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1. Resource Allocation using Binary Search This problem requires determining the minimum capacity needed to partition a set of resources into a specific number of groups. A binary search approach is suitable here. The goal is to find the smallest value x such that the items can be covered by at most k groups, where each group has a capacity lim ...

Problem 1: Optimized Dynamic Programming with Prefix Sums This problem involves a basic dynamic programming approach where we process from the end to the beginning. The naive solution has a time complexity of O(n²), but we can optimize it using prefix sums and binary search. We maintain a prefix sum array and for each position, use binary searc ...

Tree-Based Resource Distribution When distributing a fixed quantity of resources across a tree structure where each node must eventually hold an equal amount, removing any edge partitions the graph into two independent substructures. Let the total resource sum be $S$ and the number of nodes be $N$. The target allocation per node is $k = S / N$. ...

A Given a sequence ${a_n}$, select the largest possible subset such that no two selected elements sum to a prime number. Constraints: $T \leq 4$, $n \leq 750$, $a_i \leq 10^9$. Identical values can be merged into counts. Note that at most one instance of 1 may be included. This becomes a bipartite graph problem: odd numbers connect to the sourc ...