A Back-to-Basics Empirical Study of Priority Queues

The theory community has proposed several new heap variants in the recent past which have remained largely untested experimentally. We take the field back to the drawing board, with straightforward implementations of both classic and novel structures using only standard, well-known optimizations. We study the behavior of each structure on a variety of inputs, including artificial workloads, workloads generated by running algorithms on real map data, and workloads from a discrete event simulator used in recent systems networking research. We provide observations about which characteristics are most correlated to performance. For example, we find that the L1 cache miss rate appears to be strongly correlated with wallclock time. We also provide observations about how the input sequence affects the relative performance of the different heap variants. For example, we show (both theoretically and in practice) that certain random insertion-deletion sequences are degenerate and can lead to misleading results. Overall, our findings suggest that while the conventional wisdom holds in some cases, it is sorely mistaken in others

The Logarithmic Funnel Heap: A Statistically Self-Similar Priority Queue

The present work contains the design and analysis of a statistically self-similar data structure using linear space and supporting the operations, insert, search, remove, increase-key and decrease-key for a deterministic priority queue in expected O(1) time. Extract-max runs in O(log N) time. The depth of the data structure is at most log* N. On the highest level, each element acts as the entrance of a discrete, log* N-level funnel with a logarithmically decreasing stem diameter, where the stem diameter denotes a metric for the expected number of items maintained on a given level.Comment: 14 pages, 4 figure

Smooth heaps and a dual view of self-adjusting data structures

We present a new connection between self-adjusting binary search trees (BSTs) and heaps, two fundamental, extensively studied, and practically relevant families of data structures. Roughly speaking, we map an arbitrary heap algorithm within a natural model, to a corresponding BST algorithm with the same cost on a dual sequence of operations (i.e. the same sequence with the roles of time and key-space switched). This is the first general transformation between the two families of data structures. There is a rich theory of dynamic optimality for BSTs (i.e. the theory of competitiveness between BST algorithms). The lack of an analogous theory for heaps has been noted in the literature. Through our connection, we transfer all instance-specific lower bounds known for BSTs to a general model of heaps, initiating a theory of dynamic optimality for heaps. On the algorithmic side, we obtain a new, simple and efficient heap algorithm, which we call the smooth heap. We show the smooth heap to be the heap-counterpart of Greedy, the BST algorithm with the strongest proven and conjectured properties from the literature, widely believed to be instance-optimal. Assuming the optimality of Greedy, the smooth heap is also optimal within our model of heap algorithms. As corollaries of results known for Greedy, we obtain instance-specific upper bounds for the smooth heap, with applications in adaptive sorting. Intriguingly, the smooth heap, although derived from a non-practical BST algorithm, is simple and easy to implement (e.g. it stores no auxiliary data besides the keys and tree pointers). It can be seen as a variation on the popular pairing heap data structure, extending it with a "power-of-two-choices" type of heuristic.Comment: Presented at STOC 2018, light revision, additional figure

Algebras for weighted search

Weighted search is an essential component of many fundamental and useful algorithms. Despite this, it is relatively under explored as a computational effect, receiving not nearly as much attention as either depth- or breadth-first search. This paper explores the algebraic underpinning of weighted search, and demonstrates how to implement it as a monad transformer. The development first explores breadth-first search, which can be expressed as a polynomial over semirings. These polynomials are generalised to the free semi module monad to capture a wide range of applications, including probability monads, polynomial monads, and monads for weighted search. Finally, a monad trans-former based on the free semi module monad is introduced. Applying optimisations to this type yields an implementation of pairing heaps, which is then used to implement Dijkstra’s algorithm and efficient probabilistic sampling. The construction is formalised in Cubical Agda and implemented in Haskell

Selectable Heaps and Their Application to Lazy Search Trees

We show the O(log n) time extract minimum function of efficient priority queues can be generalized to the extraction of the k smallest elements in O(k log(n/k)) time. We first show the heap-ordered tree selection of Kaplan et al. can be applied on the heap-ordered trees of the classic Fibonacci heap to support the extraction in O(k \log(n/k)) amortized time. We then show selection is possible in a priority queue with optimal worst-case guarantees by applying heap-ordered tree selection on Brodal queues, supporting the operation in O(k log(n/k)) worst-case time. Via a reduction from the multiple selection problem, Ω(k log(n/k)) time is necessary. We then apply the result to the lazy search trees of Sandlund & Wild, creating a new interval data structure based on selectable heaps. This gives optimal O(B+n) lazy search tree performance, lowering insertion complexity into a gap Δi to O(log(n/|Δi|))$ time. An O(1)-time merge operation is also made possible under certain conditions. If Brodal queues are used, all runtimes of the lazy search tree can be made worst-case. The presented data structure uses soft heaps of Chazelle, biased search trees, and efficient priority queues in a non-trivial way, approaching the theoretically-best data structure for ordered data

