The ADT priority queue Orders its items by a priority value The first item removed is the one having the highest priority value

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1 The ADT priority queue Orders its items by a priority value The first item removed is the one having the highest priority value 1

2 Possible implementations Sorted linear implementations o Appropriate if the number of items in the priority queue is small o Array-based implementation o Maintains the items sorted in ascending order of priority value o Reference-based implementation o Maintains the items sorted in descending order of priority value Possible implementations Sorted linear implementations o Binary search tree implementation 2

3 Heaps 5 Heap differs from a binary tree in two significant ways Binary search trees are sorted, heaps are ordered in a weaker way. Binary search trees come in many different shapes, heaps are always complete binary tree Maxheap vs. Minheap Maxheap o A heap in which the root contains the item with the largest search key Minheap o A heap in which the root contains the item with the smallest search key 3

4 Pseudocode for the operations of the ADT heap o createheap() o // Creates an empty heap. o heapisempty() o // Determines whether a heap is empty. o heapinsert(newitem) throws HeapException o // Inserts newitem into a heap. Throws HeapException if heap is full. o heapdelete() o // Retrieves and then deletes a heap s root o // item. This item has the largest search key Heaps: An Array-based Implementation of a Heap heapdelete o Step 1: Return the item in the root o Results in disjoint heaps 4

5 Heaps: An Array-based Implementation of a Heap heapdelete o Step 2: Copy the item from the last node into the root o Results in a semiheap Heaps: An Array-based Implementation of a Heap heapdelete o Step 3: Transform the semiheap back into a heap o Performed by the recursive algorithm heaprebuild 5

6 Efficiency heapdelete is O(log n) heapinsert Strategy o Insert newitem into the bottom of the tree o Trickle new item up to appropriate spot in the tree Efficiency: O(log n) 6

7 Heapsort 13 Heapsort Strategy o Transforms the array into a heap o Removes the heap's root (the largest element) by exchanging it with the heap s last element o Transforms the resulting semiheap back into a heap 7

8 Heapsort Efficiency o Compared to mergesort o Both heapsort and mergesort are O(n * log n) in both the worst and average cases o Advantage over mergesort o Heapsort does not require a second array o Compared to quicksort o Quicksort is the preferred sorting method 8

G = {V, E} A graph G consists of two sets o A set V of vertices, or nodes o A set E of edges A subgraph o Consists of a subset of a graph s vertices and a subset of its edges Adjacent vertices o Two

9 G = {V, E} A graph G consists of two sets o A set V of vertices, or nodes o A set E of edges A subgraph o Consists of a subset of a graph s vertices and a subset of its edges Adjacent vertices o Two vertices that are joined by an edge A path between two vertices o A sequence of edges that begins at one vertex and ends at another vertex o May pass through the same vertex more than once G = {V, E} A simple path o A path that passes through a vertex only once A cycle o A path that begins and ends at the same vertex A simple cycle o A cycle that does not pass through a vertex more than once 9

10 G = {V, E} A connected graph o A graph that has a path between each pair of distinct vertices A disconnected graph o A graph that has at least one pair of vertices without a path between them A complete graph o A graph that has an edge between each pair of distinct vertices Multigraph Not a graph Allows multiple edges between vertices 10

11 Weighted graph A graph whose edges have numeric labels Undirected graph Edges do not indicate a direction Directed graph, or diagraph Each edge is a directed edge Can have two edges between a pair of vertices, one in each direction Directed path o A sequence of directed edges between two vertices Vertex y is adjacent to vertex x if o There is a directed edge from x to y 11

12 Two options for defining graphs Vertices contain values Vertices do not contain values Most common implementations of a graph Adjacency matrix Adjacency list Adjacency matrix for a graph with n vertices numbered 0, 1,, n 1 An n by n array matrix such that matrix[i][j] is o 1 (or true) if there is an edge from vertex i to vertex j o 0 (or false) if there is no edge from vertex i to vertex j 12

