#2350

Hard

auto_awesomeArray scanning plus hash lookup

LeetCode Problem Workspace

Shortest Impossible Sequence of Rolls

Find the shortest subsequence that cannot be formed from a sequence of dice rolls, with efficient array scanning and hash lookup techniques.

array hash table greedy

Problem Statement

You are given a sequence of rolls from a k-sided die and an integer k. Each roll results in a number between 1 and k. You must find the shortest subsequence of rolls that cannot be formed using any combination of the rolls up to a certain point in the sequence.

For example, given rolls = [4, 2, 1, 2, 3, 3, 2, 4, 1], and k = 4, you need to identify the shortest sequence of rolls that is impossible to form from the rolls. The task tests your ability to use array scanning and hash table techniques to efficiently find the solution.

Examples

Example 1

Input: rolls = [4,2,1,2,3,3,2,4,1], k = 4

Output: 3

Every sequence of rolls of length 1, [1], [2], [3], [4], can be taken from rolls. Every sequence of rolls of length 2, [1, 1], [1, 2], ..., [4, 4], can be taken from rolls. The sequence [1, 4, 2] cannot be taken from rolls, so we return 3. Note that there are other sequences that cannot be taken from rolls.

Example 2

Input: rolls = [1,1,2,2], k = 2

Output: 2

Every sequence of rolls of length 1, [1], [2], can be taken from rolls. The sequence [2, 1] cannot be taken from rolls, so we return 2. Note that there are other sequences that cannot be taken from rolls but [2, 1] is the shortest.

Example 3

Input: rolls = [1,1,3,2,2,2,3,3], k = 4

Output: 1

The sequence [4] cannot be taken from rolls, so we return 1. Note that there are other sequences that cannot be taken from rolls but [4] is the shortest.

Constraints

n == rolls.length
1 <= n <= 105
1 <= rolls[i] <= k <= 105

Solution Approach

Array Scanning with Hash Lookup

Iterate through the array while maintaining a hash set to track the possible sequences that can be formed from the current set of rolls. Check for each possible sequence length if it can be constructed from the existing rolls. The smallest length that cannot be formed is the solution.

Greedy Approach

Try to form subsequences starting from the shortest possible length and build upwards. Each time you confirm that a subsequence can be formed, move on to the next possible length. If a subsequence cannot be formed, return that length.

Efficient Hash Table Use

By utilizing a hash table, store the counts of each number from 1 to k as they appear in the sequence. This allows for quick checks to see if a subsequence can be formed, as opposed to checking each combination manually.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity is driven by the number of elements in the rolls and the range of k, as it involves scanning through the list and checking combinations using hash lookups. The space complexity is determined by the size of the hash table used to track possible subsequences.

What Interviewers Usually Probe

Understanding of efficient array scanning and hash table utilization.
Ability to optimize solutions using greedy algorithms.
Clear explanation of time and space complexities in terms of problem constraints.

Common Pitfalls or Variants

Common pitfalls

Overlooking the importance of using a hash table for efficient lookup.
Incorrectly assuming that all subsequences up to a certain length can always be formed.
Failing to optimize the solution for large input sizes, leading to excessive time or space complexity.

Follow-up variants

Change the size of the dice (k) to test scalability of the solution.
Introduce constraints on the number of rolls that can be made, impacting the solution's efficiency.
Consider adding additional conditions, like limiting the range of possible subsequences.

FAQ

What is the pattern to solve the Shortest Impossible Sequence of Rolls problem?

The problem relies on array scanning combined with hash lookups to efficiently track possible subsequences and identify the shortest impossible one.

How can I optimize my solution for larger inputs in the Shortest Impossible Sequence of Rolls problem?

To optimize for larger inputs, use efficient data structures like hash tables to reduce lookup times and avoid checking every possible subsequence manually.

What is the time complexity of solving the Shortest Impossible Sequence of Rolls?

The time complexity depends on the final approach, typically involving scanning the array and checking subsequences with hash lookups, making it dependent on the length of the rolls and k.

What common mistakes should I avoid in the Shortest Impossible Sequence of Rolls problem?

Avoid manual subsequence checking without using efficient data structures, and ensure you don't overlook edge cases where subsequences can't be formed.

How does GhostInterview assist with solving the Shortest Impossible Sequence of Rolls problem?

GhostInterview helps by guiding you through efficient algorithms, offering optimization tips, and ensuring you focus on the key elements of array scanning and hash table usage.

terminal

Solution

Solution 1

#### Python3

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class Solution:
    def shortestSequence(self, rolls: List[int], k: int) -> int:
        ans = 1
        s = set()
        for v in rolls:
            s.add(v)
            if len(s) == k:
                ans += 1
                s.clear()
        return ans

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