row. We can, in fact, fill $$T[2][j]$$ by scheduling one task on the first day, and $$j-1$$ on the second. Or $$2$$ tasks the first day and $$j-2$$ in the second and so on. But which of these divisions is the best? Easy! let’s try them all and see which one yields the smallest difficulty overall. Therefore we can calculate $$T[2][j]$$ as shown in Equation [eq:min_difficulty_schedule:secondRowbottomup]: $T[2][j] = \min_{1 \leq k \leq j-1} \Big( T[1][k] + \big(\ max_{k+1 \leq l \leq j }I_l \big) \Big) \label{eq:min_difficulty_schedule:secondRowbottomup}$ Equation [eq:min_difficulty_schedule:generic_bottom_up] is really saying that we can calculate the minimum difficulty of scheduling the tasks in $$I$$ up to the one having index $$j$$ by calculating the minimum among the costs of scheduling the tasks up to the index $$k \leq j-1$$ on the first day and the rest on the second. The trick here is that we have already calculated all the possible costs of scheduling all possible number of tasks, and thus all we have to do at this step is to calculate the costs of scheduling the tasks from the one having index $$k+1$$ to task $$j$$. This can be done by simply returning the maximum costs among those tasks.

This exact same reasoning can be applied to all the other rows and we can therefore come up with a general formula that can be used to fill the entire table of values as shown in Equation [eq:min_difficulty_schedule:generic_bottom_up]. $T[i][j] = \min_{i-1 \leq k \leq j-1} \Big( T[i-1][k] + \big(\ max_{k+1 \leq l \leq j }I_l \big) \Big) \label{eq:min_difficulty_schedule:generic_bottom_up}$

Clearly the solution to the entire problem is in $$T[d][|I|-1]$$: the cost of scheduling all the elements in $$I$$ in exactly $$d$$ days. Listing [list:min_difficulty_job_scheduler:solutiondpbottomup] shows an implementation of this idea.

Listing 4: Dynamic programming bottom-up solution.

using DPTable = std::vector<std::vector<int>>;

int min_difficulty_scheduler_DP_bottomup(const std::vector<int>& I, int d)
{
const int INF       = std::numeric_limits<int>::max();

return -1;

// initializing values for the first day
int maxV = std::numeric_limits<int>::min();
for (int j = 0; j < num_tasks; j++)
{
maxV    = std::max(maxV, I[j]);
T[0][j] = maxV;
}

for (int i = 1; i < d; i++)
{
for (int j = 0; j < num_tasks; j++)
{
// l is the number of tasks scheduled the previous days i-1 days
// l must be at least i-1 (it is impossible to schedule them otherwise)
for (int l = i - 1; l < j; l++)
{
// elements from [0,l] the scheduled the days before and [l+1,j] today
const auto start_task_dth_day = std::begin(I) + l + 1;
const auto end_task_dth_day   = std::begin(I) + j + 1;
}
This implementation has a space complexity of $$O(d*|I|)$$, but a closer inspection of the code and Equation [eq:min_difficulty_schedule:generic_bottom_up] should make clear that we do not really need to keep all the values for $$T$$ in memory all the time. In fact, all we need is two rows with the values for days $$i$$ and $$i-1$$. This way the complexity goes down to $$O(|I|)$$. The Listing can be easily modified so that it implements this memory saving strategy. We will leave this as an exercise for the reader.