Hi Basti! This link is not working pls help! You are commenting using your WordPress. You are commenting using your Google account. Solutions To Introduction To Algorithms 3rd -Introduction to the design and analysis of algorithms, 2 e anany v. Although these algorithms are discussed later in the book, do not miss this chance to discover them by yourself! You need to indicate how each of the three operations of the priority queue will be implemented.

There are several algorithms for this problem. Keep in mind that the words may contain multiple occurrences of the same letter.

Use binary search see Section 4. When searching in a sorted linked list, stop as soon as an element greater than or equal to the search key is encountered.

For the adjacency matrix representation: i. A graph is complete if and only if all the elements of its adjacency matrix except those on the main diagonal are equal to 1, i.

A graph has a loop if and only if its adjacency matrix has an ele- ment equal to 1 on its main diagonal, i. An undirected, without loops graph has an isolated vertex if and only if its adjacency matrix has an all-zero row. For the adjacency list representation: i. A graph is complete if and only if each of its linked lists contains all the other vertices of the graph.

An undirected, without loops graph has an isolated vertex if and only if one of its adjacency lists is empty. Mark a vertex to serve as the root of the tree, make it the root of the tree to be constructed, and initialize a stack with this vertex.

The second algorithm works as follows. Mark a vertex to serve as the root of the tree, make it the root of the tree to be constructed, and ini- tialize a queue with this vertex.

Repeat the following operations until the queue becomes empty: If there are unmarked vertices adjacent to the vertex at the front of the queue, mark all of them, attach them as children to the front vertex in the tree, and add them to the queue; then dequeue the queue. Finding the largest element is done by simply returning the value of the last element of the sorted array. The key of that vertex will be the largest element in question. Jul 17 Kenyon in Your Kitchen pm — pm Kenyon alumni working in food and drink industries are sharing their go-to happy hour pairings.

Just need a basic editor to be able… Jul 24 Kenyon in Your Kitchen pm — pm Kenyon alumni working in food and drink industries are sharing their go-to happy hour pairings. If you believe it does, draw such a stroll; if you believe it does not, explain why and indicate the smallest number of new bridges that would be required to make such a stroll possible.

II li! Icosian Game A century after Euler's discovery see Problem 4 , another famous puzzle-this one invented by the renown Irish mathematician Sir William Hamilton -was presented to the world under the name of the Icosian Game.

The game was played on a circular wooden board on which the following graph was carved: Find a Hamiltonian circuit-a path that visits all the graph's vertices exactly once before returning to the starting vertex-for this graph. Consider the following problem: Design an algorithm to determine the best route for a subway passenger to take from one designated station to another in a well-developed subway system similar to those in such cities as Washington, D.

The problem's statement is somewhat vague, which is typical of real-life problems. In particular, what reasonable criterion can be used for defining the "best" route? How would you model this problem by a graph? Rephrase the traveling salesman problem in combinatorial object terms.

Rephrase the graph-coloring problem in combinatorial object terms. Consider the following map: i' , I 26 Introduction a. Explain how we can use the graph-coloring problem to color the map so that no two neighboring regions are colored the same.

Use your answer to part a to color the map with the smallest number of colors. Design an algorithm for the following problem: Given a set of n points in the Cartesian plane, determine whether all of them lie on the same circumference. Write a program that reads as its inputs the x, y coordinates of the endpoints of two line segments P 1 Q 1 and P 2 Q 2 and determines whether the segments have a common point.

A data structure can be defined as a particular scheme of organizing related data items. The nature of the data items is dictated by a problem at hand; they can range from elementary data types e. There are a few data structures that have proved to be particularly important for computer algorithms. Since you are undoubtedly familiar with most if not all of them, just a quick review is provided here. Linear Data Structures The two most important elementary data structures are the array and the linked list.

A one-dimensional array is a sequence of n items of the same data type that are stored contiguously in computer memory and made accessible by specifying a value of the array's index Figure 1.

In the majority of cases, the index is an integer either between 0 and n - 1 as shown in Figure 1. Some computer languages allow an array index to range between any two integer bounds low and high, and some even permit nonnumerical indices to specify, for example, data items corresponding to the 12 months of the year by the month names.

