Graphs

Given an m x n 2D binary grid grid which represents a map of '1's (land) and '0's (water), return the number of islands. An island is surrounded by water and is formed by connecting adjacent lands horizontally or vertically. You may assume all four edges of the grid are all surrounded by water.

An n-bit gray code sequence is a sequence of 2n integers where: Every integer is in the inclusive range [0, 2n - 1], The first integer is 0, An integer appears no more than once in the sequence, The binary representation of every pair of adjacent integers differs by exactly one bit, and The binary representation of the first and last integers differs by exactly one bit. Given an integer n, return any valid n-bit gray code sequence.

There is a row of n houses, where each house can be painted one of three colors: red, blue, or green. The cost of painting each house with a certain color is different. You have to paint all the houses such that no two adjacent houses have the same color. The cost of painting each house with a certain color is represented by an n x 3 cost matrix costs. For example, costs[0][0] is the cost of painting house 0 with the color red; costs[1][2] is the cost of painting house 1 with color green, and so on... Return the minimum cost to paint all houses.

There are a row of n houses, each house can be painted with one of the k colors. The cost of painting each house with a certain color is different. You have to paint all the houses such that no two adjacent houses have the same color. The cost of painting each house with a certain color is represented by an n x k cost matrix costs. For example, costs[0][0] is the cost of painting house 0 with color 0; costs[1][2] is the cost of painting house 1 with color 2, and so on... Return the minimum cost to paint all houses.

Suppose you are at a party with n people labeled from 0 to n - 1 and among them, there may exist one celebrity. The definition of a celebrity is that all the other n - 1 people know the celebrity, but the celebrity does not know any of them. Now you want to find out who the celebrity is or verify that there is not one. You are only allowed to ask questions like: "Hi, A. Do you know B? " to get information about whether A knows B. You need to find out the celebrity (or verify there is not one) by asking as few questions as possible (in the asymptotic sense). You are given a helper function bool knows(a, b) that tells you whether a knows b. Implement a function int findCelebrity(n). There will be exactly one celebrity if they are at the party.

Given an integer n, return all the numbers in the range [1, n] sorted in lexicographical order. You must write an algorithm that runs in O(n) time and uses O(1) extra space.

Given an m x n matrix board where each cell is a battleship 'X' or empty '.', return the number of the battleships on board. Battleships can only be placed horizontally or vertically on board. In other words, they can only be made of the shape 1 x k (1 row, k columns) or k x 1 (k rows, 1 column), where k can be of any size. At least one horizontal or vertical cell separates between two battleships (i.e., there are no adjacent battleships).

Given two integers n and k, return the kth lexicographically smallest integer in the range [1, n].

Given an m x n binary matrix mat, return the distance of the nearest 0 for each cell. The distance between two adjacent cells is 1.

There is an m x n grid with a ball. The ball is initially at the position [startRow, startColumn]. You are allowed to move the ball to one of the four adjacent cells in the grid (possibly out of the grid crossing the grid boundary). You can apply at most maxMove moves to the ball. Given the five integers m, n, maxMove, startRow, startColumn, return the number of paths to move the ball out of the grid boundary. Since the answer can be very large, return it modulo 109 + 7.

Given a positive integer, check whether it has alternating bits: namely, if two adjacent bits will always have different values.

You are given an m x n binary matrix grid. An island is a group of 1's (representing land) connected 4-directionally (horizontal or vertical.) You may assume all four edges of the grid are surrounded by water. An island is considered to be the same as another if and only if one island can be translated (and not rotated or reflected) to equal the other. Return the number of distinct islands.

You are given an m x n binary matrix grid. An island is a group of 1's (representing land) connected 4-directionally (horizontal or vertical.) You may assume all four edges of the grid are surrounded by water. An island is considered to be the same as another if they have the same shape, or have the same shape after rotation (90, 180, or 270 degrees only) or reflection (left/right direction or up/down direction). Return the number of distinct islands.

