Problem Statement
The problem involves rendering a sorted array of n items where each item may be part of a designated group indexed from 0 up to m - 1. Items not assigned to any group are marked by -1 in the group array. Each item might also have dependent predecessor items indicated in the beforeItems list, which outlines items that must precede a given item in the sorted list.
The main challenges involve:
- Arranging items so that those belonging to the same group appear consecutively.
- Respecting the prerequisite ordering as specified by
beforeItems.
A valid solution is required where items are sorted following these rules. If there are multiple valid configurations, any one of them can be returned. However, if no arrangement satisfies the conditions, the solution should be an empty list.
Examples
Example 1
Input:
n = 8, m = 2, group = [-1,-1,1,0,0,1,0,-1], beforeItems = [[],[6],[5],[6],[3,6],[],[],[]]
Output:
[6,3,4,1,5,2,0,7]
Example 2
Input:
n = 8, m = 2, group = [-1,-1,1,0,0,1,0,-1], beforeItems = [[],[6],[5],[6],[3],[],[4],[]]
Output:
[]
Explanation:
This is the same as example 1 except that 4 needs to be before 6 in the sorted list.
Constraints
1 <= m <= n <= 3 * 104group.length == beforeItems.length == n-1 <= group[i] <= m - 10 <= beforeItems[i].length <= n - 10 <= beforeItems[i][j] <= n - 1i != beforeItems[i][j]beforeItems[i]does not contain duplicates elements.
Approach and Intuition
The resolution of this problem can be approached through the following steps:
Group Segmentation:
- Separate items based on their group assignments. Items with the same group code should be grouped together. For items not belonging to any group, handle them as individual groups for simplicity in processing.
Graph Representation for Dependencies:
- Construct a graph where each node represents an item, and directed edges represent the
beforeItemsrelationship. This structure helps in visualizing and managing the dependencies.
- Construct a graph where each node represents an item, and directed edges represent the
Topological Sorting within Each Group:
- For each group, perform topological sorting of the items considering the dependencies specified in the
beforeItemsarray. This step will arrange items in a legal order where all precedents of an item are placed before it. - If a cyclic dependency is detected within a group (i.e., topological sorting is impossible), it indicates no valid arrangement can be achieved for that group.
- For each group, perform topological sorting of the items considering the dependencies specified in the
Merging Group Orders:
- Once each group is separately sorted, concatenate the results. The explicit order between different groups doesn't usually matter unless specified by cross-group entries in
beforeItems, which should also be considered.
- Once each group is separately sorted, concatenate the results. The explicit order between different groups doesn't usually matter unless specified by cross-group entries in
Corner Cases and Validations:
- Ensure all specified conditions such as non-duplication in
beforeItems, and valid indexing is maintained. - Check if the merged result respects all inter-item dependencies across the whole set, not just within individual groups.
- Ensure all specified conditions such as non-duplication in
This algorithmic approach leverages graph theory and sorting mechanisms and must be implemented carefully to handle the complexity and scale defined by the constraints, especially given the upper limits of n and m.
Solutions
- Java
class Solution {
public int[] organizeItems(int itemsCount, int initialGroupsCount, int[] itemGroupAssociations, List<List<Integer>> dependencies) {
// Assign unique group ids to ungrouped items.
int currentGroupID = initialGroupsCount;
for (int i = 0; i < itemsCount; i++) {
if (itemGroupAssociations[i] == -1) {
itemGroupAssociations[i] = currentGroupID;
currentGroupID++;
}
}
// Create item-based dependency graph.
Map<Integer, List<Integer>> itemDependencyMap = new HashMap<>();
int[] itemInDegrees = new int[itemsCount];
for (int i = 0; i < itemsCount; ++i) {
itemDependencyMap.put(i, new ArrayList<>());
}
// Create group-based dependency graph.
Map<Integer, List<Integer>> groupDependencyMap = new HashMap<>();
int[] groupInDegrees = new int[currentGroupID];
for (int i = 0; i < currentGroupID; ++i) {
groupDependencyMap.put(i, new ArrayList<>());
}
for (int currentItem = 0; currentItem < itemsCount; currentItem++) {
for (int priorItem : dependencies.get(currentItem)) {
itemDependencyMap.get(priorItem).add(currentItem);
itemInDegrees[currentItem]++;
if (itemGroupAssociations[currentItem] != itemGroupAssociations[priorItem]) {
groupDependencyMap.get(itemGroupAssociations[priorItem]).add(itemGroupAssociations[currentItem]);
groupInDegrees[itemGroupAssociations[currentItem]]++;
}
}
}
List<Integer> sortedItems = sortGraph(itemDependencyMap, itemInDegrees);
List<Integer> sortedGroups = sortGraph(groupDependencyMap, groupInDegrees);
if (sortedItems.isEmpty() || sortedGroups.isEmpty()) {
return new int[0];
}
Map<Integer, List<Integer>> groupedItems = new HashMap<>();
for (Integer item : sortedItems) {
groupedItems.computeIfAbsent(itemGroupAssociations[item], k -> new ArrayList<>()).add(item);
}
List<Integer> combinedResults = new ArrayList<>();
for (int groupID : sortedGroups) {
combinedResults.addAll(groupedItems.getOrDefault(groupID, new ArrayList<>()));
}
return combinedResults.stream().mapToInt(Integer::intValue).toArray();
}
private List<Integer> sortGraph(Map<Integer, List<Integer>> graph, int[] inDegrees) {
List<Integer> sorted = new ArrayList<>();
Stack<Integer> zeroDegreeNodes = new Stack<>();
for (Integer node : graph.keySet()) {
if (inDegrees[node] == 0) {
zeroDegreeNodes.add(node);
}
}
while (!zeroDegreeNodes.isEmpty()) {
Integer current = zeroDegreeNodes.pop();
sorted.add(current);
for (Integer dependent : graph.get(current)) {
inDegrees[dependent]--;
if (inDegrees[dependent] == 0) {
zeroDegreeNodes.add(dependent);
}
}
}
return sorted.size() == graph.size() ? sorted : new ArrayList<>();
}
}
The Java solution provided tackles the complex problem of sorting items by groups while respecting given dependencies. This program manages to assign and sort items into groups based on their dependencies in such a way that no item precedes an item it is dependent upon.
- First, ungrouped items receive unique group IDs. This ensures that each item is assigned to a specific group, facilitating the organization process.
- Two types of dependency graphs are constructed: one for the items and one for the groups. These graphs help track dependencies and are crucial for the topological sorting that follows.
- Dependencies between items are registered, and if these items belong to different groups, the dependencies between groups are also registered.
- The actual sorting is handled by
sortGraph, a function that implements topological sorting using the Kahn's algorithm, an efficient method of sorting nodes in a directed acyclic graph based on their dependencies. - The sorting functions return lists of sorted items and groups. The function checks if any list is empty, which would indicate a cycle or error in input, resulting in returning an empty array.
- Finally, items are combined and ordered according to the sorted group order, ensuring that group dependencies are respected in the final ordering.
This code effectively manages the complexity through careful grouping and ordered processing, ensuring that all dependencies are addressed while producing the organized output of item IDs. The use of a hashmap for mapping dependencies and the stack for managing the nodes to be processed next in the topological sort are key components in handling the dynamically changing data throughout the sorting process.
- Python
class Solution:
def arrangeItems(self, num_items, num_groups, associate_group, predecessors):
new_group_id = num_groups
for idx in range(num_items):
if associate_group[idx] == -1:
associate_group[idx] = new_group_id
new_group_id += 1
dependency_graph = [[] for _ in range(num_items)]
dependency_indegree = [0] * num_items
group_dependency_graph = [[] for _ in range(new_group_id)]
group_indegree = [0] * new_group_id
for current in range(num_items):
for previous in predecessors[current]:
dependency_graph[previous].append(current)
dependency_indegree[current] += 1
if associate_group[current] != associate_group[previous]:
group_dependency_graph[associate_group[previous]].append(associate_group[current])
group_indegree[associate_group[current]] += 1
def topological_sort(graph, indegrees):
result = []
zero_indegree = [node for node in range(len(graph)) if indegrees[node] == 0]
while zero_indegree:
node = zero_indegree.pop()
result.append(node)
for neighbour in graph[node]:
indegrees[neighbour] -= 1
if indegrees[neighbour] == 0:
zero_indegree.append(neighbour)
if len(result) == len(graph):
return result
return []
items_sorted = topological_sort(dependency_graph, dependency_indegree)
groups_sorted = topological_sort(group_dependency_graph, group_indegree)
if not items_sorted or not groups_sorted:
return []
grouped_items = collections.defaultdict(list)
for item in items_sorted:
grouped_items[associate_group[item]].append(item)
final_order = []
for grp in groups_sorted:
final_order.extend(grouped_items[grp])
return final_order
The given Python solution implements a method to sort items by groups respecting dependencies. The method, arrangeItems, takes four parameters: num_items, num_groups, associate_group, and predecessors. Here's a breakdown of the process followed in the code:
Group Assignation: Each item that doesn't have an associated group (marked by
-1) is assigned a new unique group.Graph Creation: Two types of graphs are built:
- Item-level Dependencies: This graph tracks dependencies between items.
- Group-level Dependencies: This graph tracks dependencies between groups.
Topological Sort Function: A helper function,
topological_sort, is used to perform a topological sort on both item and group dependency graphs. The sort helps ensure that no item or group is placed before its prerequisites.Sorting Process:
- Perform a topological sort on the items based on their individual dependencies to get a tentative order.
- Similarly, sort the groups based on their inter-dependencies to get a group order.
Order Validation and Arrangement:
- If any of the sorts fail (i.e., if there's a cycle in the dependencies), the function returns an empty list.
- Using a sorted list of groups, the items within each group are then arranged in the order obtained from the item's topological sort.
Result Compilation: The final order of items across all groups is compiled by extending grouped lists based on the sorted groups.
This solution effectively manages dependencies at both the item and group level, ensuring that all the necessary conditions are met before an item or a group is placed in the final sequence.
No comments yet.