Trees 1: Theory, Models, Generic Heap Algorithms, Priority Queues

Trees 1: Theory, Models, Generic Heap Algorithms, Priority Queues Andy Wang Data Structures, Algorithms, and Generic Programming

Review Question • Given the following information, can I uniquely identify a tree? • Nodes listed based on an inorder traversal: • D, B, F, E, A, C • Nodes listed based on a preorder traversal: • A, B, D, E, F, C

Review Question D B F E A C A B D E F C

Review Question D B F E A C A x B x D x E x F x C x

Partially Ordered Trees • Definition: A partially ordered tree is a tree T such that: • There is an order relation >= defined for the vertices of T • For any vertex p and any child c of p, p >= c

Partially Ordered Trees • Consequences: • The largest element in a partially ordered tree (POT) is the root • No conclusion can be drawn about the order of children

Heaps • Definition: A heap is a partially ordered complete (almost) binary tree. The tree is completely filled on all levels except possibly the lowest. 4 root 3 2 1 0

Heaps • Consequences: • The largest element in a heap is the root • A heap can be stored using the vector implementation of binary tree • Heap algorithms: • Push Heap • Pop Heap

The Push Heap Algorithm • Add new data at next leaf • Repair upward • Repeat • Locate parent • if POT not satisfied • swap • else • stop • Until POT

The Push Heap Algorithm • Add new data at next leaf

The Push Heap Algorithm • Repeat • Locate parent of v[k] = v[(k – 1)/2] • if POT not satisfied • swap • else • stop

The Pop Heap Algorithm • Copy last leaf to root • Remove last leaf • Repeat • find the larger child • if POT not satisfied • swap • else • stop • Until POT

The Pop Heap Algorithm • Copy last leaf to root

The Pop Heap Algorithm • Remove last leaf

The Pop Heap Algorithm • Repeat • find the larger child • if POT not satisfied • swap • else • stop

Generic Heap Algorithms • Apply to ranges • Specified by random access iterators • Current support • Arrays • TVector<T> • TDeque<T> • Source code file: gheap.h • Test code file: fgss.cpp

Priority Queues • Element type with priority • typename T t • Predicate class P p • Associative queue operations • void Push(t) • void Pop() • T& Front()

Priority Queues • Associative property • Priority value determined by p • Push(t) inserts t, increases size by 1 • Pop() removes element with highest priority value, decreases size by 1 • Front() returns element with highest priority value, no state change

The Priority Queue Generic Adaptor template <typename T, class C, class P> class CPriorityQueue { C c; P LessThan; public: typedef typename C::value_type value_type; int Empty() const { return c.Empty(); } unsigned int Size() const { return c.Size(); } void Clear() { c.Clear(); } CPriorityQueue& operator=(const CPriorityQueue& q) { if (this != &q) { c = q.c; LessThan = q.LessThan; } return *this; }

The Priority Queue Generic Adaptor void Display(ostream& os, char ofc = ‘\0’) const { c.Display(os, ofc); } void Push(const value_type& t) { c.PushBack(t); g_push_heap(c.Begin(), c.End(), LessThan); } void Pop() { if (Empty()) { cerr << “error” << endl; exit(EXIT_FALIURE); } g_pop_heap(c.Begin(), c.End(), LessThan); c.PopBack(); } };

Trees 1: Theory, Models, Generic Heap Algorithms, Priority Queues