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Trees, Binary Search Trees, Balanced Trees, Graphs Svetlin Nakov Telerik Corporation www.telerik.com

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1. Tree-like Data Structures 2. Trees and Related Terminology 3. Implementing Trees 4. Traversing Trees 5. Balanced Trees 6. Graphs 2

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Trees, Balanced Trees, Graphs, Networks

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Tree-like data structures are Branched recursive data structures Consisting of nodes Each node can be connected to other nodes Examples of tree-like structures Trees: binary / other degree, balanced, etc. Graphs: directed / undirected, weighted, etc. Networks 4

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5 Project Manager Team Leader De- signer QA Team Leader Developer 1 Developer 2 Tester 1 Developer 3 Tester 2 Tree Graph 2 3 6 1 4 5 7 19 21 14 1 12 31 4 11 5 (20) 5 (10) 15 (15) 15 (30) 5 (5) 20(20) 10 (40) Network

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Node, Edge, Root, Children, Parent, Leaf, Binary Search Tree, Balanced Tree

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Tree data structure – terminology Node, edge, root, child, children, siblings, parent, ancestor, descendant, predecessor, successor, internal node, leaf, depth, height, subtree Height = 2 Depth 0 Depth 1 Depth 2 17 15 14 9 6 5 8 7

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Binary trees: most widespread form Each node has at most 2 children 10 17 159 6 5 8 Root node Left subtree Right child Left child 8

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Binary search trees are ordered For each node x in the tree All the elements of the left subtree of x are ≤ x All the elements of the right subtree of x are > x Binary search trees can be balanced Balanced trees have height of ~ log 2 (x) Balanced trees have for each node nearly equal number of nodes in its subtrees 9

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Example of balanced binary search tree If the tree is balanced, add / search / delete operations take approximately log(n) steps 17 19 9 6 12 25 10

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Recursive Tree Data Structure

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The recursive definition for tree data structure: A single node is tree Tree nodes can have zero or multiple children that are also trees Tree node definition in C# 12 public class TreeNode public class TreeNode { private T value; private T value; private List > children; private List > children; …} The value contained in the node List of child nodes, which are of the same type

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13 7children 19children21children14children TreeNode

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14 public TreeNode(T value) { this.value = value; this.value = value; this.children = new List >(); this.children = new List >();} public T Value { get { return this.value; } get { return this.value; } set { this.value = value; } set { this.value = value; }} public void AddChild(TreeNode child) { child.hasParent = true; child.hasParent = true; this.children.Add(child); this.children.Add(child);} public TreeNode GetChild(int index) { return this.children[index]; return this.children[index];}

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The class Tree keeps tree's root node 15 public class Tree public class Tree { private TreeNode root; private TreeNode root; public Tree(T value, params Tree [] children): this(value) public Tree(T value, params Tree [] children): this(value) { foreach (Tree child in children) foreach (Tree child in children) { this.root.AddChild(child.root); this.root.AddChild(child.root); } } public TreeNode Root public TreeNode Root { get { return this.root; } get { return this.root; } }} Flexible constructor for building trees

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16 Tree tree = new Tree (7, new Tree (7, new Tree (19, new Tree (19, new Tree (1), new Tree (1), new Tree (12), new Tree (12), new Tree (31)), new Tree (31)), new Tree (21), new Tree (21), new Tree (14, new Tree (14, new Tree (23), new Tree (23), new Tree (6)) new Tree (6))); 7 14 19 23 6 21 31 1 12 Constructing tree by nested constructors:

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DFS and BFS Traversals

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Traversing a tree means to visit each of its nodes exactly one in particular order Many traversal algorithms are known Depth-First Search (DFS) Visit node's successors first Usually implemented by recursion Breadth-First Search (BFS) Nearest nodes visited first Implemented by a queue 18

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Depth-First Search first visits all descendants of given node recursively, finally visits the node itself DFS algorithm pseudo code DFS(node){ for each child c of node for each child c of node DFS(c); DFS(c); print the current node; print the current node;} 7 14 19 23 6 21 31 1 12 1 2 3 4 5 8 6 7 9 19

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Stack: 7 Output: (empty) 20 7 14 19 23 6 21 31 1 12

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Stack: 7, 19 Output: (empty) 21 7 14 19 23 6 21 31 1 12

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Stack: 7, 19, 1 Output: (empty) 22 7 14 19 23 6 21 31 1 12