An integer has monotone increasing digits if and only if each pair of adjacent digits x and y satisfy x <= y. Given an integer n, return the largest number that is less than or equal to n with monotone increasing digits.

A virus is spreading rapidly, and your task is to quarantine the infected area by installing walls. The world is modeled as an m x n binary grid isInfected, where isInfected[i][j] == 0 represents uninfected cells, and isInfected[i][j] == 1 represents cells contaminated with the virus. A wall (and only one wall) can be installed between any two 4-directionally adjacent cells, on the shared boundary. Every night, the virus spreads to all neighboring cells in all four directions unless blocked by a wall. Resources are limited. Each day, you can install walls around only one region (i.e., the affected area (continuous block of infected cells) that threatens the most uninfected cells the following night). There will never be a tie.

On an 2 x 3 board, there are five tiles labeled from 1 to 5, and an empty square represented by 0. A move consists of choosing 0 and a 4-directionally adjacent number and swapping it. The state of the board is solved if and only if the board is [[1,2,3],[4,5,0]]. Given the puzzle board board, return the least number of moves required so that the state of the board is solved. If it is impossible for the state of the board to be solved, return -1.

You are given an n x n binary grid board. In each move, you can swap any two rows with each other, or any two columns with each other. Return the minimum number of moves to transform the board into a chessboard board. If the task is impossible, return -1. A chessboard board is a board where no 0's and no 1's are 4-directionally adjacent.

You have two types of tiles: a 2 x 1 domino shape and a tromino shape. You may rotate these shapes. Given an integer n, return the number of ways to tile an 2 x n board. Since the answer may be very large, return it modulo 109 + 7. In a tiling, every square must be covered by a tile. Two tilings are different if and only if there are two 4-directionally adjacent cells on the board such that exactly one of the tilings has both squares occupied by a tile.

An axis-aligned rectangle is represented as a list [x1, y1, x2, y2], where (x1, y1) is the coordinate of its bottom-left corner, and (x2, y2) is the coordinate of its top-right corner. Its top and bottom edges are parallel to the X-axis, and its left and right edges are parallel to the Y-axis. Two rectangles overlap if the area of their intersection is positive. To be clear, two rectangles that only touch at the corner or edges do not overlap. Given two axis-aligned rectangles rec1 and rec2, return true if they overlap, otherwise return false.

Given a positive integer n, find and return the longest distance between any two adjacent 1's in the binary representation of n. If there are no two adjacent 1's, return 0. Two 1's are adjacent if there are only 0's separating them (possibly no 0's). The distance between two 1's is the absolute difference between their bit positions. For example, the two 1's in "1001 " have a distance of 3.

On an 8 x 8 chessboard, there is exactly one white rook 'R' and some number of white bishops 'B', black pawns 'p', and empty squares '.'. When the rook moves, it chooses one of four cardinal directions (north, east, south, or west), then moves in that direction until it chooses to stop, reaches the edge of the board, captures a black pawn, or is blocked by a white bishop. A rook is considered attacking a pawn if the rook can capture the pawn on the rook's turn. The number of available captures for the white rook is the number of pawns that the rook is attacking. Return the number of available captures for the white rook.

You are given an m x n binary matrix grid, where 0 represents a sea cell and 1 represents a land cell. A move consists of walking from one land cell to another adjacent (4-directionally) land cell or walking off the boundary of the grid. Return the number of land cells in grid for which we cannot walk off the boundary of the grid in any number of moves.

You are given an m x n integer matrix grid, and three integers row, col, and color. Each value in the grid represents the color of the grid square at that location. Two squares belong to the same connected component if they have the same color and are next to each other in any of the 4 directions. The border of a connected component is all the squares in the connected component that are either 4-directionally adjacent to a square not in the component, or on the boundary of the grid (the first or last row or column). You should color the border of the connected component that contains the square grid[row][col] with color. Return the final grid.

In a warehouse, there is a row of barcodes, where the ith barcode is barcodes[i]. Rearrange the barcodes so that no two adjacent barcodes are equal. You may return any answer, and it is guaranteed an answer exists.

