#960

Hard

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Delete Columns to Make Sorted III

The problem requires minimizing deletions to ensure all strings are lexicographically sorted. Use dynamic programming for optimal solution.

array string dynamic programming

Problem Statement

You are given an array of strings strs, all of the same length. You can delete columns from the strings to make them lexicographically sorted. The goal is to determine the minimum number of column deletions required to ensure that each string is sorted in lexicographical order.

For example, given the input strs = ["babca", "bbazb"], deleting columns 0, 1, and 4 results in strs = ["bc", "az"], both of which are lexicographically sorted. Your task is to compute the fewest deletions required.

Examples

Example 1

Input: strs = ["babca","bbazb"]

Output: 3

After deleting columns 0, 1, and 4, the final array is strs = ["bc", "az"]. Both these rows are individually in lexicographic order (ie. strs[0][0] strs[1] - the array strs is not necessarily in lexicographic order.

Example 2

Input: strs = ["edcba"]

Output: 4

If we delete less than 4 columns, the only row will not be lexicographically sorted.

Example 3

Input: strs = ["ghi","def","abc"]

Output: 0

All rows are already lexicographically sorted.

Constraints

n == strs.length
1 <= n <= 100
1 <= strs[i].length <= 100
strs[i] consists of lowercase English letters.

Solution Approach

State Transition Dynamic Programming

This problem is solved using dynamic programming with a state transition approach. Define a DP table where each entry tracks the minimum number of deletions needed to ensure the strings are sorted up to a specific index. Transition between states is based on comparing the lexicographical order of the strings at each step.

Column Deletion Strategy

The key insight is to delete columns only when necessary. By iterating through all possible column deletion combinations, the DP approach determines the fewest deletions to achieve sorted strings. Consider every combination and ensure that each string remains sorted after each deletion step.

Optimizing with Lexicographical Constraints

Incorporate lexicographical order as a constraint within the DP transitions. For each string, compare its previous state with the current column's value, and delete columns that break the sorted condition. This dynamic programming solution ensures that every column deletion is strategically made to maintain order.

Complexity Analysis

Metric	Value
Time	O(N * W^2)
Space	O(W)

The time complexity of this solution is O(N * W^2), where N is the number of strings and W is the length of each string. This comes from the nested loops for checking all possible column deletions. The space complexity is O(W) due to the DP table used for storing the intermediate states of the columns.

What Interviewers Usually Probe

The candidate demonstrates understanding of dynamic programming by breaking down the problem into subproblems.
The candidate applies state transition logic in the context of lexicographical order for string arrays.
The candidate shows the ability to optimize deletions by evaluating column states.

Common Pitfalls or Variants

Common pitfalls

Overcomplicating the deletion strategy by considering unnecessary combinations of deletions.
Neglecting to consider lexicographical order while transitioning between states in dynamic programming.
Failing to properly implement the dynamic programming table, resulting in incorrect deletions or missing steps.

Follow-up variants

Adjust the problem to allow deletions based on other constraints, such as length or character range.
Extend the problem to consider lexicographical sorting within different subsets of strings.
Explore the possibility of finding a generalized solution for sorting arrays of arbitrary objects under different sorting rules.

FAQ

What is the core approach for solving the Delete Columns to Make Sorted III problem?

The core approach involves dynamic programming with a state transition technique, where each state tracks the minimum deletions needed to maintain lexicographical order.

How do you optimize the number of deletions for the problem?

The optimization is achieved by applying a dynamic programming approach that evaluates the fewest deletions necessary for sorting the strings in lexicographical order.

Can dynamic programming be used for other array-based problems?

Yes, dynamic programming is widely used in array-based problems, especially for optimization tasks that require decision-making based on prior states, like sorting or pathfinding.

What is the expected time complexity for Delete Columns to Make Sorted III?

The time complexity is O(N * W^2), where N is the number of strings and W is the length of each string.

What role does lexicographical order play in this problem?

Lexicographical order ensures that after performing column deletions, the remaining strings maintain the sorted order. This condition is central to the dynamic programming approach.

terminal

Solution

Solution 1: Dynamic Programming

We define $f[i]$ as the length of the longest non-decreasing subsequence ending at column $i$. Initially, $f[i] = 1$, and the final answer is $n - \max(f)$.

1

2

3

4

5

6

7

8

9

class Solution:
    def minDeletionSize(self, strs: List[str]) -> int:
        n = len(strs[0])
        f = [1] * n
        for i in range(n):
            for j in range(i):
                if all(s[j] <= s[i] for s in strs):
                    f[i] = max(f[i], f[j] + 1)
        return n - max(f)

Continue Practicing

#943 find the shortest superstring #902 numbers at most n given digit set #1048 longest string chain