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Stack: 7, 19 Output: 1 23 7 14 19 23 6 21 31 1 12

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Stack: 7, 19, 12 Output: 1 24 7 14 19 23 6 21 31 1 12

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Stack: 7, 19 Output: 1, 12 25 7 14 19 23 6 21 31 1 12

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Stack: 7, 19, 31 Output: 1, 12 26 7 14 19 23 6 21 31 1 12

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Stack: 7, 19 Output: 1, 12, 31 27 7 14 19 23 6 21 31 1 12

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Stack: 7 Output: 1, 12, 31, 19 28 7 14 19 23 6 21 31 1 12

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Stack: 7, 21 Output: 1, 12, 31, 19 29 7 14 19 23 6 21 31 1 12

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Stack: 7 Output: 1, 12, 31, 19, 21 30 7 14 19 23 6 21 31 1 12

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Stack: 7, 14 Output: 1, 12, 31, 19, 21 31 7 14 19 23 6 21 31 1 12

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Stack: 7, 14, 23 Output: 1, 12, 31, 19, 21 32 7 14 19 23 6 21 31 1 12

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Stack: 7, 14 Output: 1, 12, 31, 19, 21, 23 33 7 14 19 23 6 21 31 1 12

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Stack: 7, 14, 6 Output: 1, 12, 31, 19, 21, 23 34 7 14 19 23 6 21 31 1 12

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Stack: 7, 14 Output: 1, 12, 31, 19, 21, 23, 6 35 7 14 19 23 6 21 31 1 12

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Stack: 7 Output: 1, 12, 31, 19, 21, 23, 6, 14 36 7 14 19 23 6 21 31 1 12

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Stack: (empty) Output: 1, 12, 31, 19, 21, 23, 6, 14, 7 37 7 14 19 23 6 21 31 1 12 Traversal finished

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Breadth-First Search first visits the neighbor nodes, later their neighbors, etc. BFS algorithm pseudo code BFS(node){ queue node queue node while queue not empty while queue not empty v queue v queue print v print v for each child c of v for each child c of v queue c queue c} 7 14 19 23 6 21 31 1 12 5 6 7 2 3 4 8 9 1 38

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Queue: 7 Output: 7 39 7 14 19 23 6 21 31 1 12

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Queue: 7, 19 Output: 7 40 7 14 19 23 6 21 31 1 12

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Queue: 7, 19, 21 Output: 7 41 7 14 19 23 6 21 31 1 12

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Queue: 7, 19, 21, 14 Output: 7 42 7 14 19 23 6 21 31 1 12

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Queue: 7, 19, 21, 14 Output: 7, 19 43 7 14 19 23 6 21 31 1 12

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Queue: 7, 19, 21, 14, 1 Output: 7, 19 44 7 14 19 23 6 21 31 1 12

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Queue: 7, 19, 21, 14, 1, 12 Output: 7, 19 45 7 14 19 23 6 21 31 1 12

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Queue: 7, 19, 21, 14, 1, 12, 31 Output: 7, 19 46 7 14 19 23 6 21 31 1 12

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Queue: 7, 19, 21, 14, 1, 12, 31 Output: 7, 19, 21 47 7 14 19 23 6 21 31 1 12

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Queue: 7, 19, 21, 14, 1, 12, 31 Output: 7, 19, 21, 14 48 7 14 19 23 6 21 31 1 12

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Queue: 7, 19, 21, 14, 1, 12, 31, 23 Output: 7, 19, 21, 14 49 7 14 19 23 6 21 31 1 12

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Queue: 7, 19, 21, 14, 1, 12, 31, 23, 6 Output: 7, 19, 21, 14 50 7 14 19 23 6 21 31 1 12

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Queue: 7, 19, 21, 14, 1, 12, 31, 23, 6 Output: 7, 19, 21, 14, 1 51 7 14 19 23 6 21 31 1 12

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Queue: 7, 19, 21, 14, 1, 12, 31, 23, 6 Output: 7, 19, 21, 14, 1, 12 52 7 14 19 23 6 21 31 1 12

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Queue: 7, 19, 21, 14, 1, 12, 31, 23, 6 Output: 7, 19, 21, 14, 1, 12, 31 53 7 14 19 23 6 21 31 1 12

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Queue: 7, 19, 21, 14, 1, 12, 31, 23, 6 Output: 7, 19, 21, 14, 1, 12, 31, 23 54 7 14 19 23 6 21 31 1 12

