CS350 - Exam 4 (100 Points)

Transcription

1 Fall 0 Name CS0 - Exam (00 Points).(0 points) Re-Black Trees For the Re-Black tree below, inicate where insert() woul initially occur before rebalancing/recoloring. Sketch the tree at ALL intermeiate steps showing any rebalancing/recoloring that occurs at that step. Note: Use ashe circles for re noes an ouble circles for black noes as shown in the figure. Any subtrees that are unchange may simply be represente by a triangle chil showing the color an alue of the root of the subtree

2 .(0 points) AA Trees For the AA tree below, perform remoe(0) using the successor. List in orer ALL intermeiate steps, een if the operation oes not moify the tree, noting which operation is being performe on which (subtree root) noe rerawing the tree wheneer it is moifie. Note: Any subtrees that are unchange may simply be represente by a triangle chil

3 .( points) Binary Heaps (a) For the following array, list in orer which alues (i.e. not inices) woul be processe uring the builheap() operation. DO NOT perform the builheap() operation on this array, simply state the alues that woul be processe (b) Gien the min binary heap below, show ALL intermeiate steps for the operation insert(). Note: Any subtrees that are unchange may simply be represente by a triangle chil

4 (c) Gien the min binary heap below, show ALL intermeiate steps for the operation remoemin(). Note: Any subtrees that are unchange may simply be represente by a triangle chil

5 () Draw a min binary heap containing the alues -9 which woul cause the worst case runtime for insertion of the alue into the heap. Note: You o not nee to show either the insertion process to buil the original heap or the insertion process for the aitional alue. (e) Write C++-ish pseuocoe for a percolatedown() routine for a min binary heap containing integers (implemente similarly to the lab) which takes a single parameter ix for the inex into the binary heap s backing array. You may assume that there is a heapsize fiel containing the number of elements in the heap an the backing array fiel is heaparray[]. DO NOT assume that there are methos to fin inices of chilren or parent alues. oi percolatedown(int ix) {

6 .( points) Hash Tables (a) Insert the following keys with gien hash slot alues into the three hash tables using the inicate methos for resoling collisions. List all collisions by noting which alue collies with which other alue. key slot B I N A R Y H E P (i) Linear Probing (ii) uaratic Probing (iii) Chaining (b) What is the loa factor of the aboe tables?

7 (c) Briefly explain the properties of a goo hash function an how you woul etermine if a gien hash function is goo or not. () Write C++-ish pseuocoe for a remoe() routine which takes a single parameter x for the alue to be remoe from the hash table. Assume, similar to the lab implementation, that the hash table uses chaining for collisions an alues are store in Noe objects with ata an next fiels. You may also assume that there is a hash() metho which returns the hash slot for the alue an a fin() metho which returns a boolean alue inicating if a specifie alue is present in the table. oi remoe(t x) {

8 .( points) Graphs (a) Represent the graph below using an ajacency list (b) When is it more beneficial to use an ajacency list representation instea of an ajacency matrix representation for a graph?

9 (c) Run Dijkstra s algorithm on the graph below using ertex as the source. At each step, show the final priority queue in the left table an only complete shortest path istances in the right table circling the current ertex that was processe. Be sure to show the initialization in the first set of tables. On the blank graphs, use soli lines to inicate eges on foun shortest paths an ashe lines to inicate eges on potential paths currently uner exploration. 7 Complete Distances Complete Distances

10 Complete Distances Complete Distances Complete Distances

11 Complete Distances Complete Distances Complete Distances

12 6.( points) Runtime (a) In the table below, gie the asymptotic run times, using O() notation, for the aerage an worst cases for the following ata structures fin(x) ag case fin(x) worst case insert(x) ag case insert(x) worst case remoe(x) ag case remoe(x) worst case AA Tree Binary Heap Hash Table (linear probing) Hash Table (chaining) (b) Briefly explain which ata structure you enjoye most uring this class.