This feature positively distinguishes arrays from linked lists see below. It is also assumed that every element of an array occupies the same amount of computer storage. Arrays are used for implementing a variety of other data structures. Promi- nent among them is the string, a sequence of characters from an alphabet termi- nated by a special character indicating the string's end.

Strings composed of zeros and ones are called binary strings or bit strings. Strings are indispensable for pro- cessing textual data, defining computer languages and compiling programs written in them, and studying abstract computational models.

Operations we usually per- form on strings differ from those we typically perform on other arrays say, arrays of numbers.

They include computing the string length, comparing two strings to determine which one precedes the other according to the so-called lexicographic order, i. A linked list is a sequence of zero or more elements called nodes each containing two kinds of information: some data and one or more links called pointers to other nodes of the linked list. A special pointer called "null" is used to indicate the absence of a node's successor.

In a singly linked list, each node except the last one contains a single pointer to the next element Figure 1. To access a particular node of a linked list, we start with the list's first node and traverse the pointer chain until the particular node is reached.

Thus, the time needed to access an element of a singly linked list, unlike that of an array, depends on where in the list the element is located. On the positive side, linked lists do not require any preliminary reservation of the computer memory, and insertions and deletions can be made quite efficiently in a linked list by reconnecting a few appropriate pointers.

We can exploit flexibility of the linked list structure in a variety of ways. For example, it is often convenient to start a linked list with a special node called the header.

This node often contains information about the linked list such as its current length; it may also contain, in addition to a pointer to the first element, a pointer to the linked list's last element. Another extension is the structure called the doubly linked list, in which every node, except the first and the last, contains pointers to both its successor and its predecessor Figure 1. The array and linked list are two principal choices in representing a more abstract data structure called a linear list or simply a list.

The basic operations performed on this data structure are searching for, inserting, and deleting an element. Two special types of lists, stacks and queues, are particularly important. A stack is a list in which insertions and deletions can be done only at the end. This end is called the top because a stack is usually visualized not horizontally but vertically akin to a stack of plates whose "operations" it mimics very closely.

As a result, when elements are added to pushed onto a stack and deleted from popped off it, the structure operates in the "last-in-first-out" LIFO fashion, exactly as the stack of plates does if we can remove only the top plate or add another plate to top of the stack. Stacks have a multitude of applications; in particular, they are indispensable for implementing recursive algorithms.

A queue, on the other hand, is a list from which elements are deleted from one end of the structure, called the front this operation is called dequeue , and new elements are added to the other end, called the rear this operation is called enqueue. Consequently, a queue operates in the "first -in-first-out" FIFO fashion akin, say, to a queue of customers served by a single teller in a bank.

Queues also have many important applications, including several algorithms for graph problems. Many important applications require selection of an item of the highest prior- ity among a dynamically changing set of candidates. A data structure that seeks to satisfy the needs of such applications is called a priority queue. A priority queue is a collection of data items from a totally ordered universe most often, inte- ger or real numbers.

The principal operations on a priority queue are finding its largest element, deleting its largest element, and adding a new element. Of course, a priority queue must be implemented so that the last two operations yield an- other priority queue.

Straightforward implementations of this data structure can be based on either an array or a sorted array, but neither of these options yields the most efficient solution possible. A better implementation of a priority queue is based on an ingenious data structure called the heap.

We discuss heaps and an important sorting algorithm based on them in Section 6. Graphs As mentioned in the previous section, a graph is informally thought of as a collec- tion of points in the plane called "vertices" or "nodes," some of them connected by line segments called "edges" or "arcs. We call the vertices u and v endpoints of the edge u, v and say that u and v are incident to this edge; we also say that the edge u, v is incedent to its endpoints u and v.

A graph G is called undirected if every edge in it is undirected. If a pair of vertices u, v is not the same as the pair v, u , we say that the edge u, v is directed from the vertex u, called the edge's tail, to the vertex v, called the edge's head. We also say that the edge u, v leaves u and enters v.

