Solve pair-counting and account-merging problems

Problem A — Count qualifying products

You are given two integer arrays A and B, and an integer T.

For each element a in A, count how many elements b in B satisfy:

$a \times b \ge T$

Return an array ans of length len(A) where ans[i] is the count for A[i].

Input/Output

• Input: integers arrays A, B, integer T (all non-negative).
• Output: integer array ans.

Constraints (typical interview scale)

• 1 <= len(A), len(B) <= 1e5
• 0 <= A[i], B[i] <= 1e5
• 0 <= T <= 1e10

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Problem B — Merge user accounts by shared identifiers

You are given a list of accounts. Each account is a list of strings:

• The first string is a user name.
• The remaining strings are identifiers (e.g., emails).

Two accounts belong to the same real user if they share at least one identifier (directly or transitively). Merge all accounts that belong to the same user.

Output format

Return the merged accounts as a list where each merged account is:

• [name, id1, id2, ...]
• Identifiers must be unique and sorted lexicographically.
• You may output merged accounts in any order.

Notes

• If multiple accounts with different names get connected via identifiers, assume the name for the merged result can be taken from any one of the connected accounts (state your choice).

Constraints (typical interview scale)

• 1 <= number_of_accounts <= 1e4
• Total number of identifiers across all accounts <= 1e5

Count Products Meeting a Threshold

For each a in A, count how many b in B satisfy a * b >= T. Inputs are non-negative integers.

Constraints

• len(A), len(B) can be up to 1e5
• Values are non-negative

Examples

Example 1

Input: ([2, 4, 0], [1, 3, 5], 10)
Output: [1, 2, 0]

Example 2

Input: ([0, 1], [0, 2], 0)
Output: [2, 2]

Hints

1. Sort B and binary-search the smallest b that works for each a.

Merge Accounts by Shared Identifiers

Each account is [name, id1, id2, ...]. Merge accounts connected by shared identifiers. Identifiers in each merged account are unique and sorted. If connected accounts have different names, choose the lexicographically smallest name; sort merged accounts deterministically by name and identifiers.

Constraints

• Total identifiers <= 1e5

Examples

Example 1

Input: ([['John', 'a@mail', 'b@mail'], ['John', 'b@mail', 'c@mail'], ['Mary', 'z@mail']],)
Output: [['John', 'a@mail', 'b@mail', 'c@mail'], ['Mary', 'z@mail']]

Example 2

Input: ([['B', 'x'], ['A', 'x', 'y'], ['C', 'q']],)
Output: [['A', 'x', 'y'], ['C', 'q']]

Hints

1. Use union-find on identifiers.
2. Sort the final identifiers to make output deterministic.