Freaks City

This article addresses the problem of finding the coefficient of the $x^n y^m$ term in the expansion of the polynomial $(ax + by)^k$. The solution involves applying the binoimal theorem and calculating combinations modulo 10007. Binomial Theorem Application The binomial theorem states that $(x+y)^k = \sum_{i=0}^{k} \binom{k}{i} x^{k-i} y^i$. In ...

Unique Substrings in Wraparound String Given a string p, find the number of unique non-empty substrings of p that are also substrings of the infinite wraparound string "abcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyz...". The infinite string repeats the alphabet sequence cyclically. DP Solution: We'll use a DP array where dp[i] r ...

0/1 Knapsack Given N items and a knapsack with capacity V, each item can only be selected once. Item i has volume v[i] and value w[i]. Determine which items to select to maximize total value without exceeding the knapsack's volume. Constraints: 0 < N, V ≤ 1000 0 < v[i], w[i] ≤ 1000 Time Complexity: O(N × V) int n, m; int f[100010], w[10 ...

Sorting & Searching Fundamentals Sorting rearranges a array into ascending or descending order. Searching finds the index of a given element. JavaScript provides sort() for sorting and indexOf() for searching, but understanding underlying algorithms is essential for performance tuning and problem-solving. Bubble Sort Array.prototype.bubbleS ...

Given a string s, the objective is to locate and return longest substring that reads the same forwards and backwards. Examples Input: s = "babad" Output: "bab" Note: "aba" is also a valid answer. Input: s = "cbbd" Output: "bb" Input: s = "a" Output: "a" Input: s = "a ...

Given a m x n grid filled with non-negative numbers, find a path from the top-left corner to the bottom-right corner wich minimizes the sum of all numbers along its path. You can only move either down or right at any point in time. Approach: Dynamic Programming This problem is a classic example of dynamic porgramming. The key is to build a solu ...

1648. Sell Diminishing-Valued Colored Balls The objective is to maximize profit when selling balls whose values decrease by 1 after each sale. The optimal approach is a greedy strategy where we always sell the currently most valuable balls available. By sorting the inventory in descending order, we can visualize the stock as columns. We process ...

Understanding the Floyd-Warshall Algorithm The Floyd-Warshall algorithm is a classic dynamic programming approach used to compute the shortest paths between all pairs of vertices in a weighted graph. It operates efficiently on dense graphs where the number of edges is close to the square of the number of vertices. With a time complexity of \\(O ...

Challenging problems require innovative approaches to solve efficiently. Short Colorful Strip Given that n equals m, the final configuration must be a permutation of n. Key observations: When coloring an interval, the smallest color within that interval is always colored first. The coloring operation requires all points in the covered interval ...

Longest Increasing Subsequence (LIS) Problem: Given an unsorted array, find the length of the longest increasing subsequence. Dynamic Programming Approach: Define DP array: Let lis[i] represent the length of the longest increasing subsequence ending at index i. Transition: For each i, iterate through all j < i, and if nums[i] > nums[j], ...