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Queue: 7, 19, 21, 14, 1, 12, 31, 23, 6 Output: 7, 19, 21, 14, 1, 12, 31, 23, 6 55 7 14 19 23 6 21 31 1 12

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Queue: 7, 19, 21, 14, 1, 12, 31, 23, 6 Output: 7, 19, 21, 14, 1, 12, 31, 23, 6 56 7 14 19 23 6 21 31 1 12 The queue is empty stop

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DFS traversal of binary trees can be done in pre- order, in-order and post-order Pre-order: left, root, right 6, 9, 12, 17, 19, 25 In-order: root, left, right 17, 9, 6, 12, 19, 25 Post-order: left, right, root 6, 12, 9, 25, 19, 17 17 199 6 12 25 57

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What will happen if in the Breadth-First Search (BFS) algorithm a stack is used instead of queue? An iterative Depth-First Search (DFS) – in-order 58 BFS(node){ queue node queue node while queue not empty while queue not empty v queue v queue print v print v for each child c of v for each child c of v queue c queue c}DFS(node){ stack node stack node while stack not empty while stack not empty v stack v stack print v print v for each child c of v for each child c of v stack c stack c}

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Live Demo

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AVL Trees, B-Trees, Red-Black Trees, AA-Trees

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Ordered Binary Trees (Binary Search Trees) For each node x the left subtree has values ≤ x and the right subtree has values > x Balanced Trees For each node its subtrees contain nearly equal number of nodes nearly the same height Balanced Binary Search Trees Ordered binary search trees that have height of log 2 (n) where n is the number of their nodes Searching costs about log 2 (n) comparisons 61

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62 33 18 1524 3172029 54 4260 37435985

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Balanced binary search trees are hard to implement Rebalancing the tree after insert / delete is complex Well known implementations of balanced binary search trees AVL trees – ideally balanced, very complex Red-black trees – roughly balanced, more simple AA-Trees – relatively simple to implement Find / insert / delete operations need log 2 (n) steps 63

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B-trees are generalization of the concept of ordered binary search trees B-tree of order d has between d and 2*d keys in a node and between d+1 and 2*d+1 child nodes The keys in each node are ordered increasingly All keys in a child node have values between their left and right parent keys If the b-tree is balanced, its search / insert / add operations take about log(n) steps B-trees can be efficiently stored on the disk 64

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B-Tree of order 2, also known as 2 - 3 - 4 -tree: 65 1721 71118202631 24568912162223252729303235

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.NET Framework has several built-in implementations of balanced search trees: SortedDictionary SortedDictionary Red-black tree based map of key-value pairs OrderedSet OrderedSet Red-black tree based set of elements External libraries like "Wintellect Power Collections for.NET" are more flexible http://powercollections.codeplex.com http://powercollections.codeplex.com 66

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Definitions, Representation, Traversal Algorithms

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Set of nodes with many-to-many relationship between them is called graph Each node has multiple predecessors Each node has multiple successors 7 19 21 14 1 12 31 4 11 Node with multiple predecessors Node with multiple successors 68

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Node (vertex) Element of graph Can have name or value Keeps a list of adjacent nodes Edge Connection between two nodes Can be directed / undirected Can be weighted / unweighted Can have name / value A Node A Edge B 69

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Directed graph Edges have direction Undirected graph Undirected edges 7 19 21 1 12 4 3 22 2 3 G J F D A E C H 70

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Weighted graph Weight (cost) is associated with each edge G J F D A E C H Q K N 10 4 14 6 16 9 8 7 5 22 3 71

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Path (in undirected graph) Sequence of nodes n 1, n 2, … n k Edge exists between each pair of nodes n i, n i+1 Examples: A, B, C is a path H, K, C is not a path G C B A HN K 72

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Path (in directed graph) Sequence of nodes n 1, n 2, … n k Directed edge exists between each pair of nodes n i, n i+1 Examples: A, B, C is a path A, G, K is not a path G C B A HN K 73

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Cycle Path that ends back at the starting node Example: A, B, C, G, A Simple path No cycles in path Acyclic graph Graph with no cycles Acyclic undirected graphs are trees G C B A HN K 74

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Unconnected graph with two connected components Two nodes are reachable if Path exists between them Connected graph Every node is reachable from any other node G J F D A Connected graph G J F D A E C H 75

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Graphs have many real-world applications Modeling a computer network like Internet Routes are simple paths in the network Modeling a city map Streets are edges, crossings are vertices Social networks People are nodes and their connections are edges State machines States are nodes, transitions are edges 76

