MIT OpenCourseWare http://ocw.mit.edu6.006 Introduction to AlgorithmsSpring 2008For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Lecture 3 Ver 2.0 Scheduling and Binary Search Trees 6.006 Spring 2008 Lecture 3: Scheduling and Binary Search Trees Lecture Overview • Runway reservation system – Definition – How to solve with lists • Binary Search Trees – Operations Readings CLRS Chapter 10, 12. 1-3 Runway Reservation System Airport with single (very busy) runway (Boston 6 1)• → • “Reservations” for future landings • When plane lands, it is removed from set of pending events • Reserve req specify “requested landing time” t • Add t to the set of no other landings are scheduled within < 3 minutes either way. – else error, don’t schedule Example 37 4146 49 56time (mins)nowx x x xFigure 1: Runway Reservation System Example Let R denote the reserved landing times: R = (41, 46, 49, 56) Request for time: 44 not allowed (46�R) 53 OK 20 not allowed (already past) | R |= n Goal: Run this system efficiently in O(lg n) time 1Lecture 3 Ver 2.0 Scheduling and Binary Search Trees 6.006 Spring 2008 Algorithm Keep R as a sorted list. init: R = [ ] req(t): if t < now: return "error" for i in range (len(R)): if abs(t-R[i]) <3: return "error" %\Theta (n) R.append(t) R = sorted(R) land: t = R[0] if (t != now) return error R = R[1: ] (drop R[0] from R) Can we do better? • Sorted list: A 3 minute check can be done in O(1). It is possible to insert new time/plane rather than append and sort but insertion takes Θ(n) time. • Sorted array: It is possible to do binary search to find place to insert in O(lg n) time. Actual insertion however requires shifting elements which requires Θ(n) time. • Unsorted list/array: Search takes O(n) time • Dictionary or Python Set: Insertion is O(1) time. 3 minute check takes Ω(n) time What if times are in whole minutes? Large array indexed by time does the trick. This will not work for arbitrary precision time or verifying width slots for landing. Key Lesson: Need fast insertion into sorted list. New Requirement Rank(t): How many planes are scheduled to land at times ≤ t? The new requirement necessitates a design amendment. 2Lecture 3 Ver 2.0 Scheduling and Binary Search Trees 6.006 Spring 2008 Binary Search Trees (BST) 4949797949467949464164insert 49insert 79insert 46insert 41insert 64BSTBSTBSTBSTrootall elements > 49 off to the right, in right subtreeall elements < 49, go into left subtreeBSTNILFigure 2: Binary Search Tree Finding the minimum element in a BST Key is to just go left till you cannot go left anymore. 79494179494646Figure 3: Delete-Min: finds minimum and eliminates it All operations are O(h) where h is height of the BST. 3Lecture 3 Ver 2.0 Scheduling and Binary Search Trees 6.006 Spring 2008 Finding the next larger element next-larger(x) if right child not NIL, return minimum(right) else y = parent(x) while y not NIL and x = right(y) x = y; y = parent(y) return(y); See Fig. 4 for an example. What would next-larger(46) return? 79494146Figure 4: next-larger(x) What about rank(t)? Cannot solve it efficiently with what we have but can augment the BST structure. 79494643 6483621311what lands before 79?keep track of size of subtrees, during insert and deleteFigure 5: Augmenting the BST Structure Summarizing from Fig. 5, the algorithm for augmentation is as follows: 1. Walk down tree to find desired time 2. Add in nodes that are smaller 3. Add in subtree sizes to the left In total, this takes O(h) time. 4Lecture 3 Ver 2.0 Scheduling and Binary Search Trees 6.006 Spring 2008 49461 + 2 + 1 + 1 = 57964subtreesubtreeFigure 6: Augmentation Algorithm Example All the Python code for the Binary Search Trees discussed here are available at this link Have we accomplished anything? Height h of the tree should be O(log(n). 46434955Figure 7: Insert into BST in sorted order The tree in Fig. 7 looks like a linked list. We have achieved O(n) not O(log(n)!! ..|Balanced BSTs to the rescue...more on that in the next lecture!
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