13 Adjacency matrix for a weighted graph with n vertices numbered 0, 1,, n 1 An n by n array matrix such that matrix[i][j] is o The weight that labels the edge from vertex i to vertex j if there is an edge from i to j o if there is no edge from vertex i to vertex j Adjacency list An adjacency list for a graph with n vertices numbered 0, 1,, n 1 o Consists of n linked lists o The ith linked list has a node for vertex j if and only if the graph contains an edge from vertex i to vertex j o This node can contain either o Vertex j s value, if any o An indication of vertex j s identity 13

14 Adjacency list for an undirected graph Treats each edge as if it were two directed edges in opposite directions Depth-first search (DFS) traversal Proceeds along a path from v as deeply into the graph as possible before backing up Does not completely specify the order in which it should visit the vertices adjacent to v A last visited, first explored strategy 14

15 Breadth-first search (BFS) traversal Visits every vertex adjacent to a vertex v that it can before visiting any other vertex A first visited, first explored strategy An iterative form uses a queue A recursive form is possible, but not simple TOPOLOGICAL SORTING Simple algorithms for finding a topological order topsort1 o Find a vertex that has no successor o Remove from the graph that vertex and all edges that lead to it, and add the vertex to the beginning of a list of vertices o Add each subsequent vertex that has no successor to the beginning of the list o When the graph is empty, the list of vertices will be in topological order 15

16 TOPOLOGICAL SORTING Simple algorithms for finding a topological order topsort2 o A modification of the iterative DFS algorithm o Strategy Push all vertices that have no predecessor onto a stack» If all adjacent vertex of top item have been visited, pop, add top item to the front of output list» Otherwise, select one unvisited adjacent vertex, push Each time you pop a vertex from the stack, add it to the beginning of a list of vertices When the traversal ends, the list of vertices will be in topological order SPANNING TREES You can determine whether a connected graph contains a cycle by counting its vertices and edges A connected undirected graph that has n vertices must have at least n 1 edges A connected undirected graph that has n vertices and exactly n 1 edges cannot contain a cycle A connected undirected graph that has n vertices and more than n 1 edges must contain at least one cycle 16

17 THE DFS SPANNING TREE To create a depth-first search (DFS) spanning tree Traverse the graph using a depth-first search and mark the edges that you follow When the traversal is complete, the graph s vertices and marked edges form the spanning tree To create a breath-first search (BFS) spanning tree Traverse the graph using a bread-first search and mark the edges that you follow When the traversal is complete, the graph s vertices and marked edges form the spanning tree MINIMUM SPANNING TREES Minimum spanning tree A spanning tree for which the sum of its edge weights is minimal Prim s algorithm Finds a minimal spanning tree that begins at any vertex Strategy o Find the least-cost edge (v, u) from a visited vertex v to some unvisited vertex u o Mark u as visited o Add the vertex u and the edge (v, u) to the minimum spanning tree o Repeat the above steps until there are no more unvisited vertices 17

18 MINIMUM SPANNING TREES Prim s algorithm Strategy o Find the least-cost edge (v, u) from a visited vertex v to some unvisited vertex u o Mark u as visited o Add the vertex u and the edge (v, u) to the minimum spanning tree o Repeat the above steps until there are no more unvisited vertices SHORTEST PATHS Dijkstra s shortest-path algorithm Determines the shortest paths between a given origin and all other vertices Uses o A set vertexset of selected vertices o An array weight, where weight[v] is the weight of the shortest (cheapest) path from vertex 0 to vertex v that passes through vertices in vertexset 18

ADT Priority Queue. Heaps. A Heap Implementation of the ADT Priority Queue. Heapsort

ADT Priority Queue. Heaps. A Heap Implementation of the ADT Priority Queue. Heapsort ADT Priority Queue Heaps A Heap Implementation of the ADT Priority Queue Heapsort 1 ADT Priority Queue 3 The ADT priority queue Orders its items by a priority value The first item removed is the one having