1203. Sort Items by Groups Respecting Dependencies

Problem Description

In this problem, we have n items and m groups. Items are assigned to groups and some items must come before others. We need to find a sorted list of the items such that items from the same group are adjacent in the sorted list, and any item that must come before another item does so in this list.

Specifically:

• Items are identified by integers from 0 to n - 1.
• Groups are identified by integers from 0 to m - 1.
• Each item may belong to one group or no group at all (indicated by -1).
• Some items have dependencies, i.e., they must come before certain other items in the sorted list, and these dependencies are given as beforeItems, where beforeItems[i] indicates the list of items that should come before item i.

The goal is to return a sorted list of items fulfilling all the given criteria, or an empty list if no such arrangement is possible.

Intuition

To solve this problem, we need a way to deal with groupings and item dependencies at the same time. One effective approach is to use topological sort, which is used in dependency resolution algorithms where you need to find a linear ordering of nodes based on their dependencies.

Here's the intuition broken down:

1. Handle Group Assignments:

   • Items without a group should be treated as if they are in their own unique group. This can be done by assigning new group numbers starting from m to n - 1 to items without a group.
   • Create a list of items for each group so that we can process them together later.

2. Topological Sorting:

   • Perform a topological sort on the groups themselves to ensure that groups with dependencies are ordered correctly.
   • Perform a topological sort for the items within each group to ensure that items with dependencies are ordered correctly.

3. Creating a Dependency Graph and a Degree List:

   • For each item, have a list (or adjacency list) to represent its dependencies (edges towards other items and groups).
   • For each item and group, calculate the number of incoming edges (in-degree), which represents how many dependencies they have.

4. Building the Solution:

   • Use topological sorting to order the groups. If the groups can't be sorted due to circular dependencies, return an empty list.
   • For each group, use topological sorting to order the items within that group. If the items can't be sorted due to circular dependencies, again return an empty list.
   • If both topological sorts succeed, combine the sorted items from each group according to the sorted group order to form the final list.

5. Resolving Possible Multiple Valid Solutions:

   • The problem statement allows for any valid solution if there is more than one. Hence, we don't need to find all possible solutions but just need to return one if it exists.

The provided Python code is a direct implementation of this intuition. It manages the group assignments, constructs the dependency graphs and in-degree lists, performs topological sorting, and assembles the final sorted list of items based on the sorting of groups and items within those groups.

Learn more about Depth-First Search, Breadth-First Search, Graph and Topological Sort patterns.

Solution Approach

The solution approach implements a topological sort to handle dependency resolution between items and groups. Here is a step-by-step implementation walkthrough:

1. Assign New Groups to Ungrouped Items:

   • Imagine each ungrouped item as its own group. Start assigning new group identifiers, beginning with m and incrementing for each new item without a group.
   • Use this process to create a group_items list that holds lists of items, each representing a group.

2. Creating Graphs and Degree Lists:

   • Prepare two sets of graphs (item_graph and group_graph) and two degree lists (item_degree and group_degree). These data structures are used to represent dependencies between items and dependencies between groups, respectively.
   • Graphs are represented using adjacency lists. In-degree lists (item_degree and group_degree) are used to keep track of the number of incoming edges (dependencies) for each node (item or group).

3. Adding Edges to Graphs:

   • Iterate over the groups and beforeItems list to add dependencies to the graphs.
   • If the current item and the before item belong to the same group, add a dependency edge in the item_graph and increase the item's in-degree.
   • If they belong to different groups, add a dependency edge in the group_graph and increase the group's in-degree.

4. Topological Sort Function:

   • Define the topo_sort() function that performs a topological sort based on a degree list and graph.
   • Use a queue to process the nodes that have an in-degree of 0 (no dependencies).
   • For each processed node, decrease the in-degree of its dependent nodes. If a node's in-degree reaches 0, add it to the queue.
   • If the number of processed nodes is equal to the total items, this implies a valid topological order is found. Otherwise, it implies a cycle, and the function returns an empty list.

5. Sorting Groups:

   • Perform a topological sort on the groups using the group_graph and group_degree.
   • If the topological sort does not return a valid order, it indicates that a cycle exists between groups and there is no valid solution, so return an empty list.

6. Sorting Items Within Each Group:

   • Iterate over each group based on the order determined by the group topological sort.
   • Perform a topological sort on the items of the current group based on the item_graph and item_degree.
   • If the topological sort returns a valid order for items within the group, append this order to the final answer.

7. Final Answer:

   • After processing all groups and their corresponding items, assemble the final sorted list of items. Ensure that if there is a valid topological order for both items and groups, this list is returned.

By using this approach, the implementation handles both items' and groups' internal dependencies. The defined topo_sort function processes items within groups and the groups themselves, resolving these dependencies and obtaining the final sorted list of items.

Example Walkthrough

Let's consider a small example with n = 5 items and m = 2 groups to walk through the solution approach.

Suppose we have the following group assignment:

• Item 0 belongs to group 0.
• Item 1 belongs to group 0.
• Item 2 belongs to group 1.
• Item 3 has no group (indicated by -1).
• Item 4 has no group (indicated by -1).

