Problem 1: Unique Snowflake Collection

Problem Statement: At n different times, snowflakes of various shapes fall (represented by distinct integers). We want to collect snowflakes from time a to time b such that no duplicate shapes are collected, and the total number of snowflakes collected is maximized.

Solution Approach: Two Pointers Technique (O(n) time complexity)

• Initialize left and right pointers at the start of the sequence
• Expand the right pointer while the current snowflake hasn't been collected
• If a duplicate is found, record the current window size (right-left+1) as a candidate for maximum
• Move the left pointer forward to exclude the leftmost snowflake
• Continue until the right pointer reaches the end of the sequence

Optimizaton Notes:

• Implement custom max function for performance
• Use fast input/output methods
• Apply discretization to reduce value range

Problem 2: Sequence Duplicate Detection

Problem Statement: Given a sequence defined by a0=1 and ai+1=(A×ai+ai mod B) mod C, find the index of the first duplicate element.

Solution Approach: Iterative Generation with Hash Map Tracking

• Initialize a hash map to track seen elements >li>Generate sequence elemants starting from a0- For each generated element:
  • If not in hash map, add it with a seen flag
  • If already in hash map, return current index as answer

Problem 3: Word Frequency Counting

Problem Statement: Count the frequency of each word in a given document.

Solution Approach: Aho-Corasick Automaton

• Build an automaton structure where each node represents a character in a word
• Track the count of words ending at each node
• Construct failure links for efficient pattern matching
• Process the document through the automaton
• For each word, its frequency is the sum of counts in its failure tree subtree

Problem 4: Second Minimum Spanning Tree

Problem Statement: Find the second minimum spanning tree of a graph, where its total weight is strictly greater than the minimum spanning tree but minimal among all such possibilities.

Solution Approach: Modified Kruskal's Algorithm with Preprocessing

• First, find the minimum spanning tree using Kruskal's algorithm
• Preprocess the tree to record the maximum and second-maximum edge weights between all pairs of nodes
• For each non-tree edge, calculate the potential new tree weight by replacing the maximum edge on the path between its endpoints
• The minimum of these potential weights gives the second minimum spanning tree

Problem 5: Maximum Cycle Value in Directed Graph

Problem Statement: In a directed graph, for each node i, find the minimum value in a cycle that passes through node x, and then determine the maximum of these minimum values across all nodes.

Solution Approach: Bidirectional Dijkstra

• Build two versions of the graph: original and reversed
• From node x, perform Dijkstra's algorithm on both graphs to find shortest paths to all other nodes
• For each node i, the sum of distances from x to i and from i back to x forms a cycle passing through x
• The maximum of these sums across all nodes is the solution

Problem 6: Constrained Color Spanning Tree

Problem Statement: Given an undirected weighted graph with edges colored either black or white, find the minimum weight spanning tree that contains exactly 'need' white edges.

Solution Approach: Binary Search with Modified Kruskal

• Use binary search on the weight adjustment value for white edges
• For each candidate adjustment value:
  • Modify white edge weights by adding the adjustment value
  • Run Kruskal's algorithm to find the minimum spanning tree
  • Count the number of white edges in the resulting tree
• Select the adjustment value that yields a tree with exactly 'need' white edges
• Final tree weight is the sum minus adjustment value multiplied by 'need'