Problem A. Pyramid of Unit Cubes

Transcription

1 Problem A Pyramid of Unit Cubes Consider a N-level pyramid built of unit cubes. An example for N=3 can be seen in the image below. Formally, a pyramid of size N has N levels, where the i-th level (counting from the top) contains an i by i grid of unit cubes. You have K cubes. First, you select a suitable pyramid size as follows: If K is exactly the number of cubes necessary to build a pyramid of size N for some N, you pick that size. Otherwise, you pick the smallest pyramid size you cannot build. Now you start building the pyramid in a systematic bottom-up way. First you build the complete bottom level, then you build the level above that, etc. When building a level, also proceed in a systematic way, starting the next row only when the previous one is full. For example, for 2 cubes you should get the following incomplete pyramid:

2 Given an integer K specifying the number of cubes you have, return the surface area of the possibly incomplete pyramid you will build according to the instructions above. The first line of the input gives an integer T, which is the number of test cases. Each test case contains an integer K ( N,000,000,000). For each test case, output the surface area of the possibly incomplete pyramid you will build according to the instructions above. Sample

3 Problem B Spreading News You are the manager of a company, and you want all of your employees to be notified of an important news item as quickly as possible. Your company is organized in a tree-like structure: each employee has exactly one direct supervisor, no employee is his own direct or indirect supervisor, and every employee is your direct or indirect subordinate. You will make a phone call to each of your direct subordinates, one at a time. After hearing the news, each subordinate must notify each of his direct subordinates, one at a time. The process continues this way until everyone has heard the news. Each person may only call direct subordinates, and each phone call takes exactly one minute. Note that there may be multiple phone calls taking place simultaneously. Compute the minimum amount of time, in minutes, required for this process to be completed. Employees will be numbered starting from, while you will be numbered 0. Furthermore, every supervisor is numbered lower than his or her direct subordinates.s First line of the input contains T the number of test cases. Each test case contains an integer N ( N 70) denoting the number of employees in your company including you. Next line contains N- integer. The i th integer denote the supervisor of i th employee(i starts from ). Look you(employee 0) do not have any supervisor. For each test case output the minimum amount of time, in minutes, required. Sample

4 Problem C Tree Normalization Trees are important data structures in programming. In this problem, you will be given a tree that was constructed from a fully connected undirected acyclic graph with exactly N nodes. First, a distinct number between 0 and N-, inclusive, was assigned to each node. Then, one of the nodes was selected to be the root of the tree. Finally, each non-root node was assigned its neighbor closest to the root as its parent. You will be given the tree as an array, where the i-th element is the parent of the i-th node, or - if the i-th node is the root (indices are 0-based). A tree is considered equivalent to this tree if it can be constructed from the same original graph using the method described above. This means you can renumber the nodes and select a different node as the root (see examples for clarification). an array containing the equivalent tree that comes first lexicographically. First line of the input contains T the number of test cases. Each test case starts with a line containing N ( N 0) the number of nodes in the tree. Next line contains N integer where i th integer is the parent of i-th node or - if the i-th node is the root. For each test case, output the lexicographically equivalent tree in one line (N integers separated by a single space). Sample

5 Problem D Teleporting The Kingdom of Byteland has a big capital city that is the center of its industry and social life. To connect every city to the capital, the King decided to develop a network of roads. Each city, except for the capital, was responsible for building a single bidirectional road from itself to another city. For each city X, the following algorithm was used to determine the destination of the road built by that city: Measure the Euclidean distances from all the cities to the capital, and consider only those which are strictly closer to the capital than city X (include the capital in that list). If there is more than city, pick the city which is closest to city X. If there are multiple such cities, pick the city among them with the smallest X coordinate. If there are still multiple such cities, pick the city among them with the smallest Y coordinate. After all the roads were built, some people started to complain that they had to go through too many other cities to get to the capital. The King decided to construct M teleports to solve this problem. Each teleport would provide an instant connection between the city where it is placed and the capital. Let's define the inconvenience of a city as the minimal number of roads one needs to follow to get from that city to the capital, or to a city with a teleport. For example, the inconvenience of the capital, and of all cities with teleports, is 0. The inconvenience of a city that doesn't have a teleport but is directly connected to the capital or to a city with a teleport is, and so on. Note that the shortest route from a city to the capital may involve traveling further away from the capital to reach a teleport. The inconvenience of the whole Kingdom is the maximum inconvenience among its cities. You will be given the co-ordinates of the cities. Distribute the M teleports in such a way that minimizes the inconvenience of the kingdom and output that minimum inconvenience values. First line of the input contains T the number of test cases. First line of each test case contains M(0 M 4)the number teleports. Next line contains N( N 0) the number of cities. Each of the next N line contains the co-ordinates of N cities. The co-ordinates are denoted by 2 integers X and Y separated by a single space. The first co-ordinates denote the co-ordinates of the capital. For each test case output the minimum inconvenience values.

6 Sample

7 Problem E Binary Search Tree A binary tree is either empty or it consists of a root node and two binary trees, called the left subtree and the right subtree of the root node. Each node of our binary trees will contain one lowercase letter. We say that a binary tree is a binary search tree (BST) if and only if for each node the following conditions hold: All letters in the left subtree of the node occur earlier in the alphabet than the letter in the node. All letters in the right subtree of the node occur later in the alphabet than the letter in the node. Examples of BSTs with 4 nodes: c b a /\ /\ \ b d a d c / / /\ a c b d A pre-order code of a BST is a String obtained in the following way: The pre-order code of an empty BST is an empty string. The pre-order code of a non-empty BST is obtained in the following way: Let L and R be the preorder codes of the left and right subtree, respectively. Then the pre-order code of the whole BST is the concatenation of the letter in its root node, L and R (in this order). The pre-order codes for the trees above are "cbad", "badc" and "acbd", respectively. Consider all BSTs with exactly N nodes containing the first N lowercase letters. Order these trees alphabetically by their pre-order codes. Our sequence of BSTs is one-based, i.e., the index of the first tree in this sequence is. Compute the pre-order code of the BST at the specified index M in this sequence. First line of the input contains T the number of test cases. Each test case consists of two integers N( N 9) and M( M 2,000,000,000). For each test case output the pre-order code of the BST at the specified index M in this sequence. If M is larger than the number of BSTs with exactly N nodes, output a blank line.

8 Sample ba cbad abcdeohgfniljkm dcab dcba cbafed abcdekhfgji ihagfedbcjkl