1152. Analyze User Website Visit Pattern - Detailed Explanation

Problem Statement

You are given three arrays of the same length:

• username: an array of strings where each element represents a user’s name.

• timestamp: an array of integers where each element represents the time a website was visited.

• website: an array of strings where each element represents the website visited.

Each index i indicates that the user username[i] visited website[i] at time timestamp[i].

Task:
Find the 3-sequence pattern (i.e. a list of three websites in the order they were visited) that is visited by the largest number of users.

• For each user, consider all possible sequences of three websites they visited in chronological order. (If a user visited fewer than 3 websites, ignore that user.)

• Count each 3-sequence only once per user—even if the same user has the same sequence more than once.

• If there is more than one 3-sequence with the same highest frequency, return the lexicographically smallest sequence (compare element by element).

Example 1:

Input:

username = ["joe", "joe", "joe", "james", "james", "james", "mary", "mary", "mary"]
timestamp = [1, 2, 3, 4, 5, 6, 7, 8, 9]
website = ["home", "about", "career", "home", "cart", "maps", "home", "about", "career"]

Explanation:

• For joe (visits sorted by timestamp): the only 3-sequence is ["home", "about", "career"].
• For james: the only 3-sequence is ["home", "cart", "maps"].
• For mary: the only 3-sequence is ["home", "about", "career"].

The sequence ["home", "about", "career"] is visited by 2 users (joe and mary) while ["home", "cart", "maps"] is visited by only 1 user.
Output: ["home", "about", "career"]

Example 2:

Input:

username = ["u1", "u1", "u1", "u2", "u2", "u2", "u2"]
timestamp = [10, 20, 30, 5, 15, 25, 35]
website = ["a", "b", "a", "a", "b", "a", "c"]

Explanation:

• u1 visits (in order): ["a", "b", "a"] → Only one 3-sequence: ["a", "b", "a"].
• u2 visits (in order): ["a", "b", "a", "c"] → Possible sequences include:
  ["a", "b", "a"], ["a", "b", "c"], and ["a", "a", "c"] (among others).
  Count each unique sequence per user.

After counting unique sequences across users, the one with the highest frequency is chosen (with lexicographical order used as a tie-breaker).

Constraints:

• The arrays username, timestamp, and website have the same length.
• The length of these arrays is at least 3.
• There is no guarantee that the given data is sorted by timestamp; you must sort the visits by timestamp per user.

Hints to Approach the Problem

1. Group and Sort:
   Start by grouping the website visits by user. For each user, sort their visits based on the timestamp so that you can extract the 3-sequence in the correct order.

2. Generate 3-Sequences:
   For each user, generate all possible combinations (in order) of 3 websites. Use a set per user to avoid counting duplicate sequences for the same user.

3. Count Frequencies:
   Use a map (or dictionary) to count how many unique users have each 3-sequence. Then, determine the 3-sequence with the highest frequency.

4. Lexicographical Order:
   When two sequences have the same frequency, compare them lexicographically to choose the smallest one.

Approaches to Solve the Problem

Approach 1: Brute Force with Grouping and Counting

Steps:

• Grouping:
  Traverse all input arrays simultaneously to group the data by user. For each user, maintain a list of (timestamp, website) pairs.

• Sorting:
  For every user group, sort the list of visits by timestamp.

• Generate Sequences:
  For each user, use three nested loops (or a combinatorial helper) to generate all possible 3-sequences. Add these sequences to a set for that user (to avoid duplicates).

• Frequency Counting:
  Use a hash map to count each unique 3-sequence (keyed by a tuple or list of three websites) over all users. Increment the count only once per user.

• Determine Result:
  Iterate over the map to find the sequence with the maximum count. In case of a tie, choose the lexicographically smallest sequence.

Time Complexity:

• Grouping and sorting: O(n log n) overall (n being the total number of visits).
• Generating sequences per user: For a user with k visits, there are O(k³) sequences; given small constraints (e.g., n ≤ 50), this is acceptable.

Space Complexity:

• Additional space is used for the grouped data, sets, and map. Overall, it is O(n).

Approach 2: Optimized Counting (Still Based on Grouping)

Idea:
Even though the brute force generation of 3-sequences is acceptable under the given constraints, you can still optimize by:

• Reducing duplicate generation by using a set per user.
• Merging the counting step while iterating over the user’s sequences.

This approach is largely similar to Approach 1 since the constraints are small, but it emphasizes careful handling of duplicates and lexicographical comparisons.

Detailed Walkthrough (Visual Example)

Consider the first example:

Input:

username = ["joe", "joe", "joe", "james", "james", "james", "mary", "mary", "mary"]
timestamp = [1, 2, 3, 4, 5, 6, 7, 8, 9]
website = ["home", "about", "career", "home", "cart", "maps", "home", "about", "career"]

1. Grouping and Sorting:

   • joe:
     Visits: [(1, "home"), (2, "about"), (3, "career")]
     Sorted order remains: ["home", "about", "career"]

   • james:
     Visits: [(4, "home"), (5, "cart"), (6, "maps")]
     Sorted order: ["home", "cart", "maps"]

   • mary:
     Visits: [(7, "home"), (8, "about"), (9, "career")]
     Sorted order: ["home", "about", "career"]

2. Generate 3-Sequences per User:

   • For joe, there is only one possible 3-sequence: ("home", "about", "career").
   • For james, one 3-sequence: ("home", "cart", "maps").
   • For mary, one 3-sequence: ("home", "about", "career").

3. Frequency Counting:

   • Count ("home", "about", "career"): appears for joe and mary → frequency = 2.
   • Count ("home", "cart", "maps"): appears for james → frequency = 1.

4. Result Determination:
   The most frequent 3-sequence is ("home", "about", "career").

Code Implementation

Python Code