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This is a hard difficulty Coding & Algorithms question, commonly asked during Take-home Project rounds at Uber.

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This question is commonly asked for Software Engineer candidates at Uber during technical interviews.

Solve two online assessment problems

Quick Overview

This two-part prompt evaluates algorithmic problem-solving skills: the first problem tests optimization under index-based movement and number-theoretic constraints (prime indices ending with digit 3), while the second assesses tree traversal and multiset/path frequency reasoning for palindrome-anagram checks on rootward paths.

Solve two online assessment problems

Company: Uber

Role: Software Engineer

Category: Coding & Algorithms

Difficulty: hard

Interview Round: Take-home Project

An online assessment contained two coding problems: 1. **Maximize skipped positions** You are given an array of length `n` indexed from `0` to `n - 1`. Start at index `0` and reach index `n - 1`. At any index `i`, you may: - move to `i + 1`, if it exists; or - jump to any index `j > i` such that `j` is a prime number and the decimal representation of `j` ends with `3`. If you jump from `i` to `j`, you gain `j - i - 1` points, equal to the number of positions skipped. Moving one step to the right gives `0` points. Compute the maximum total score achievable when you reach the last index. 2. **Count palindromic-anagram ancestors in a rooted tree** An undirected tree with `tree_nodes` nodes is rooted at node `0`. The nodes are numbered from `0` to `tree_nodes - 1`. Each node `i` stores a lowercase character `arr[i]`. You are also given an array `queries` of length `m`. For each query node `q = queries[i]`, consider every node `v` on the path from `q` to the root, including `q` itself. Count how many such nodes `v` have the property that the characters on the path from `q` up to `v` can be rearranged to form a palindrome. Return an integer array of length `m`, where the `i`-th value is the answer for `queries[i]`. Example: - `tree_nodes = 4` - `arr = ['z', 'a', 'a', 'a']` - edges: `(0, 1)`, `(0, 2)`, `(1, 3)` - `queries = [3]` Output: `[3]` Explanation: For query node `3`, the path segments ending at nodes `3`, `1`, and `0` all have character counts that can be rearranged into a palindrome.

Quick Answer: This two-part prompt evaluates algorithmic problem-solving skills: the first problem tests optimization under index-based movement and number-theoretic constraints (prime indices ending with digit 3), while the second assesses tree traversal and multiset/path frequency reasoning for palindrome-anagram checks on rootward paths.

Part 1: Maximize Skipped Positions

You stand at index 0 of an array of length n and want to reach index n - 1. From index i, you may either move to i + 1 (if it exists) or jump to any index j > i such that j is prime and its decimal representation ends with 3. A jump from i to j earns j - i - 1 points, equal to the number of skipped positions. Moving one step earns 0 points. Compute the maximum total score possible by the time you reach index n - 1. The array values themselves do not matter, so the function receives only n.

Constraints

1 <= n <= 10^6
Indices are 0-based: 0 to n - 1
A jump destination must be prime and must end with digit 3

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Part 2: Count Palindromic-Anagram Ancestors in a Rooted Tree

You are given an undirected tree with tree_nodes nodes, rooted at node 0. Node i stores a lowercase character arr[i]. For each query node q, consider every node v on the path from q up to the root, including q itself. For each such v, look at the characters on the path segment from q up to v, inclusive. Count how many of those path segments can be rearranged to form a palindrome. Return an array of answers in the same order as queries. A multiset of characters can be rearranged into a palindrome if at most one character has odd frequency.

Constraints

1 <= tree_nodes <= 2 * 10^5
0 <= len(queries) <= 2 * 10^5
arr[i] is a lowercase English letter
edges forms a valid tree rooted at node 0

Examples

Input: (1, ['a'], [], [0])

Expected Output: [1]

Explanation: The only path segment is ['a'], which is already a palindrome.

Input: (4, ['z', 'a', 'a', 'a'], [(0, 1), (0, 2), (1, 3)], [3])

Expected Output: [3]

Explanation: For query node 3, the segments 3->3, 3->1, and 3->0 all have at most one odd character count.

Input: (3, ['a', 'b', 'a'], [(0, 1), (1, 2)], [2, 1, 0])

Expected Output: [2, 1, 1]

Explanation: For node 2, segments 'a' and 'aba' work. For node 1, only 'b' works. For node 0, only 'a' works.

Input: (5, ['a', 'b', 'b', 'a', 'c'], [(0, 1), (0, 2), (1, 3), (1, 4)], [3, 4, 2])

Expected Output: [2, 1, 1]

Explanation: Query 3 has valid segments 'a' and 'aba'. Query 4 has only 'c'. Query 2 has only 'b'.

Input: (4, ['a', 'a', 'b', 'b'], [(0, 1), (1, 2), (2, 3)], [3, 3])

Expected Output: [4, 4]

Explanation: All four ancestor segments for node 3 can be rearranged into palindromes, and the query is repeated.

Hints

A string can be permuted into a palindrome iff at most one character count is odd. Represent odd/even counts with a 26-bit mask.
During a DFS from the root, maintain counts of prefix masks along the current root-to-node path so each query can be answered from the current path state.