Analysis of Algorithms

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1 Algorithm An algorithm is a procedure or formula for solving a problem, based on conducting a sequence of specified actions. A computer program can be viewed as an elaborate algorithm. In mathematics and computer science, an algorithm usually means a small procedure that solves a recurrent problem. Algorithms are widely used throughout all areas of IT (information technology). A search engine algorithm, for example, takes search strings of keywords and operators as input, searches its associated database for relevant web pages, and returns results. Analysis of Algorithms Why performance analysis? There are many important things that should be taken care of, like user friendliness, modularity, security, maintainability, etc. Why to worry about performance? The answer to this is simple, we can have all the above things only if we have performance. Given two algorithms for a task, how do we find out which one is better? One naive way of doing this is implement both the algorithms and run the two programs on your computer for different inputs and see which one takes less time. There are many problems with this approach for analysis of algorithms. 1) It might be possible that for some inputs, first algorithm performs better than the second. And for some inputs second performs better. 2) It might also be possible that for some inputs, first algorithm perform better on one machine and the second works better on other machine for some other inputs. Asymptotic Analysis is the big idea that handles above issues in analyzing algorithms. In Asymptotic Analysis, we evaluate the performance of an algorithm in terms of input size (we don t measure the actual running time). We calculate, how does the time (or space) taken by an algorithm increases with the input size. For example, let us consider the search problem (searching a given item) in a sorted array. One way to search is Linear Search (order of growth is linear) and other way is Binary Search (order of growth is logarithmic). To understand how Asymptotic Analysis solves the above mentioned problems in analyzing algorithms, let us say we run the Linear Search on a fast computer and Binary Search on a slow computer. For small values of input array size n, the fast computer may take less time. But, after certain value of input array size, the Binary Search will definitely start taking less time compared to the Linear Search even though the Binary Search is being run on a slow machine. The reason is the order of growth of Binary Search with respect to input size logarithmic while the order of growth of Linear Search is linear. So the machine dependent constants can always be ignored after certain values of input size.

2 Growth Rates lgorithms analysis is all about understanding growth rates. That is as the amount of data gets bigger, how much more resource will my algorithm require? Typically, we describe the resource growth rate of a piece of code in terms of a function. To help understand the implications, this section will look at graphs for different growth rates from most efficient to least efficient. Constant Growth Rate A constant resource need is one where the resource need does not grow. That is processing 1 piece of data takes the same amount of resource as processing 1 million pieces of data. The graph of such a growth rate looks like a horizontal line Logarithmic Growth Rate A logarithmic growth rate is a growth rate where the resource needs grows by one unit each time the data is doubled. This effectively means that as the amount of data gets bigger, the curve describing the growth rate gets flatter (closer to horizontal but never reaching it). The following graph shows what a curve of this nature would look like.

3 Linear Growth Rate A linear growth rate is a growth rate where the resource needs and the amount of data is directly proportional to each other. That is the growth rate can be described as a straight line that is not horizontal.

4 Log Linear A log linear growth rate is a slightly curved line. the curve is more pronounced for lower values than higher ones Quadratic Growth Rate A quadratic growth rate is one that can be described by a parabola.

5 Cubic Growth Rate While this may look very similar to the quadratic curve, it grows significantly faster

6 Exponential Growth Rate An exponential growth rate is one where each extra unit of data requires a doubling of resource. As you can see the growth rate starts off looking like it is flat but quickly shoots up to near vertical (note that it can't actually be vertical) Depth First Search (DFS) algorithm traverses a graph in a depthward motion and uses a stack to remember to get the next vertex to start a search, when a dead end occurs in any iteration. Tree Algorithms Vocabulary and Definitions Node Edge A node is a fundamental part of a tree. It can have a name, which we call the key. A node may also have additional information. We call this additional information the payload. While the payload information is not central to many tree algorithms, it is often critical in applications that make use of trees.

7 Root An edge is another fundamental part of a tree. An edge connects two nodes to show that there is a relationship between them. Every node (except the root) is connected by exactly one incoming edge from another node. Each node may have several outgoing edges. The root of the tree is the only node in the tree that has no incoming edges. In Figure Figure 2, / is the root of the tree. Path A path is an ordered list of nodes that are connected by edges. For example, Mammal Carnivora Felidae Felis Domestica is a path. Children The set of nodes cc that have incoming edges from the same node to are said to be the children of that node. In Figure Figure 2, nodes log/, spool/, and yp/ are the children of node var/. Parent A node is the parent of all the nodes it connects to with outgoing edges. In Figure 2 the node var/ is the parent of nodes log/, spool/, and yp/. Sibling Nodes in the tree that are children of the same parent are said to be siblings. The nodes etc/ and usr/ are siblings in the filesystem tree. Subtree A subtree is a set of nodes and edges comprised of a parent and all the descendants of that parent. Leaf Node A leaf node is a node that has no children. For example, Human and Chimpanzee are leaf nodes in Figure 1. Level The level of a node nn is the number of edges on the path from the root node to nn. For example, the level of the Felis node in Figure 1 is five. By definition, the level of the root node is zero. Height The height of a tree is equal to the maximum level of any node in the tree. The height of the tree in Figure 2is two. With the basic vocabulary now defined, we can move on to a formal definition of a tree. In fact, we will provide two definitions of a tree. One definition involves