Given an n x n binary matrix grid, return the length of the shortest clear path in the matrix. If there is no clear path, return -1. A clear path in a binary matrix is a path from the top-left cell (i.e., (0, 0)) to the bottom-right cell (i.e., (n - 1, n - 1)) such that: All the visited cells of the path are 0. All the adjacent cells of the path are 8-directionally connected (i.e., they are different and they share an edge or a corner). The length of a clear path is the number of visited cells of this path.

A stepping number is an integer such that all of its adjacent digits have an absolute difference of exactly 1. For example, 321 is a stepping number while 421 is not. Given two integers low and high, return a sorted list of all the stepping numbers in the inclusive range [low, high].

Given 2 integers n and start. Your task is return any permutation p of (0,1,2.....,2^n -1) such that : p[0] = start p[i] and p[i+1] differ by only one bit in their binary representation. p[0] and p[2^n -1] must also differ by only one bit in their binary representation.

Given a m x n binary matrix mat. In one step, you can choose one cell and flip it and all the four neighbors of it if they exist (Flip is changing 1 to 0 and 0 to 1). A pair of cells are called neighbors if they share one edge. Return the minimum number of steps required to convert mat to a zero matrix or -1 if you cannot. A binary matrix is a matrix with all cells equal to 0 or 1 only. A zero matrix is a matrix with all cells equal to 0.

Given an n x n integer matrix grid, return the minimum sum of a falling path with non-zero shifts. A falling path with non-zero shifts is a choice of exactly one element from each row of grid such that no two elements chosen in adjacent rows are in the same column.

You have a grid of size n x 3 and you want to paint each cell of the grid with exactly one of the three colors: Red, Yellow, or Green while making sure that no two adjacent cells have the same color (i.e., no two cells that share vertical or horizontal sides have the same color). Given n the number of rows of the grid, return the number of ways you can paint this grid. As the answer may grow large, the answer must be computed modulo 109 + 7.

Given an n x n binary grid, in one step you can choose two adjacent rows of the grid and swap them. A grid is said to be valid if all the cells above the main diagonal are zeros. Return the minimum number of steps needed to make the grid valid, or \-1 if the grid cannot be valid. The main diagonal of a grid is the diagonal that starts at cell (1, 1) and ends at cell (n, n).

Given n points on a 2D plane where points[i] = [xi, yi], Return the widest vertical area between two points such that no points are inside the area. A vertical area is an area of fixed-width extending infinitely along the y-axis (i.e., infinite height). The widest vertical area is the one with the maximum width. Note that points on the edge of a vertical area are not considered included in the area.

Given an integer n, find a sequence that satisfies all of the following: The integer 1 occurs once in the sequence. Each integer between 2 and n occurs twice in the sequence. For every integer i between 2 and n, the distance between the two occurrences of i is exactly i. The distance between two numbers on the sequence, a[i] and a[j], is the absolute difference of their indices, |j - i|. Return the lexicographically largest sequence. It is guaranteed that under the given constraints, there is always a solution. A sequence a is lexicographically larger than a sequence b (of the same length) if in the first position where a and b differ, sequence a has a number greater than the corresponding number in b.

You have a cubic storeroom where the width, length, and height of the room are all equal to n units. You are asked to place n boxes in this room where each box is a cube of unit side length. There are however some rules to placing the boxes: You can place the boxes anywhere on the floor. If box x is placed on top of the box y, then each side of the four vertical sides of the box y must either be adjacent to another box or to a wall. Given an integer n, return the minimum possible number of boxes touching the floor.

You are given two integers m and n. Consider an m x n grid where each cell is initially white. You can paint each cell red, green, or blue. All cells must be painted. Return the number of ways to color the grid with no two adjacent cells having the same color. Since the answer can be very large, return it modulo 109 + 7.

You are given an m x n integer matrix points (0-indexed). Starting with 0 points, you want to maximize the number of points you can get from the matrix. To gain points, you must pick one cell in each row. Picking the cell at coordinates (r, c) will add points[r][c] to your score. However, you will lose points if you pick a cell too far from the cell that you picked in the previous row. For every two adjacent rows r and r + 1 (where 0 = 0. -x for x < 0.