And we have these dependencies in beforeItems:

• beforeItems[0] = [2] (Item 2 must come before Item 0).
• beforeItems[1] = [] (No dependencies for Item 1).
• beforeItems[2] = [] (No dependencies for Item 2).
• beforeItems[3] = [1,4] (Items 1 and 4 must come before Item 3).
• beforeItems[4] = [] (No dependencies for Item 4).

Step 1: Assign New Groups to Ungrouped Items

Items 3 and 4 don't belong to any group, so we assign them to new groups:

• Item 3 will belong to group 2 (since m = 2).
• Item 4 will belong to group 3.

The group_items list will be:

• Group 0: [0, 1]
• Group 1: [2]
• Group 2: [3]
• Group 3: [4]

Step 2: Creating Graphs and Degree Lists

We create two separate graphs (item_graph and group_graph) and degree lists (item_degree and group_degree) for dependency representation.

Our graphs and degree lists start off empty, but will be populated as follows:

• item_graph: {0: [], 1: [], 2: [], 3: [], 4: []}
• item_degree: {0: 0, 1: 0, 2: 0, 3: 0, 4: 0}
• group_graph: {0: [], 1: [], 2: [], 3: []}
• group_degree: {0: 0, 1: 0, 2: 0, 3: 0}

Step 3: Adding Edges to Graphs

We add the necessary edges from the beforeItems specification:

• Add an edge from group 1 to group 0 because Item 2 (in group 1) must come before Item 0 (in group 0). This also increases the in-degree of group 0.
• For Item 3, add edges from Items 1 and 4 to Item 3 in item_graph. This increases the in-degree of Item 3 by 2.

Updated graphs and in-degree lists after this step:

• item_graph: {0: [], 1: [], 2: [], 3: [1, 4], 4: []}
• item_degree: {0: 0, 1: 0, 2: 0, 3: 2, 4: 0}
• group_graph: {0: [1], 1: [], 2: [0, 3], 3: [2]}
• group_degree: {0: 1, 1: 0, 2: 0, 3: 1}

Step 4: Topological Sort Function

We aim to sort both the groups and the items within groups topologically using the topo_sort() function.

Step 5: Sorting Groups

Using topo_sort() on groups, we can find an order, for example, [1, 3, 2, 0], based on the dependencies (group 1 has no dependencies, followed by group 3, and so on).

Step 6: Sorting Items Within Each Group

Following the sorted groups, we proceed to sort the items within each group:

• Group 1: Item 2 has no dependencies.
• Group 3: Item 4 has no dependencies.
• Group 2: Item 3 can be sorted directly as its dependencies are from other groups.
• Group 0: Item 1 has no dependencies, and Item 0 comes after Item 2 from another group.

Step 7: Final Answer

After sorting, we combine the lists of items within the sorted groups to get the final answer, which could be [2, 4, 3, 1, 0].

This reflects that the items within the same group are adjacent, and dependencies are preserved. If any step in this process had found a cycle (a dependency that can't be resolved), we would have returned an empty list instead.

Solution Implementation

Time and Space Complexity

Time Complexity

The algorithm involves several steps with different complexities. Below is the breakdown:

1. Initialization of Data Structures: Initializing group_items, item_degree, group_degree, item_graph, and group_graph involves iterating over n and m, resulting in O(n + m) time.

2. Setting up degrees and graphs:

   • For each item from 0 to n, the algorithm checks the groups of items in beforeItems list, which, in the worst case, may include all other items, leading to O(n^2) time.
   • This is done for items and groups, so the worst-case time complexity here is O(n^2 + m^2), considering groups could be as numerous as items.

3. Topological Sort:

   • The function topo_sort performs a BFS-like algorithm which, in the worst case, processes each vertex and each edge once. For items, there are n vertices and, in the worst case, n*(n-1) edges for a dense graph. This amounts to O(n + n*(n-1)) which is O(n^2).
   • For groups, there are n + m vertices and, in the worst case, (n + m)*(n + m - 1) edges. The complexity for groups, therefore, is O((n + m) + (n + m)*(n + m - 1)) which simplifies to O(n^2 + m^2).

Combining these parts, the overall worst-case time complexity of the algorithm is O(n^2 + m^2).

Space Complexity

The space complexity is determined by the amount of memory allocated for variables and data structures in the algorithm, which relates to the following:

1. Data Structures for Graphs and Degrees:

   • group_items: O(n + m)
   • item_degree, group_degree: O(n + m)
   • item_graph, group_graph: These are adjacency lists, which, in the worst case, can store n*(n-1) edges for items and (n + m)*(n + m - 1) edges for groups, resulting in O(n^2) and O((n + m)^2) space respectively.

2. Queue in topo_sort: In the worst case, where all vertices are put into the queue before getting dequeued, we need O(n) space for items and O(n + m) space for groups.

3. Res and ans Lists: These lists will contain exactly n and n + m elements respectively, providing a space complexity of O(n) and O(n + m).

Combining the space complexities for the data structures and the temporary lists, the overall worst-case space complexity of the algorithm is O(n^2 + m^2).

Learn more about how to find time and space complexity quickly using problem constraints.