📘 algorithms

1. **Problem Statement:** We are given an array of integers, including both positive and negative numbers. We need to find the contiguous subarray within this array that has the ma

1. **Problem a: Time Complexity Analysis of CleanAndNormalize** The pseudo-code has nested loops: outer loop runs $n$ times, inner loop runs from $i$ to $n$, averaging about $\frac

1. The problem is to analyze the time complexity of the canonical encoding algorithm for a tree with $n$ nodes. 2. The given complexity is $O(n \log n)$, derived from sorting the c

1. **Problem Statement:** We have a variant of the 0-1 knapsack problem where each item $a_i$ can be rejected, fully accepted, or half-accepted. The capacity is $S$, and $V(T,m)$ i

1. **Problem Statement:** We have a variant of the 0-1 knapsack problem with capacity $S$ and $n$ items. Each item $a_i$ can be accepted fully, rejected, or half-accepted. Half-acc

1. **Problem Statement:** We want to find the maximum sum of a decreasing subsequence ending at each index $i$ in an array $A$ of distinct positive integers. 2. **Definition:** Let

1. **Problem Statement:** We have $n$ items to sort using BucketSort with $k = \frac{n}{\log \log n}$ buckets. The input distribution is non-uniform: half the items are uniformly d

1. **Problem Statement:** We have $n$ items to sort using BucketSort with $k = \frac{n}{\log \log n}$ buckets. The input distribution is non-uniform: half the items are uniformly d

1. **Problem statement:** We want to analyze the running time of a dynamic programming algorithm Alg that fills a table of size $n^2$ for an input array of size $n$. 2. **Understan

1. **Problem Statement:** We want to analyze the running time of a dynamic programming algorithm Alg that fills a table of size $n^2$ for an input array of size $n$. 2. **Understan

1. **Problem Statement:** Determine if the sorting algorithm A described is stable. 2. **Recall the definition of stability in sorting:** A sorting algorithm is stable if it preser

1. **Problem Statement:** Traverse the given 6x6 maze from the start cell S at (0,2) to the goal cell G at (5,5) using Depth-First Search (DFS). Walls are blocked cells and cannot

1. **Problem statement:** Solve the recurrence relation $T(n) = T(n - 2) + n^2$ using the Master Theorem step-by-step. 2. **Note:** The Master Theorem applies to recurrences of the

1. **Problem Statement:** Compare the time complexities of BottomUpSort (Merge Sort) and SelectionSort for $n = 32,768$. 2. **Formulas and Rules:**

1. The problem is to solve the "tree from Bottova," which likely refers to a specific mathematical or algorithmic problem involving a tree structure named Bottova. 2. Since the exa

1. Let's define Bubble-Down time for a node of height $h$ as the number of comparisons or swaps needed until the node reaches a leaf. 2. Since at each step in Bubble-Down the node

1. The Bubble-Down operation (also known as Sift-Down) for a node of height $h$ in a heap involves comparing and possibly swapping the node with its children down the tree until th

1. **Problem Statement:** We want to find a contiguous subarray of length $\log n$ in an array of $n$ numbers (where $n$ is a power of 2) that has the maximum sum. 2. **Naive Appro

1. **Stating the problem:** We analyze a variant of the Median-of-Five algorithm where the input of size $n$ is partitioned into $\frac{n}{3}$ blocks of size 3, instead of $\frac{n

1. **Problem statement:** We have a variant of the Median-of-Five algorithm using $n/3$ blocks of size 3 instead of $n/5$ blocks of size 5. Our goals are: - a) Derive a recursive t

1. **Problem statement:** We consider a Median-of-Five variant where the input of size $n$ is partitioned into $\frac{n}{3}$ blocks of size 3 instead of 5. 2. **Part (a) - Deriving