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Adjacency list Each node holds a list of its neighbors Adjacency matrix Each cell keeps whether and how two nodes are connected Set of edges 01010010 1000 0100 1 2 3 4 1 234 {1,2} {1,4} {2,3} {3,1} {4,2} 1 {2, 4} 2 {3} 3 {1} 4 {2} 2 41 3 77

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public class Graph { int[][] childNodes; int[][] childNodes; public Graph(int[][] nodes) public Graph(int[][] nodes) { this.childNodes = nodes; this.childNodes = nodes; }} Graph g = new Graph(new int[][] { new int[] {3, 6}, // successors of vertice 0 new int[] {3, 6}, // successors of vertice 0 new int[] {2, 3, 4, 5, 6}, // successors of vertice 1 new int[] {2, 3, 4, 5, 6}, // successors of vertice 1 new int[] {1, 4, 5}, // successors of vertice 2 new int[] {1, 4, 5}, // successors of vertice 2 new int[] {0, 1, 5}, // successors of vertice 3 new int[] {0, 1, 5}, // successors of vertice 3 new int[] {1, 2, 6}, // successors of vertice 4 new int[] {1, 2, 6}, // successors of vertice 4 new int[] {1, 2, 3}, // successors of vertice 5 new int[] {1, 2, 3}, // successors of vertice 5 new int[] {0, 1, 4} // successors of vertice 6 new int[] {0, 1, 4} // successors of vertice 6}); 0 6 4 1 5 2 3 78

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Depth-First Search (DFS) and Breadth-First Search (BFS) can traverse graphs Each vertex should be is visited at most once 79 BFS(node) { queue node queue node visited[node] = true visited[node] = true while queue not empty while queue not empty v queue v queue print v print v for each child c of v for each child c of v if not visited[c] if not visited[c] queue c queue c visited[c] = true visited[c] = true} DFS(node) { stack node stack node visited[node] = true visited[node] = true while stack not empty while stack not empty v stack v stack print v print v for each child c of v for each child c of v if not visited[c] if not visited[c] stack c stack c visited[c] = true visited[c] = true}

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80 void TraverseDFSRecursive(node) { if (not visited[node]) if (not visited[node]) { visited[node] = true visited[node] = true print node print node foreach child node c of node foreach child node c of node { TraverseDFSRecursive(c); TraverseDFSRecursive(c); } }} vois Main() { TraverseDFS(firstNode); TraverseDFS(firstNode);}

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Live Demo

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Trees are recursive data structure – node with set of children which are also nodes Binary Search Trees are ordered binary trees Balanced trees have weight of log(n) Graphs are sets of nodes with many-to-many relationship between them Can be directed/undirected, weighted / unweighted, connected / not connected, etc. Tree / graph traversals can be done by Depth-First Search (DFS) and Breadth-First Search (BFS) 82

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Questions? http://academy.telerik.com

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1. Write a program to traverse the directory C:\WINDOWS and all its subdirectories recursively and to display all files matching the mask *.exe. Use the class System.IO.Directory. 2. Define classes File { string name, int size } and Folder { string name, File[] files, Folder[] childFolders } and using them build a tree keeping all files and folders on the hard drive starting from C:\WINDOWS. Implement a method that calculates the sum of the file sizes in given subtree of the tree and test it accordingly. Use recursive DFS traversal. 84

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3. Implement the recursive Depth-First-Search (DFS) traversal algorithm. Test it with the sample graph from the demonstrations. 4. Implement the queue-based Breath-First-Search (BFS) traversal algorithm. Test it with the sample graph from the demonstrations. 5. Write a program for finding all cycles in given undirected graph using recursive DFS. 6. Write a program for finding all connected components of given undirected graph. Use a sequence of DFS traversals. 85

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7. Write a program for finding the shortest path between two vertices in a weighted directed graph. Hint: Use the Dijkstra's algorithm. Dijkstra's algorithmDijkstra's algorithm 8. We are given a set of N tasks that should be executed in a sequence. Some of the tasks depend on other tasks. We are given a list of tasks { t i, t j } where t j depends on the result of t i and should be executed after it. Write a program that arranges the tasks in a sequence so that each task depending on another task is executed after it. If such arrangement is impossible indicate this fact. Example: { 1, 2 }, { 2, 5 }, { 2, 4 }, { 3, 1 } 3, 1, 2, 5, 4 86

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