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ITCharge Algorithm Solutions

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https://algo.itcharge.cn

A collection of algorithm solutions and complexity analyses for LeetCode problems and fundamental data structures, including topics such as insertion sort, array manipulation, and string processing.

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What's inside ITCharge Algo

  1. Introduction to Counting Dynamic Programming

    Counting DP is a technique used to determine the number of feasible solutions that satisfy specific conditions. Unlike optimization DP, which seeks the best solution, Counting DP focuses on the total count of all valid solutions, ensuring that no solutions are repeated or omitted. It is typically preferred over pure combinatorial mathematics for large-scale problems due to better time and space complexity and the avoidance of calculating high-order combinations.

  2. Overview of the KMP String Matching Algorithm

    The Knuth-Morris-Pratt (KMP) algorithm is used for efficient string matching. Unlike the Brute Force approach, which resets the text pointer upon a mismatch, KMP utilizes a precomputed 'next' array (also known as a partial match table or prefix table). This allows the algorithm to skip unnecessary comparisons by leveraging information about the pattern's own prefixes and suffixes, ensuring the text pointer always moves forward.

  3. Overview of Digit Dynamic Programming (Digit DP)

    Digit DP is a counting-based dynamic programming technique used to solve problems involving the digits of numbers. It is typically used to find the count of numbers in a range [left, right] that satisfy specific conditions, or to find the k-th smallest number satisfying such conditions.

    Key characteristics include:

    1. Large search ranges (e.g., up to $10^9$) that make naive iteration impossible.
    2. Constraints based on the digits of the numbers.
    3. Use of the 'prefix sum' idea: to find the count in [left, right], calculate the count for [0, right] and subtract the count for [0, left - 1].

    Digit DP is most commonly implemented using memoized recursion (DFS) because it simplifies the management of multiple state parameters.

  4. Overview of Horspool String Matching Algorithm

    The Horspool algorithm is a simplified version of the Boyer-Moore algorithm used for finding a substring (pattern) within a larger string (text). It relies exclusively on a simplified 'bad character rule'.

    Key logic during matching:

    1. Compare the pattern with the text from right to left.
    2. If a mismatch occurs, look at the character in the text that aligns with the last character of the current pattern window.
    3. Shift the pattern to the right based on a precomputed shift table (bc_table):
      • If the character exists in the pattern (excluding the last character), shift to align the rightmost occurrence of that character with the end of the window.
      • If the character does not exist in the pattern, shift the pattern by its entire length.

  5. Overview of common sorting algorithms

    Summary of common sorting techniques:

    • Bubble Sort: Repeatedly swaps adjacent elements to move the largest element to the end.
    • Selection Sort: Repeatedly selects the minimum element from the unsorted part and moves it to the sorted part.
    • Insertion Sort: Inserts unsorted elements into their correct position within the sorted part.
    • Shell Sort: A variation of insertion sort that uses gaps to sort elements far apart first.
    • Merge Sort: A divide-and-conquer algorithm that splits the array in half, sorts them, and merges them back.
    • Quick Sort: Picks a pivot element and partitions the array into two sub-arrays based on the pivot.
    • Heap Sort: Uses a heap data structure to repeatedly extract the maximum/minimum element.
    • Counting Sort: Counts occurrences of each unique element to determine positions.
    • Bucket Sort: Distributes elements into buckets and sorts each bucket individually.
    • Radix Sort: Sorts elements digit by digit (positional sorting).

  6. Overview of Two Pointer techniques for Linked Lists

    The Two Pointers technique is used to solve complex linked list problems efficiently by reducing time and space complexity. It is categorized into three main types:

    1. Fast-Slow Pointers (Different Starting Points): The fast pointer starts $n$ steps ahead. When the fast pointer reaches the end, the slow pointer is at a specific position relative to the end (e.g., the $n$-th node from the end).
    2. Fast-Slow Pointers (Different Step Lengths): Both start at the same point, but the fast pointer moves twice as fast as the slow pointer. Used for finding the middle of a list or detecting cycles.
    3. Separate Pointers: Pointers move independently across different lists or from different starting points. Used for merging sorted lists or comparing lists.

  7. Overview of Binary Indexed Tree (Fenwick Tree)

    A Binary Indexed Tree (BIT), or Fenwick Tree, is a data structure used to efficiently calculate prefix sums and range sums of a sequence. It supports dynamic single-point updates and prefix sum queries, both in O(log n) time, with a space complexity of O(n). It leverages the binary representation of indices to decompose prefix sums into multiple sub-interval sums.

  8. Overview of the Sunday String Searching Algorithm

    The Sunday algorithm is an efficient string searching method that improves upon basic matching by using a 'bad character' shift table. Unlike Boyer-Moore, it matches from left to right. When a mismatch occurs, the algorithm looks at the character in the text immediately following the current match window (T[i + m]) to determine how far to shift the pattern string forward.

    There are two shift scenarios:

    1. If the character T[i + m] exists in the pattern, the pattern is shifted so that its rightmost occurrence of that character aligns with T[i + m].
    2. If the character T[i + m] does not exist in the pattern, the pattern is shifted by m + 1 positions (where m is the pattern length).

  9. Overview of Interval Dynamic Programming (Interval DP)

    Interval DP is a dynamic programming method where the state is defined by the left and right endpoints of a range $[i, j]$. It is typically used to solve problems where an operation is performed within a range to minimize cost or maximize value. The core strategy is to solve for the smallest intervals first and iteratively merge them to find the optimal solution for larger intervals, eventually reaching the full range.

  10. Overview of the Sliding Window Algorithm

    The Sliding Window algorithm is an application of the two-pointer technique used on arrays or strings to maintain a window (fixed or variable length) that dynamically tracks an optimal solution within a range. It is primarily used to find continuous subarrays or substrings that satisfy specific conditions, effectively optimizing nested loops into a single loop to reduce time complexity.

  11. Introduction to Probability and Expectation DP

    Probability DP (or Expectation DP) is a class of dynamic programming used to solve problems involving probabilities and mathematical expectations. These methods leverage the linear properties of probability and expectation to establish recurrence relations. The primary challenges in these problems are deriving the state transition equations and applying foundational probability theory.

  12. Overview of Hash Table design and collision strategies

    A Hash Table maps keys to storage locations using a hash function to achieve efficient lookup, insertion, and deletion. High performance depends on two factors:

    1. Hash Function Construction: The goal is to distribute keys uniformly to minimize collisions. Common methods include:

      • Division method (除留余数法)
      • Mid-square method (平方取中法)
      • Radix conversion (基数转换法)
      • Direct addressing (直接定址法)
      • Others: Digital analysis, folding, random number, product, and dot product methods.

    2. Collision Resolution: When different keys map to the same address (Hash Collision), two main strategies are used:

      • Open Addressing (开放地址法): Finds another empty slot via probing (Linear, Quadratic, or Pseudo-random probing).
      • Separate Chaining (链地址法): Stores colliding elements in a linked list at the same slot; this is more flexible and widely used in practice.