You are given an n x n integer matrix. You can do the following operation any number of times: Choose any two adjacent elements of matrix and multiply each of them by -1. Two elements are considered adjacent if and only if they share a border. Your goal is to maximize the summation of the matrix's elements. Return the maximum sum of the matrix's elements using the operation mentioned above.

A room is represented by a 0-indexed 2D binary matrix room where a 0 represents an empty space and a 1 represents a space with an object. The top left corner of the room will be empty in all test cases. A cleaning robot starts at the top left corner of the room and is facing right. The robot will continue heading straight until it reaches the edge of the room or it hits an object, after which it will turn 90 degrees clockwise and repeat this process. The starting space and all spaces that the robot visits are cleaned by it. Return the number of clean spaces in the room if the robot runs indefinetely.

You are given a 0-indexed binary matrix grid. In one operation, you can flip any 1 in grid to be 0. A binary matrix is well-isolated if there is no 1 in the matrix that is 4-directionally connected (i.e., horizontal and vertical) to another 1. Return the minimum number of operations to make grid well-isolated.

You are given an integer n representing the number of playing cards you have. A house of cards meets the following conditions: A house of cards consists of one or more rows of triangles and horizontal cards. Triangles are created by leaning two cards against each other. One card must be placed horizontally between all adjacent triangles in a row. Any triangle on a row higher than the first must be placed on a horizontal card from the previous row. Each triangle is placed in the leftmost available spot in the row. Return the number of distinct house of cards you can build using all n cards. Two houses of cards are considered distinct if there exists a row where the two houses contain a different number of cards.

You are given an integer n. You roll a fair 6-sided dice n times. Determine the total number of distinct sequences of rolls possible such that the following conditions are satisfied: 1. The greatest common divisor of any adjacent values in the sequence is equal to 1. 2. There is at least a gap of 2 rolls between equal valued rolls. More formally, if the value of the ith roll is equal to the value of the jth roll, then abs(i - j) > 2. Return the total number of distinct sequences possible. Since the answer may be very large, return it modulo 109 + 7. Two sequences are considered distinct if at least one element is different.

You are given an m x n integer matrix grid, where you can move from a cell to any adjacent cell in all 4 directions. Return the number of strictly increasing paths in the grid such that you can start from any cell and end at any cell. Since the answer may be very large, return it modulo 109 + 7. Two paths are considered different if they do not have exactly the same sequence of visited cells.

A valid cut in a circle can be: A cut that is represented by a straight line that touches two points on the edge of the circle and passes through its center, or A cut that is represented by a straight line that touches one point on the edge of the circle and its center. Some valid and invalid cuts are shown in the figures below. Given the integer n, return the minimum number of cuts needed to divide a circle into n equal slices.

You are given a positive integer n. Each digit of n has a sign according to the following rules: The most significant digit is assigned a positive sign. Each other digit has an opposite sign to its adjacent digits. Return the sum of all digits with their corresponding sign.

You are given a m x n matrix grid consisting of non-negative integers where grid[row][col] represents the minimum time required to be able to visit the cell (row, col), which means you can visit the cell (row, col) only when the time you visit it is greater than or equal to grid[row][col]. You are standing in the top-left cell of the matrix in the 0th second, and you must move to any adjacent cell in the four directions: up, down, left, and right. Each move you make takes 1 second. Return the minimum time required in which you can visit the bottom-right cell of the matrix. If you cannot visit the bottom-right cell, then return -1.

You are given an m x n binary matrix grid. An island is a group of 1's (representing land) connected 4-directionally (horizontal or vertical.) You may assume all four edges of the grid are surrounded by water. The area of an island is the number of cells with a value 1 in the island. Return the maximum area of an island in grid. If there is no island, return 0.

