Computer Science

1.4.1. Stack

1.4.2. Heap

1. Fibonacci Heap
   
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   [2025-12-31 Wed 20:57] -> Dijkstra's Algorithm
   

   
   

   • A collection of trees satisfying the minimum-heap property:
     • The key of a child is always greater than or equal to the key of a parent. The minimum element is always at the root of one of the trees.
   • used as a min-priority queue
     • an abstract data type that provides 3 basic operations: add_with_priority(), decrease_priority() and extract_min().
   • typically used to implement the priority queue element of Dijkrsta's algorithm
   • While Fibonacci heaps have very good theoretical complexities, in practice, other heap types such as pairing heaps are faster. This is because even in the simplest implementation, Fibonacci heaps require four pointers for each node, other heaps need two or three.

   Compared with binomial heaps, the structure of a Fibonacci heap is more flexible. The trees do not have a prescribed shape and in the extreme case the heap can have every element in a separate tree. This flexibility allows some operations to be executed in a lazy manner, postponing the work for later operations. For example, merging heaps is done simply by concatenating the two lists of trees, and operation decrease key sometimes cuts a node from its parent and forms a new tree.

   • amortized runtime analysis performed on potential value given by: φ = t+2m
     • where t is the number of trees and m is the number of marked nodes
   • Operations
     • find-min O(1)
     • merge O(1)
     • insert O(1)
     • delete-min O(log n)
     • decrease-key O(k) + c(-k + 4)
       • where k is the number of new trees inserted during the series of cascading cuts

   Although the total running time of a sequence of operations starting with an empty structure is bounded by the bounds given above, some (very few) operations in the sequence can take very long to complete (in particular, delete-min has linear running time in the worst case). For this reason, Fibonacci heaps and other amortized data structures may not be appropriate for real-time systems.

   • [1403.0252] A Back-to-Basics Empirical Study of Priority Queues
     • Recent experimental results suggest that the Fibonacci heap is more efficient in practice than most of its later derivatives, including quake heaps, violation heaps, strict Fibonacci heaps, and rank pairing heaps, but less efficient than pairing heaps or array-based heaps.
2. Brodal Queue
   
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   • O(1) for insertion, find-minimum, meld (merge two queues) and decrease-key
   • O(log(n)) for delete-minimum and general deletion.
   • First heap variant to achieve these bounds without resorting to amortization of operational costs.

1.4.3. Array

1. Vector
   
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2. String
   
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1.4.4. Dictionary

1.4.5. Hash Table

1.5.1. Risch algorithm

1.8.1. Algorithmic Information Theory

1. Kolmogorov Complexity
   
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   • https://www.cs.cmu.edu/~venkatg/teaching/15252-sp20/notes/Kolmogorov-Complexity.pdf

2.1.1. Anti-aliasing