In terms of asymptotic space and time complexity, what is the most efficient priority-queue? Specifically I am looking for priority queues which minimize the complexity of inserts, it's ok if deletes are a little slower.
If you're looking for a survey of priority-queues which minimises complexity of deletes over inserts, see: Does there exist a priority queue with $O(1)$ extracts?.
Insert: $\mathcal{O}(1)$
Find-min: $\mathcal{O}(1)$
Decrease-key: $\mathcal{O}(1)$
Delete: $\mathcal{O}(\log \log n)$
Space: $\mathcal{O}(n)$
THEOREM 1. We can implement a priority queue that with n integer keys in the range $[0 , N )$ in linear space supporting find-min, insert, and dec-key in constant time, and delete in $\mathrm{\mathcal{O}(log\ log\ min \{n, N\})}$ time.
Which is established with a combination of:
LEMMA 3. Let $\tau(n, N)$ be the delete time for a priority queue for up to $n$ integers in the range $[0 , N)$ supporting insert and dec-key in constant time. Then $\tau ( n, N ) \le τ ( N, N)$. This holds whether $\tau$ is amortized or worst-case.
and:
THEOREM 6. Theorem 6. We can implement a priority queue that with $n$ integer keys in the range $[0 , N)$ in linear space supporting find-min, insert, and dec-key in constant time, and delete in $\mathrm{\mathcal{O}(1 + log\ log\ n − log\ log\ q)}$ time for a key of rank $q$.
Like with anything in CS, there is no "best something". There are always trade offs. But, perhaps this section of Wikipedia's article on Fibonacci heap could help you:
Fibonacci heap: Amortized $\mathcal{O}(\log\ n)$ delete and delete_min, amortized $\mathcal{O}(1)$ decrease_key and $\mathcal{O}(1)$ the rest.
Brodal queue: Worst-case $\mathcal{O}(\log\ n)$ delete and delete_min, $\mathcal{O}(1)$ the rest.
Pairing heap: Amortized $\mathcal{O}(\log\ n)$ delete and delete_min, unknown decrease_key, but bounded by $\mathcal{\Omega}(\log \log\ n)$, amortized $2^{\mathcal{O}(\sqrt{\log\log\ n})}$, $\mathcal{O}(1)$ the rest.
Finally a simplification of the Brodal queue[1] has been released[2], which unlike[3]; are worst-case values (rather than amortised).
Insert: $\mathcal{O}(1)$
Find-min: $\mathcal{O}(1)$
Decrease: $\mathcal{O}(1)$
Meld: $\mathcal{O}(1)$
Delete-min: $\mathcal{O}(log\ n)$, that's equivalent to $\approx 70\ log\ n$ here, which is a smaller constant than Brodal.
$\mathcal{O}(log\ M)$ insert, search, delete [see 'Lemma 3.1']
$\dagger\ \mathcal{O}(1)$ findMin, findMax, extractMin, extractMax, predecessor, successor
$2M + \mathcal{O}(log\ M)$ bits of ordinary memory and $m$ bits of Yggdrasdil memory [this second type of memory is defined in the paper]
(where $M$ is the bounded integer universe: $M = [0, \cdots, M - 1]$
$\dagger$ Can someone confirm this result, as I am assuming it from 'Theorem 3.1'; i.e.: that "update" in 'Lemma 3.3' only refers to random inserts and deletes rather than extract{Min,Max}.
Brodnik, Andrej, Svante Carlsson, Michael L. Fredman, Johan Karlsson, and J. Ian Munro. “Worst Case Constant Time Priority Queue.” Journal of Systems and Software 78, no. 3 (2005): 249 – 256. doi:10.1016/j.jss.2004.09.002. (I read the 6 page version)
$\boldsymbol{\mathcal{O}(1)}$ insertion
$\boldsymbol{\mathcal{O}(log\ \textbf{min}\{w_x, q_x\})}$ get-min, extract-min, delete, and decrease of an element $\boldsymbol{x}$
Where $w_x$ (respectively, $q_x$) is the number of elements that were accessed after (respectively, before) the last access to $x$ and are still in the priority queue at the time when the corresponding operation is performed.
A. Elmasry, A. Farzan, and J. Iacono, “A priority queue with the time-finger property,” J. of Discrete Algorithms, vol. 16, pp. 206–212, Oct. 2012. 1
