Two-Level Heaps: A New Priority Queue Structure with Applications to the Single Source Shortest Path Problem

  • K. Subramani
  • Kamesh Madduri
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5573)


The Single Source Shortest Paths problem with positive edge weights (SSSPP) is one of the more widely studied problems in Operations Research and Theoretical Computer Science [1,2] on account of its wide applicability to practical situations. This problem was first solved in polynomial time by Dijkstra [3], who showed that by extracting vertices with the smallest distance from the source and relaxing its outgoing edges, the shortest path to each vertex is obtained. Variations of this general theme have led to a number of algorithms, which work well in practice [4,5,6]. At the heart of a Dijkstra implementation is the technique used to implement a priority queue. It is well known that using Dijkstra’s approach requires Ω(nlogn) steps on a graph having n vertices, since it essentially sorts vertices based on their distances from the source. Accordingly, the fastest implementation of Dijkstra’s algorithm on a graph with n vertices and m edges should take Ω(m + n·logn) time and consequently the Dijkstra procedure for SSSPP using Fibonacci Heaps is optimal, in the comparison-based model. In this paper, we introduce a new data structure to implement priority queues called Two-Level Heap (TLH) and a new variant of Dijkstra’s algorithm called Phased Dijkstra. We contrast the performance of Dijkstra’s algorithm (both the simple and the phased variants) using a number of data structures to implement the priority queue and empirically establish that Two-Level heaps are far superior to Fibonacci heaps on every graph family considered.


Short Path Random Graph Priority Queue Short Path Problem Graph Instance 
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. MIT Press, Cambridge (2001)zbMATHGoogle Scholar
  2. 2.
    Goldberg, A.V.: Scaling algorithms for the shortest paths problem. SIAM Journal on Computing 24(3), 494–504 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dijkstra, E.W.: A note on two problems in connexion with graphs. Numerische Mathematik 1, 269–271 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Raman, R.: Recent results in single-source shortest paths problem. SIGACT news 28, 81–87 (1997)CrossRefGoogle Scholar
  5. 5.
    Ahuja, R.K., Mehlhorn, K., Orlin, J.B., Tarjan, R.E.: Faster algorithms for the shortest path problem. Journal of the ACM 37(2), 213–223 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Pallottino, S.: Shortest path methods: Complexity, interrelations and new propositions. NETWORKS: Networks: An International Journal 14, 257–267 (1984)zbMATHGoogle Scholar
  7. 7.
    Park, J., Penner, M., Prasanna, V.: Optimizing graph algorithms for improved cache performance. In: Proc. Int’l Parallel and Distributed Processing Symp. (IPDPS 2002), Fort Lauderdale, FL (April 2002)Google Scholar
  8. 8.
    Demetrescu, C., Goldberg, A., Johnson, D.: 9th DIMACS implementation challenge – Shortest Paths,
  9. 9.
    Chakrabarti, D., Zhan, Y., Faloutsos, C.: R-MAT: A recursive model for graph mining. In: Proc. 4th SIAM Intl. Conf. on Data Mining, Florida, USA (April 2004)Google Scholar
  10. 10.
    Cherkassky, B., Goldberg, A., Radzik, T.: Shortest paths algorithms: theory and experimental evaluation. Mathematical Programming 73, 129–174 (1996)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Goldberg, A.: Network optimization library,

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • K. Subramani
    • 1
  • Kamesh Madduri
    • 2
  1. 1.LDCSEEWest Virginia UniversityMorgantownUSA
  2. 2.Computational Research DivisionLawrence Berkeley National LaboratoryBerkeleyUSA

Personalised recommendations