Given a reference of a node in a [connected](https://en.wikipedia.org/wiki/Connectivity(graphtheory)#Connectedgraph) undirected graph. Return a [deep copy](https://en.wikipedia.org/wiki/Objectcopying#Deep_copy) (clone) of the graph. Each node in the graph contains a value (int) and a list (List[Node]) of its neighbors. class Node { public int val; public List neighbors; } Test case format: For simplicity, each node's value is the same as the node's index (1-indexed). For example, the first node with val == 1, the second node with val == 2, and so on. The graph is represented in the test case using an adjacency list. An adjacency list is a collection of unordered lists used to represent a finite graph. Each list describes the set of neighbors of a node in the graph.

You are given an m x n grid rooms initialized with these three possible values. -1 A wall or an obstacle. 0 A gate. INF Infinity means an empty room. We use the value 231 - 1 = 2147483647 to represent INF as you may assume that the distance to a gate is less than 2147483647. Fill each empty room with the distance to its nearest gate. If it is impossible to reach a gate, it should be filled with INF.

You are given an m x n grid where each cell can have one of three values: 0 representing an empty cell, 1 representing a fresh orange, or 2 representing a rotten orange. Every minute, any fresh orange that is 4-directionally adjacent to a rotten orange becomes rotten. Return the minimum number of minutes that must elapse until no cell has a fresh orange. If this is impossible, return -1.

There is an m x n rectangular island that borders both the Pacific Ocean and Atlantic Ocean. The Pacific Ocean touches the island's left and top edges, and the Atlantic Ocean touches the island's right and bottom edges. The island is partitioned into a grid of square cells. You are given an m x n integer matrix heights where heights[r][c] represents the height above sea level of the cell at coordinate (r, c). The island receives a lot of rain, and the rain water can flow to neighboring cells directly north, south, east, and west if the neighboring cell's height is less than or equal to the current cell's height. Water can flow from any cell adjacent to an ocean into the ocean.

Given an m x n matrix board containing 'X' and 'O', capture all regions that are 4-directionally surrounded by 'X'. A region is captured by flipping all 'O's into 'X's in that surrounded region.

There are a total of numCourses courses you have to take, labeled from 0 to numCourses - 1. You are given an array prerequisites where prerequisites[i] = [ai, bi] indicates that you must take course bi first if you want to take course ai. For example, the pair [0, 1], indicates that to take course 0 you have to first take course 1. Return true if you can finish all courses. Otherwise, return false.

There are a total of numCourses courses you have to take, labeled from 0 to numCourses - 1. You are given an array prerequisites where prerequisites[i] = [ai, bi] indicates that you must take course bi first if you want to take course ai. For example, the pair [0, 1], indicates that to take course 0 you have to first take course 1. Return the ordering of courses you should take to finish all courses. If there are many valid answers, return any of them. If it is impossible to finish all courses, return an empty array.

You have a graph of n nodes labeled from 0 to n - 1. You are given an integer n and a list of edges where edges[i] = [ai, bi] indicates that there is an undirected edge between nodes ai and bi in the graph. Return true if the edges of the given graph make up a valid tree, and false otherwise.

You have a graph of n nodes. You are given an integer n and an array edges where edges[i] = [ai, bi] indicates that there is an edge between ai and bi in the graph. Return the number of connected components in the graph.

In this problem, a tree is an undirected graph that is connected and has no cycles. You are given a graph that started as a tree with n nodes labeled from 1 to n, with one additional edge added. The added edge has two different vertices chosen from 1 to n, and was not an edge that already existed. The graph is represented as an array edges of length n where edges[i] = [ai, bi] indicates that there is an edge between nodes ai and bi in the graph. Return an edge that can be removed so that the resulting graph is a tree of n nodes. If there are multiple answers, return the answer that occurs last in the input.

A transformation sequence from word beginWord to word endWord using a dictionary wordList is a sequence of words beginWord -> s1 -> s2 -> ... -> sk such that: Every adjacent pair of words differs by a single letter. Every si for 1 "hot " -> "dot " -> "dog " -> cog ", which is 5 words long.