A graph whose every edge is directed is called directed. Directed graphs are also called digraphs. It is normally convenient to label vertices of a graph or a digraph with letters, integer numbers, or, if an application calls for it, character strings Figure 1.

The graph in Figure 1. The digraph in Figure 1. Our definition of a graph does not forbid loops, or edges connecting vertices to themselves. Unless explicitly stated otherwise, we will consider graphs without loops. We get the largest number of edges in a graph if there is an edge connecting each of its 1 V 1 vertices with all IV I - 1 other vertices. A graph with every pair of its vertices connected by an edge is called complete.

Whether we are dealing with a dense or sparse graph may influence how we choose to represent the graph and, consequently, the running time of an algorithm being designed or used. Graph representations Graphs for computer algorithms can be represented in two principal ways: the adjacency matrix and adjacency lists.

The adjacency matrix of a graph with n vertices is an n-by-n boolean matrix with one row and one column for each of the graph's vertices, in which the element in the ith row and the jth column is equal to 1 if there is an edge from the ith vertex to the jth vertex, and equal to 0 if there is no such edge.

For example, the adjacency matrix for the graph in Figure 1. Note that the adjacency matrix of an undirected graph is always symmetric, i. The adjacency lists of a graph or a digraph is a collection of linked lists, one for each vertex, that contain all the vertices adjacent to the list's vertex i. Usually, such lists start with a header identifying a vertex for which the list is compiled.

For example, Figure 1. To put it another way, adjacency lists indicate columns of the adjacency matrix that, for a given vertex, contain 1 's. If a graph is sparse, the adjacency list representation may use less space than the corresponding adjacency matrix despite the extra storage consumed by pointers of the linked lists; the situation is exactly opposite for dense graphs.

In general, which of the two representations is more convenient depends on the nature of the problem, on the algorithm used for solving it, and, possibly, on the type of input graph sparse or dense. Weighted graphs A weighted graph or weighted digraph is a graph or digraph with numbers assigned to its edges. These numbers are called weights or costs. An interest in such graphs is motivated by numerous real-life applications, such as finding the shortest path between two points in a transportation or communication network or the traveling salesman problem mentioned earlier.

Both principal representations of a graph can be easily adopted to accommo- date weighted graphs. If a weighted graph is represented by its adjacency matrix, then its element A[i, j] will simply contain the weight of the edge from the ith to the jth vertex if there is such an edge and a special symbol, e. Such a matrix is called the weight matrix or cost matrix. This approach is illustrated in Figure l.

For some applications, it is more convenient to put O's on the main diagonal of the adjacency matrix. Adjacency lists for a weighted graph have to include in their nodes not only the name of an adjacent vertex but also the weight of the corresponding edge Figure 1. Paths and cycles Among many interesting properties of graphs, two are impor- tant for a great number of applications: connectivity and acyclicity.

Both are based on the notion of a path. A path from vertex u to vertex v of a graph G can be de- tined as a sequence of adjacent connected by an edge vertices that starts with u and ends with v. If all vertices of a path are distinct, the path is said to be simple.

The length of a path is the total number of vertices in a vertex sequence defining the path minus one, which is the same as the number of edges in the path. For ex- ample, a, c, b, f is a simple path oflength 3 from a to fin the graph of Figure 1. In the case of a directed graph, we are usually interested in directed paths. A directed path is a sequence of vertices in which every consecutive pair of the vertices is connected by an edge directed from the vertex listed first to the vertex listed next.

For example, a, c, e, f is a directed path from a to fin the graph of Figure 1. A graph is said to be connected if for every pair of its vertices u and v there is a path from u to v. Informally, this property means that if we make a model of a connected graph by connecting some balls representing the graph's vertices with strings representing the edges, it will be a single piece.

If a graph is not connected, such a model will consist of several connected pieces that are called connected components of the graph. For example, the graphs of Figures 1. Sa are connected, while the graph in Figure 1. Graphs with several connected components do happen in real-life applica- tions. Introduction to the Design and Analysis of Algorithms, 2nd Edition. Anany Levitin, Villanova University. Availability This title is out of print.

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