Why Some Heaps Support Constant-Amortized-Time Decrease-Key Operations, and Others Do Not

  • John Iacono
  • Özgür Özkan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8572)


A lower bound is presented which shows that a class of heap algorithms in the pointer model with only heap pointers must spend \(\Omega \left( \frac{\log \log n}{\log \log \log n} \right)\) amortized time on the Decrease-Key operation (given O(logn) amortized-time Extract-Min). Intuitively, this bound shows the key to having O(1)-time Decrease-Key is the ability to sort O(logn) items in O(logn) time; Fibonacci heaps [M. .L. Fredman and R. E. Tarjan. J. ACM 34(3):596-615 (1987)] do this through the use of bucket sort. Our lower bound also holds no matter how much data is augmented; this is in contrast to the lower bound of Fredman [J. ACM 46(4):473-501 (1999)] who showed a tradeoff between the number of augmented bits and the amortized cost of Decrease-Key. A new heap data structure, the sort heap, is presented. This heap is a simplification of the heap of Elmasry [SODA 2009: 471-476] and shares with it a O(loglogn) amortized-time Decrease-Key, but with a straightforward implementation such that our lower bound holds. Thus a natural model is presented for a pointer-based heap such that the amortized runtime of a self-adjusting structure and amortized lower asymptotic bounds for Decrease-Key differ by but a O(logloglogn) factor.


Terminal Structure Marked Node Fibonacci Heap Bucket Sort Leftmost Child 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. MIT Press (2009)Google Scholar
  2. 2.
    Dijkstra, E.W.: A note on two problems in connexion with graphs. Numerische Mathematik 1, 269–271 (1959)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Elmasry, A.: Pairing heaps with O(log log n) decrease cost. In: SODA, pp. 471–476 (2009)Google Scholar
  4. 4.
    Elmasry, A.: Pairing Heaps with Costless Meld. In: de Berg, M., Meyer, U. (eds.) ESA 2010, Part II. LNCS, vol. 6347, pp. 183–193. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  5. 5.
    Fredman, M.L.: A Priority Queue Transform. In: Vitter, J.S., Zaroliagis, C.D. (eds.) WAE 1999. LNCS, vol. 1668, pp. 244–258. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  6. 6.
    Fredman, M.L.: On the Efficiency of Pairing Heaps and Related Data Structures. J. ACM 46(4), 473–501 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Fredman, M.L., Sedgewick, R., Sleator, D.D., Tarjan, R.E.: The Pairing Heap: A New Form of Self-Adjusting Heap. Algorithmica 1(1), 111–129 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Fredman, M.L., Tarjan, R.E.: Fibonacci heaps and their uses in improved network optimization algorithms. J. ACM 34(3), 596–615 (1987)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Iacono, J.: Improved Upper Bounds for Pairing Heaps. In: Halldórsson, M.M. (ed.) SWAT 2000. LNCS, vol. 1851, pp. 32–45. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  10. 10.
    Pettie, S.: Towards a Final Analysis of Pairing Heaps. In: FOCS, pp. 174–183 (2005)Google Scholar
  11. 11.
    Sleator, D.D., Tarjan, R.E.: Self-Adjusting Binary Search Trees. J. ACM 32(3), 652–686 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Sleator, D.D., Tarjan, R.E.: Self-Adjusting Heaps. SIAM J. Comput. 15(1), 52–69 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Stasko, J.T., Vitter, J.S.: Pairing Heaps: Experiments and Analysis. Commun. ACM 30(3), 234–249 (1987)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • John Iacono
    • 1
  • Özgür Özkan
    • 1
  1. 1.New York UniversityUSA

Personalised recommendations