The Bucharest University of Economic Studies. Data Structures. Heap Priority Queues Maps and Sets

Transcription

1 The Bucharest University of Economic Studies Data Structures Heap Priority Queues Maps and Sets

2 Heap A heap is a specialized tree-based data structure that satisfies the property: if A is parent for B than between the values of A and B a semantic relation of total order is applied.

3 Binary-Heap A Binary-Heap structure is a complete binary tree which can be linearized.

4 Binary-Heap Total order semantics: ==,!=, >=, <=, >, < Equality Completeness Consistence Transitivity

5 Binary-Heap Equality for x and y, values of two nodes in a parent-descendant relation, if: (x==y) x and y are identical;

6 Binary-Heap Completeness for each x and y, values of two nodes, one of the following relation is true and unique: (x<y) (x == y) (x>y)

7 Binary-Heap Consistence for any x, y, values of two nodes, relation: (x>y) is identical with (y<x); (x>=y) is identical with (y<=x) identical with ((x>y) (x==y)); (x<=y) identical with (y>=x) identical with ((x<y) (x==y)); (x!=y) identical with!(x==y);

8 Binary-Heap Transitivity for any x, y, z, values of different nodes: if (x<y) and (y<z) then (x<z) is true;

9 Binary-Heap If node A is parent of node B, then: -if value(a) >= value(b) => max Heap -if value(a) <= value(b) => min Heap The maximum or minimum value is always accessible with O(1) complexity time.

10 Binary-Heap void processing() { while(true) { task x = GetNextTask(tasks); process(x); } }

11 GetNewTask() Binary-Heap Stacks the newest task Queues the oldest task Priority Queues the most urgent

12 Binary-Heap

13 Binary-Heap Tree linearization: node i = left descendent, 2i + 1 right descendent, 2i + 2 i 1 parent, 2

14 Heap - Insertion Insert element

15 Heap - Insertion

16 Heap - Insertion

17 Heap - Insertion

18 Heap - Insertion Pseudocode: 1. Insert the new node in the first valid (available) location; 2. while(new element doesn t comply with the heap restriction) switch between the new and the parent node;

19 Heap - Deletion Delete

20 Heap - Deletion

21 Heap - Deletion

22 Heap - Deletion

23 Heap - Deletion

24 Heap - Deletion

25 Heap - Deletion Pseudocode: 1. saving the value of the root, the deleted node; 2. switch values between the root and the last element from the structure; 3. delete last element from the structure; 4. while(the new root doesn t comply with the heap condition) switch between the new element and the child node;

26 Heap Sort

27 Priority Queues - PQs insert Priority queue delete Min/Max Insertion = re-heapification upward Deletion = re-heapification downward

28 Priority Queues - PQs Other operations: Increasing the priority of an element; Decreasing the priority of an element; Deleting an element; Searching for min; Searching for max; Heapsort;

29 Data collections Containers with the following characteristics: Unique values; Ordered elements; Null elements;

30 Data collections Sets container with unique values stored: Ordered: uses tree structures with order relation; Unordered: hash tables; Maps container with elements of type <key, value> stored: Ordered: uses tree structures with order relation; Unordered: hash tables; operator [];

31 Sets Pointing The elements are referred using the key and not the absolute position in the set; Order All the elements are following a strict order; Sets The value of the element is also the key used for referring it; Unique values There are no identical elements in the set; Allocation Memory allocation is dynamically managed by an allocator object type

32 Maps Pointing The elements are referred using the key and not the absolute position in the set; Order All the elements are following a strict order; Maps Each element has a key associated with the mapped value; in this case the value of the element is different than the value of the key; Unique values There can be elements that have the same mapped value but no elements that can have the same key; Allocation Memory allocation is dynamically managed by an allocator object type

33 C++ / Java Implementations Structure C++ Java Priority queue std : : priority_queue Maps std::map<char,float> mymap; HashMap mymap; Sets std::set<int> myset; HashSet myset;

34 Map Java Interface HashMap Fast access to elements; traversal order doesn t matter; TreeMap LinkedHashMap Sorted values inside the collection Fast access and sorted values

35 Set Java Interface HashSet Fast access to elements TreeSet Partial order between elements LinkedHashSet Fast access and ordered values

36 Resources Ion IVAN, Marius POPA, Paul POCATILU, coordonatori Structuri de date, Editura ASE, Bucuresti, 2008, ISBN Ion SMEUREANU Programarea in limbajul C/C++, Editura CISON, 2001 Michael MAIN Data Structures & Other Objects Using Java, Editura Pearson, 2012, ISBN: