Priority Queues. Priority Queue Specification. Priority Queue ADT. Implementation #1

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1 Priority Queues A Greedy Approximation Algorithm for Computing Shortest Superstrings Lecture 13 Riley 7.9, Cormen et al. parts of 11 Priority Queues A queue in which all items are assigned a priority by which they are ordered. First in line stands the item with the most extreme (highest or lowest) priority. (Not the item that has spent the longest time in the queue as is the case in ordinary queues). However, to solve ties, the time in the queue is used. The meaning of priority differs. Depends on the context; choose a suitable concrete definition! Lecture 13 - Tomas Johansson 2 Priority Queue ADT Priority Queue Specification Lecture 13 - Tomas Johansson 3 Lecture 13 - Tomas Johansson 4 Implementation #1 Use a list and add new items at the end. Items are somehow tagged with a priority. Could use pairs with the priority and the item. dequeue searches through the entire list. Time complexity of operations: When there are k items in the queue it takes O(1) time to insert (enqueue) new items. Good! O(k) time to dequeue. Bad.. Assume objects were entered in the order of A2, B3, C1, D1, E2, F3, G1, H3. Lecture 13 - Tomas Johansson 5 Lecture 13 - Tomas Johansson 6 1

2 Implementation #2 Use a list but keep it sorted at all time according to priority. enqueue searches through the list comparing priorities until the proper place for the new item has been found. dequeue takes out the first element of the list. Time complexity of operations: When there is k elements in the queue it takes O(k) time to insert (enqueue) new items. Bad.. O(1) time to dequeue. Good! Lecture 13 - Tomas Johansson 7 Assume objects were entered in the order of A2, B3, C1, D1, E2, F3, G1, H3. Lecture 13 - Tomas Johansson 8 Better implementations? If we insert (enqueue) n elements into a priority queue and then dequeue them this will take O(n 2 ) time regardless of whether implementation 1 or 2 is used. O(n 2 ) time is not very practical (see Problem 2 of Homework1..) Is it possible to improve this? YES! Use a tree structure called Heap. Heaps will be covered in a couple of lectures, but first what is a tree structure?! Lecture 13 - Tomas Johansson 9 A tree is a connected acyclic graph. A graph is a set of nodes and a set of arcs (or edges). An arc connects a pair of nodes to each other. Two nodes connected by an arc are adjacent. There is a path between any pair of nodes. A path is a sequence of nodes where each successive node (except for the first) is adjacent to its predecessor in the sequence. There are no cycles/loops in a tree. Lecture 13 - Tomas Johansson 10 Tree! Not a tree! (But a forest) Not a tree! The trees we will encounter are directed and rooted. Directed means that arcs are one-way only. Paths go in the direction of arcs. Rooted One of the nodes is the root of the tree. There is exactly one path from the root to another node. Nodes without outgoing arcs are called leafs. Lecture 13 - Tomas Johansson 11 Lecture 13 - Tomas Johansson 12 2

3 Rooted trees Lecture 13 - Tomas Johansson 13 Lecture 13 - Tomas Johansson 14 Some properties Nodes have at most one incoming arc. All nodes have one except for the root that has none. The number of arcs is always one less than the number of nodes. A tree becomes disconnected if an arc is removed. Binary In a binary tree each node is either a leaf or it has exactly two (2) outgoing arcs. Binary trees can be used to represent trees with other out-degrees: A B C A B C Lecture 13 - Tomas Johansson 15 Lecture 13 - Tomas Johansson 16 Some terminology Inheritance tree in UML Lecture 13 - Tomas Johansson 17 Lecture 13 - Tomas Johansson 18 3

4 =(1-2)*(3+4) Expression =1-2 =2-1 Heap A heap is a binary tree with the following properties: All levels are full, except for the lowest level when nodes are added from the left The greatest/smallest value in any subtree is stored in that subtree s root. Observation: The height of a heap is O(lg n) =(5+6) / (7 - ((8+9) * 3)) Lecture 13 - Tomas Johansson 19 Lecture 13 - Tomas Johansson 20 Priority Queue using heap Greedy algorithms Insertion as well as removing elements in a heap can be done in O(lg n) time. Inserting and removing n objects took O(n 2 ) time using a linked list, now it only takes O(n lg n) time! Greed, for the lack of a better word, is good! Greed is right! Greed works! -Gordon Gecko, Wall Street, Lecture 13 - Tomas Johansson 21 Lecture 13 - Tomas Johansson 22 Change-making problem Give change for the amount n with the least number of coins of the denomination d 1 >d 2 > >d m USA example: d 1 =25, d 2 =10, d 3 =5, d 4 =1 What is the solution for the problem instance of n=48? Strategy: Always use the largest possible coin! Greedy algorithm A greedy algorithm always makes the locally optimal choice hoping to achieve the globally optimal solution In USA, the greedy strategy works! Butwhatifd 1 =7, d 2 =5, d 3 =1, and n=11? Obviously, greed does not always work Lecture 13 - Tomas Johansson 23 Lecture 13 - Tomas Johansson 24 4

5 More on the change-making problem: What this country needs is an 18-cent piece Jeffrey Shallit, University of Waterloo, Abstract: We consider sets of coin denominations which permit change to be made using as few coins as possible, and explain why the United States should be using the 18-cent piece. but the greedy algorithm does not produce optimal results using the optimal denominations. Approximation algorithms Some problems become very difficult (wrt time/memory) as the input size grows Finding the optimal solution is not always feasible Approximation algorithms give the suboptimal solution, but they are faster/cheaper than finding the optimal solution Lecture 13 - Tomas Johansson 25 Lecture 13 - Tomas Johansson 26 A Greedy Approximation Algorithm for computing Shortest Superstrings The problem of computing the shortest (common) superstring: Given a set S of strings, find the shortest string that contains all the strings in S as sub-strings. Example: { ABB, CAB, ICA } has ICABB as its shortest superstring. Applications: Data compression and DNA sequencing. Lecture 13 - Tomas Johansson 27 Shortest Superstring Bad news: The problem is NP-hard. Means there are (probably) no fast solutions. One solution: Try all possible orderings of the input strings. For each ordering, place one string after another and have them overlap as much as possible. This gives us the shortest superstring for that particular ordering. Keep the shortest of all superstrings obtained; this is the overall shortest superstring. For n unique input strings there are n! orderings Lecture 13 - Tomas Johansson 28 Shortest Superstring Good news: There are fast algorithms that compute provably good approximations! Better than nothing A pretty good solution: A greedy algorithm(!) Let S be the set of input strings. Repeat the following as long as S has more than one element: Check which pair of strings s 1 and s 2 in S that overlap the most.» This is where the algorithm is greedy. Form one string s 3 from s 1 and s 2 by aligning them with maximum overlap. Remove s 1 and s 2 from S, and add s 3. Report the one string remaining in S as the result. Has been proved: The solution computed by the greedy algorithm is at most twice (2) as long as the optimal solution(!) Lecture 13 - Tomas Johansson 29 Implementation of greedy algorithm VectorPair -overlap:vector -all:vector +VectorPair(Vector v1, Vector v2) +VectorPair(Vector v) -align(vector v1, Vector v2):void +sizeoverlap():int +sizeall():int +overlapasvector():vector +allasvector():vector Lecture 13 - Tomas Johansson 30 5