Optimal Algorithms for Ranked Enumeration of Answers to Full Conjunctive Queries

We study ranked enumeration of join-query results according to very general orders defined by selective dioids. Our main contribution is a framework for ranked enumeration over a class of dynamic programming problems that generalizes seemingly different problems that had been studied in isolation. To this end, we extend classic algorithms that find the k-shortest paths in a weighted graph. For full conjunctive queries, including cyclic ones, our approach is optimal in terms of the time to return the top result and the delay between results. These optimality properties are derived for the widely used notion of data complexity, which treats query size as a constant. By performing a careful cost analysis, we are able to uncover a previously unknown tradeoff between two incomparable enumeration approaches: one has lower complexity when the number of returned results is small, the other when the number is very large. We theoretically and empirically demonstrate the superiority of our techniques over batch algorithms, which produce the full result and then sort it. Our technique is not only faster for returning the first few results, but on some inputs beats the batch algorithm even when all results are produced.Comment: 50 pages, 19 figure

Reitinhakualgoritmien toiminnan optimointi tieaineistolla

Tiivistelmä. Nopeimman reitin haku tieaineistolla on osa monien ihmisten arkipäivää esimerkiksi käytettäessä mobiililaitteita. Optimaalisen reitin haku kahden pisteen välillä on aihe, josta on olemassa paljon aiempaa tutkimustietoa. Klassisia algoritmeja, jotka ratkaisevat nopeimman reitin ongelman graafissa ja joita voidaan hyödyntää myös tieaineistolla ovat Dijkstran algoritmi ja A* -algoritmi. Uusimmissa tutkimuksissa on kehitetty nopeita algoritmeja, joille tarvittava reittiaineisto esikäsitellään nopeuden parantamiseksi. Tämän työn keskeisenä tutkimuskysymyksenä on selvittää, miten klassisia reitinhakualgoritmeja voidaan optimoida Suomen kattavalla Digiroad-tieaineistolla aiemmasta tutkimuksesta löytyvin menetelmin. Työssä tutkitaan tietorakenteiden optimointia tieaineistolle, keskinopeuden käyttöä heuristiikkana matka-ajan optimoinnissa sekä miten tieaineiston metadataa voidaan hyödyntää algoritmien optimoinnissa. Saatuja tuloksia verrataan esikäsittelyä hyödyntäviin algoritmeihin. Kirjallisuuskatsaus sisältää käsittelyn klassisista reititysalgoritmeista, tieaineiston erityispiirteistä sekä miten esikäsittelyä hyödyntävien menetelmien kehitys on vaikuttanut tutkimukseen. Lisäksi käsitellään tietorakenteiden roolia reititysalgoritmien toiminnassa sekä millaisia tuloksia eri optimointimenetelmien yhdistelmillä on saatu. Tämän työn tutkimusmenetelmänä on suunnittelutiede (Design Science). Menetelmän avulla luotu artefakti lukee Digiroad-aineistoa ja toimii yhtenäisenä alustana kolmelle erilaisille koejärjestelylle, jotka vastaavat esitettyihin tutkimuskysymyksiin. Jokaisessa koejärjestelyssä kerättiin tietoa algoritmien toiminnassa suorituskyky ja laatumittareilla käyttäen viittäsataa satunnaisesti valittua pisteparia. Tietorakenteiden ja algoritmien koejärjestelyn keskeisenä tuloksena havaittiin A* -algoritmin suoriutuvan 3,59 kertaa nopeammin kuin Dijkstran algoritmi Digiroad-aineistolla. Lisäksi tietorakenteiden käytännön toteutuksella on suuri vaikutus algoritmien nopeuteen. Binäärikeon toimintaa optimoimalla voitiin parantaa reitityksen nopeutta A* -algoritmilla 6,43 kertaisesti verrattuna perinteiseen binäärikekoon. Keskinopeutta käsittelevässä koejärjestelyssä tutkittiin A* -algoritmin heuristiikkaa ja millä arvioidulla keskinopeudella saadaan parhaat tulokset heuristiikassa linnuntietä hyödyntäen. Paras tasapaino reititysnopeuden ja optimaalisesta reitistä poikkeaman välillä saavutettiin arvoilla 70–90 km/h. Tällöin poikkeama oli 0,03–1,61 % optimaalisesta ratkaisusta reititysajan ollessa 3,54–14,84 kertaa parempi kuin referenssinä toimineessa Dijkstran algoritmissa. Kolmannessa koejärjestelyssä hyödynnettiin Digiroad-aineiston metatietoja luomalla niiden pohjalta erilaisia kaksitasoisia hierarkkisia algoritmeja käyttäen pohjana A* -algoritmia ja aiempien koejärjestelyjen tuloksia tavoitteena parantaa reititysaikoja. Tulokset voidaan jakaa kahteen ryhmään. Laadukkaimmilla algoritmeilla saavutettiin 14,25–18,27 kertainen parannus Dijkstran algoritmiin verrattuna poikkeaman optimaalisesta reitistä ollen 0,79–0,92 %. Nopeimmilla algoritmeilla saavutettiin 26,36–37,54 kertainen parannus Dijkstran algoritmiin, mutta poikkeama optimaalisesta reitistä oli tällöin 2,60–3,21 %. Tutkituilla algoritmeilla voidaan lähestyä nopeudeltaan yksinkertaisia esikäsittelyä hyödyntäviä algoritmeja, jos poikkeama optimaalisesta reitistä on hyväksyttävissä