8 nodes and edges. The second definition, which will prove to be very useful, is a recursive definition. Definition One: A tree consists of a set of nodes and a set of edges that connect pairs of nodes. A tree has the following properties: One node of the tree is designated as the root node. Every node, except the root node, is connected by an edge from exactly one other node, where the parent is root node.. A unique path traverses from the root to each node. If each node in the tree has a maximum of two children, we say that the tree is a binary tree. Figure 3 illustrates a tree that fits definition one. The arrowheads on the edges indicate the direction of the connection. Definition Two: A tree is either empty or consists of a root and zero or more subtrees, each of which is also a tree. The root of each subtree is connected to the root of the parent tree by an edge. Figure 4 illustrates this recursive definition of a tree. Using the recursive definition of a tree, we know that the tree in Figure 4has at least four nodes, since each of the triangles representing a subtree must have a root. It may have many more nodes than that, but we do not know unless we look deeper into the tree.

9 BST Basic Operations The basic operations that can be performed on a binary search tree data structure, are the following Insert Inserts an element in a tree/create a tree. Search Searches an element in a tree. Preorder Traversal Traverses a tree in a pre-order manner. Inorder Traversal Traverses a tree in an in-order manner. Postorder Traversal Traverses a tree in a post-order manner. We shall learn creating (inserting into) a tree structure and searching a data item in a tree in this chapter. We shall learn about tree traversing methods in the coming chapter. Insert Operation The very first insertion creates the tree. Afterwards, whenever an element is to be inserted, first locate its proper location. Start searching from the root node, then if the data is less than the key value, search for the empty location in the left subtree and insert the data. Otherwise, search for the empty location in the right subtree and insert the data. Traversal is a process to visit all the nodes of a tree and may print their values too. Because, all nodes are connected via edges (links) we always start from the root (head) node. That is, we cannot randomly access a node in a tree. There are three ways which we use to traverse a tree

10 In-order Traversal Pre-order Traversal Post-order Traversal Generally, we traverse a tree to search or locate a given item or key in the tree or to print all the values it contains. In-order Traversal In this traversal method, the left subtree is visited first, then the root and later the right sub-tree. We should always remember that every node may represent a subtree itself. If a binary tree is traversed in-order, the output will produce sorted key values in an ascending order. We start from A, and following in-order traversal, we move to its left subtree B. B is also traversed in-order. The process goes on until all the nodes are visited. The output of inorder traversal of this tree will be D B E A F C G Algorithm Until all nodes are traversed Step 1 Recursively traverse left subtree. Step 2 Visit root node. Step 3 Recursively traverse right subtree.

11 Pre-order Traversal In this traversal method, the root node is visited first, then the left subtree and finally the right subtree. We start from A, and following pre-order traversal, we first visit A itself and then move to its left subtree B. B is also traversed pre-order. The process goes on until all the nodes are visited. The output of pre-order traversal of this tree will be A B D E C F G Algorithm Until all nodes are traversed Step 1 Visit root node. Step 2 Recursively traverse left subtree. Step 3 Recursively traverse right subtree. Post-order Traversal In this traversal method, the root node is visited last, hence the name. First we traverse the left subtree, then the right subtree and finally the root node.

12 We start from A, and following Post-order traversal, we first visit the left subtree B. B is also traversed post-order. The process goes on until all the nodes are visited. The output of postorder traversal of this tree will be D E B F G C A Algorithm Until all nodes are traversed Step 1 Recursively traverse left subtree. Step 2 Recursively traverse right subtree. Step 3 Visit root node. Graph Data Structure A graph is a pictorial representation of a set of objects where some pairs of objects are connected by links. The interconnected objects are represented by points termed as vertices, and the links that connect the vertices are called edges. Formally, a graph is a pair of sets (V, E), where V is the set of vertices and Eis the set of edges, connecting the pairs of vertices. Take a look at the following graph

13 In the above graph, V = {a, b, c, d, e} E = {ab, ac, bd, cd, de} Graph Data Structure Mathematical graphs can be represented in data structure. We can represent a graph using an array of vertices and a two-dimensional array of edges. Before we proceed further, let's familiarize ourselves with some important terms Vertex Each node of the graph is represented as a vertex. In the following example, the labeled circle represents vertices. Thus, A to G are vertices. We can represent them using an array as shown in the following image. Here A can be identified by index 0. B can be identified using index 1 and so on. Edge Edge represents a path between two vertices or a line between two vertices. In the following example, the lines from A to B, B to C, and so on represents edges. We can use a two-dimensional array to represent an array as shown in the following image. Here AB can be represented as 1 at row 0, column 1, BC as 1 at row 1, column 2 and so on, keeping other combinations as 0. Adjacency Two node or vertices are adjacent if they are connected to each other through an edge. In the following example, B is adjacent to A, C is adjacent to B, and so on. Path Path represents a sequence of edges between the two vertices. In the following example, ABCD represents a path from A to D.

