Lectures 8 and 9: Trees and Heap Sort COMS10007 - Algorithms Dr. - PowerPoint PPT Presentation

Lectures 8 and 9: Trees and Heap Sort COMS10007 - Algorithms Dr. Christian Konrad 19.02.2020 Dr. Christian Konrad Lectures 8 and 9: Trees and Heap Sort 1 / 20

In-class Test In-class Test: When? March 10th, 1pm (during exercise classes) Where? Ivy Gate G.01 and Ivy Gate 1.01, two groups Your timetables have been updated accordingly! How long? 50 mins What should I expect? All lectures and exercise sheets are relevant (Peak Finding is excluded). Example in-class test uploaded to unit webpage You are allowed extra time? Get in touch with me (email) Dr. Christian Konrad Lectures 8 and 9: Trees and Heap Sort 2 / 20

Sorting Algorithms seen so far Sorting Algorithms seen so far Insertion-Sort: O ( n 2 ) in worst, in place, stable Merge-Sort: O ( n log n ) in worst case, NOT in place, stable Heap Sort (best of the two) O ( n log n ) in worst case, in place, NOT stable Uses a heap data structure (a heap is special tree) Data Structures Data storage format that allows for efficient access and modification Building block of many efficient algorithms For example, an array is a data structure Dr. Christian Konrad Lectures 8 and 9: Trees and Heap Sort 3 / 20

Trees Definition: A tree T = ( V , E ) of size n is a tuple consisting of V = { v 1 , v 2 , . . . , v n } and E = { e 1 , e 2 , . . . , e n − 1 } with | V | = n and | E | = n − 1 with e i = { v j , v k } for some j � = k s.t. for every pair of vertices v i , v j ( i � = j ), there is a path from v i to v j . V are the nodes/vertices and E are the edges of T . � � ✗ Dr. Christian Konrad Lectures 8 and 9: Trees and Heap Sort 4 / 20

Rooted Trees Definition: (rooted tree) A rooted tree is a triple T = ( v , V , E ) such that T = ( V , E ) is a tree and v ∈ V is a designed node that we call the root of T . Definition: (leaf, internal node) A leaf in a tree is a node with exactly one incident edge. A node that is not a leaf is called an internal node . Dr. Christian Konrad Lectures 8 and 9: Trees and Heap Sort 5 / 20

Children, Parent, and Degree Further Definitions: The parent of a node v is the closest node on a path from v to the root. The root does not have a parent. The children of a node v are v ’s neighbors except its parent. The height of a tree is the length of a longest root-to-leaf path. The degree deg( v ) of a node v is the number of incident edges to v . Since every edge is incident to two vertices we have � deg( v ) = 2 · | E | = 2( n − 1) . v ∈ V The level of a vertex v is the length of the unique path from the root to v plus 1. Dr. Christian Konrad Lectures 8 and 9: Trees and Heap Sort 6 / 20

Properties of Trees Property: Every tree has at least 2 leaves Proof Let L ⊆ V be the subset of leaves. Suppose that there is at most 1 leaf, i.e., | L | ≤ 1. Then: � � � deg( v ) = deg( v ) + deg( v ) v ∈ V v ∈ L v ∈ V \ L ≥ | L | · 1 + ( | V | − | L | ) · 2 = 2 | V | − | L | ≥ 2 n − 1 , a contradiction to the fact that � v ∈ V deg( v ) = 2( n − 1) in every tree. Dr. Christian Konrad Lectures 8 and 9: Trees and Heap Sort 7 / 20

Binary Trees Definition: ( k -ary tree) A (rooted) tree is k-ary if every node has at most k children. If k = 2 then the tree is called binary. A k ary tree is full if every internal node has exactly k children, complete if all levels except possibily the last is entirely filled (and last level is filled from left to right), perfect if all levels are entirely filled. complete 3-ary tree full 3-ary tree perfect binary tree Dr. Christian Konrad Lectures 8 and 9: Trees and Heap Sort 8 / 20

Height of Perfect and Complete k -ary Trees Height of k -ary Trees The number of nodes in a perfect k -ary tree of height i − 1 is i − 1 k j = k i − 1 � k − 1 . j =0 In other words, a perfect k -ary tree on n nodes has height: k i − 1 n = k − 1 k i = n ( k − 1) + 1 i = log k ( n ( k − 1) + 1) = O (log k n ) . Similarly, a complete k -ary tree has height O (log k n ). Remark: The runtime of many algorithms that use tree data structures depends on the height of these trees. We are therefore interested in using complete/perfect trees. Dr. Christian Konrad Lectures 8 and 9: Trees and Heap Sort 9 / 20

Priority Queues Priority Queue: Data structure that allows the following operations: Build(.): Create data structure given a set of data items Extract-Max(.): Remove the maximum element from the data structure others... Sorting using a Priority Queue Dr. Christian Konrad Lectures 8 and 9: Trees and Heap Sort 10 / 20

From Array to Tree Interpretation of an Array as a Complete Binary Tree Easy Navigation: Parent of i : ⌊ i / 2 ⌋ Left/Right Child of i : 2 i and 2 i + 1 Dr. Christian Konrad Lectures 8 and 9: Trees and Heap Sort 11 / 20

Heap Property The Heap Property Key of nodes larger than keys of their children Heap Property → Maximum at root Important for Extract-Max(.) Dr. Christian Konrad Lectures 8 and 9: Trees and Heap Sort 12 / 20

The Heapify Operation Constructing a Heap: Build(.) Given a binary tree, transform it into one that fulfills the Heap Property 1 Traverse tree with regards to right-to-left array ordering 2 If node does not fulfill Heap Property: Heapify() Dr. Christian Konrad Lectures 8 and 9: Trees and Heap Sort 13 / 20

Lectures 8 and 9: Trees and Heap Sort COMS10007 - Algorithms Dr. - PowerPoint PPT Presentation

Lectures 8 and 9: Trees and Heap Sort COMS10007 - Algorithms Dr. Christian Konrad 19.02.2020 Dr. Christian Konrad Lectures 8 and 9: Trees and Heap Sort 1 / 20 In-class Test In-class Test: When? March 10th, 1pm (during exercise classes)

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