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algorithms

Fun programming puzzles and my algorithmic solutions

Dynamic Programming

  • Pots of Gold
    Given an array of numbers (pots of gold in a line), two players alternate turns choosing from one of the two ends. Whoever gets the larger sum wins. Design an algorithm that plays optimally with the smallest time complexity. Return the scores of the two players

  • pots [9, 7, 5, 3, 1, 0, 2, 4, 6, 8] should return (25, 20)

  • pots [1, 3, 4, 12, 5, 7] might return (22, 10)

  • Longest Palindrome Substring
    Find the longest substring which is a palindrome. If there are multiple longest palindromes, find the one which starts at the earliest index.

    • "xyabcdeedcbaz" should return "abcdeedcba"
    • "xyabcbazdefedw" should return "abcba"

  • Longest Increasing Subsequence
    Given a sequence of numbers, find the longest subsequence with strictly increasing numbers.

  • [10, 1, 3, 9, 2, 7, 5, 6] returns length 4 ([1, 2, 5, 6] or [1, 3, 5, 6])

  • [5, 4, 3, 2, 1] returns length 1 ([1] or [2] or [3] or [4] or [5])

Arrays

  • Maximum Subarray Problem
    Given an array of numbers, find the contiguous subarray with the largest sum. If multiple subarrays have the same value, return the shortest subarray (the subarray with the least number of elements). If multiple subarrays are equally short, return the subarray starting at the earliest index. If there are no positive elements, return None. Otherwise, return the subarray.

  • [3, 2, -6, 2, 9, -12, 10] should return [2, 9]

  • [-4, 12, 3, -5, 4, 3, -2] should return [12, 3, -5, 4, 3]

  • [4, 5, -12, 9] should return [9]

  • [3, 4, -11, 2, 5] should return [3, 4]

  • Maximum Subarray Difference
    Given an array, find two disjoint contiguous subarrays such that the difference between their sums is a maximum.

  • [4, 3, -1, -4, 1, -5, 8, 4, -2, 3, -1] should return 22

  • [-1, 3, 2, 5, 2, -5, 1, -4, -3, -1, 2] should return 24

  • In-Place Array Reversal
    Given an array, reverse the elements using only constant space

  • [1, 2, 3, 4, 5] should return [5, 4, 3, 2, 1]

  • [1, 2, 3, 4] should return [4, 3, 2, 1]

  • Move Zeros to End
    Given an array of numbers, move all the zeros to the end, using constant space

  • [1, 2, 0, 3, 0, 0, 4, 5, 0] => [1, 2, 3, 4, 5, 0, 0, 0, 0]

  • [1, 2, 3] => [1, 2, 3]

  • Next Greater Number
    Given an array, construct an array which, for every index in the original array, has the next greatest number in the original array. If no greater number exists, have that element be None/null.

    • [4, 5, 7, 4, 2, 7, 9, 3, 5] => [5, 7, 9, 7, 7, 9, None, 5, None]

Sorted Arrays

  • Merge Two Sorted Arrays
    • A = [1, 2, 4, 6], B = [2, 3, 5] => [1, 2, 2, 3, 4, 5, 6]
    • A = [1, 1, 2, 4], B = [3, 5, 6] => [1, 1, 2, 3, 4, 5, 6]
  • Merge K Sorted Arrays (in nk log(k) time)
    • [1, 2, 3], [4, 5, 6], [2, 7] => [1, 2, 2, 3, 4, 5, 6, 7]
    • [2, 4], [1, 3, 5, 7], [6, 9], [8] => [1, 2, 3, 4, 5, 6, 7, 8, 9]
  • Find the Median
    Find the median of two sorted arrays of perhaps unequal lengths M and N.
    • A = [3, 5, 7], B = [4, 6, 8, 10] => 6
    • A = [1, 2, 3], B = [4, 5, 6] => 3.5
  • Symmetric Difference
    Given two sorted lists, return a list of elements not in the intersection
    • A = [1, 2, 3, 4], B = [1, 3, 5] => [2, 4, 5]
    • A = [2, 3], B = [1, 3, 3, 3, 3] => [1, 2]

Trees (Recursion)

Although the below inputs are arrays, assume we have functions that convert the arrays (in the specified order) to Node(element, left_subtree, right_subtree).

  • Binary Search Trees (BST)
    Given a tree, determine whether the tree satisfies the BST property: every node on the right subtree must be greater than the current node and every node on the left subtree must be less than the current node
    • (inorder) [[[2, 4, 5], 9, 10], 12, [None, 15, 17]] should return True
    • (inorder) [[1, 3, 5], 7, [9, 11, 10]] should return False
  • Tree Reversal
    Given a tree, switch the left and right children for each node.
    • (preorder) [2, [3, [4, 5, None], 6], [7, 8, 9]] should return [2, [7, 9, 8], [3, 6, [4, None, 5]]]
    • (preorder) ['A', ['B', ['C', 'D', None], None], None] should return ['A', None, ['B', None, ['C', None, 'D']]]

Linked Lists

  • Linked List Reversal
    • (A -> B -> C -> None) should returns (C -> B -> A -> None)
  • Cyclic Linked List
    Determine if the given linked list contains a cycle (loops back to a node that was present earlier in the linked list).
    • A -> B -> C -> (loop back to B) => True
    • A -> B -> C -> None => False
  • Maximum Sum Path
    Given two sorted linked lists, construct the linked list with the maximum sum path. The path must start at the beginning of one of the two linked lists and may only crossover when the value of the two linked lists is the same. Only constant extra space may be used. If there are multiple maximum sum paths, return the path that has the lowest number at the divergence.
    • A = [1, 5, 6, 9], B = [2, 3, 6, 7, 8] should return [1, 5, 6, 7, 8]
    • A = [2, 4, 6], B = [1, 3, 5] should return [2, 4, 6]

Strings

  • Reverse the Words
    • "hello world" should return "world hello"
    • "these words reversed" should return "reversed words these"
  • Matching Delimiters
    Given a string, determine if the delimiters (), {}, [] are matching and valid. That is, there are equal numbers of opening/closing delimiters and for each closing delimiter, it should match the most recent unmatched opening delimiter.
    • "(this[is{true}])" should return True
    • "([)]" should return False
    • "(this[is]false" should return False
  • Valid Interleaving
    Given strings A, B, and C, determine if C is an valid interleaving of A and B. A valid interleaving is one where each character in C from A is in the same order as it was in A, and each character in C from B is in the same order as it was in B.
    • A = 'abcd', B = 'efgh', C = 'aefbgchd' should return True
    • A = 'abc', B = 'abd', C = 'abcdab' should return False
  • Duplicate Characters
    Given a UTF-8 string (or just a string with letters and numbers), find the most frequent character. If there is a tie, return the character that appears first in the string.
    • "abcdae' should return "a"
    • "a2a2b3" should return "a"
  • First Nonrepeated Character
    Given a UTF-8 string, find the first nonrepeated character. If no characters are nonrepeated, return None.
    • "abcdefabde" should return "c"
    • "abcabc" should return None

Sorting

  • Quicksort
  • In-Place Quicksort
  • Merge sort
  • Bucket Sort