14 Basic Operations Following are basic primary operations of a Graph Add Vertex Adds a vertex to the graph. Add Edge Adds an edge between the two vertices of the graph. Display Vertex Displays a vertex of the graph

15 As in the example given above, DFS algorithm traverses from S to A to D to G to E to B first, then to F and lastly to C. It employs the following rules. Rule 1 Visit the adjacent unvisited vertex. Mark it as visited. Display it. Push it in a stack. Rule 2 If no adjacent vertex is found, pop up a vertex from the stack. (It will pop up all the vertices from the stack, which do not have adjacent vertices.) Rule 3 Repeat Rule 1 and Rule 2 until the stack is empty. Step Traversal Description Initialize the stack. 1.

16 2. Mark S as visited and put it onto the stack. Explore any unvisited adjacent node from S. We have three nodes and we can pick any of them. For this example, we shall take the node in an alphabetical order. 3. Mark A as visited and put it onto the stack. Explore any unvisited adjacent node from A. Both Sand D are adjacent to A but we are concerned for unvisited nodes only. 4. Visit D and mark it as visited and put onto the stack. Here, we have B and C nodes, which are adjacent to D and both are unvisited. However, we shall again choose in an alphabetical order.

17 We choose B, mark it as visited and put onto the stack. Here Bdoes not have any unvisited adjacent node. So, we pop Bfrom the stack. 5. We check the stack top for return to the previous node and check if it has any unvisited nodes. Here, we find D to be on the top of the stack. 6. Only unvisited adjacent node is from D is C now. So we visit C, mark it as visited and put it onto the stack. 7. Greedy Algorithms Huffman coding is a lossless data compression algorithm. The idea is to assign variablelength codes to input characters; lengths of the assigned codes are based on the frequencies of corresponding characters. The most frequent character gets the smallest code and the least frequent character gets the largest code. The variable-length codes assigned to input characters are Prefix Codes, means the codes (bit

18 sequences) are assigned in such a way that the code assigned to one character is not prefix of code assigned to any other character. This is how Huffman Coding makes sure that there is no ambiguity when decoding the generated bit stream. Let us understand prefix codes with a counter example. Let there be four characters a, b, c and d, and their corresponding variable length codes be 00, 01, 0 and 1. This coding leads to ambiguity because code assigned to c is prefix of codes assigned to a and b. If the compressed bit stream is 0001, the de-compressed output may be cccd or ccb or acd or ab. See this for applications of Huffman Coding. There are mainly two major parts in Huffman Coding 1) Build a Huffman Tree from input characters. 2) Traverse the Huffman Tree and assign codes to characters. Steps to build Huffman Tree Input is array of unique characters along with their frequency of occurrences and output is Huffman Tree. 1. Create a leaf node for each unique character and build a min heap of all leaf nodes (Min Heap is used as a priority queue. The value of frequency field is used to compare two nodes in min heap. Initially, the least frequent character is at root) 2. Extract two nodes with the minimum frequency from the min heap. 3. Create a new internal node with frequency equal to the sum of the two nodes frequencies. Make the first extracted node as its left child and the other extracted node as its right child. Add this node to the min heap. 4. Repeat steps#2 and #3 until the heap contains only one node. The remaining node is the root node and the tree is complete. Let us understand the algorithm with an example: character Frequency a 5 b 9 c 12 d 13 e 16 f 45 Step 1. Build a min heap that contains 6 nodes where each node represents root of a tree with single node. Step 2 Extract two minimum frequency nodes from min heap. Add a new internal node with frequency = 14. Now min heap contains 5 nodes where 4 nodes are roots of trees with single element each, and one heap node is root of tree with 3 elements character Frequency

19 c 12 d 13 Internal Node 14 e 16 f 45 Step 3: Extract two minimum frequency nodes from heap. Add a new internal node with frequency = 25 Now min heap contains 4 nodes where 2 nodes are roots of trees with single element each, and two heap nodes are root of tree with more than one nodes. character Frequency Internal Node 14 e 16 Internal Node 25 f 45 Step 4: Extract two minimum frequency nodes. Add a new internal node with frequency = 30 Now min heap contains 3 nodes. character Frequency Internal Node 25 Internal Node 30 f 45 Step 5: Extract two minimum frequency nodes. Add a new internal node with frequency = 55

20 Now min heap contains 2 nodes. character Frequency f 45 Internal Node 55 Step 6: Extract two minimum frequency nodes. Add a new internal node with frequency = 100 Now min heap contains only one node. character Frequency Internal Node 100 Since the heap contains only one node, the algorithm stops here. Steps to print codes from Huffman Tree: Traverse the tree formed starting from the root. Maintain an auxiliary array. While moving to the left child, write 0 to the array. While moving to the right child, write 1 to the array. Print the array when a leaf node is encountered. The codes are as follows: character code-word

21 f 0 c 100 d 101 a 1100 b 1101 e 111

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