Problem Description

Alice has a collection of cards, each with a number written on it. To win the game, Alice needs to organize these cards into several groups where each group must have the following properties:

1. Each group contains exactly groupSize cards.
2. The groupSize cards in each group must be consecutive cards by their numbers.

The challenge is to determine if it is possible for Alice to rearrange her cards to meet the criteria above. We are given an integer array hand, which represents the cards Alice has (where each element is the number on the card), and an integer groupSize. The goal is to return true if Alice can rearrange the cards into the desired groups, or false if she cannot.

Intuition

The intuition behind the solution is to count each card and then try to form groups starting with the lowest card value. Here's the thinking process:

1. Count the occurrence of each card value using a Counter data structure. This lets us know how many times each card appears in the hand.

2. Sort the hand array so that we can process the cards from the smallest to the largest. Sorting is crucial because we want to form groups starting with the smallest cards available.

3. Iterate through the sorted cards. For each card value v that still has occurrences left in the counter (i.e., cnt[v] is not zero):

   a. Try to make a group of groupSize starting from v. Check each card value x from v to v + groupSize - 1 to ensure they exist (i.e., their count is greater than zero in the counter).

   b. If any card value x in this range is not found (i.e., cnt[x] is zero), it means we can't form a group starting from v, and we return false.

   c. If the card is found, decrement the count for that card in the counter since it's now part of a group.

4. After attempting to form groups for all card values, if no problems arise, it means it is possible to rearrange the cards into groups satisfying Alice's conditions, and we return true.

By following this process, we can efficiently determine whether or not it is possible to organize Alice's hand into groups of groupSize consecutive cards.

Learn more about Greedy and Sorting patterns.

Solution Approach

The implementation of the solution leverages a few key ideas in order to check if the cards in Alice's hand can be rearranged into groups of consecutive numbers. Here is a step-by-step breakdown of the solution with reference to the algorithms, data structures, or patterns used:

1. Use of Counter from the collections library: The solution begins by counting the frequency of each card using the Counter data structure. This allows us to keep track of how many copies of each card we have and efficiently update the counts as we form groups.

2. Sorting the cards: Before we can form our groups, we sort the array hand. This is essential because we need to look at the smallest card first and attempt to group it with the next groupSize - 1 consecutive card numbers.

3. Iterating through sorted cards: We begin a for loop through each card value v in the sorted hand.

   for v in sorted(hand):

4. Forming groups by checking card availability: For each card v that is present in the counter (i.e., cnt[v] > 0), we look for the next groupSize consecutive numbers. We iterate from v up to v + groupSize, checking if each consecutive card x is available.

   for x in range(v, v + groupSize):

5. Checking and decrementing count: Each time we find the card x is available (i.e., cnt[x] > 0), we decrement its count in our counter to indicate that it is now part of a group. If x is not available (i.e., cnt[x] == 0), we cannot form the required group and return False.

   if cnt[x] == 0:
       return False
   cnt[x] -= 1

6. Optimizing by removing zero counts: Once a card's count reaches zero, we remove it from the counter. This is an optimization step that reduces the size of the counter object as we continue through the cards. It's not strictly necessary for the correctness of the algorithm but can improve performance by reducing the lookup time for the remaining items.

   if cnt[x] == 0:
       cnt.pop(x)

7. Success condition: If we successfully iterate through all the card values without finding a group that cannot be formed (that is, we never return False), it means that it is possible to rearrange the cards as desired. In that case, the function returns True.

By employing this solution approach, we are able to effectively determine the possibility of arranging Alice's cards into the specified groups, exploiting the capabilities of the Counter data structure for efficient counting and updating, as well as the ordered processing of the cards based on their values.

Example Walkthrough

Let's say Alice has the following cards in her hand: [1, 2, 3, 6, 2, 3, 4], and the group size she wants to form is 3. We want to determine if she can organize her cards into groups where each group contains exactly three consecutive cards.

1. Count the occurrences of each card value using Counter:

   • The count will be {1: 1, 2: 2, 3: 2, 4: 1, 6: 1}.

2. Sort the hand array:

   • The sorted hand is [1, 2, 2, 3, 3, 4, 6].

3. Iterate through the sorted cards:

   • Start with the smallest value: 1.
   • Attempt to find 1, 2, 3 for the first group. All are present, so decrement their counts: {2: 1, 3: 1, 4: 1, 6: 1}.
   • The next smallest is 2 (since 1 is no longer there).
   • Attempt to find 2, 3, 4 for the next group. All are present, so decrement their counts: {3: 0, 4: 0, 6: 1}.
   • Counter now contains only {6: 1}, which is not enough to form a group of three consecutive numbers.

4. Encountering an issue forming groups:

   • We are left with the card 6 which cannot form a group with two other consecutive numbers.
   • Therefore, it is not possible for Alice to organize her cards into the required groups.

Following the steps outlined in the solution approach, we checked each card and its availability to form groups, updated the counts, and concluded that the arrangement is not possible. We would return False in this case.

Solution Implementation

Time and Space Complexity

The given Python function isNStraightHand checks whether an array of integers (hand) can be partitioned into groups of groupSize where each group consists of consecutive integers.

Time Complexity

The time complexity of the function can be broken down as follows:

1. Counting Elements (Counter): Constructing a counter for the hand array takes O(N) time, with N being the length of the hand array.

2. Sorting: The sorted(hand) function has a time complexity of O(N log N) since it sorts the array.

3. Iteration Over Sorted Hand: The outer loop runs at most N times. However, within this loop, there is an inner loop that runs up to groupSize times. This results in a total of O(N * groupSize) operations in the worst case.

Therefore, the overall worst-case time complexity of the function is O(N log N + N * groupSize). Since groupSize is a constant, it can be simplified to O(N log N).

Space Complexity

The space complexity pertains to the additional memory used by the algorithm as the input size grows. For this function:

1. Counter Space Usage: The Counter used to count instances of elements in hand will use O(N) space.

2. Sorted Array Space: The sorted() function creates a new list and thus, requires O(N) space as well.

Since both the Counter and the sorted list exist simultaneously, the overall space complexity would also be O(N).

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