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CSCI 6212 Design and Analysis of Algorithms Heapsort

CSCI 6212 Design and Analysis of Algorithms Heapsort. Dr. Juman Byun The George Washington University. Space Complexity. Complete Binary Tree. (Binary) Max-Heap. 16. 14. 10. 8. 7. 9. 3. 2. 4. 1. (Binary) Max-Heap. 1. 16. 2. 3. 14. 10. 4. 5. 6. 7. 8. 7. 9. 3. 8. 9.

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CSCI 6212 Design and Analysis of Algorithms Heapsort

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  1. CSCI 6212 Design and Analysis of AlgorithmsHeapsort • Dr. Juman Byun • The George Washington University

  2. Space Complexity

  3. Complete Binary Tree

  4. (Binary) Max-Heap 16 14 10 8 7 9 3 2 4 1

  5. (Binary) Max-Heap 1 16 2 3 14 10 4 5 6 7 8 7 9 3 8 9 10 2 4 1

  6. To get array indices of relative positions • Parent(i) • 1 return floor(i/2) • Left(i) • 1 return 2i • Right(i) • 1 return 2i + 1

  7. Max-Heap in Array 16 14 10 8 7 9 3 2 4 1

  8. 16 Max-Heapify(A,i) 4 10 14 7 9 3 2 8 1

  9. 16 4 10 14 7 9 3 2 8 1 Max-Heapify(A,i) 1 l = LEFT(i) 2 r = RIGHT(i) 3 if l <= A.heap-size and A[l] > A[i] 4 largest = l 5 else largest = i 6 if r <= A.heap-size and A[r] > A[largest] 7 largest = r 8 if largest != i 9 exchange A[i] with A[largest] 10 Max-Heapfiy(A, largest)

  10. Max-Heapify(A,2) 16 1 4 10 2 3 14 7 9 3 4 5 6 7 2 8 1 10 8 9 Max-Heapify(A,i) 1 l = LEFT(i) 2 r = RIGHT(i) 3 if l <= A.heap-size and A[l] > A[i] 4 largest = l 5 else largest = i 6 if r <= A.heap-size and A[r] > A[largest] 7 largest = r 8 if largest != i 9 exchange A[i] with A[largest] 10 Max-Heapfiy(A, largest)

  11. Max-Heapify(A,2) 16 1 4 10 2 3 14 7 9 3 4 5 6 7 2 8 1 10 8 9 Max-Heapify(A,i) 1 l = LEFT(i) 2 r = RIGHT(i) 3 if l <= A.heap-size and A[l] > A[i] 4 largest = l 5 else largest = i 6 if r <= A.heap-size and A[r] > A[largest] 7 largest = r 8 if largest != i 9 exchange A[i] with A[largest] 10 Max-Heapfiy(A, largest)

  12. Max-Heapify(A,2) 16 1 4 10 2 3 14 7 9 3 4 5 6 7 2 8 1 10 8 9 Max-Heapify(A,i) 1 l = LEFT(i) 2 r = RIGHT(i) 3 if l <= A.heap-size and A[l] > A[i] 4 largest = l 5 else largest = i 6 if r <= A.heap-size and A[r] > A[largest] 7 largest = r 8 if largest != i 9 exchange A[i] with A[largest] 10 Max-Heapfiy(A, largest)

  13. Max-Heapify(A,2) 16 1 14 10 2 3 4 7 9 3 4 5 6 7 2 8 1 10 8 9 Max-Heapify(A,i) 1 l = LEFT(i) 2 r = RIGHT(i) 3 if l <= A.heap-size and A[l] > A[i] 4 largest = l 5 else largest = i 6 if r <= A.heap-size and A[r] > A[largest] 7 largest = r 8 if largest != i 9 exchange A[i] with A[largest] 10 Max-Heapfiy(A, largest)

  14. Max-Heapify(A,4) 16 1 14 10 2 3 4 7 9 3 4 5 6 7 2 8 1 10 8 9 Max-Heapify(A,i) 1 l = LEFT(i) 2 r = RIGHT(i) 3 if l <= A.heap-size and A[l] > A[i] 4 largest = l 5 else largest = i 6 if r <= A.heap-size and A[r] > A[largest] 7 largest = r 8 if largest != i 9 exchange A[i] with A[largest] 10 Max-Heapfiy(A, largest)

  15. Max-Heapify(A,4) 16 1 14 10 2 3 8 7 9 3 4 5 6 7 2 4 1 10 8 9 Max-Heapify(A,i) 1 l = LEFT(i) 2 r = RIGHT(i) 3 if l <= A.heap-size and A[l] > A[i] 4 largest = l 5 else largest = i 6 if r <= A.heap-size and A[r] > A[largest] 7 largest = r 8 if largest != i 9 exchange A[i] with A[largest] 10 Max-Heapfiy(A, largest)

  16. Max-Heapify(A,9) 16 1 14 10 2 3 8 7 9 3 4 5 6 7 2 4 1 10 8 9 Max-Heapify(A,i) 1 l = LEFT(i) 2 r = RIGHT(i) 3 if l <= A.heap-size and A[l] > A[i] 4 largest = l 5 else largest = i 6 if r <= A.heap-size andA[r] > A[largest] 7 largest = r 8 if largest != i 9 exchange A[i] with A[largest] 10 Max-Heapfiy(A, largest)

  17. Build-Max-Heap(A) • 1 A.heap-size = A.length • 2 for i = floor(A.length/2) downto 1 • 3 Max-Heapify(A,i) 16 1 14 10 2 3 8 7 9 3 4 5 6 7 2 4 1 10 8 9

  18. 16 Heapsort(A) 4 10 14 7 9 3 Build-Max-Heap(A) 2 8 1

  19. 16 Heapsort(A) 14 10 8 7 9 3 Build-Max-Heap(A) 2 4 1

  20. 16 Heapsort(A) 14 10 8 7 9 3 Build-Max-Heap(A) for i = A.length downto 2 exchange A[1] with A[i] 2 4 1 1 2 3 4 5 6 7 8 9 10

  21. Heapsort(A) Build-Max-Heap(A) for i = A.length downto 2 exchange A[1] with A[i] 16 1 14 10 2 3 8 7 9 3 4 5 6 7 2 4 1 8 9 10

  22. Heapsort(A) Build-Max-Heap(A) for i = A.length downto 2 exchange A[1] with A[i] 1 1 14 10 2 3 8 7 9 3 4 5 6 7 2 4 16 8 9 10

  23. Heapsort(A) Build-Max-Heap(A) for i = A.length downto 2 exchange A[1] with A[i] A.heap-size = A.heap-size - 1 1 1 14 10 2 3 8 7 9 3 4 5 6 7 2 4 16 8 9 10

  24. Heapsort(A) Build-Max-Heap(A) for i = A.length downto 2 exchange A[1] with A[i] A.heap-size = A.heap-size - 1 Max-Heapify(A,1) 14 1 8 10 2 3 4 7 9 3 4 5 6 7 2 1 16 8 9 10

  25. Running Time of Heapsort(A) Build-Max-Heap(A) for i = A.length downto 2 exchange A[1] with A[i] A.heap-size = A.heap-size - 1 Max-Heapify(A,1) O( lg n )

  26. Running Time of Heapsort(A) Build-Max-Heap(A) for i = A.length downto 2 exchange A[1] with A[i] A.heap-size = A.heap-size - 1 Max-Heapify(A,1) O( n lg n )

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