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Analysis of Algorithms CS 477/677

This lecture discusses the mid-term results for CS 477/677, including interval trees and their operations, properties of intervals, designing interval trees, and interval search.

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Analysis of Algorithms CS 477/677

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  1. Analysis of AlgorithmsCS 477/677 Instructor: Monica Nicolescu Lecture 15

  2. Mid-Term Results CS 477 min = 33 max = 110 average = 76.3 CS 477/677 - Lecture 15

  3. Mid-Term Results CS 677 min = 70 max = 119 average = 96 CS 477/677 - Lecture 15

  4. Interval Trees Def.:Interval tree = a red-black tree that maintains a dynamic set of elements, each element x having associated an interval int[x]. • Operations on interval trees: • INTERVAL-INSERT(T, x) • INTERVAL-DELETE(T, x) • INTERVAL-SEARCH(T, i) CS 477/677 - Lecture 15

  5. i i i i j j j j i j j i Interval Properties • Intervals i and j overlap iff: low[i] ≤ high[j] and low[j] ≤ high[i] • Intervals i and j do not overlap iff: high[i] < low[j] or high[j] < low[i] CS 477/677 - Lecture 15

  6. Interval Trichotomy • Any two intervals i and j satisfy the interval trichotomy: exactly one of the following three properties holds: • i and j overlap, • i is to the left of j (high[i] < low[j]) • i is to the right of j (high[j] < low[i]) CS 477/677 - Lecture 15

  7. high[int[x]] max max[left[x]] max[right[x]] Designing Interval Trees • Underlying data structure • Red-black trees • Each node x contains: an interval int[x], and the key: low[int[x]] • An inorder tree walk will list intervals sorted by their low endpoint • Additional information • max[x] = maximum endpoint value in subtree rooted at x • Maintaining the information max[x] = Constant work at each node, so still O(lgn) time CS 477/677 - Lecture 15

  8. [16, 21] 30 [25, 30] 30 [26, 26] 26 [17, 19] 20 [19, 20] 20 [8, 9] 23 [15, 23] 23 [5, 8] 10 [6, 10] 10 [0, 3] 3 Designing Interval Trees • Develop new operations • INTERVAL-SEARCH(T, i): • Returns a pointer to an element x in the interval tree T, such that int[x] overlaps with i, or NIL otherwise • Idea: • Check if int[x] overlaps with i • Max[left[x]]≥ low[i] • Go left • Otherwise, go right low high [22, 25] CS 477/677 - Lecture 15

  9. [5, 8] 10 [0, 3] 3 [15, 23] 23 [8, 9] 23 [19, 20] 20 [6, 10] 10 [26, 26] 26 [25, 30] 30 [16, 21] 30 [17, 19] 20 x = NIL Example x i = [11, 14] i = [22, 25] x x CS 477/677 - Lecture 15

  10. INTERVAL-SEARCH(T, i) • x ← root[T] • while x ≠ nil[T] and i does not overlap int[x] • do if left[x] ≠ nil[T] and max[left[x]] ≥ low[i] • then x ← left[x] • else x ← right[x] • return x CS 477/677 - Lecture 15

  11. Readings • Chapters 14, 15 CS 477/677 - Lecture 15

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