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LCC Data Structures and Algorithms

website·Indexed 29 days ago

https://trayimurti.github.io/lcc/

A collection of educational resources on data structures and algorithms, featuring detailed guides on graph algorithms (BFS, Dijkstra's, Floyd-Warshall, SPFA, Prim's), minimum spanning trees, and dynamic programming. Includes solved competitive programming problems from INOI 2012, 2013, and 2014.

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What's inside LCC

  2. Overview of Dijkstra's Algorithm for shortest paths

    Dijkstra's algorithm is used to find the shortest paths from a single starting node to all other nodes in a graph. It is generally more efficient than the Bellman-Ford algorithm and is suitable for processing large graphs. A critical constraint of Dijkstra's algorithm is that it cannot be used on graphs containing negative weight edges.

  3. Overview of Depth-First Search (DFS) Graph Traversal

    Depth-First Search (DFS) is a graph traversal algorithm that starts at a designated starting node and explores as far as possible along each branch before backtracking. It visits all nodes reachable from the starting node by following the edges of the graph.

  4. Overview of the Bellman-Ford Algorithm

    The Bellman-Ford algorithm computes the shortest paths from a single starting node to all other nodes in a graph. Unlike some other shortest-path algorithms, it can process graphs with negative edge weights. While it cannot find a shortest path if the graph contains a cycle with a negative total length (as the path could infinitely decrease), the algorithm is capable of detecting the presence of such negative cycles.

  5. Understand Spanning Tree fundamentals

    A spanning tree is a subgraph that includes all nodes of the original graph and a subset of its edges such that the subgraph remains connected and acyclic (no cycles). The total weight of a spanning tree is calculated as the sum of the weights of all edges included in the tree.

  6. Understand basic Graph components and terminology

    A graph is a mathematical structure used to represent relationships between objects. It consists of two primary components: Nodes (also called Vertices), which represent the objects, and Edges, which represent the relationships between those objects.

  7. Understand Topological Sorting in Directed Graphs

    A topological sort is a linear ordering of nodes in a directed graph such that for every directed edge from node 'a' to node 'b', node 'a' comes before node 'b' in the ordering.

    Key constraints and properties:

    • Requirement: A directed graph has a topological sort if and only if it is acyclic (a Directed Acyclic Graph or DAG).
    • Cycles: If a graph contains a cycle, a topological sort is impossible because no node in the cycle can precede all others.
    • Uniqueness: Topological sorts are not necessarily unique; a single graph may have multiple valid topological orderings.

  8. Understand the Union-Find Structure (UFS)

    A union-find structure maintains a collection of disjoint sets, ensuring that no element belongs to more than one set. It supports two primary operations, both with a time complexity of O(log n):

    1. Unite: Joins two separate sets into a single set.
    2. Find: Identifies the representative element of the set containing a specific element.

  9. Find the shortest contiguous segment with sum K (Problem Definition)

    Given a list of integers L = [A1, A2, A3, …, AN] where each element Ai ≥ 0, the goal is to find the shortest contiguous segment in the list that adds up to a given integer K. If multiple segments exist with the same minimum length, any one of them can be returned. For example, if L = [2, 1, 3, 1, 4, 4, 1, 8, 1, 7] and K = 9, a valid answer is the segment indices {6, 7} (which corresponds to elements [1, 8]).

  10. Understand the concept of Data Structures

    In computer science, a data structure is a specialized format for organizing, managing, and storing data to enable efficient access and modification. It consists of a collection of data values, the relationships between those values, and the specific functions or operations that can be applied to the data, effectively forming an algebraic structure.

  11. Understand the concept of Algorithms

    An algorithm is a finite sequence of well-defined instructions used to solve a class of problems or perform computations, such as data processing or automated reasoning. A valid algorithm must be expressible within a finite amount of space and time using a formal language. It starts from an initial state and input, proceeds through a finite number of states, and terminates at a final state producing an output. While most are deterministic, randomized algorithms incorporate random input. This differs from a heuristic, which is a problem-solving approach that may not be fully specified or guarantee an optimal result.

  12. Implement Minimum Spanning Tree using Kruskal's Algorithm

    Kruskal's algorithm is used to build a Minimum Spanning Tree (MST) by greedily adding edges to a graph. The process starts with a spanning tree containing only the graph's nodes and no edges. The algorithm then iterates through all edges sorted by weight in ascending order, adding an edge to the tree only if it does not create a cycle.